United States
Environmental Protection
Anency
Office of Radiation Programs
Las Vegas Facility
P.O. Box 18416
Las Vegas NV 89114
EPA520/682014
August 1982
&EPA
A Basic Technique and
Models for Determining
Exposure Rates over
UraniumBearing Soils

EPA520/682014
August 1982
A BASIC TECHNIQUE AND MODELS
FOR DETERMINING EXPOSURE RATES
OVER URANIUMBEARING SOILS
George V. OkszaChocimowski
August 1982
Office of Radiation ProgramsLas Vegas Facility
U.S. Environmental Protection Agency
Las Vegas, Nevada 89114

DISCLAIMER
This report has been reviewed by the Office of Radiation Programs  Las
Vegas Facility, U. S. Environmental Protection Agency, and approved for
publication. Mention of trade names or commercial products does not
constitute endorsement or recommendation for their use.

PREFACE
The Office of Radiation Programs of the U.S. Environmental Protection
Agency carries out a national program designed to evaluate the exposure of man
to ionizing and nonionizing radiation, and to promote development of controls
necessary to protect the public health and safety and assure environmental
quality.
Exposures by direct external gamma irradiation from nuclides in the
uranium238 decay chain, naturally present in the environment  as in
commercial grade ore deposits  or in byproducts  as in the tailings piles of
uranium mills  represent an element of risk that must be quantitatively
assessed to determine the need for remedial action and the setting of
necessary controls. This report illustrates the application of basic
theoretical methods and models for the prediciton of exposure rates at the
locations of concern, as an initial step for the required risk assessment.
Readers of this report are encouraged to inform the Office of Radiation
Programs of any errors or omissions. Comments or requests for further
information are invited.
Wayne A. Bliss
Acting Director
Office of Radiation Programs, LVF

ABSTRACT
The application of simple computerimplemented analytical procedures to
predict exposure rates over uraniumbearing soil deposits is demonstrated in
this report. The method is based, conceptually, on the energydependent
pointsource buildup factor and, operationally, on two consecutive
integrations. The dependence of photon fluxes on spatial variables is
simplified by an analytical integration over the physical dimensions of the
deposit, represented as a slab bearing homogeneously distributed nuclides of
the uranium238 decay chain, at equilibrium, and covered with a sourcefree
overburden slab; both slabs being of variable thickness but of infinite area!
extent. The resultant analytical expression describes flux as function of
energydependent parameters, thickness of the source slab, and depth of
overburden, and is equated analytically to exposure rates bearing the same
dependence. Elementary computer techniques are then employed to integrate
numerically the exposure rates corresponding to the specific energies of
uranium238 decay chain, for chosen thicknesses of the overburden and
uraniumbearing slabs. The numerical integration requires the use of buildup
factors, attenuation and absorption coefficients expressed as continuous
functions of energy by curvefitting equations included in the report.
As direct application of the method, maximum exposure rates over
uraniumbearing deposits are calculated. In addition, the dependence of
exposure rates on the thickness of the uraniumbearing slab and depth of
overburden is reduced to a simple model. These results, valid for uranium
mill tailings piles, are compared to those obtained by other authors, and then
applied to detemine changes in exposure rates due to radon gas emanation from
source materials.

CONTENTS
Preface iii
Abstract iv
Figures vi
Tables viii
Acknowledgment ix
1. Introduction 1
2. Analytical Bases and Development 4
3. Implementation 19
4. Results 26
I. Previous Results and Models 26
II. Comparison with Present Results 28
III. Models Based on Present Techniques and
Comparison with Previous Models 31
5. Applications 48
References 54
Appendices
A. Choice of Empirical Function to Represent GammaRay
Buildup 56
B. Simplifying Assumptions 61
C. Exposure Rates and Flux Equations 67
D. Decay Scheme and Energy Spectrum 83
E. Choice of Medium Representing Uranium Mill Tailings 91
F. Dose Buildup Coefficients for Taylor's and Berger's
Formulas 96
I. Taylor's Coefficients 96
II. Berger's Coefficients 101
G. Ancillary CurveFitting Equations 107
H. Computer Implementation 121
I. Sample Calculations for a Monoenergetic Case 127
J. Comments on CurveFitting Exposure Rate Models 131
K. Interrlationship of Exposure Rates 137
L. Radon Distribution Through Overburden 140
M. Effects of Radon Diffusion on Exposure Rates 145

FIGURES
Number Page
1 Section of overburden and tailings slab 12
2 Depthdependent relaxation parameter L(d) 35
3 Comparison of depthdependent relaxation parameter L(d)
with Schiager (1974) relaxation constant 38
4 Relative decrease in exposure rates with increasing thickness
of cover slab, applying depthdependent relaxation
parameter and Schiager (1974) relaxation constant 39
5 Relative increase in exposure rate according to present
model, compared to Schiager (1974) model 41
6 Effects of increasing source slab thickness on exposurerates
comparing present results with Schiager1s 42
7 Relative decrease in exposure rates with increasing thickness
of overburden, for emanation power E=20 and various radon
diffusion coefficient values 51
8 Effects of radon emanation in reducing maximum exposure
rate, for E=20 and various values of radon diffusion
coefficient D 53
1C Geometry for flux calculations 68
1D Uranium238 Decay Series 84
1E Magnitude of [A/(l+ซi) + (1A)/(1+02)] for various
energies in various media 95
1F Taylor's Dose Buildup coefficient A, for apoint
isotropic source in water, as function of energy 98
2F Taylor's Dose Buildup coefficient aj, for a point
isotropic source in water, as function of energy 99
3F Taylor's Dose Buildup coefficient c^, fฐr a point
isotropic source in water, as function of energy 100
4F Comparison of 20 MFP and 7 MFP Berger's buildup
factors as function of distance, for a 0.255 MeV source 102
VI

Figures (continued)
Number Page
5F Effective Buildup Factors for the surface of an
infinitely thick source slab, based on Berger's
7 MFP coefficients, as function of energy 106
1G Mass attenuation coefficients for various materials 109
2G Mass attenuation coefficient for water, as function
of energy 114
3G Mass attenuation coefficient for air, as function of
energy 116
4G Mass energyabsorption coefficient for air, as
function of energy 118
5G Graphical representation of the Second Order
Exponential Integral ฃ2, as function of
generalized argument 120
1H Example of computer implementation basic
operational scheme 126
11 Massattenuation coefficient and buildup for
aluminum, as function of energy 129
1L Tailings and cover configuration 140
2L Distribution of free radon in tailings and cover
for 9 different values of D, with a cover thickness
d=30 cm, and E=0.2 144
1M Schematic representation of numerical integration
method, applied to a cover d=100 cm 145
2M Relative decrease in exposure rates, as function of
increasing overburden slab thickness for emanation
power E=20 and different values of radon diffusion
coefficient 147
Vll

TABLES
Number Page
1 Analytical Expressions of Flux at the Surface of a
Tailings Pile, Based on Taylor's Buildup Formula 14
2 Proportional Increase in Exposure Rates, with Respect to
Maximum, as Function of Increasing Source Slab Thickness 33
3 Proportional Decrease in Exposure Rates, with Respect to
Maximum, as Function of Increasing Cover Slab Thickness 34
4 Comparison of Present Maximum Exposure Rates and Models
with Previously Published Models and Values 47
1A Dose Buildup Factor (B) for a Point Isotropic Source 58
2A Comparison of Taylor's and Berger's Buildup Factors
with Tabulated Values of Buildup for Eight Energies 60
1D Volumetric Source Strengths SV(E) for Energies
E<_ 0.5 MeV 85
2D Volumetric Source Strengths S,/(E) for Energies
E^O.5 MeV ; 87
1E BuscaglioneManzini Coefficients for Taylor Dose
Buildup Factor Formula 93
1F Values of Bwc, C and D for Energies 0.255 MeV to
1.0 MeV 104
1G Effects on Flux and Exposure Rates of Using Water and
Aluminum Buildup Factors and Attenuation Coefficients 112
1J Thickness of Overburden Slab Which Must not be
Exceeded with the Use of a Constant L 136
vm

ACKNOWLEDGMENT
The author gratefully acknowledges the assistance and advice of several
individuals in the preparation of this report. Special recognition is
extended to Dr. Harold L. Beck, of the Environmental Measurements Laboratory,
New York, of the Department of Energy, for constructive criticism ultimately
leading to a better study; to Mr. David E. Bernhardt, of the Office of
Radiation Programs, Evaluation Branch  Las Vegas, of the Environmental
Protection Agency, for significant contributions in problems of format and
presentation; to Mr. Thomas R. Morton, of the Eastern Environmental Radiation
Facility (EERF), Environmental Studies Branck, for valuable suggestions
concerning needed expansion of the present work; and to Dr. Ross A Scarano of
the Nuclear Regulatory Commission, Uranium Recovery Licensing Branch, Division
of Waste Management, for useful comments on applications of the present
technique.
Although the scope of the present work and other limitations did not
permit pursuing every suggested improvement to a logical conclusion, the
author appreciates the interest and recognizes the assistance of the above
named individuals, but accepts full responsibility for the contents of this
report.
IX

Introduction
238
External gamma exposure rates over soils containing nuclides of the U
decay chain (such as uranium mill tailings piles) have been evaluated with
models and techniques differing in generality and level of sophistication.
The resultant range of estimates reflects the diversity of approaches. Some
of the higher predictions are unquestionably due to simple methods incorpo
rating, necessarily, conservative assumptions. More reliable methods, based
on thorough analytical treatment and processing of abundant input data,
frequently require complex programming and extensive computer use, in excess
of resources and time allotted by many facilities to specific projects. It
follows that the immediate practical value of such evolved techniques is
limited to that of the published final results, which may not be directly
applicable to the needs of potential users.
Such limitations and drawbacks were an important factor motivating the present
work, extended to serve a threefold purpose, as described below:
1) to demonstrate the reliability of a method, based on the "buildup
factor" concept, requiring limited programming and computer use while
avoiding many of the inaccuracies or uncertainties inherent in
simplified models;
2) to apply this method in generating simple models relating exposure
rates to depths of uraniumbearing soil deposits and cover material;

3) to illustrate the usefulness of these simple models under conditions
of greater complexity  specifically, by examining the reduction in
exposure rates due to radon exhalation from uraniumbearing soil and
the effects of radon penetration into the overburden. Additional
analysis was required to realize this last objective.
The proposed method, models, their application, results and comparisons with
results obtained by other authors are described or presented in the main text
of the report. Analytical treatments, assumptions, curvefitting equations,
ancillary tables and graphs are discussed in appendices, referenced in the
main text.
This report evolved from an exploration of simple, analytically based tech
niques whereby results from previous methods could be critically examined.
Given the exploratory nature of the original study, the use of substitutions,
extrapolations and approximations in applying the method was due to unavail
ability of other data or convenience rather than the rigorous analysis on
which the method is based. Nevertheless, their use may be justified by their
contribution to the effectiveness of the technique, demonstrated by results
which are in close agreement with previously published models and values,
particularly with some that "have been reproduced by a number of other
investigators" (Beck, 1981).
In view of the relative simplicity of the method, such close agreement may
exceed expectations fostered by reliance on more complex techniques. The

element of fortuity cannot be entirely denied, in that the substitutions,
extrapolations and approximations used in implementing the method undoubtedly
produced errors that were mutually compensatory to a large extent, as
evidenced by the results. To further support the validity of the latter,
extensive appendices were included with the report. These provide a detailed
description of the analytical bases of the method, the logical foundation of
assumptions, substitutions, etc., complete presentation of the data base and
treatment (including curvefitting errors), computer implementation and sample
calculations. Additional appendices contain some basic but relevant comments
on the models derived from the results, their interrelationship and applica
tion.

Analytical Bases and Development
In principle, the assessment of exposure rates from any radioactive source
requires identifying the energies of photons reaching the point of concern and
calculating the photon flux corresponding to each of these energies. In
common practice, the first requirement is reduced to equating the photon
energies at the assessment point to the energies of photons emitted by the
source. The second requirement entails determining the effects of distance
and the attenuation capabilities of a specified medium in reducing the
probability that a photon of a given energy, from a source of known
configuration, will reach the point of interest. For a point source, such
determination ultimately results in
ep(E)r
 S(E)   (1)
where<ฃ(E) = flux of photons of energy E at assessment point, photons/cm sec
S(E) = pointsource rate of emission of photons of energy E, or "point
source strength", photons/sec
p p
4nr = surface of a sphere of radius r, cnr
exp[p(E)r] = exponential attenuation term, function of distance r, absorbing
medium, and photon energy E, dimensionless
u(E) = total linear attenuation coefficient of absorbing medium for
photons of energy E, cm
r = distance between pointsource and assessment point, cm
E = energy with which photons are emitted by the point source, MeV
The linear attenuation coefficient y(E) represents the probability that a
photon of energy E will interact with the medium in any one of several
4

possible ways per each unit of distance it travels in this medium. Since any
detectable interaction of a photon with the medium involves a detectable
energy loss and/or change of direction, the use of y(E) in (1) implies that
any photon emitted with energy E that interacts with the medium will not
contribute to the flux of photons of energy E at the point of concern. The
exclusion of such "secondary" or "scattered" photons, of energies less than E,
facilitates the calculation and definition of a "primary exposure rate",
limited to those ("primary") photons that succeed in reaching the point of
interest without any prior interaction. The calculation requires the use of
from Equation (1) in the following general expression for exposure rates,
X(E) = f.\
uen(E)
(2)
air
where X(E) = exposure rate from photons of energy E, in R/s
F^ = conversion constant
= 1.824401368 x 10~8 g . R/Mev
E = gamma energy, in MeV
= "flux" of gammas of energy E, in gammas/(cm .s)
y (E) = energy dependent mass energy absorption coefficient for air, in
I 9
p~ cm^/g
air
Photons excluded from the primary flux by an interaction with the medium
are not exempt from subsequent absorption and scattering events, and have a
finite probability of reaching the point of assessment after successive
scatterings. Because of the large number of possible occurrences of every
type of interaction, the photons scattered to this point compound a complex
aggregate of "secondary fluxes" of virtually every energy below the energy of

emission E. The difficulty of individually calculating each of these fluxes
is a serious obstacle to the determination of the corresponding exposure rates
[see Equation (2)], a significant concern since the latter contribute
substantially to the total exposure rate at the point of interest. To
circumvent these difficulties, the total net effect of secondary radiations
may be equated to a nominal increase of the primary flux, by a socalled
"buildup factor B", based on experimental and theoretical results, so that
v(E)r
S(E) B(E)
The values tabulated for B depend on the energy of emission, on the source
configuration, on the absorbing medium and, to some extent, on the effect
being observed. Thus, there are slight differences between energy buildup,
energyabsorption buildup, and dose buildup factors for the same energy,
medium and configuration. Since dose in air is proportional to exposure,
"pointsource dose buildup factors," valid for infinite media, are used in the
present work. These may be used to illustrate the above description of
buildup factors as
o/p\ _ i + secondary dose rate due to point source emitting photons of energy E /*\
* ~ primary dose rate due to point source emitting photons of energy E
= energydependent pointsource dose buildup factor, for unspecified
infinite medium, dimensionless
Applying B(E) in (3) results in a flux<^>(E), nominally of photons retaining
their initial energy E, which can be used in (2) to calculate exposure rates
including both primary and scattered photons.

Tabulations of buildup factor values at various distances from a source in
an infinite medium have been available since 1954, or earlier, for each set of
conditions specifying either a point isotopic or monodirectional plane source,
one of nine source energies, and one of seven* infinite medium materials. To
facilitate analytical treatment and interpolation for untabulated energies,
several empirical functions have been fitted to these tables. These include
"linear," "quadratic" and "cubic" forms, as well as other polynomial fits
containing exponential terms, all of them with fitting coefficients valid for
a specific source energy. The fitting coefficients in some of these forms
apply only within specified distances from the source, which prompted the
selection of a form employing coefficients of greater generality, such as
Taylor's Dose Buildup Factor Formula (see Appendix A),
a,(E)p(E)r a2(E)p(E)r
BT(E,yr) = A(E)e x +[lA]e * (5)
where Br(E,ijr) = energy and distance dependent buildup factor, Taylor's
Formula, dimensionless
A(E),o1(E),a2(E) = Taylor's energydependent fitting parameters,
dimensionless
y(E) = energydependent attenuation coefficient, cm
r e distance, cm
*Author's note. Four additional materials are included in Trubey (1966).
7

Taylor's fitting parameters A(E), a^(E) and (^(E) "are not available
below 0.5 MeV" (Trubey, 1966), which excludes a range of lower energies
238
comprising roughly 15% of the total energy emitted by the U decay chain at
radioactive equilibrium. Part of this range may be covered by the use of
Berger's Buildup Factor Formula,
BB(E,yr) = 1 + C(E)u(E)reD(E)^E)r (6)
where C(E), D(E) = energydependent fitting parameters, dimensionless
Berger's fitting parameters are available for energies equal to or greater
than 0.255 MeV, excluding energies corresponding to only 3% or 4% of the total
energy emitted by the decay chain. Buildup at these energies can be
tentatively estimated by a specialized application of Berger's formula, as
discussed in the appropriate section.
Although Taylor's fitting coefficients apply over a smaller energy range than
Berger's parameters, the latter have the disadvantage of being valid only for
specified distances from the source of emission. Thus, a set of Berger's
parameters is applicable for up to seven "mean free path" lengths (yr = 7),
another for up to 20 MFPs, etc. This restricts the application of Berger's
formula to special cases, as will be seen, while Taylor's formula is not
subject to such restrictions.
The wide range of applicability of Taylor's fitting parameters makes this
formula suitable for analytical treatments involving distributed gammaray
238
sources of variable dimensions, a useful generalization of U decay chain
deposits when studied as a set. For present purposes, such a generalized
8

repository is represented as a smooth, flat, moisturefree soil slab of
uniform, specified (variable) thickness but infinite in area, containing, in
uniform distribution and radioactive equilibrium, all the nuclides of the
OOQ
uranium decay chain from U to stable lead. This nuclidebearing soil
slab is covered with an infinitely wide slab of sourcefree overburden, of a
uniform, specified (variable) thickness, having the same buildup and
attenuation properties as the slab beneath. The bases for these simplifying
assumptions are discussed, to some extent, in Appendix B.
The physical model outlined above requires some qualifications affecting
the course of subsequent analysis and import of the results, as follows:
1) The thicknesses specified for the nuclidebearing slab or overburden slab
need not be limited to finitude. Infinitely thick source slabs without
overburden are included in the analysis.
238
2) The radioactive equilibrium of the U decay chain nuclides and their
uniform distribution in the source slab allow equating the activity per
unit volume of any such nuclide to that of the parent. This is assumed to
238 3
be "1 pCi of U per cm " in uraniumbearing soils, or "1 pCi of
pOC . "3
Ra per cm " in uranium mill tailings piles, if the absence of the
226Ra progenitors 238U through 230Th in tailings is taken into
account. However, these last mentioned nuclides are of small consequence
in exposure rate calculations. To simplify the study, both uranium
bearing soils and (conservatively) tailings piles are assumed to contain
3 ' 238
activity concentrations of 1 pCi/cm of every nuclide from U to
stable lead.

3) The repository is nominally "moisture free", for purposes of analysis,
since the present method can not determine the buildup and attenuation
effects of water independently from the material in which it is entrapped.
The consequences of including or increasing soil moisture must be learned
indirectly, as results from the attendant increase in soil density. Cal
culations implementing the analysis assume a soil density of 1.6 g/cm ,
corresponding to the densities of "dry packed tailings" and "moist packed
earth" studied by Schiager (1974) and that of soil containing "10% water"
by Beck (1972).
4) To facilitate analytical application of Taylor's and Berger's Buildup
Factor Formulas, the buildup properties of both the source slab and over
burden material are assumed to be sufficiently similar as to be charac
terized by the same set of energydependent fitting parameters A, alt
aa, etc. This similarity may be expected to extend to other properties
of relevance, such as densities and massattenuation coefficients, or
linear attenuation coefficients; it is so assumed in the calculations
implementing the analysis. Nevertheless, the analysis does not require
equal linear attenuation coefficients for source slab and overburden, thus
they are allowed to differ in the analytical development, as a concession
to greater generality.
5) The assumption of a sourcefree overburden does not consider the migratory
222
capabilities of Rn gas, which will permeate the overburden slab
transforming it into a "secondary" repository of uranium decay chain
222
nuclides, from Rn to stable lead. Consequently, the direct results
of the analysis and subsequent numerical treatment will apply strictly to
uraniumbearing soils (or uranium mill tailings piles) covered with
10

overburden impervious to radon gas penetration. Nevertheless, the numerical
models developed for this specialized case are useful in dealing with radon
permeable overburdens, as demonstrated in another section.
238
Any single decay of a nuclide in the U series may be accompanied by
the release of photons of specific energies, characteristic of the decaying
nuclide, with probability of emission varying according to photon energy.
These probabilities of emission, or "intensities", have been determined for
all the characteristic photon energies observed in the decay of each uranium
series nuclide, and tabulated as dimensionless decimal fractions or
percentages with the implicit units of "number of photons of energy E emitted
per decay" of the nuclide of interest. The product of the latter and a known
amount of this nuclide (in activity units of "decays per unit time) produces a
set of "source strengths" or "emission rates" of "photons of energy E emitted
per unit time" by the given quantity of the decaying nuclide, for all the
energies E characterizing this decay. Applying this process to each of the
pOQ
"ฐU decay chain nuclides, uniformly distributed throughout the source slab
3 23
with equilibrium activities of 1 pCi/cm (or 3.7 x 10 decays/cm per
second), generates an ensemble of "volumetric source strengths S (E)" 
o
bearing units of "photons of energy E emitted per cnr per second"  for
every photon energy E released in the chain.
The product of any given volumetric source strength S (E) and an
infinitesimal volume element dV is analogous to a joint source, emitting
Sy(E)dV photons of energy E per second, not unlike the joint sources of
Equations (1) and (3). To exploit this analogy in the context of the present
physical model it is helpful to examine Figure 1.
11

Point of assessment,
on surface of
r, distance between dV
and point of
assessment
dV,
volume element,
(dV = r2sineded

ut(E)(rdsece)  uc(E)dsece
SV(E)B(E)    dV (7)
whered

Table 1. Analytical expressions of flux <(E) at the surface of an infinitely wide uraniumbearing
soil slab or tailings pile of uniform thickness, covered with an overburden of uniform depth, based
on Taylor's buidup factor formula and valid for all E ^ 0.5 MeV. An equation originating from Berger's
form of the buildup factor, for all E < 0.5 MeV, has been included for the conditions of maximum flux.
Tailings Pile or UraniumBearing Soil of Finite
Thickness "t".
t = independent variable
Tailings Pile or UraniumBearing Soil of Infinite
Thickness.
t = oo
Cover Material
or Overburden
of Finite
Thickness "d".
d = independent
variable
SV(E)A(E)
General Case:
finite, variable t,d
2pt(E)
Special Case: infinite t; finite, variable d
for studying effects of cover thickness in reducing

based on Taylor's form of the buildup factor, limiting the application of
Berger's form to the case of "maximum flux", as explained below.
For each of the four cases covered in Table 1, a relationship between 0.5 MeV. These exposure rates correspond to fluxes determined
with Taylor's coefficients (see "special cases" in Table 1), and roughly 85%
238
of the total energy emitted by the U decay chain at equilibrium, a factor
that supports the inferred relationships. Emphasizing the variable of
concern, they may be expressed as
ฃx(E.d)
all E > 0.5 MeV
ฃX(E.O)
all E > 0.5 MeV
ฃx(E,t)
all E > 0.5 MeV
ฃX(E,)
all E > 0.5 MeV
ฃx(E,d)
all E
ฃx(E,0)
all E
ฃx(E.t)
all E
ฃX(E.)
all E
_ X(d)
X(0)
x(t)
X()
(8)
and ...
ซ.ll^.rv^fcj_ti _ i i r~ \//j_\
(9)
where X(E,d) = exposure rate due to photons of energy E, with cover of depth d,
assuming infinitely thick source slab (see Table 1).
15

X(E,t) = exposure rate due to photons of energy E, with a source slab
of thickness t, assuming absence of cover (see Table 1).
X(E,0) = X(E,oฐ) = maximum exposure rate due to photons of energy E, with a bare
source slab of infinite thickness (see Table 1).
X(d) = total exposure rate, with cover of depth d, assuming
infinitely thick source slab (simplified notation).
X(t) = total exposure rate, with a source slab of thickness t,
assuming absence of cover (simplified notation).
*
X(0) = X(ฐฐ) = maximum total exposure rate, with a bare source slab of
infinite thickness (simplified notation)
A corollary assumption implicit in (8) and (9) is that the partial sum of
exposures due to all E < 0.5 MeV depends on d and t in exactly the same manner
as the partial sum of exposures due to all E ^ 0.5 MeV. This may be only
approximately true. As Beck (1981) points out, "low energy photons will
clearly be attenuated and absorbed at a faster rate than higher energy photons
[with increasing depth of overburden]," although recognizing that the error
(overestimate) is "relatively small since the low energy sources contribute
only a small fraction of the exposure" allows retaining Equations (8) and (9)
as valid approximations.
With numerical implementation, the ratios in (8) and (9) can express the
dependence of total exposure rate on d and t without specifying the magnitude
of the maximum total exposure rate X(0) = X(ซ)  i.e. the case of a bare
source slab of infinite thickness. This allows for a separate evaluation of
the maximum exposure rate, without jeopardizing the reliability of the above
dependence by the inclusion of terms of conceivably lesser accuracy. Such a
separate evaluation would consist of a summation of maximum exposure rates
16

corresponding to all energies emitted in the source slab, both above and below
0.5 MeV. In the present context, this means adding maximum exposure rates
obtained using Taylor's coefficients to the somewhat more tentative maximum
rates based on Berger's buildup formula, despite integration inconsistencies
discussed in a previous paragraph. The relevant flux formulas are presented
in Table 1 ("maximum flux, exposure rate, case") with further details given in
"Implementation."
The formulas in Table 1 do not include the minor contributions of
"skyshine" (see Appendix B). Determination of this component by the buildup
factor method would require buildup coefficients for "air", unavailable in the
consulted references. On the premise that the "skyshine" effect is minor for
bare source slabs, and totally negligible for covered slabs, exposure rates
obtained from Table 1 and Equation (2) are valid for the airground
interface. A simple modifying factor was sought to convert these to the
corresponding exposure rates at one meter above ground, for a closer
comparison with previously published values. Such a conversion should,
ideally, account for the energydependent buildup and attenuation capabilities
of the intervening meter of air. However, the unavailability (or
nonexistence) of buildup factor coefficients for "air" leads to a simpler
approach, limited to attenuation effects.
The modifying factor is expressed as the ratio of exposure rate at one
meter above ground, including air attenuation effects, to the corresponding
exposure rate at ground level, for photons of a given energy E. Since the
correction is intended primarily for the case of maximum exposure rate, the
source slab is assumed to be infinitely thick, with an air "cover" of
generalized height h (1 meter, in this case). The assumption of infinite
17

thickness also simplifies analysis while remaining consistent with situations
encountered in practice, since exposure rates from an "infinitely" thick slab
compare closely to those from any slab over 1 to 2 feet in thickness, as shown
in "Results." The analysis involves the use of yet another version of the
buildup factor, the "linear form",
BL(E,ur) = 1 + 0(E)v(E)r (10)
Leaving pertinent details to Appendix C, the analytical process results in the
modifying factor
(11)
The product of (11) and any of the flux formulas in Table 1 represents the
corresponding flux at a height h above ground level. Setting h = 100 cm and
applying the results in (2) produces exposure rates corrected for air
attenuation at one meter above ground, facilitating comparison to previous
results. The exclusion of buildup effects in (11) implies a slight
underestimation of these exposure rates, just as neglect of "skyshine" effects
produces a similarly small underestimation of exposure rates at ground level.
However, these two effects are not cummulative, and may be balanced, to some
extent, by the overestimate in lowenergy exposure rates described in
Appendix G.
18

Implementation
Determination of total exposure rates X(E), at ground level and at one
meter above the surface, requires establishing the values of S , pt, yc,
yairป (uen/p)air' A> ฐ1ป ฐ2ป C and D for every ener9^ E borne ^
photons emitted in the source slab. Some of these parameters, in conjunction
with the cover and source slab thicknesses d and t, produce the argument of
the second order integral ^ which must also be quantitatively determined
[see Table 1 and Equations (2) and (11)]. All but one of these parameters may
be expressed as piecewise continuous functions of energy or of the argument,
in the case of Ep, by means of curvefitting equations. The only exception
is the volumetric source strength Sv, which is not a continuous function of
energy, and entails a tabulation.
The tabulation consists of "source terms" S (E), in units of "photons of
energy E per cm per second," corresponding to all the possible photon
energies E accompanying decay of source slab nuclides. Quantitatively, these
entries represent the products of intensities, in "photons of energy E emitted
per decay," and the rate of decay equivalent to an assumed equilibrium
activity of 1 pCi/cm (or 3.7 x 10 decays/cm3 per second) of each
nuclide in the uranium series from 238U to 206Pb. Omitted from
consideration are 218Po, 210Bi and the branch decay nuclides 218At,
Tl and Tl, since they are not photon emitters or have an extremely
low probability of emitting gammas (see decay scheme in Appendix D).
The need to rely on tabulated values necessitates the use of numerical
integration techniques in obtaining a total exposure rate X(E). For ease of
implementation, the 282 volumetric source strengths S (E) calculated with
19

decay data from Kocher (1977) are distributed between two tables, in Appendix
D. One of these, with 105 SV(E) values for energies E<0.5 MeV, is meant for
applications of Berger's buildup formula (see Table 1). The remaining 177
entries, for E>0.5 MeV, serve as input to the various expressions in Table 1
derived with Taylor's buildup coefficients.
Values of Taylor's buildup parameters A, aj, and

Values attained by [B](E) in the energy range 0.5 MeV <_ E ฃ 3.0 MeV, for each
of the materials under consideration, are compared graphically in Appendix E.
The comparison establishes that Taylor's buildup parameters for either "water"
or "ordinary concrete" generate the highest values of buildup [B](E) in the
range of energies examined. This necessitates additional criteria to effect a
choice.
The decisive selection criterion originates from the need to extend analysis
to energies below 0.5 MeV, lower limit of applicability for Taylor's buildup
coefficients. Berger's coefficients for "water" are available for energies
down to 0.255 MeV, while those for "ordinary concrete" and other materials do
not exist for energies under 0.5 MeV (Trubey, 1966). By elimination, the
buildup properties of "soil" are maximized by using "water" buildup
coefficients.
On the basis of the above selection, Taylor's buildup coefficients A(E),
oj(E) and a2(E) for water (i.e. "soil") are represented in Appendix F by
the corresponding number of energydependent curvefitting equations. A
different method is applied to Berger's coefficients, since these are used
only for conditions of maximum flux and exposure rate. Referring to the
appropriate equation in Table 1, these conditions can be seen to result in a
"buildup term" jl+C(E)/[D(E)l] j totally independent of spatial parameters
d and t. This allows expressing the entire "buildup term," in a compound
manner, as a single energydependent variable, and representing it accordingly
by a curvefitting equation.
21

The dependence of this compound "buildup term" on C(E) and D(E) allows it to
be represented in different ways, corresponding to the manner in which the
energydependence of C and D is expressed. The latter varies according to
what range of distances between point source and detector requires application
of Berger's buildup factor formula, with 7 MFP*, 10 MFP, 15 MFP and 20 MFP
fits reported by Trubey (1966). A discussion in Appendix F suggests that
Berger's coefficients C and D based on a 7 MFP fit are appropriate for 0.255
MeV sources in an infinitely thick soil slab having "water" buildup
properties. For 0.255 MeV gammas, 7 MFP in water are approximately 55 cm,
which matches closely the slab thickness equivalent to an "infinite" slab when
exposure rates are calculated with Taylor's coefficients, as will be seen in
"Results".
Accordingly, the bracketted "buildup term" { 1 + C(E)/[D(E)1]2 j is
represented by a curvefitting equation using C(E) and D(E) values based on a
7 MFP fit. This selection provides the added advantage of greater accuracy,
as discussed in Appendix F, and a correspondingly more solid base for
extrapolations. Since the parameters C(E) and D(E) are not available for
energies below 0.255 MeV, some judicious extrapolation is required to cover
the remainder of the photon energies emitted in uraniumbearing soils.
In lieu of extrapolation, an energydependent "correction term" is added, for
energies below 0.255 MeV, to the curvefitting equation describing the
"buildup term" as function of energy. The net effects of the correction
include a buildup of "1.0" at E = 0.01 MeV and a maximum buildup occurring at
E 0.12 MeV, meeting constraints set in Appendix F. The energy of maximum
* MFP  meanfreepath lengths, as multiples of u(E)r = 1
22

buildup reflects mathematical convenience, without benefit of new or special
insights into buildup in water (or soil) at small energies. Nevertheless, a
rough analysis by Evans (1972) suggests that the assumption of maximum buildup
at 0.12 MeV is not in great error.*
Conservative maximization of the buildup properties of soil was an
important factor in selecting water as a soil surrogate. The effect of this
selection on flux may be gaged by examining Equation (12). Assuming the
attenuation coefficient vt(E) for soil to be known, and to be equally valid
and applicable to all possible soil surrogates, maximum buildup at a given
energy would inevitably lead to maximum flux (see Appendix E). Furthermore,
if this were true .for all the energies of concern, a maximum total exposure
rate would be equally certain.
A reasonable estimate of ut(E) as function of energy may be obtained from a
graph in Appendix G, showing the energydependent behavior of massattenuation
coefficients p/p of typical soil components, including water. For energies E>
0.23 MeV, the y/p coefficients of these materials lie within a narrow band of
values, with a maximum difference of about 15% (between H^O and Fe).
Consequently, the product of any such coefficient times the density of soil,
o
assumed to be 1.6 gm/cm , will represent ut(E) with a maximum possible
* Evans' estimate of buildup, for point sources in an infinite medium, as
function of Compton scattering, total attenuation and total absorption linear
coefficients, respectively os, p0 and ua.is
B = 1 + y (vQr) ["The Atomic Nucleus",Chapter 25, Eqn. (4.18)]
a
The energydependence of os, MO and ya implies a maximum B in the range
0.06 MeV < E < 0.09 MeV, in water, and in the range 0.09 MeV < E < 0.15 MeV
in aluminum (indicated as an alternative replacement for "soil" by Beck, 1981).
23

error of 15%. Taking into account that Si and 0 constitute 75% of soil, by
weight (Hammond, 1966), reduces this maximum error to the probable range of
10%  12%.
However, buildup and attenuation are not independent effects (see Evans's
approximation, footnote of preceding page) and thus the choice of p/p coeffi
cient may not be entirely arbitrary. Since water was used to represent the
buildup properties of soil, the corresponding choice of (U/P)^ Q for the
massattenuation coefficient of soil would maintain consistency. Some
consequences of this consistency are viewed in Appendix 6, with emphasis on
resultant compensating errors. The rest of this appendix is allocated to
curvefitting equations for (v/p)u n> (U/P),,,. and (VU^/P)..:.., as functions of
\\n\j air en air
energy, and for the 2nd order exponential function E2(x) as function of the
argument.
In both Appendix F and Appendix G, the accuracy of the various curve
fitting equations is emphasized by reference to maximum errorsoffit (at any
point) ranging from 0.5% to ฑ1.74%. The only exception is the 3% error
estimated at E = 2.45 MeV in the curve fit for o^(E), a parameter of small
magnitude always added to "1.0", which effectively reduces this maximum error
to approximately 0.1% (see Table 1). Consequently, curvefitting inadequacies
must be eliminated as a potential source of major error  the above piecewise
continuous functions of energy appear to be viable alternatives to inter
polating subroutines commonly used in computer implementation.
The present scheme of computer implementation, designed to obtain total
exposure rates based on Table 1, Equation (2), tables and energydependent
parameters in the various appendices, etc., is outlined in Appendix H. An
24

application of this scheme to the case of a monoenergetic emitter ( K)
uniformly distributed throughout a bare, infinitely thick source slab is
presented in Appendix I, as illustration.
25

Results
The theoretical and empirical foundations of the buildup factor concept
have been extensively discussed in the leading section of this report, to
provide the necessary solid basis for subsequent analytical development. The
resultant mathematical formulations in Table 1, Equations (2), (11), etc. are,
in their context, generally valid and represent an equally reliable
operational base for quantitative implementation of the method. However, the
translation from generality to specificity required in the implemental process
incorporates approximations, simplifications and extrapolations of unverified
effect on accuracy of results. Above all, the unavailability of buildup and
attenuation parameters for soil and their substitution with the corresponding
coefficients for water indicates that the method, however analytically sound,
produces results that must be regarded as only tentatively valid. To test
their validity and, by implication, that of the techniques employed, these
results may be usefully compared with the results and models of previous
investigators, such as Beck (1972) and Schiager (1974).
I. Previous Results and Models
Beck (1972) employs a polynomial series approximation to the Boltzmann
238 232
transport equation to determine exposure rates due to U, Th and
40
K decay chain gamma emitters, distributed uniformly in the ground with
infinite halfspace geometry. Two of his results are particularly relevant to
present purposes. Using the simplified notation of Equations (8), (9) and
(47C), they are:
26

exposure rate at 1 meter above the surface of a bare, infinitely thick source
40
slab containing a uniform distribution of K, source of 1.464 MeV gammas,
Xlm(~) = 0.179 uR/h per pCi/g (14)
total exposure rate at 1 meter above the surface of a bare, infinitely thick
source slab containing, in uniform distribution and radioactive equilibrium,
poo
all the nuclides of the U decay chain through
Xlm(ป) = 1.82 pR/h per pCi/g (15)
Schiager (1974) draws from experimental data available to him to propose a
buildup factor for calculations involving tailings piles which, in present
notation, is
B = e[nt/(l + wt)]
(16)
where u = 0.11 cm" , attenuation coefficient for "dry packed tailings" or
"moist packed earth" of density 1.6 g/cm
With other correction factors, Schiager1s model of total exposure rate as
function of a bare tailings slab thickness t may be expressed, in the
simplified notation of Equation (9), as
X(t) = 0.92[lE2(ut)]e[gt/(1+yt)]yR/h per pCi/g (17)
It follows from the above that, for bare, "infinitely" thick tailings, the
total exposure rate is
27

X(ฐฐ) = 0.92 e vR/h per pCi/g (18)
= 2.5 uR/h per pCi/g
In addition, Schiager (1974) includes a graph of decreasing exposure rate as
function of increasing thickness of overburden. Using the simplified notation
of Equation (8), this is interpreted as
X(d) e"d/L (19)
Xlo) "
where L = soil relaxation length with respect to exposure rate, cm
= 14 cm, in Schiager's graph (1974)
Schiager's equations are intended for tailings, thus primarily for Ra and
238
daughters rather than for the more inclusive U decay chain. However, the
Ra decay chain comprises roughly 98% of the total energy emitted by the
238
U chain (see Appendix D). Neglecting this minor difference, Schiager's
results may be compared to those of the present report.
II. Comparison With Present Results
The first comparison is useful in testing the accuracy of the
approximation for usoi(E) in Appendix G, namely
"soil
. P
Psoi1 (20)
o
where psoj = 1.6 g/cnr
28

To that effect, Beck's result for the monoenergetic 1.464 MeV gammas from
K, in Equation (14), is contrasted to that produced by the present simpler
method, detailed in Appendix I,
XlmH = 0.172 uR/h per pCi/g (21)
The present result is less than 4% smaller than Beck's corresponding value
in Equation (14), suggesting that use of water buildup parameters with the
approximation in (20) and Appendix G is not unreasonable for energies above
0.25 MeV. By implication, the use of (20) for energies below 0.25 MeV should
produce conservative results (see pertinent discussion in Appendix G).
Another valuable comparison involves maximum total exposure rates at one
meter above the airground interface of a tailings pile or uraniumbearing
poo
soil, containing all the uranium series nuclides from U (inclusive)
210
through Po in radioactive equilibrium and uniform distribution.
Calculation 2 in Appendix H represents the computerimplemented application of
the present method, generating
Xlm(ป) = 1.96 yR/hr per pCi/g (22a)
This value is 7.7% higher than Beck's in (15). However, the entire energy
poo
spectrum for the U decay chain was employed in arriving at the result in
(22a), whereas Beck explicitly excluded xrays and low intensity gammas from
his calculations. Eliminating the contributions of "the same to (22a) permits
a more valid comparison:
XlmH = 1.89 uR/hr per pCi/g (22b)
29

The reduced exposure rate is less than 4% higher than Beck's in (15).
Recalling the result in (21), it can be tentatively concluded that the present
method estimates exposure rates within ฑ4% of Beck's results, when adjusted
for proper comparison.
The corresponding maximum total exposure rate at ground surface, including
poo
all energies in the U decay chain, is
X(ป) = 2.06 yR/hr per pCi/g (23)
This result is 5% higher than the exposure rate at one meter above ground,
Equation (22a), whereas Beck (1972) mentions a corresponding difference of
only 2%.
Schiager (1974) evidently ignores these minor differences, describing his
results as "exposure rate over the slab", applicable to "a point near the
surface." His maximum exposure rate of "2.5 pR/h per pCi/g" is 21% greater
than the ground surface maximum in (23) and 28% greater than the maximum at
one meter above ground, in (22a). A comparison of Schiager's maximum to those
produced by reduced spectra, i.e., excluding xrays and low intensity gammas,
leads to still greater differences, as may be expected. Thus, Schiager's
maximum is 32% greater than the corresponding value in (22b) and 37% greater
than Beck's maximum in (15), both for 1 meter above ground.
Based on the above discussion, the most suitable application for Schiager's
model is in describing exposure rates at ground level. Nevertheless, his
maximum exposure rate appears exceedingly conservative when compared to the
various maxima obtained by Beck and the present method. The latter results
30

may be alleged to be mutually supportive, to an extent limited by the
substitutions, approximations and other inadequacies of the present method.
By contrast, Beck underestimates the maximum exposure rate, by excluding
xrays and gammas with intensities less then 0.1% (1972). This eliminates
over 200 entries from the tables in Appendix D, pertaining to xrays, weak
gammas represented by their summed intensities and average energy, and gammas
of effectively low intensity due to alternate decay modes with low branching
234
ratios (Pa)*. Although individually insignificant, their summed products
of energy times intensity represent a potential 4.6% increment to the total
energy emitted in Beck's source spectrum. Being fairly representative of the
spectrum as a whole, with energies ranging from 0.01 MeV to 1.93 MeV, these
omitted photons may proportionately increase Beck's maximum exposure rate of
"1.82 uR/h per pCi/g" to as much as "1.9 pR/h per pCi/g."
III. Models Based on Present Techniques
and Comparison with Previous Models
The discrepant estimates of maximum exposure rate in the preceding section
indicate the existence of uncertainties in the bases and processes of such
estimation. These uncertainties contributed to the rejection of models
explicitly postulating numerical values of maximum exposure rate, in favor of
expressions describing the dependence of ratios X(d)/X(0) and X(t)/X(a,) on
varying d and t. Such "relative effect" models have the advantage of
substantially reducing potential error, through mutual cancellation of terms,
while avoiding commitment to a maximum value.
* The tables in Appendix D do not differentiate between xrays, weak gammas
and gammas of intensity greater than 0.1%. To verify the assertion motivating
this footnote, reference to Kocher (1977) is suggested.
31

Accordingly, Calculations 3) and 4) in Appendix H were repeated with several t
and d values, generating X(t) and X(d) ground surface exposure rates which
were then normalized with respect to X(ซ) = x(0) = 1.765 yR/h per pCi/g,
maximum exposure rate, at ground surface, due to gammas of energies above 0.5
MeV. The resultant sets of ratios X(t)/X(ซ>) and X(d)/X(0), displayed in
Tables 2 and 3, are expected to apply at ground surface and at one meter above
OOQ
ground level, for exposure rates due to the entire U energy spectrum,
i.e. including the 15 of total energy emitted in the range E<0.5 MeV.
The results in Table 3 [from Calculation 4)] are particularly useful in
the development of mathematical models. The depthdependence of the ratio
X(d)/X(0) has been often expressed as a decreasing exponential function with
an argument "d/L", where d( is the depth of cover and L is the "relaxation
length" [see Equation (19)]. This relaxation length represents the thickness
of cover required to reduce the exposure rate by a factor of "e", and is
assumed to be constant for a given material, e.g. Schiager estimates it to be
=14 cm, for soil, in (19). The ratios X(d)/X(0) in Table 3 allow testing the
accuracy of this assumption by rearranging (19) to produce
L or L(d) : (24)
In^ [from Table 3]
*(0) ^
The results of (24) are included in Table 3. They indicate that L is by no
means a constant, but a well defined function of d. Furthermore, when
graphed (Figure 2) they suggest that the increase in L(d) as d increases is
not a transient phenomenon for the range 1 cm ฃ 100 cm, but that the trend
will continue for higher d. It is clear, however, that L(d) is a slowly
varying function, particularly as d increases  thus statistical fluctuations
32

Table 2. Proportional Increase in Exposure Rates*, with Respect to Maximum,
as Function of Increasing Thickness of the Uranium Bearing Slab
Thickness t of
UraniumBearing
Slab, cm
1
2
3
4
5
6
7
8
9
10
15
20
30
40
50
60
70
80
90
100
03
Exposure Rate X (t) *
Due to Slab of thick
ness t, yR/h per pCi/g
0.31176
0.51318
0.66856
0.79662
0.90348
0.99993
1.0848
1.1560
1.2192
1.2754
1.4765
1.5914
1.6993
1.7391
1.7544
1.7601
1.7630
1.7641
1.7645
1.7647
1.7649
Ratio of Exposure rate X(t)
to Maximum Exposure Rate
X(ฐฐ): X(t)/X(ฐฐ), dimensionless
0.17665
0.29077
0.37881
0.45137
0.51192
0.56657
0.61464
0.65500
0.69080
0.72264
0.83660
0.90170
0.96283
0.98536
0.99406
0.99753
0.99896
0.99955
0.9998
0.99991
1.0
* Tabulated exposure rates X(t) [including X(ซ) = 1.7649 yR/h per pCi/g]
represent summations of exposure rates due to all gammas of energy greater
than 0.5 MeV, using Taylor's buildup factor parameters. Since energies E >0.5
MeV comprise over 85% of the total energy emitted by the 238U decay chain at
equilibrium, the resulting ratios are expected to apply to exposure rates due
to the entire 238U energy spectrum [with X(ซ) = 2.06 yR/h per pCi/g,per example]
33

TableS. Proportional Decrease in Exposure Rates1), With Respect to Maximum,
as Function of Increasing Thickness of the Overburden Slab, d
2}
Thickness d '
of Overburden
Slab, cm
0
1
2
3
4
5
6
7
8
9
10
15
20
30
40
50
60
70
80
90
100
Exposure Rate '
X(d), With Cover
Slab of Thick
ness d, vR/h/pC1/g
1.7649
1.4531
1.2517
2.0963
0.9683
0.8614
0.7650
0.6801
0.6089
0.5457
0.4895
0.2884
0.1735
6.559 x ID'2
2.583 x 10"2
1.048 x 102
4.348 x ID'3
1.837 x 10"3
7.881 x IQ4
3.422 x 10"4
1.502 x 10"4
Ratio of Exposure
Rate X(d) to Maxi
mum Exposure Rate
X(0), or X(d)/x(o),
dimensionless
1.0
0.8234
0.7092
0.6212
0.5486
0.4881
0.4334
0.3854
0.3450
0.3092
0.2774
0.1634
0.0983
3.717 x 102
1.464 x 10"2
5.937 x 10~3
2.464 x 103
1.041 x 103
4.465 x 10"4
1.939 x 104
8.510 x 10"5
Depthdependent
Relaxation Length
L(d),cm
5.145
5.832
6.301
6.663
6.971
7.177
7.341
7.517
7.668
7.797
8.280
8.622
9.112
9.469
9.753
9.990
10.193
10.371
10.529
10.671
^Tabulated exposure rates X(d) [including X(0) = 1.7649 yR/h per pCi/g]
represent summations of exposure rates due to all gammas of energy greater
than 0.5 MeV, using Taylor's buildup factor parameters. Since energies E >0.5
MeV comprise over 85% of the total energy emitted by the 238U decay chain at
equilibrium, the resulting ratios are expected to apply to exposure rates due
to the entire 238u energy spectrum [with X(0) = 2.06 yR/h per pCi/g,as example]
2'Limited to d ฃ 100 cm because of exponentially increasing computer
"roundoff" error.
34

14
12
10
L(d)
L(d) resultant from present computer  implemented
method (see fable 3) :  \
L(d) = d
This curve can be closely
approximated by
= d0e
where
e+i
)],
cm
= ..1 cm
(maximum observed error in fit
is approximately 1.0% at any point
1 10
Depth of Cover, d (cm)
Figure 2 . Depthdependent relaxation parameter L(d), as obtained by the present
computer implemented model. Accompanying the graph is a curvefitting equation by
the present author, which replicates the graphed results with a maximum observed
error of 1.0% , at any point.
100
35

and equipment inadequacies may frustrate experimental verification of the
functional behavior under many conditions. The present author represents L(d)
as
L(d)= d0e1/4ln[2e2(geM)] . in cm (25)
do
where L(d) = depthdependent relaxation length with respect to exposure
rate, cm
d = depth of cover, in cm
dQ = 1 cm
The resemblance of (25) to a theoretically derived function requires special
emphasis of the fact that it is merely a convenient fit of computer output
data. In the process of obtaining this "pseudoanalytical formula," fitting
coefficients corresponding to powers of the natural base "e" were found to
produce optimum results  replicating the values l_(d) in Table 3 with a maximum
curvefitting error, at any point, of 1.0%*. This discouraged the use of
simpler, but less accurate formulations of the type L(d) = a + b ln(d/dQ),
as discussed in Appendix J.
Replacing L in (19) with the depthdependent L(d) in (25) produces
X(d) .
(26)
* The import of the small curvefitting errors mentioned throughout this
section is discussed in the closing paragraphs of same.
36

This expression matches the corresponding X(d)/X(0) values in Table 3 with a
maximum observed curvefitting error of about 1.1%, at any point.
Graphical comparisons of Equations (25) and (26) with the models of Equation
(19) may be found in Figures 3 and 4. Both these figures demonstrate the
conservatism of L and X(d)/X(0) from Schiager (1974) in contrast to those of
the present method.
The ratios X(d)/X(0) in Table 3 may be theoretically related to the ratios
X(t)/X(co) in Table 2 by a relationship derived in Appendix K, for the special
cases t = d. which is
. for t  d (27)
X(ซ) X(0)
Applying Equation (26) to the above expression summarizes the ratios X(t)/X(ป)
as function of source slab thickness t,
(28)
To test the validity of (28) it is necessary to compare the values X(t)/X(ซ>)
obtained by this equation to the values in Table 2, which were obtained
independently [Calculation 3)] from those of Table 3. This comparison yields
a maximum curvefitting error of less than 1%, at any point (maximum error:
0.7%, at t = 5 cm).
37

CO
00
30
20
10
9
8
7
Relaxation constant L =14 cm (present author's interpretation) from Schiager (1974)
Depthdependent relaxation "parameter" L(d), from present
computer implemented method,
L(d>
(cm)
2
L(d)=d0e1/4ln[2e2(f e+0]
, in cm
where do= 1 cm
(This curvefitting equation replicates computerresults
with a maximum observed error of 1.0 % at any point)
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
Cover Slab Depth d , cm
Figure 3 . Comparison of proposed depthdependent relaxation "parameter" L(d), from results obtained
with the present computer implemented model, with the relaxation constant from Figure 4 in Schiager
(1974), attributed to Throckmorton(1973), as interpreted by the present author.

1.0
10
1
10
2
X(d)
X(0)
10
3
10
4
10
5
Model from Schiager (1974), interpreted
X(d)_ e/L
X(0)
\ where L ^ 14 cm
x
X
X,
Results obtained
by oresent computer 
implemented method
(Table 3), closely fitted
(with a maximum observed error
of 1.1 *) by
T:rr^ 1
X(d)_ 0 U/4ln[2e2(^e + l)])
\
where d0= 1 cm
i.e. leading to use of /d0 as a dimensinless
variable.
10 20 30 40 50 60 70
Cover Slab Depth d, in cm
so
90
100
Figure 4 . Relative decrease in exposure rate, with respect to
maximum exposure rate possible, as function of increasing thickness
of the overburden slab, as obtained by the present proposed computer
implemented method (Table 3), fitted by applying the proposed model
of the depthdependent relaxation length L , and compared to a model
from Schiager (1974).
39

The results in Table 2 or from Equation (28) can be compared with those of
Schiager (1974), obtained for tailings. A formalized expression of Schiager's
ratios X(t)/X(ฐฐ) may be produced by dividing Equation (17) by Equation (18),
both from Schiager's model,
(29)
"V /
where y = Schiager's linear attenuation coefficient in dry tailings
=0.11 cm"1
Graphs of X(t)/X(oo) and X(t) from (29) are presented in Figures 5 and 6,
respectively, along with the corresponding graphed values from Table 2 and
Equation (28) for the present model. The latter strongly support Schiager's
statement that For any situation involving tailings depths of more than 1 or
2 feet...the external exposure rate over the tailings can be calculated as
follows:
X(PR/h) = 0.92eCRa(pCi/g) = 2.5 CRa(pCi/g)"
where CRa = Radium concentration, pCi/g (in Schiager's notation)
In other words, maximum exposure rates [see Equation (18)] should be closely
approximated with source slab thicknesses of 1 or 2 feet. In meeting this
criterion, the present model is superior to Schiager's, as may be verified by
comparing ratios X(t)/X(oo) from Equations (28) and (29), graphed in Figure 5.
The present model predicts 97% of maximum exposure rate at t = 30.5 cm (1
foot) and 99.8% at t = 61 cm (2 feet), whereas Schiager's corresponding values
are not quite 4/5 and 9/10, respectively. To attain 97% of maximum exposure
40

x(t)
.8
.6
.5
.3
.2
Present Computer Implemented Model.
This Curve Can Be Closely Approximated By:
t/tp
Wher* te = 1 cm
( Maximum error
in curve fit
is 0.7% ) /
/
/
/ *    Schiager's Model . _MlL [i _
X (oo) L
_../
.__//..
/
where M o.n cm
I
10 20 30 40 50
Thickness t of Uranium Bearing Soil Slab, in cm
60
70
Figure 5 . Relative increase in exposure rate, with respect to maximum exposure rate possible,
as function increasing thickness of the uraniumbearing soil slab, as obtained by the present
computerimplemented method (Table 2), approximated by a curvefitting equation with a maximum
observed error in fit of 0.7 % , and compared to Schiager's model (1974).
41

2.6 
2.4 
2 2~
MaPฐsed Model =
Present Proposed Model X(t)=X(oo)
i
(Schiager's) X(t) = [1  E2(>Jt)l e 1+Mt X(oo) ^^ ^
where t0 = l cm
p = 0.11 cm"
i.e.L. is dimensionless
*
"*
a = activity, Pc'/g, Ra or U
The above expression is NOT an analytically derived function, but a curvefitting
equation with constant coefficients expressed as powers of natural base 6 =2.718....
These fitting coefficients were found to produce optimum resultsthe ratios X(t)/X(

rate, Schiager's model requires t=280 cm (9 feet), and as much as t=3700 cm
(120 feet) to reach 99.8%.
The two models show closer agreement when comparing exposure rates (Figure 6),
rather than the above X(t)/X(ฐฐ) ratios. However, this relative agreement is
limited to small source slab thicknesses t<40 cm, and merely reflects the
difference in maximum exposure rates X(ฐฐ)  Schiager's maximum of "2.5 pR/h
per pCi/g" is substantially higher than Beck's or the present model's maxima.
o
In summation, the present model expresses the dependence of exposure rate
on source slab thickness t in a manner consistent with Schiager's
observations, as quoted, has a reliable analytical foundation, and the support
of a method of implementation that produced maximum exposure rates within ฑ 4%
of Beck's results. One additional advantage of this model is the analytically
demonstrable relationship to the dependence of exposure rates on overburden
depth d [see Equations (26), (27) and (28)], which leads to an internally
consistent comprehensive model of exposure rate as function of d and t,
X(d.t)=Xmaxe
_ e
V

This comprehensive model consists, primarily, of the product of curvefitting
equations (26) and (28), describing the magnitude of ratios X(d)/X(0) and
X(t)/X(ฐฐ) as functions of their respective arguments. It may be recalled that
the choice of ratios to represent exposure rate dependence on d and t was
intended to reduce potential errors, while avoiding commitment to explicit
values of maximum exposure rate X(0) = X(ซ>). The resultant flexibility of
(26) and (28) allowed the formulation of the comprehensive model in (30)(see
Appendix J), while qualifying it to incorporate, within reason, different
values of maximum exposure rate Xmax. This is a distinct advantage of the
proposed model(s), since uncertainties in buildup and attenuation properties
of soil indicate that the present author's maximum exposure rates in (22a) and
(23) require further substantiation.
In that context, the ฑ 4% difference* between Beck's results (1972) and those
of the present study, although indicative of general agreement, nevertheless
represents a residual conflict that cannot be readily resolved. The
inadequacies of the present method do not allow proposing the resultant maxima
in preference to Beck's. On the other hand, Beck's exclusion of xrays and
lowintensity gammas leads to an underestimation of maximum exposure rate, by
up to 4.6% in terms of Beck's results. Consequently, assigning a specific
value to maximum exposure rate may be premature, a range of values being more
representative of persisting uncertainties. Prudence dictates that the limits
of such range be realistic but conservative. Two sets of limits are required
for the cases of current interest.
* Excluding the contribution of xrays and low intensity gammas from the
present results.
44

For the case of maximum exposure rate at one meter above ground, the present
estimate in (22a) provides an upper limit. The lower limit of the range was
generated by conservatively increasing Beck's result in (15) by 4.6% . Thus,
at one meter above ground,
1.90 uR/h per pCi/g <_ Xmax <_ 1.96 yR/h per pCi/g (31)
For maximum exposure rate at ground surface, the upper limit was obtained from
(23), i.e. an increase of 5% over the corresponding value in (31). Since Beck
estimated a difference of only 2% between exposure rates at ground level and
at one meter above ground, the lower limit in (31) was increased
proportionately, for consistency with his study. Thus, for ground surface
exposure rates,
1.94 yR/h per pCi/g <_ Xmax <_ 2.06 yR/h per pCi/g (32)
The value ranges (31) and (32) imply potential maximum errors of 3% and 6% ,
respectively, which may be assumed to represent the net effect of different
soilsurrogate materials, approximations, etc., in the two studies, but
excluding the effect of different spectra. These maximum potential errors
delimit the liability of using maxima from (31) or (32)* in the comprehensive
model of Equation (30). Including the combined curvefitting errors of (26)
and (28), a total of 1.8% , this model should express exposure rate as
function of d and t with a maximum possible error of less than 8% , for any
set of d and tvalues not exceeding 100 cm, severally.
* Implicit in the process of setting ranges (31) and (32) is the constraint
that a maximum exposure rate at ground surface, chosen from (32), should be 2%
to 5% greater than the corresponding maximum exposure rate at one meter above
ground level, from (31).
45

As pointed out in Figures 2 through 6, the models in (25), (26), (28) and,
by extension, the comprehensive model in (30), are based on results obtained
by computerized techniques. These are virtually indispensable for the
efficient performance of repetitive mathematical tasks, but introduce small
inaccuracies in the process of "rounding off" results to a prescribed number
of digits. The effect of such "computer round off errors" grows progressively
larger with increasing d and t, ultimately compromising the validity of all
results corresponding to d and t greater then 100 cm. Such effect is
immaterial in modeling X(t)/X(ป), but very significant in studying the
efficacy of cover thickness d in reducing exposure, as described by
X(d)/X(0). Since "small" round off errors in X(d) may represent differences
of orders of magnitude, the modeling of X(d)/X(0) was not extended beyond
results verifiable by Equation (27) and comparison of Tables 2 and 3, values
corresponding to larger d being left to extrapolation.
In the latter context, the graph of L(d) in Figure 2 suggests that any
expression providing an accurate fit to the values graphed should be
applicable, with resonable expectations of accuracy, to a range of cover
depths d extending beyond 100 cm. Since Equation (25) meets such requirement
with a maximum error of 1% , at any point, corresponding expectations of
generality accrue to this equation and the adjunct Equation (26), representing
X(d)/X(0). Such presumed generality does not negate the possibility of
increased error for values of d greatly in excess of 100 cm; it merely
restates that errors of such magnitude as to invalidate Equation (26)  and
thus (28) and (30)  cannot be anticipated on the basis of the graph in Figure
2 and the key equation (25). In that vein, the aforementioned equations are
included in the comparison of general models summarized by Table 4.
46

TABLE 4 . Comparison of Maximum Exposure Rates and Hodels Based on Present Technique With Previously Published Models and Values.
Schiager's Models _ CurveFitting Hodels. From Values Obtained by Present Technique
Exposure rate
tailings slab thickness t (cm)
and overburden depth d (cm) 
comprehensive model ^1 chain
As collated from Schiager (1974)
u r i
X(d.t) = 0.92 [l  E2(,jt)] e
X(d.t) =Xmax e
3(<e+l)J
Ratio of exposure rate due to
bare tailings slab t cm thick
to exposure rate due to infi
nitely thick slab  238U chain
Ratio of exposure rate due to
slab covered with overburden
d cm thick to exposure rate
due to bare slab  U chain.
As adapted from Schiager (1974)
= 1 e
As implied by Figure 4 in Schiager (1974)
H= e"/L
X(0)
X(0)
= e
.{_^__
t = tailings slab thickness,
in cm
d = overburden depth, in cm
t0 = d0 = 1 cm
Xmax = X(ป) = X(0) = maximum
exposure rate, obtained
with t = and d = 0
Relaxation length , in cm,
with respect to exposure rate
238U chain.
From Figure 4 in Schiager (1974)
L = 14 cm
L(d)=doe/4ln[2e2(
e
1)]
Overburden is assumed to be
impervious to radon gas, in
these models.
Schiager (1974)
Beck (1972)
Values Obtained by Present Technique
Maximum exposure rate at ground
surface  U chain.
Maximum exposure rate at 1 m
238
above ground surface Uchain
Same as above, excluding weak
238
gammas and Xrays  U chain.
Xmax = 0.92 e
per pCi/g
2.5 juR/h per pCi/g
Xmax = 2.06 >jR/h per pCi/g
XITMY = K% ^R/h per pC1/9
max '
X =1.89 ^R/h per pCi/g
Exposure rate over bare, infi
nitely thick deposit of 40K,at
1 m above surface.
Xmax =0.179 pR/h per pCi/g
Xmax = 0.172 >jR/h per pC /g

Applications
One of the primary purposes of this report is to demonstrate the
application of simple mathematical models, developed in the originating study,
to conditions of somewhat greater complexity than those envisioned in the
course of such development. It should be recalled that the analysis and
implementation ultimately yielding Equations (25), (26), (28) and the
comprehensive model in (30) were made possible by a number of simplifying
assumptions (Appendix B), which admit of conditions that are, generally,
improbable but conceptually not impossible. The relevant exception to this
generality is the assumption that radon will not emanate from the tailings or
uraniumbearing soil, implying a lack of motivity conceptually improbable and
generally impossible for a noble gas in a porous medium.
To illustrate one of the consequences of this faulty assumption, it suffices
to apply Equation (30) to the case of a bare tailings slab. Since the Xmax
222
value in (30) was obtained assuming that Rn does not diffuse out of the
source material, it follows that (30) will overestimate exposure rate.
The reverse is true when Equations (25), (26), and (30) are applied to
determine the shielding effects of cover. Unless the overburden is
ppp
impermeable to Rn, the exposure rates from a tailings pile covered with
overburden of thickness d will be substantially underestimated  by orders of
magnitude if d>100 cm. This is due to the fact that radon gas may be
generally expected to diffuse into the cover material, generating a source of
gamma rays with considerably less shielding than the thickness of the
overburden would indicate. Fortunately, models developed in the preceding
48

sections may be used to provide a more realistic estimate of exposure rates
due to a covered pile.
The first step in such determination is establishing the distribution of
222
Rn in the tailings and cover material. This will depend on the thickness
of cover d, the radon emanation power E, and the diffusion coefficient of
"free" radon in soil, D. Applying Pick's law to the general diffusion
equation, with the boundary conditions and treatment of Appendix L, results in
the following two equations:
in overburden C_ (z) = Ee~adsinh [a(dz)] (33)
(for z>0) CTOTAL
in tailings Ct (z) = 1  Eea(z~d)cosh(ad) (34)
(for z<0) 1TOTAL
where Cc (z) = Rn concentration in overburden, in pCi/g of free
I U I AL
radon, per pCi/g of Ra in tailings, as function of
distance z above tailingscover interfere.
(z) = Rn concentration in tailings, in pCi/g of both free
I U I AL ?9fi
and bound radon, per pCi/g of Ra in tailings, as
function of distance z below tailingscover interface.
z = generalized distance, normal to tailingscover interface
where z>0, above tailingscover interface,
z = 0, at tailingscover interface,
z<0, below tailingscover interface
777
E = emanation power, fraction of Rn free to diffuse out
of soil grains, dimensionless
49

/D , where \?2? = decay constant of 222Rn,
Rn *"Rn e i
= 2.1 x IQV1
D = diffusion coefficient of
Rn, in cm /s
d = depth of cover, cm
222
With the assumption that Rn is in radioactive equilibrium with all
daughter nuclides throughout the overburden and tailings, the distributions
given in (33) and (34) permit establishing exposure rates above the cover, by
the use of numerical integration techniques applying the comprehensive model
of Equation (30).
The techniques employed take advantage of the fact that the concentration of
nuclides increases with decreasing z, and of the linear relationship between
concentration and exposure rate, e.g. a concentration of 0.1 pCi/g will lead
to an exposure rate onetenth of that in (30). By representing the
concentrations in (33) and (34) as a set of discrete increments AC
corresponding to distance increments Az, an ensemble of infinitely thick slabs
with different nuclide concentrations AC is generated. All but one of these
slabs are represented as having sourcefree overburdens of thicknesses equal
to multiples of Az, according to the number of Az increments required to reach
the depth corresponding to a specific AC. This allows direct application of
Equation (30) to each of these slabs to calculate an element of exposure rate
AX (Appendix M).
The sum of all such elements AX results in a total exposure rate X
corresponding to a set of conditions comprising a given thickness of cover d,
a diffusion coefficient D, and an emanation power E. Setting E = 0.2, a set
50

of graphs for different D was obtained, describing the effect of increasing d
in terms of X(d)/X(0), in Figure 7.
1.0
10
i
io2
E=0.2
X(d)
X(0)
ID'3
_JOD 0.02
"^jD0,01
1D= 0.005
i i
(D= 0.002
DOiOOl
10
D=o,qp<
0=0.00(2
10"
D=0
10 20 30 40 50 60 70 80 90 100
Cover Slab Depth d, in cm
Figure 7 . Relative decrease in exposure rates, with respect to
maximum exposure rate possible, as function of increasing thickness
d of the overburden slab, for emanation power E = 20% and different
value of radon diffusion coefficient in soil, D, in the range 0.02
cm2/s > D > 0.0002 cm2/s.
51

In the case of a bare, infinitely thick tailings slab, the distribution of
radon is governed by Equation (34) with d = 0, which produces
C. (z) = 1  Eeฐz , for z < 0 (35)
1TOTAL
Applying to (35) the technique outlined in Appendix M, the effect of radon
emenation in reducing maximum exposure rates can be estimated. For the
specific case of E = 0.2 and 0.0001 cm2/s ฃ D ฃ 0.05 cm2/s, the process
yields results that may be approximated by the curvefitting Equation (36) and
Figure 8.
Xm,x (D) 0.21
maX =0.75/A\ (36)
= \V
where X_,v = exposure rate (maximum) over a bare, infinitely thick
maxE=0.2
tailings pile with an emanation power E = 0.2, as
function of diffusion coefficient D, in yR/h per pCi/g
(of 226Ra)
X = exposure rate (maximum) over a bare, infinitely thick
maxE=0
tailings pile without radon emanation, in uR/h per
pCi/g.
= Xmav in Equation (30)
ill a X
2
D = radon diffusion coefficient, cm /s
D = reference constant
= 1 cm2/s
52

.9
max
D, radon diffusion coefficient, cm Is
Figure 8. Exposure rate over a bare, infinitely thick tailings
pile reduced by radon emanation effects, as function of diffusion
coefficient D, for 0.0001 cm2/s < D < 0.05 cm2/s, and E = 0.2.
53

REFERENCES
Beck, 1972 Harold L. Beck, "The Physics of Environmental Radiation
Fields", The Natural Radiation Environment II, Adams, T. A. S.,
Lowder, W. M., and Gesell, T., Eds.Report CONF720805 (U.S.
E.R.D.A., Washington)
Beck, 1981 Personal Communication from H. L. Beck, D.O.E., Environmental
Measurements Laboratory, to D. E. Bernhardt, O.R.P.L.V.F.,
U.S.E.P.A., May 27, 1981
Evans, 1972 Robley D. Evans, "The Atomic Nucleus," McGrawHill Book Company.
Copyright 1955. Fourteenth printing May 26, 1972.
Ford, Bacon & Davis, 1977 Phase IITitle I, Engineering Assessment of
Inactive Uranium Mill Tailings, for U.S. Energy
Research and Development Administration (Grand
Junction, Colorado) Contract No. E(05l)1658,
Salt Lake City, Utah, 1977
G.E.I.S. Uranium Milling, 1979 Generic Environmental Impact Statement on
Uranium Milling. NUREG0511, Volume I,
Project M25, U.S. Nuclear Regulatory
Commission, Office of Nuclear Material
Safety and Safeguards, April 1979
Glasstone and Sesonske, 1967 Samuel Glasstone and Alexander Sesonske
Nuclear Reactor Engineering, Van Nostrand
Reinhold Company, 1967
Hammond, 1966 C. R. Hammond, "The Elements" Chemical Rubber Co. Handbook
of Chemistry and Physics, 47th. Edition, 19661967
Handbook of Mathematical Functions National Bureau of Standards, Applied
Mathematical Series, 55, U.S. Department
of Commerce, June 1964
ISIS, 1975 Hugh T. McFadden, "Interactive Statistical Instructional System
User's Guide" Computing Center, Lehigh University, June 1975
Kocher, 1977 D. C. Kocher, "Nuclear Decay Data for Radionuclides Occurring
in Routine Releases From Nuclear Fuel Cycle Facilities"
ORNL/NUREG/TM102, Oak Ridge National Laboratory, Oak Ridge,
Tennessee, August 1977
54

Morgan and Turner, 1967
K. Z. Morgan and T. E. Turner, editors Principles
of Radiation Protection John Wiley and Sons, Inc.,
T_7
Radiological Health Handbook
Bureau of Radiological Health and Training
Institute, Environmental Control
Administration. U.S. Department of Health,
Education, and Welfare, Public Health
Service, Rockville, Maryland, January 1970
Schiager, 1974
Keith T. Schiager, "Analysis of Radiation Exposures on or
Near Uranium Mill Tailings Piles", Radiation Data and
Reports. Volume 15, No. 7, RDDRA 4 15 (7) 375476 (1974),
U.S. Environmental Protection Agency, Office of Radiation
Programs, July 1974
Trubey, 1966
D. K.Trubey, "A Survey of Empirical Functions Used to Fit
GammaRay Buildup Factors", ORNLRSIC10 Oak Ridge National
Laboratory, Radiation Shielding Information Center, February
1966
55

Appendix A
Choice of Empirical Function to Represent GammaRay Buildup
Determination of external exposure rates from any radioactive source
generally requires a calculation of photon fluxes at the points of interest.
The latter procedure accounts for the interactions of electromagnetic
radiation with the materials it encounters between the point of emission and
the receptor. The effects of such interaction can be described in terms of
the two related concepts of "attenuation" and "buildup".
By ascribing to each photon an "identity" characterized by energy and
direction, the process of "attenuation" can be defined essentially as one of
"identity Joss", in which scattering and absorption interactions with matter
alter the direction and reduce the energy of the original or "primary"
radiation. For a well collimated beam, attenuation of primary photons
approximates a net loss of photons, since scattering would effectively remove
them from the narrow beam. The photon intensity drops exponentially with
distance, and is fairly easy to calculate, for such conditions. However, for
the more common "poor geometry" or "broad beam" situations, such calculation
would result in a sizable underestimation of photon flux at the point of
concern.
Calculation of gammaray exposure rates from sources distributed in
absorbing media must include the effects of "secondary" radiation, consisting
mostly of Comptonscattered photons with the addition of annihilation
radiation from pairproduction, and of Xrays resulting from photoelectric
56

interactions and bremsstrahlung. Determination of this extra contribution, or
"buildup", requires the solution of the Boltzmann transport equation for
photons, an extremely involved calculation that has been carried out by
several different techniques, with varying success. The most publicized of
these techniques, the "method of moments", has ultimately produced "buildup
factors" for point isotropic sources of up to nine energies between 0.255 and
10.0 MeV, embedded in infinite media consisting of water or one of six
elements with Z ranging from 13 to 92.
Paraphrasing Trubey (1966), a "buildup factor" may be defined as "the
ratio of any quantity of interest, characteristic of the total gammaray flux,
at a chosen point in a given medium, to the same quantity characteristic of
the unscattered flux at that same point". Thus, there exist energyflux
buildup, energyabsorption buildup, and dose (or dose rate) buildup factors
(Glasstone and Sesonske, 1967). The differences between the various buildup
factors are often neglected, but may be significant in critical calculations.
In addition to source energy and medium composition buildup factors are
also dependent on spatial coordinates, as implied by the definition and the
columnar arrangement of Table 1A. Since the latter pertains to isotropic point
sources in infinite media, such dependence is sufficiently expressed by
tabulated values corresponding to one single spatial variable "r", distance
from the point source. All other geometries would require an integration over
the dimensions of the source, with the spatiallydependent buildup factor
included in the integrand. This clearly necessitates expressing the buildup
factor as an explicit function of spatial coordinates.
57

Table 1A Dose Buildup Factor (B) for a Point Isotropic Sorts
Material
U'ater
Aluminum
Iron
Tin
Tungsten
Lead
Uranium
MoV
0.255
0.5
1.0"
2.0
3.0
4.0
fi.O
8.0
10.0
0.5
1.0
2.0
3.0
4.0
6.0
8.0
10.0
0.5
1.0
2.0
3.0
4.0
6.0
8.0
10.0
0.5
1.0
2.0
3.0
4.0
6.0
8.0
10.0
0.5
1.0
2.0
3.0
4.0
6.0
8.0
10.0
0.5
1.0
2.0
3.0
4.0
5.1097
6.0
8.0
10.0
0.5
1.0
2.0
3.0
4.0
fi.O
8.0
10.0
ur*
1
3.09
2.52
2.13
1.33
1 .69
1.53
1.46
1 .38
1.33
2.37
2.02
1.75
1.64
1.53
1.42
1.34
1.28
1.98
1.87
1.76
1.55
1.45
1.34
1.27
1.20
1.56
1.64
1.57
1.46
1.38
1.26
1.L9
1.14
1.28
1.44
1.4?
1.36
1.29
1.20
1.14
1.11
1.24
1.37
1.39
1.34
1.27
1.21
1.13
1.14
1.11
1.17
1.31
1.33
1.29
1.24
1.16
1 .12
1.09
2
7.14
5.14
3.71
2.77
2.42
2.17
1.91
1.74
1.63
4.24
3.31
2.61
2.32
2.08
1.85
1.68
1.55
3.09
2.89
2.43
2.15
1.94
1.72
1.56
1.42
2.03
2.30
2.17
1.96
1.81
1.57
1.42
1.31
1.50
1.83
1. 85
1.74
1.62
1.43
1.32
1.25
1.42
1.69
1.76
1.68
1.56
1.46
1.40
1.30
1 .23
1.30
1.56
L.64
I..58
1.50
1 .36
1.27
1.20
4
23.0
14.3
7.68
4.88
3.91
3.34
2.76
2.40
2.19
9.47
6.57
A. 62
3.78
3.22
2.70
2.37
2.12
5.98
5.39
4.13
3.51
3.03
2.58
2.23
1.95
3.09
3.74
3.53
3.13
2.82
2.37
2.05
1.79
1.84
2.57
2.72
2.59
2.4!
2.07
1.81
1.64
1.69
2.26
2.51
2.43
2.25
2.08
1.97
1.74
1.58
1.48
1.93
2.23
2.21
2.09
I .35
1.66
1.51
7
72.9
38.3
If). 2
8.46
6.23
5.13
3.99
3.34
2.97
21.5
13.1
8.05
6.14
5.01
4.06
3.45
3.01
11.7
10.2
7.25
5.85
4.91
4.14
3.49
2.99
4.57
6.17
5.87
5.28
4.82
4.17
3.57
2.99
2.24
3.62
4.09
4.00
4.03
3.60
3.05
2.62
2.00
3.02
3.66
2.75
3.61
3.44
3.34
2.89
2.52
1.67
2.50
3.09
3.27
3.21
2.96
7.61
2.26
10
166
77.6
27.1
12.4
8.63
6.94
5.18
4.25
3.72
38.9
21.2
11.9
8.65
6.88
5.49
4.58
3.96
19.2
16.2
10.9
8.51
7.11
6.02
5.07
4.35
6.04
8.85
8.53
7.91
7.41
6.94
6.19
5.21
2.61
4.64
5.27
5.92
6.27
6.29
5.40
4.65
2.27
3.74
4.84
5.30
5.44
5.55
5.69
5.07
4.34
1.85
2.97
3.95
4.51
4.66
4.80
4.36
3.73
15
456
178
50.4
19.5
12.8
9.97
7.09
5.66
4.90
80.8
37.9
!8.7
13.0
10.1
7.97
6.56
5.63
35.4
28.3
17.6
13.5
11.2
9.89
8.50
7.54
8.64
13.7
13.6
13.3
13.2
14.8
15.1
12.5
3.12
6.25
8.07
9.66
12.0
15.7
15.2
14.0
2.65
4.81
6.87
8.44
9.80
11.7
13.8
14.1
12.5
2.03
3.67
5.36
6.97
8.01
10.8
11 .2
10.5
70
932
334
82.2
27.7
17.0
12.9
3.85
6.95
5.98
141
53.5
26.3
17.7
13.4
10.4
3.52
7.32
55.6
42.7
25.1
19.1
16.0
14.7
13.0
12.4
__
18.8
19.3
20.1
21.2
29.1
34.0
33.4
_
(7.35)
(10.6)
14.1
20.9
36.3
41.9
39.3
(2.73)
5.86
9.00
12.3
16.3
23.6
32.7
44.6
39.2
__

(6.48)
9.68
12.7
73.0
2S.O
7S.5
* jur=mass absorption coefficient (p/p) X distance (cm) X shield density (g/cm2)
From the Radiological Health Handbook (1970)
58

There exist many expressions, or "forms", of the buildup factor as
function of source energy (E) and distance from the source (r). Three of the
best known are the "linear", "quadratic", and " cubic " forms of the buildup
factor, polynomials of the 1st, 2nd, and 3rd degree in r, respectively, with
energydependent coefficients. Two other polynomial forms, "Berger's" and
"Taylor's", include exponential terms with products of distance and
energydependent parameters both as coefficients preceding the exponential
functions and/or as function arguments. All but one of the five forms have
one common characteristic: that the energydependent fitting parameters are
valid up to a certain distance from the point source, and have to be replaced
with others once that distance is significantly exceeded. The attendant
discontinuities plus the fact that each succeeding set of parameters renders a
given form increasingly less accurate suggest the need for other choices for a
general treatment.
The sole exception to the above mentioned drawbacks is provided by
Taylor's Form of the buildup factor, which can be written
BT(E,yr) = A(E)e'ซi(E)w(E)r+[1.A]eoa(E)p(E)r (iA)
where Bf(E,iir) = energy and distance dependent buildup factor, dimensionless
A(E),a,(E),a2(E) = energydependent fitting parameters, dimensionless
u(E) = energydependent attenuation coefficient, cm'1
r = distance, cm
The energydependent parameters A, a,, and a2 are expected to retain their
validity to a great extent at most distances from the source, producing
buildup factors (thus, exposure formulas) of consistent accuracy. Table 2A
59

illustrates this consistency as contrasted to that of Berger's Form, which
is sometimes used as a standard of comparison (Trubey,1966). There is little
variation between the mean percentage deviations of Taylor's Dose Formula at 7
MFP (mean free paths) and the corresponding values at 20 .MFP, where 1 MFP =
yr. This is particularly true of water and the six pure elements originally
examined by the "method of moments" (seeTable 1A) and considerably less so for
the various types of concrete, which are mixtures.
Table 2A Comparison of Average Percentage Deviation of Dose Buildup
Factors for a Point Isotropic Source, Obtained Using Taylor's and
Berger's Formulas Versus Tabulated Buildup Factors, for Eight Energies
(Trubey. 1966).
Medium
Water
Aluminum
Iron
Tin
Tungsten
Lead
Uranium
Ordinary concrete
Ferrophos . concrete
Magnetite . concrete
Barytes concrete
20 MFP
Berger*
4.0
25
2.1
13
17
23
1.6
32
32
2.9
2.6
Mean Percent
Range
Taylor
36
2.8
25
1.9
1.6
0.8
0.8
2.9
2.6
4.2
34
age Deviation
7 MFP
Berger**
1.2
0.7
05
0.2
0.3
07
0.4
2.0
1.4
0.9
0.6
Range
Taylor
37
25
2.5
17
1.2
0.5
0.5
4.0
33
4.8
37
*20MFP parameters used.
**7~MFP parameters used.
(From ORNLRSIC10, "A Survey of Empirical Functions Used to Fit GammaRay
Buildup Factors," by D. K. Trubey dated February 1966, Oak Ridge National
Laboratory.
60

Appendix B
Simplifying Assumptions
The choice of Taylor's form of the buildup factor in Appendix A was
influenced by the need of dealing with extended sources of highly variable
dimensions, characteristic of uranium238 decay chain deposits when studied as
a group. Such a general study is greatly simplified by a number of
assumptions, presumed to apply for most soils containing U2^8 and/or
daughters, but with express emphasis on uranium mill tailing piles.
Assumption 1. Infinite Planar Extent of Tailings
a. Uranium mill tailings piles normally extend over tens of
thousands  often hundreds of thousands  of square meters
(Ford, Bacon and Davis, 1977).
b. External exposures on the surface of tailings piles are
usually characterized by "worst case" conditions  i.e., at
the center of the pile surface, ignoring "edge effects."
c. The major component of such exposures would be due to
photons traveling through soil, mostly. If exposures were
limited to these photons, a detector at the center of the
pile surface, a short distance above the airtailings
interface, would not distinguish between a large, though
finite, area and one of infinite extent.
d. However, photons scattering through air can reach a given
point from much greater distances than by traveling through
61

soil, and thus the exposure rate detected over tailings
piles must include a "skyshine" component of photons from
sources near the pile surface but distant from the detector
(Beck, 1981). This component would increase as the pile
surface area increases, a dependence that becomes more pro
nounced for decreasing depths of overburden.
e. The assumption of infinitely wide areas (e.g., Beck, 1972)
would not detract from the accuracy of calculations dealing
with the exposure component in c), while conservatively
maximizing the minor contribution of "skyshine,"*in d).
For the purposes of simplifying calculation and comparison
with the results of other investigators (Beck, 1972;
Schiager, 1974) infinitely wide tailings piles were
assumed for this study.
Assumption 2. Finite Depth of Tailings
The effect of different thicknesses of mill tailings on the
exposure and dose rates is one of the objects of the present
study.
Assumption 3. Smooth, Flat Interfaces
a. Realistically, tailingsground, overburdentailings, air
tailings interfaces can be expected to be neither smooth
nor flat.
* Author's note: "skyshine" contributions are not included in this study.
62

b. Roughness at the airtailings interface would "tend to
increase the field close to the interface by a slight
amount." (Beck, 1972)
c. Thus, the assumption of smooth, flat interfaces leads to
exposure rates, etc., being underestimated, slightly, for
most surfaces. Severe roughness would presumably result in
greater error.
d. Smooth, flat interfaces are assumed in the present study,
which greatly simplifies analysis. Since this assumption
is routinely made in studies of this nature, comparison of
results is also facilitated. Nevertheless, it represents a
drawback of this and similar methods.
Assumption 4. Absence of Soil Moisture
a. Increasing soil moisture from 0% to 25% by weight will not
substantially affect gammaray transport (Beck, 1972).
b. However, increases in soil moisture would always result in
increases of in situ soil density, "which for the uniformly
distributed sources reduces the source activity per gram
and thus...fluxes, exposure rates, etc." (Beck, 1972)
c. The present proposed method accommodates small, uniform,
changes in soil density with extreme ease and, with con
sistent use of either "in situ" or "laboratory" soil
densities, produces valid results.
63

d. Thus, "absence of soil moisture" is not a strict require
ment of the proposed method; it i,s merely a convenient
choice, since the density of "dry packed tailings" studies
in this case corresponds to the density of "moist packed
earth" studied by other investigators (Schiager, 1974) thus
simplifying comparison of results.
Assumption 5. No Radon Emanation
a. Over 95% of the total photon energy emitted in the 238u
decay chain originates from 222Rn anc daughters.
b. However, 222pn js a noble gas which can emanate into the
soil or tailings air, diffusing through the soil and cover
material, and eventually, into the atmosphere. Typically
20% of the 222RU is free t0 diffuse in this manner, thus
effectively reducing the source of gamma rays within the
tailings while simultaneously creating a source of gamma
rays within the cover material.
c. To facilitate comparison with results obtained by other
researchers, who assumed "no radon emanation," the same
simplifying assumption is made for the present method.
This is roughly equivalent to assuming that cover material
is impermeable to radon diffusion and may lead to over
estimating, by orders of magnitude, the effective shield
ing capabilities of cover, as discussed in Appendices J,L
and M.
64

Assumption 6. Radioactive Equilibrium
For simplicity, all the members of the ^38y decay chain
are assumed to be in radioactive equilibrium, notwithstanding
the capability mentioned in 5c. Thus, source concentrations in
pCi/g as used in this report refer to "pCi of 2^U per gram
of soil," etc., which reflects standard practice.
Assumption 7. Uniform Distribution of Nuclides in Tailings
a. A large volume of tailings may be expected to contain many
local inhomogeneities.
b. However, a detector is affected by gammas from many points
in the pile, which reduces in some degree, the effect of
local differences.
c. For most sites, the assumption of uniform distribution has
been found to be a valid approximation (Beck, 1972).
Assumptions 8,9. Uniform Distribution of Overburden Material
Replicating Assumptions 1 and 2, the overburden material is
assumed to be of infinite planar extent but of some given,
finite, thickness. The latter can be "zero" for the common
case of "no overburden present."
Assumption 10. Identity of Buildup Factor Parameters for Tailings and
Overburden
Tailings and overburden material are assumed to be identical
insofar as buildup factor parameters are concerned.
65

In summation, a typical uraniumbearing soil or uranium mill tailings pile is
represented as a flat slab of finite thickness but infinite in area,
containing in uniform distribution and radioactive equilibrium, the nuclides
of the uranium chain from either 238y or 226Ra* ^0 stable lead. The soil
or tailings slab is covered with a similar slab of sourcefree overburden, in
the more general case.
*The differences in the energies emitted in these two cases is minimal
66

Appendix C
Exposure Rates and Flux Equations
The determination of exposure rates to photons from any radioactive source
entails, basically,a conversion from photon flux. For photons of a specific
energy E, the correspondence of exposure rate and flux may be expressed by
(10
air
where X(E) = exposure rate from photons of energy E, in R/s
FX = conversion constant
= 1.824401368 x 108 g . R/Mev
E = gamma energy, in MeV
(E) = "flux" of gammas of energy E, in gammas/(cm2s)
uen(E)j = energy dependent mass energy absorption coefficient
~p for air, in cm^/g
air
An obviously necessary input to the above equation is calculation of the
photon flux at the point of interest. For gamma rays of a specified energy,
from extended sources, such calculation would consider primarily the geometric
aspects of source distribution and overall source configuration, as affected
by the spatial dependence of the buildup factor.
In the case under study, the extended source consists of uranium decay
chain nuclides, at radioactive equilibrium, dispersed uniformly throughout an
infinitely wide tailings slab of finite thickness covered with a sourcefree
overburden slab. With these basic premises and Figure 1C, general equations
for the monoenergetic photon flux at any point "o" in the overburden, at a
distance "d" from the oberburdentailings interface, are developed in the
following pages.
67

Volume Element dV. with source strength Sv photons/cm3sec
URANIUM
BEARING
SOIL OR
TAILINGS
SLAB
Point 0 in Cover Slab
I (Overburden), at a Distance d
I From OverburdenTailings Interface
Figure 1C. Geometry for flux calculations with a slabdistributed
source (uraniumbearing soil or uranium mill tailings) covered with
a sourcefree overburden slab.
68

Consider a generalized volume element dV within the tailings slab, in Figure
1, of specific source strength Svป at a distance r from some unspecified
point 0 in the overburden or cover material. Taking buildup into account,
plus the generally assumed different attenuation capabilities of the two
media, the flux contribution from dV at point 0 can be basically expressed
(Morgan and Turner, 1967) as:
. svB erseceuc sece ^
4Trr2
where <ฃ = "flux", photons/cm2. sec
Sv = source strength per unit volume, photons/cm3. sec
B = buildup factor, dimensionless
ut,uc = attenuation coefficients for uranium bearing soil (or
tailings) and cover material , respectively, cm1
dV = volume element, cm3, equal to "r2sirie de d dr" (see Figure ic)
To obtain the total flux <ฃ of photons of a given energy at 0, equation (2C)
must be integrated over the tailings and cover slabs dimensions. Such
integration must include the buildup factor, as already discussed, and
necessitates adapting the chosen buildup form to suit the geometric
#
configuration. Referring to Equation (1A), Assumption 10 and Figure 1C, the
spatial dependence of Taylor's Form of the buildup factor can be described,
* See Appendix B
69

for the present case, by Equation (3C)below.
BT(r,e) = Agauitfrdsecejojucdsece +[i_A]ea2lJt (rdsece )a2ycdsece
(3C)
Equation (3C) can now replace the generalized "B" in Equation (2C) and the
resultant expression integrated. Prior to doing so, however, the integrand
can be simplified by multiplying Equation (3C) by the exponential term in
Equation (2C)
/ \ wtCrdseceJyrdsece ... , , / \ , /, *\f / \ / \ /A r\
BT(r,e)x e u ' u = Af^e) g^r) + (!A)f2(e)g2(r) (4C)
where f^e) = e Utuc)(l+ai)dsece
9i(r) = eVt(l+ci1)r
and f2(e) = e("tMc)(l+ซ2)dsece
g2(r) =eซd+ซ2)r
With these transformations, the integration of Eqn. (2C) can be indicated
as
(lA)F2(e)g2(r)] (5.c)
dV
v
where dV = r2 sineded(>dr
Therefore,
2n Tr/2 (t+d)sece
= d si
4ir J J
nede [Af^ejg^r) + (lA)f2(e)g2(r)]dr (6C)
J
oo d sece
The integration with respect to r produces the following two terms
70

A eCisece(1.eTisece) (1.A)eC2sece(1.eT2sece)
Ti/t * T2/t
where (^ = ;ucd(l+cn) , C2 =
TI = yutt(l+cn) , T2 =
(7C)
To integrate with respect to e, the above is multiplied by sins and the
product expressed as the sum of four separate integrals
A
Ti/t
f 'Cl
C, sece
sinede.
W/2
(Cj+Tjsece
e sinede
..o
(8C)
T2/t
ir/2
C2sece
ซ/
(C2+T2)sece
e sinede
sinede
Lo o
To perform the integrations, a substitution is required, with the
corresponding changes in the limits of integration
y = sece
dy = sec26sine de (9C)
= y2 sine de
thus^= sinede
y2
As e varies from 0 to w/2 , y = sece varies from 1 to 
Equation (8C)can now be rewritten
A
Ti/t
"Ciy
(CiHTi)y
6 Ju
ฐy
y2
T2/t
[Kc*y
J ^
.1
dy
I
i
(C2+T2)y
dy
(10C)
1 1
The form of Equation (10C)leads directly to an evaluation in terms of the
familiar 2nd order exponential integral E2:
71

j  E2(Ci+Ti)j + 1M)[E2(C2)  E2(C2+T2)
Vt L ^ l' ^ 1 17J j2/t
The last integration, with respect to f, merely introduces a factor of 2* into
the numerator of Equation (6C) which now becomes
Sv/
E2(C2+T2)]j
Replacing Clf C2 , Tj and T2 with their equivalences, defined in Equation (7C)
permits rewriting Eqn. (12C) in a more meaningful form.
(13C)
The integration performed above was strictly geometric, involving
only the physical dimensions of the tailings slab and cover; it was not
affected procedurally by the energydependence of the buildup factor,
attenuation, and source strength parameters. Nevertheless, the
energydependence of these parameters cannot be neglected; it is obvious that
they must all correspond to some definite energy E in any given particular
case, or Equation (13C) would be invalidated. More relevantly, this
correspondence must extend to the resultant flux , now specifically limited
to photons of one single energy. Thus, a more accurate rendition of Eqn. (13
C ) would be as follows:
E2
^
{pc(E)d[l+a2(E)]}E2[w:(E)d+rt(E)t][l+o2(E)]]
72

where <(E) = "flux" of photons of energy E, photons/cm2. sec.
SV(E) = volumetric source strength, photons of energy E/cm3.sec.
yt(E) = attenuation coefficient of uraniumbearing soil or tailings
material for energy E, cm1
MC(E) = attenuation coefficient of cover material for energy E, cm"1
AfEJ.o^E), } Taylor's form buildup factor parameters for photons of
and

energy "E", for the "bare U.B.S. or tailings" case. The conditions of maximum
surface "flux" are obtained by postulating an infinite thickness, "t=ป".
Although infinitely thick tailings piles have not been reported to date,
"fluxes" corresponding to such a "worst case" are approached asymptotically
with "sufficiently large" but finite values of "t".
SV(E)
A(E) lA(E)
l+a2(E)J
(16C)
To isolate the effects of varying cover depth "d" on surface "fluxes", a
constant thickness "t" must be maintained in Equation (14) while altering d.
Setting "t=ป" again, as a convenient example, produces
SV(E)A(E)
Sv(E)[lA(E)]
2pt(E)[l+a2(E)]
\
pc(E)d[Ha2(E)]\
The energydependence of buildup, attenuation and sourcestrength
parameters has been repeatedly noted in Equations (14C) through (17C) to
stress the fact that their output is, in each case, a monoenergetic flux. By
direct application of Equation (Cl) to such resultant singleenergy flux(es)
the corresponding exposure rate(s) can then be computed. However, the
exposure rate attributable to photons of one specific energy would obviously
not suffice to describe conditions at a uranium mill tailings pile,
characterized by a complex spectrum of emission energies. The flux and then
the exposure rate corresponding to each and every energy produced by the
nuclide inventory of the pile would have to be calculated singly, followed by
a process intergrating all exposures. Note, however, that the integration
cannot be performed analytically, since SV(E) is not a continuous function
of energy; a numerical integration, best done with a computer, is required.
74

where X = exposure rate due to photons of
ir 2E all energies, in R/s.
This expression introduces a serious problem, namely that the buildup
parameters A(E), o1(E) and a2(E) of Taylor's form, upon which the analytical
development is based, "are not available below 0.5 MeV" (Trubey, 1966). This
means that up to 15% of the total photon energy emitted in a pile at
radioactive equilibrium would be left unrepresented, unless some means to
extend analysis below 0.5 MeV is found. One viable technique requires use of
Berger's Form of the buildup factor,
BB(E,yr) = 1 + C(E)yr
where C(E), D(E) = energy dependent fitting parameters, dimensionless
Applying Equation (18C) to the conditions of Figure 1, Assumption 10, etc.,
produces the following expression for the spatial dependence of Berger's Form
R / *\ i r r i A ^ A A Qn D[yt(rdsece)+ydsece]
BB(r,0) = 1 + C [yt(rdsece) + ycdsece]e (19C)
Replacing the generalized "B" in Equation (2C) with the above expression and
carrying out the multiplication produces
> + Cf(r,e)e(D1)f(r'e)]dr (20C)
where f(r,e) = Mtr(u(.yc)dsece
To integrate the bracketed expression with respect to r, note that
75

dr
, thusdr=?l
(21C)
which permits expressing the integrals as
(Dl)f
^ If e df
This produces the following two terms
e(ฐ1)f [ (Dl)f l]
liCTr L J
(22C)
constant
(23C)
With the limits of integration made explicit, the first term of (23C) becomes
(d+t)sece
i ucdsece/, Pttsece\
(24C)
e
= J_eucdsece/ Pttsece\
dsece
Continuing with this first term, the integration with respect to e can be
indicated as
'ycdsece f*(ytt+ycd)sece
ie sinede  e sinede
f
Lo o
With the substitution V = sece ,
(25C)
(26C)
thus
dY = sec2e sine de
* Y2 sine de
dX = Sln9de
X2
with the corresponding change of limits, (25C) can be rewritten as
dy . fe(ytt^cd)y
dy
(27C)
76

As with Equation (10C), the above integration results in two 2nd order
exponential integrals
(28C)
Integrating this with respect to $ introduces a factor of 2n. Multiplying the
product of 2ir and (28C) by the constant term of Equation(20C) yields the first
term of the integration of(20C)with respect to r, e, ^>.
(1st. Term) = 2v_ E (M) .
(29C)
Now the process of evaluating the second term of Equation(23C)is undertaken:
(d+t)sece
(Dl) utr(up)dSece j (M)
Expression (30C)results in a 4term polynomial
,, J
jte) + T2(e)
. _ . .
where Ti(e) = e
T2(e) = e
T3(6) =e(
t + wcd)sece
jucd)sece
The integration with respect to 0 is indicated below
IT/2 TT/2 TT/2 71/2
Ti(0)sin0d0 +  T2(0)sin0d0 +  T3(0)sin0d0 +
oooo
77
T
I
" i
dsece
(30C)
(31C)

For the terms including TI(B ) and 13(9), the following substitution is
useful
y = sece (32C)
dy = sece(sinesece)do
y
For the terms including T2(e) and T4(e)> the corresponding substitution is
y = sece (33C)
dy = sec2e sinede
dy = sinede
y2
These substitutions necessitate a change in limits of integration, from "0 to % "
to "1 to ป". The integration with respect to e now produces a polynomial in terms
of 1st. and 2nd. order exponential integrals,
(Dl)(ytt+ycd) E:[ (Dl)(ytt+ycd)]
E2 [(Dl)(ytt+ycd)]
(Dl)ycd
E2 [(Dl)ycd]
(34C)
The above expression can be simplified by making use of the following
relationships
X
or
E,(X) = e
Ei(X) = E2(X) + e'x
(35C)
78

With the transformations in (35C) the first and third terms in brackets in
become, respectively
E2 [(Dl)(ytt+ycd)J  e
(Dl)(ptt+pcd)
(36C)
and E2
(Dl)ycdJ
(Dl)ycd
Cancelling like terms, this becomes
(Dl)wcd (Dl)(utt+ucd)
ut(Dl)2
 e
or C e
(Dl)ycd
(37C)
Integrating with respect to ty, etc. results in the second term of the
integration of Equation (20C)with respect to r, e,
Term) = ^ e^^fle^v]
SVC (Dl
(38C)
2yt(Dl)2
Adding the 1st term from Equation (29C) and expressing the energydependence
of relevant parameters produces
_ ME)
C(E)
 E2[yt(E)t
uc(E)dr CD(E)1]
le
(39C)
2
79

For the important case of an "infinitely thick" tailings slab (t = ป) without
cover material (d=0), Equation (39C) reduces to
SV(E)
C(E)
(40C)
With the values of "surface flux " obtained through Equations (14C) ,
(15C),(16C),(17C) or (40C), applied in Equation (1C), the exposure rates
at ground surface can be determined, for gamma radiation of a specific energy.
However, much of the published data refers to exposure rates at a specific
height (typically 1 meter) above ground surface. Accordingly, a modifying
factor was sought, to relate "surface exposure rates" obtained from the above
equations to the corresponding rates at one meter above ground, thus
facilitating comparison with previous results.
This modifying factor can be expressed as a ratio of exposure rate at a
height "h" above ground, including buildup and attenuation effects, to the
corresponding exposure rate at ground level, for photons of a given energy E.
The source of the emissions is assumed to be an infinitely thick slab with an
air "cover" of thickness "h". The assumption of infinite thickness is meant
to simplify analysis, based on yet another version of the buildup factor, the
"linear" form,
BL(E,yr) = 1 + a(E)y(E)r (41C)
where a(E) = energy  dependent fitting parameter, dimensionless
Replacing uc and d in (2C) with wair and h, respectively, plus including
the above formula for B, with the necessary specifications ซt(E) (for
tailings) and <*air(E) results in an integrable expression. The details of
the integration are given in Morgan and Turner (1967) and shall not be
repeated here, with only the results being presented, below.
' 80

The flux of photons of energy E, at a height h above ground level, is
(42C)
The second term within brackets is subject to the following relationship
= e
air
where, for h = o,
"air1"
and E2(ya.rh)
h=o
= 1
h=o
= 1
Thus, for h=o, or "ground level case", (42C) reduces to
(43C)
The modifying factor is obtained by dividing (42C) by(43C),
ฐairuairh
(44C)
The second term of (44C) may be eliminated if buildup in one meter of air is
neglected, i.e. the case of BL = 1, unit buildup, implying that a r = Q
[see Equation (41C)]. This reduces (44C) to the following expression, with
energy dependences indicated,
(45C)
81

Although based on flux ratios, the modifying factor FM(E) is directly
applicable to exposure rates, as an examination of Equation (1C) can verify,
due to mutually cancelling terms.
With this modification, the numerical integrations resulting in "total"
exposure rates at ground level and at one meter above ground level can be
represented by Equations (46C) and (47C), respectively.
XZE< ฃ/xii?ii,) g . (46c)
5
m
^d ;_ E FxEi*(Ei)^ni^ .E2[pa1r(El)loOcm] (47C)
J air
m
t'
j=i L r Jair
where i = 1, 2, ...n, indices of discrete energies below 0.5 MeV.
j = 1, 2, ...m, indices of discrete energies above 0.5 MeV.
The indices i and j in the above equations refer to discrete energies below
and above 0.5 MeV, respectively, corresponding to the choice of buildup form:
the first summation terms in both (46C) and (47C) indicate "the sums of
exposure rates, at ground level and at 1 meter above the surface, due to gamma
emissions of energies up to 0.5 MeV , calculated on the basis of Berger's
buildup factor" ; the second summation terms in both equations signify similar
processes employing Taylor's form of the buildup factor, for energies greater
than 0.5 MeV.
82

Appendix D
Decay Scheme and Energy Spectrum
The typical uranium bearing soil slab subject of this study is assumed to
contain 238(j jn radioactive equilibrium with all decay daughters through
210po, as shown in Figure 1D. Several branching decays have been omitted,
namely 218/\t and 206y (neither of which is a gamma emitter) and 210ji.
None of the mentioned nuclides is produced in more than 0.02% of decays of the
parent nuclide; the "main branch" nuclides 214pt)> 210p0 and 214p0)
respectively, being assumed to correspond to 100% of the parent
disintegrations, for simplicity. Consequently, the only branching included in
the decay scheme is that of 234pa_metastable (1.17 minutes) and 234pa (5,7
hours).
With the decay scheme of Fig. 1D and the radionuclide decay data of
Kocher (1977), a complete spectrum of gamma emission energies present in a
uraniumbearing soil can be compiled. Postulating a "Base Case" of "1 pCi per
cubic cm", and making use of Kocher's intensities, an energydependent "source
term" SV(E) is found for each energy E, to implement Eqns. (14C) through
(17C).(40C) and finally (43C) and (44C). In agreement to the form of these
last two equations, the SV(E) values are distributed between two tables.
Table 1D contains SV(E) terms for energies up to 0.5 MeV, for a total of
n=105 values, while Table 2D consists of the remaining m=177 values, for
energies over 0.5 MeV, where "m" and "n" refer to indices in (43C) and (44C)
83

ATOMIC WGT.
ELEMENT
ATOMIC NO.
HALFLIFE
U238
Th234
Pam234 (99.8756)
Pa234 (0.13SO
U234
Th230
Ra226
Rn222
Po218
Pb214
Bi214
Po214
Pb210
Bi210
Po210
Pb206
Figure 1D. Uranium238 Decay Series
84

Table 1D Volumetric
Source Strength Sv(E)
for Energies E<0.5 MeV
From Kocher (1977)
f
1
/;
4
''.'I
6
..,
8
9
i. 0
1. .1.
i ';>
!3
i. 1
i.5
1. 6
1.7
1 9
j c
2 0
.' j
"> ':>
23
24
> 1:1:
26
27
28
29
30
3 :l.
3 2
33
34
35
36
37
38
39
40
4 .1.
42
43
44
45
46
47
48
49
50
51
52
Gamma Energy,
in MeV, E
.> 13000E01
t49600E ():!.
,13300E01
,63282E01
,92367E 01
,92792E01
, .1 1.28.1
, 76 OOOE 0 1
, 1360 or;: 01
,43450E01
,632001;;: oi
,699ooi;;:oi
,805ooEoi
,94665E0.i
t98439E" 0.1.
,99700Eoi
, 10340
.11100
ซ 12530
,13128
, 1 3 4 3 /'
v 137 70
.14030
.14410
,15020
, 15 2 70
,15930
.17080
> 1 7 4 6 0
.,18600
, 19360
,19970
.200!;:0
,20290
,22020
,22687
';j ") /. j:; ;
,24540
v2'lB90
,26710
,27210
,28610
,28960
,29370
,31250
,31630
,32070
,32830
,32830
,33030
.35180
,36960
Intensity, or
fraction of
decays producing
gammas of energy
E, dimension! ess
,870ooi;;:oi
,70000E 03
.9QOOOE0.1
,390001;;: o:i.
,2570oi:;:oi
,3ooooi;;:oi
* 24900 E 02
,200001;;: 02
, :!. n;:520E02
,i5600i;:: 05
,403()0i;;:04
,29900E05
, 50 700 E 05
,20150E03
,32630E03
,61100E04
 15600E05
* 15210E03
, .' 1300E04
.26000E03
.2/300E.". 05
, 19500E05
,120901;;: 04
,49400E 05
,26 OOOE 05
vV3459E04
,.?4900E05
,6240oi;;: 05
,247001: 05
,24700E05
,702ooE 05
, 9.1 OOOE 05
,14 040!. 04
, 14300E04
, 29900 E 05
.84500E04
,49400!: 04
,1 1. 700E04
,40300E04
., 22100E05
,15600E04
,i820oi;:: 05
,14040E05
,41600E04
,37700E05
,15600E05
,15600E05
,3/.A)OE04
,37700E04
, 11440E04
.74100E05
,36400E04
Volumetric
Source
Strength, in
"gammas of
energy E per
cm3 per sec"
SV(E).
,32190E02
, 25900 E ()4
,36260E02
, 14430E02
,95090E03
,11100E02
,92130E04
, 74000E04
,54334E04
,57/20E 07
, !. 49 HE 05
, 1. 1063E06
, i.8759E.06
< /4555E05
, L2073E04
,22607E.05
, 5 77 20 E 07
.56277E05
.52910E06
,96200!;;: 05
,1.()101E06
.72:! DOE 07
,44733!:06
,'!8278E06
, 96200E 07
.34580E0:vi
,3:51 13 E 06
,23088E06
,91390E07
v91390E"07
, 259 74 E 06
33670E06
< 51 94 8 c 06
,52910E06
, 1 1 063E06
,31265E05
, 18278E05
,43290E06
, :i.491:iE05
,8:1. 770E07
,577201;;: 06
,67340E07
,51948E07
.15392E05
,13949E...06
,57720E07
,57720E07
, 13949E05
,13949E05
,4232HF06
,27417E06
,13468E05
NUCLIDE
238u
234
Th
234_
Pa
85

Table 1D (Continued)
53
54
.'.'.i .'.';
56
57
58
59
60
61
l\ ">
6 3
64
66
67
6 8
6 9
70
7 1
'' 2
73
74
75
76
7'7
79
80
ol
82
83
84
85
86
87
88
89
90
9 .1.
92
93
94
95
96
97
98
99
100
1 0 1
102
103
104
1.05
Gamma Energy,
in MeV, E
> 37220
,40980
,42690
,44690
'M5860
,461.80
<. 46750
,47210
,47350
,48000
> 48250
, ;i 3 60 Or gj.
, 94 A 631: <;:>
, 9 8 4 .' 9 !. ' ) 1
, 13000! 01
,5 52201: 01
, 1 :>1 40
, 1 :>300I 01
,6'8 10400
, .1. 1800 E 02
,40000E03
,85()OOE01
.38000E02
,70000E03
^SIOOOE 02
, 18000E02
, 29900E02
< i.3::>ooi;;:02
,32800E01
.. 1 3595
, 1 100 OF 01
,63300E01
, 10700
,4 76 00 !> 0:1.
,. 7 4 7 () 0 1:;. () ;
,55.100E 02
, 32000E02
.> 1 9200
,37081
,33800E02
,44000E ;)2
,50000E03
,52000E OP
.35800E02
>60000E02
,26900E02
,18000E02
,36000E02
,41000E02
, 16700E02
,iioooi;;:02
,31800E02
,21800E02
, 13300E02
,1 1800E02
,24300
,40500E01
Volumetric
Source
Strength, in
"gammas of
energy E per
cm* per sec"
SV(E).
!i1i?^l^
, 18759E06
, 57720;:; 07
,72i50E 06
* 76960E07
,18278E06
, 11544E06
,86580E07
< 1 3949;;:;o6
, 13949E06
>. 1 6259E03
, 4 2 7 5 3 ;;;:  o ^
, 69470E04
"X o A i:i r\ <;. .... i"; '".'
f ..' '...' v *.:> \. i... \. .*..
, 4 366 OE 04
, 1 4800E04
,3:l.450E02
, :!. 4060E03
, 25900E04
,29970E03
,66600E()4
,1 1063 E 03
v 50320E 04
,:l.2:i36E02
.50300E02
,40700E03
,2342 IE 02
,395901;;. 02
,17612E02
,2/639E02
,20 38 7 [ 03
, 1 1840E03
, 71040E02
,13 720 E01
 :l.2506E'03
,16280E03
, 18500E04
,19240E03
, 132 46 E 03
, 22200E03
,99530E04
,66600E04
, 13320E03
,15:l.70E03
,6:!.790E04
,40700E04
,1 1766E03
,80660E04
,49210E04
,43660E04
.89910E02
,14985E02
NUCLIDE
234Pa(eont.)
234 Pa m
u
23ฐTh
226Ra
2MPb
Bi
J10Pb
86

Table 2D Volumetric
Source Strength Sy(E)
for Energies E>0.b MeV
From Kocher (1977)
V
!
2
..y
4
5
*
...,
8
9
.1. 0
.1 .!
., ..
i. ...:
13
1 4
15
.; 6
17
.1. 8
19
20
22
23
24
"'5
2 '';'
? >'
28
""' iV
30
31
32
33
.'54
35
36
37
3d
. "> '*"'
40
41
42
43
44
45
4 6
47
48
49
50
Gamma Energy,
in MeV, E
,50680
,51360
,52060
, 52 1.00
,52800
,53320
,53 7:1.0
,55/00
, 56650
,56926
,56926
,57410
, 58.V. O
, 59650
t. :'i f"f ::> '.
,61. 140
,61620
,62350
, 6 7750
,6306 0
.63450
,63970
, 64320
,64620
,65320
,63500
v 66060
j 6 6 4 6 O
, 6 6 6 '' ' !
,66980
68330
,68550
, 69250
,69*10
,70600
, 7 1 1 2 0
,73300
..73840
v .^281.
.74650
.75480
,76000
* 76636
, /' 6 6' .:' 6
, 76870
,77790
,78080
,78310
, 78627
,79360
Intensity, or
fraction of
decays producing
gammas of energy
E, dimensionless
, 182 OOE 04
, 15600E04
< 136 OOE 04
, 156 OOE 04
.50700E03
, 260 00!!!.' 05
,20800E05
> 3 2 5 0 0 E  0 5
> 2 21. OOE 04
, 1 3520E03
,40300E04
.. 26000E04
. 19500E05
: 78000E03
, 1 6900 E 04
, 10400E04
,260()OE05
.13000E05
3S400E !)5
'"i \ * I ) l' i j ( 1 !"i
,37700E05
. 27300E05
,286 OOE 05
,2 86 OOE 05
, 1 6900E04
,79300E05
.... ..., .., (..; ^j:: ....,s
, :i 95 OOE 04
, 19500E04
, 19500E04
31 ;:;OOE05
, 3 rf 1 0 0 1.  0 5
, ! 6 9 0 0 1'..' 0 4
,5 98 OOE 04
, 41600 E 04
,208 OOE 05
, :! ::. 050E 03
; 1.04 OOE 04
, 39000E04
, 1 1700 E 04
. 18.2 00!!" 04
.20800E05
, 10400E04
,26000E05
. 7:i 500E05
,26000E05
,20800E04
,63700E05
,20800E04
.. 19500E04
Volumetric
Source
Strength, in
"gammas of
energy E per
cm* per sec"
,67340!: Of
,57720i: ..{
,5/720i;;0,.
,57720E06
, 18759E06
,96200E07
,76960E07
* 1 2025 i::. 06
,81 770 iV 06
,50024E05
* :!. 191 1E05
..96 20 0!'" 06
.. 721 5 OK: 07
,2 886 OF 06
,62530E06
,384 HOE 06
,96200E07
ป 481 OOE 07
,32708E06
, :!. 9 7 2 1 E  0 6
,:!.3949E06
,10 10 IE 06
,1()582E06
,10382E06
, 6 2530 E () 6
,29341 E0 6
, 13949E06
,72.I.50E06
* 72:!. 30!:.'.' 06
,72.I50E06
, !. 1 54 4c 06
. 12987E06
, 6 2 5 3 0 i'."  0 6
.22126h: 05
, 1 5 3 92 E 05
,76960E07
, 40885E 05
, 3848 OE 06
,14430E05
,43290E06
,67340E06
,76960E07
,38480E06
,96 2 OOE 07
.26455E06
,96200E07
,769601 06
.23569E06
,/6960E06
.72150E06
NUCLIDE
234
Pa
87

Table 2D(Continued)
51
52
53
54
55
56
57
58
59
60
61
6 2
6 3
64
65
66
67
68
69
'0
M
i ,
:'3
'4
V5
' f.)
.,...,
78
79
80
8 :i.
32
83
84
85
fk>
87
80
89
90
91
92
93
94
95
96
97
98
99
.too
Gamma Energy,
in MeV, E
,79620
,80450
.80550
.808:1.0
,81250
,81940
,82470
,82630
,83110
,84190
,84480
,87290
,87670
< 08 051
,88051
,88324
,89860
,90480 .
,92000
,92460
,92670
,92670
,94600
, 9 4 9 0 0
* 97880
,97880
,98050
,98050
,98340
1 ,0227
1 ,0283
1 ,0449
:!. ,0744
1*0825
1 , 1085
1,1223
1 , 1260
1 ,1.531
1 , :!. 7 1 3
1 ,2080
1,2175
1,2409
1,2510
1 ,2771
1,2928
1,3530
1 ,3584
1,3941
1,3997
1,4270
Intensity, or
fraction of
decays producing
gammas of energy
E, dimensionless
,44200E"04
,50700E05
, 4 29 00 E 04
, 63700 E 04
,63700EOS
,286<)oi;:: 04
, 46800 E 04
,41600E04
,72800E04'
,18200E05
,76700E05
, 15600E05
,35100E()4
,53300E04
, 84 500 E 04
, 1 5600E03
,52000E04
,63700E05
,50700E05
,364()()E.04
, 14300E03
,6 630 OF 04
, ;;!60ooi:;:o3
, 10140E03
, 18200E04
, :!.8200L":04
,28600E04
, 19500E04
.31200E04
,46800E05
, :iOO:l.OE () 4
, 63 700 E 05
,22100E05
,9R80()E05
,37700E05
,63700E05
, 10140E04
,29900E05
,32500E05
. ,37700E05
,49400E05
,27300E05
,37700E05
,16900E05
,89700E05
, 221 00 E 04
,26000E05
,50700E04
,27300E05
,27300E05
Volumetric
Source
Strength, in
"gammas of
energy E per
cm3 per sec"
SV(E)ซ
.16.;;54E" 05
, :i.87!::;cn';:06
, 15873E05
,23569F.... ()5
, 23 56 9 E 06
>10582E05
, 17316E05
,15392E05
,26936E05
,67340E07
, 28379E06
,5//'20!.0/
, 12987E05
,19 72 IE 05
,31265i:05
, 577201 :05
,1.9240!:05
v23569E06
, 18759E06
,13468E05
,52910E05
,24531E05
. 96200E05
,37518E05
,6734()E06
,67340E06
,10582E05
,72150E06
, 1 1544E05
,17316E06
,3703?E()6
,23569!;;;. ..06
,81770E07
.36556E06
,13949E..06
.23569E06
,3751 8E06
,11063E06
,12025E06
, :l 3949E06
,18278E06
,10101E06
,13949E06
,62530E07
,33189E06
.81770E06
,96200E07
,18759E05
,10101E06
,101 OIF 06
NUCLIDE
234Pa

Table 2D (Continued)
1 0 1
102
103
1.04
105
106
.1.07
108
109
1. 1 0
1 1 1
1 1 2
1 1 3
1 1. 4
1 1 5
1 1 6
1 1 7
1 1 8
1 1 9
.1.20
1 2 '1
1 2 2
1 2 3
124
125
126
127
128
129
130
J. 3 1
332
133
134
135
136
137
138
139
140
1 4 1
142
1.43
144
145
146
147
.1.48
149
150
Gamma Energy,
in MeV, E
1 ,4461
1 ,4526
1,4600
1 ,4937
1 ,5160
1 ,5801
1,5854
1 ,5938
1,6280
1 ,6382
1 ,6560
1,6685
1 ,6863
1 ,6940
1,6998
1 , 7560
1 ,7722
1,7969
1 ,891 1
1,8975
1,9050
1 ,9265
1 ,0061.
,76636
1 ,0010
,93050
,51200
,53369
, ''j 8 0 1 b
,78591
, b .:; 7 i.. ,.;.
,51100
,60932
,66545
,70311
,71986
, 75284
,76836
,78610
,80617
,82118
,90425
,93405
,96408
1,0520
1*0700
1,1203
1 ,1337
1,1552
1,2077
Intensity, or
fraction of
decays producing
gammas of energy
E, dimension! ess
.72800E05
, 1 5600E04
*37700E05
,27300E05
,50700E05
,20800E05
.20800E05
.50700E05
,16900E05
,32500E05
, 195 OOE 05
*13780E04
,50700E05
, 1 6 900 E 04
,19500E05
* 3 .12 OOE 05
, 1950GE05
,37/OOE05
,24700E05
,'>0800E05
,351 OOE 05
,!;7200E05
,19240E04
* ?0673E02
,:;8846E02
,:<6952E02
^'OOOOE03
,. 9000E02
> 36400E02
,10 9 OOE. Ol
, 59000E02
,:i.4700E02
,46.1.80
,15600E01
,47222E~02
,4 03 OOE 02
.13300E02
,48800E01
,310 OOE 02
,12300E01
,15000E02
,10500E02
,3. 1.6 OOE 01
,38300E02
,31500E02
, 2 85 OOE 02
,15000
* 255 OOE 02
,16900E02
.46000E02
Volumetric
Source
Strength, in
"gammas of
energy E per
arj3 per sec"
*26936E06
, 1 3949E 06
,10 10 IE () 6
,18759E06
*76960E07
,76960E07
,18759E06
.62530E07
,12025E06
,72150E07
.50986E06
,18759E06
.62530E06
,72150E07
,1 1544E06
,72150E07
,13949E06
,91390E07
,76960E07
,12987E06
,21 164E06
.7.1. 188 E 06
..76490E04
, 2 1 7 73 E O 3
.13672E03
,2 96 OOE 04
,70300E04
, 1 3 4 6 8 E  0 3
,40330F <> 3
v 2 1 830 E 03
,54390E04
*17087E01
,57720E03
,17472E03
,149. HE 03
ป492.10E"04
, 180561;:: 02
,11470E03
. 45510 E 03
.55500E04
,38850E04
. 1169 2 E 02
,14 17 IE 03
, 1 1655E03
.10545E03
,55500E02
ป94350E04
.62530E04
.17020E03
NUCLIDE
234B
D^M /^*ซ ป 
rQ (Cont. ;
234pam
Rn222
Pb
214Bi
89

Table 2D(Continued)
Gamma Energy,
in MeV, E
:l.5:l l , 23 8:1.
152 l,28:i.O
153 1,3038
15 A 1 ,3777
155 .1. ,38 S3
156 1,4015
157 1,4080
.i58 1,5092
159 :l ,5385
160 1,5433
161 1,5832
162 'I ,5*47
'i 6 3 .'!. v 5 9 9 3
:i 64 1 ,661.3
' ^5 1,6840
1. 6 6 '!. , 7 7 9 6
1.6 7 1 ,7645
;.68 !. ,8384
16v 1 < 8474
170 1 ..8737
I.'' i. 1,9763
I V7 3, 1183
i >3 7, 204i
'; /' /I 7 ... ' ' 9 X 4
173 2,4477
i 76 i , 0843
77 ,79200
Intensity, or
fraction of
decays producing
gammas of energy
E, dimensionless
, 59200E01
, :i.4700i;::o:i.
< 12J.OOE 02
> 40200E.' 01
i 7 8 0 0 0 i::"  0 ::.:'
, 13900E0:i
,24800E01
, 21 900 E 01
,41 OOOE02
, 3 5 0 0 o ฃ 0 2
, 7 2 0 0 0 i:: 0 2
,26500E(72
, 334 O () J7 () 2
, 1 1.500 E01
, 2 3 6 0 0 E  0 2
, "? 0 5 0 0 !:."  0 :i
, .'. !Vr 9 0 0
. 'X8300!::  02
, 21 2 OOF (> 1
, 22600EO2
. 1 7700 E" 02
> 1 2100E' OJ
, 49900E'0';
., 79400E09
, i 5 5 0 0 E 0 !
, "z,P, 70 ('/'"' (> 1
, . 0000i:: '03
Vol umetri c
Source
Strength, in
"gammas of
energy E per
cm3 per sec"
SV(E).
, 21 9041::. ' 02
,543901;' 03
,4477()E04
, :l 48 / 4 1:7 02
.. 2 8 8 6 O i:;. () 3
,51430E03
. 9 j ;:',;;)!::.... (>;x
^ fi 'j C> 30:: () 3.
, j. 5 j 7 0 E!  0 3
, 1 2 ''50 E 03
,26.440E 03
,98050E04
i '"''x '''', o\'.:' ...,, '.i
,42550E03
,87320E04
'i '; '.' J { ! '! i ' 07
:. 58830E02
 1417 E03
,78440!;;: 03
,83620E04
,.6549 (> :;  0 ^
,447V()E03
v i 3 ^ /, 3 ;:;; .... (i ;;.;
, 1 1 988:;77 >...;;
.. 57330F03
, ;[ 4 3 [ v r  0 3
. 370001:; 05
NUCLIDE
21V
Dl (Cent.)
Po214
90

Appendix E
Choice of Medium Representing Uranium Mill Tailings
The usefulness of equations based on Taylor's or Berger's buildup factor
is closely connected to the availability of parameters corresponding to either
form for a given transport medium. These parameters have been obtained for
elements such as tin, lead, etc., for water and for homogeneous mixtures of
well defined composition, such as the various types of concrete in Table 2A
but not for "soil" or "uranium mill tailings". This omission is due, in all
probability, not only to the complexity of the projected task, but also to the
envisioned lack of generality of the presumptive results (no two soils or
tailings piles are more than vaguely similar in composition). Consequently,
any relatively simple method based on the buildup factor concept must
incorporate the parameters of one of the materials of Table 2A,which entails a
choice. The choice must be made realistically but conservatively, i.e., a
material representing "tailings" should produce a greater, rather than smaller
"buildup" of secondary radiation, regardless of any other characteristic.
The selection is facilitated by Equation (16C) describing the "flux" of
photons, emitted with some energy E > 0.5 MeV, at the surface of a bare
tailings slab of infinite thickness, i.e., a "worst case" condition.
A (E)=Mi) LliiL. + OiME) 1
2wt(E)Ll+o1(E) l+a2(E)J
Assuming the coefficient yt(E) for "tailings" to be already known as a
function of energy, and bearing in mind that the values of SV(E) have been
tabulated in Tables 1D and 2D,the flux ^ (E) for a given energy is directly
91

proportional to the magnitude of the bracketed term in the above expression.
This bracketed term is solely a function of the Taylor parameters A, o^ , and
02 which, in turn, depend exclusively on the choice of material, for a given
energy E. Obviously, that transport medium which produces the largest values
of the term in brackets would represent the most conservative choice.
The values of Taylor's parameters A, 04, and o2 are given in Table 1E for
each of the 11 media previously listed in Table 2A, and for energies ranging
from 0.5 MeV to 3.0 MeV (this upper bound exceeding the highest gamma energy
?38
observed in the U decay chain). Based on these values, the magnitude of
the term [A/(l+o1) + (lA)/(l+a2)] has been plotted, for each material, over
the indicated energy range, in Figure 1E.
It is clear from this plot that either "water" or "ordinary concrete"
would produce the highest values of buildup, necessitating additional criteria
to effect a selection. In this regard, an important consideration is the need
for extending analysis below the 0.5 MeV limit existing for Taylor's buildup
factor. Since Berger's coefficients for 0.255 MeV exist for "water", but
appear to be unavailable for "ordinary concrete" (Trubey, 1966), the choice of
"water" parameters for energies above and below 0.5 MeV would be consistent
and obvious.
The extent to which fluxes and exposure rates may be overestimated on the
basis of the above selection cannot be precisely determined. By comparing
the buildup in water to that in aluminum, Beck (1981) suggests that results
obtained with the present selection may be high by 5 to 10%, at 1 meter, and
even more at greater distances. However, the choice of water introduces
compensating errors, alleviating, at least in part, the mentioned drawback, as
discussed in Appendix G.
92

Table 1E BuscaglioneManzini* Coefficients
for Taylor Dose Buildup Factor Formula
Material
Water
Aluminum
Iron
Tin
Tungsten
Lead
Uranium
Bb (MeV)
0.5
1
2
3
0.5
1
2
3
0.5
1
2.
3
0.5
1
2
3
05
1
2
3
0.5
l
2
3
0.5
i
2
3
A
100.81*5
19.601
12.612
11.110
38.911
28.782
16.981
10.583
31.379
2i*. 957
17.622
13218
11.1*1*0
11.1*26
8.783
5.^00
2.655
3.231*
350V
l*. 722
1.677
2.981*
51*21
5580
1. 1*1*1*
2.081
3287
i*.883
ซi
0.12687
0.09037
0.05320
0.03550
0.10015
0.06820
0.01*588
0.01*066
0.0681*2
0.06086
.0.01*627
0.01*1*31
0.01800
0.01*266
0.0531*9
0.071*1*0
0.0171*0
..0.01*751*
0.06053
0.061*68
0.03081*
0.03503
0.031*82
0.051*22
0.021*59
0.03862
0.03997
0.01*950
%
 0.10925
 0.02522
0.01932
0.03206
 0.06312
 0.02973
0.00271
0.02511*
 0.0371*2
 0.021*63
 0.00526
 0.00087
0.03187
0.01606
0.01505
0.02080
0.1131*0
0.13058
0.08862
0.011*01*
0.3091*1
0.131*86
0.01*379
0.00611
0.35167
0.22639
0.08635
0.00981
93

Table 1E( continued)
Material
Ordinary
Concrete
Ferrophos
phorous
Concrete
Magnetite
Concrete
Barytes
Concrete
Eb (MeV)
0.5
1
2
3
0.5
1
2
3
0.5
l
2
3
05
1
2
3
A
38.225
25.507
18.089
13640
61.341
46.087
14.790
10.399
75.471
49.916
14.260
8.160
33026
23.014
9.350
6.269
,
0.14824
0.07230
0.04250
0.03200
0.07292
0.05202
0.04726
0.04290
0.07479
0.05195
0.04692
0.04700
0.06129
0.06255
0.05700
o.o6o64
*
 0.10579
 0.01843
0.00849
0.02022
 0.05265
 0.02845
0.00867
0.02211
 0.05534
 0.02796
0.01531
0.04590
 0.02883
 0.02217
0.03850
0.04440
*From "A Survey of Empirical Functions Used to Fit GammaRay
Buildup Factors." By O.K. Trubey, ORNLRSIC10, Published
February 1966.
94

As Function of Energy E and Transport
Medium. !
Energy, MeV
Figure 1E Magnitude of [A/(!+ซ,) + UA)/(l+a2)] as function of energy and
choice of transport medium representing "uranium mill tailings"
95

Appendix F
Dose Buildup Coefficients for Taylor's and Berger's Formulas
The choice of Taylor's (and Berger's) "water parameters as conservative
substitutes for the unavailable "U.B.S. or mill tailings" coefficients was
based, primarily, on a visual inspection of Figure 1E drawn using known values
of A, a1 , and c^ at energies of 0.5, 1, 2, and 3 MeV (Table 1E). These four
values are obviously insufficient for meeting the requirements of Equations
(43C) and (44C) and equations leading thereto Tables 1D and 2D identify 282
different gamma energies from nuclides in the 23^U decay chain, ranging
roughly from 0.01 to 2.45 MeV. Fortunately, both Taylor's and Berger's
coefficients are smooth functions of energy, which enabled the present author
to obtain the necessary curvefitting expressions.
I. Taylor's Coefficients
For Taylor's coefficients, required for 177 gamma energies from 0.5 to
2.45 MeV (Table 2D), the following equations apply:
(Figure 1F) A(E) = exp
f 
\b 
E
where a = 0.560 423 309 6
b = 0.266 709 Oil 9
c = 2.211 317 385
(Figure 2F) OI(E) = ax + bl InE
96

where a} = 0.090 035
b: = 0.053 141 671 84
(Figure 3F) a2(ฃ) = ^ + c2E + d2 (3F)
where a2 = 0.113 514 887 2
b2 = 0.098 224 139 43
C2 = 0.004 721 763 81
d2 = 0.082 863 985 76
Note that the energy range for which Taylor's coefficients are valid comprises
85% of the energy emitted in the 238(j decay chain, and that they are valid
generally, i.e. without regard to the number of mean free paths involved (see
Table 2A , supra). Thus the brunt of calculations concerning the effects of
varying U.B.S. or tailings slab thickness, cover material thickness, and
relaxation lengths is aptly borne by these coefficients.
97

100
90
80
70
60
50
40
AfE)
30
20
r
TAYLOR'S DOSE BUILDUP COEFFICIENT A , FOR A POINT
ISOTROPIC SOURCE IN WATER , As Function of Energy E.
0.560 423 309 6
0.266709011 9  E
+ 2.211 317 385'
A(E) = e
For 0.5 MeV< ES 2.45 MeV
0.85% Error at 3 Mev
1.0
2.0
ENERGY, MeV
3.0
Figure 1F. Taylor's Dose Buildup coefficient A, for a point isotropic
source in water, as function of gamma energy E.
98

.02
.04
.06
.08
.10
.12
TAYLOR'S DOSE BUILDUP COEFFICIENT CL , FOR
A POINT ISOTROPIC SOURCE IN WATER, 83
Function of Energy E , for 0.5 MeV< E< 2.45 MeV
Estimated 3% Error (Maximum) at E =2.45 MeV
0.5
1.0
ENERGY,MeV
2.0
3.0
Figure 2F.Taylor's Dose Buildup coefficient QI , for a point
isotropic source in water, as function of energy E.
99

.04
.03
.02
.01
.01
.02
.03
a2(E)
.04
.05
.06
.07
.08
.09
.10
.11
TAYLOR'S DOSE BUILDUP COEFFICIENT Q? ,
FOR A POINT ISOTROPIC SOURCE IN WATER^.
as Function of Energy E,
for 0.5 MeV < E < 2.45 MeV
C,= 
d2=
0.113 514 887 2
0.09822413943
0.004 721 763 81
0.082 863 985 76
0.5
1.0 2.0
ENERGY,MeV
3.0
Figure 3F.Taylor's Dose Buildup coefficient Q2, for a point
isotropic source in water, as function of energy E.
100

II. Berger's Coefficients
For Berger's factors, required for 105 gamma energies from 0.01 to 0.5 MeV
(Table 1D) a more limited application is appropriate. Note that Berger's
parameters are, spatially, less generally valid than Taylor's, i.e. an element
of uncertainty is introduced as a given number of MFP's is exceeded (see Table
2A & supra). This is an undesirable effect in calculating relaxation lengths
or any thicknessdependent quantity. Fortunately, Berger's factors need apply
to only 15% of the total energy released by the 238U decay chain, which
suggests that simplified approaches would not result in gross overall error.
Specifically, the best use of Berger's factors is thought to be one that
bypasses  or ignores the problems of discontinuity inherent to dealing with
varying thicknesses of tailings, and limits their application to one simple
case. This simple case is that of the bare tailings slab of "infinite"
thickness, a "maximum flux" or "worst case" condition expressed by Equation
(40C) repeated below.
\
2)
2ut(E)l
This equation may be viewed as producing a (somewhat) tentative corrective
term to be added to the corresponding "worst case" fluxes (and exposures)
obtained via Taylor's parameters and Equation (16C), with other results
adjusted accordingly when pertinent. Spatial dependence being absent from
Equation (40C), the energy dependence of Berger's factors may be dealt with in
a "compound" manner, defining a "Berger's effective buildup factor for worst
case conditions", or BWC(E).
, . C(E) (4F)
101

Berger's Form parameters C(E) and D(E) have been calculated by A. B. Chilton
for yr ฃ 7, yr ฃ 10, and yr <_ 20 (Trubey, 1966.). The choice of one set of
parameters over another appears to be moot, since the slab under consideration
is assumed to be of infinite area! extent, regardless of the manner in which
the slab "infinite" thickness can be represented. Choosing the C(E) and D(E)
parameters for the 20 MFP case may seem, at first regard, a slightly better
option, since they apply over a greater range. To offset this presumed
advantage, the corresponding parameters for the 7 MFP fit generally produce
more conservative buildup values. This may be verified by comparing, in Figure
4F, the buildup factors at various distances from a 0.255 MeV point source in
an infinite water medium, obtained with Berger's formula using both 7 MFP and
20 MFP coefficients. Further analysis is suggested by these considerations.
Figure 4F. Buildup
factors as functions of
distance, in meanfree
paths, from a 0.255 MeV
isotropic point source
in an infinite water
medium, calculated with
Berger's buildup formula
using C and 0 buildup
coefficients based on 20
MFP and 7 curve fits.
The latter result in
higher buildup for dis
tances over 4 MFP's.
100
t

For distances up to 4 MFP's from the 0.255 MeV point source, buildup factors
obtained with C and D coefficients based on the 20 MFP fit are slightly higher
than the corresponding buildup based on 7 MFP fit coefficients. This
p
observation achieves significance when the buildup term [1+C/(D1) ], for
the bare, infinitely thick source slab case (see Table 1), is evaluated with
both sets of C and D parameters: the 20MFPfit values for C and D result in
a buildup term that is also higher (by 8% or 9%) than the value produced by
the 7MFPfit coefficients, implying that buildup from sources within a short
distance from a receptor will override all buildup effects from more distant
sources. This "distance of observable effect" must be, indeed, rather short,
since the 7MFPfit buildup factor at 10 MFP's from the 0.255 MeV source
exceeds the 20MFPfit buildup factor by 87% (see Figure 4F), and by roughly
400% at 20 MFP's, while the above comparison of buildup terms for infinite
source slabs obviously negates such enormous differences. Consequently, the
B (E) term in Equation (4F) may be gainfully evaluated with C(E) and D(E)
we
coefficients applicable to distances of less than 10 MFP's, with 7MFPfit
parameters being a natural choice. This is an important factor considering
that the 20 MFP fit at 0.255 MeV ("dose buildup", point source in water)
produces a maximum error of 30%, whereas that for a 7 MFP fit is only 10%
(Trubey, 1966). Since the parameters C(E) and D(E) are not available for
energies below 0.255 MeV, some judicious extrapolation is required to cover
the remainder of the gamma energy range, in which light the choice of 7 MFP
coefficients appears judicious by offering less possibility of serious error.
The values acquired by B..r(E) as function of energy, based on Berger's
WC
parameters C(E) and D(E), for the 7 MFP fit, are given in Table 1F.
103

Table 1F. Values of Bwr, C* and D* for Energies 0.255 MeV to 1.0 MeV
WC*
Energy E
(MeV)
0.255
0.5
1.0
Dimensionless parameters
C(E)
1.7506
1.3245
1.0622
D(E)
0.2609
0.2078
0.1052
B (E)  1 * C(E)
[D(E)1]2
4.2046
3.1105
2.3266
* Coefficients C and D from Trubey (1966)
The corresponding curvefitting equation for Bwc as function of energy follows:
/a. \
(Figure 4F) Bwr(E) = 1.0 + exp
 b,
(5F)
we'
where a3 = 204.525 558 5
b3 = 17.131 305 11
c, = 12.221 355 02
Although the values used in the curve fit ranged from 0.255 MeV to 1.0 MeV,
inclusive, the range of Equation (5F) applicable to the purposes of this
study is determined as 0.185 MeV < E < 0.5 MeV. The upper bound is prescribed
by the availability of Taylor's coefficients (preferred to Berger's) for
energies E ^ 0.5 MeV. The setting of the lower bound at a value below 0.255
MeV is based on necessity, and requires additional explanation.
Generally, some extrapolation of a curve fitting equation may be regarded
as valid, to the extent that it does not conflict with accepted facts. Such a
pop
conflict occurs at the lowest energies of the U chain decay spectrum. To be
specific, at E = 0.01 MeV, Equation (5F) produces B.,.., = 17.6 . This is clearly
WLr
a fallacy, since at this energy the mass energyabsorption coefficient %?/
approaches the value of the massattenuation coefficient vyp , suggesting that
104

B (0.01 MeV) =* 1.0 . Consequently, some correction is required if Equation
we
(5F) is to apply to energies below 0.255 MeV. The correction should produce a
net effective buildup factor meeting mainly the following constraints:
1. At E = 0.01 MeV, the net effective buildup factor should have a value of 1.0
2. For the range 0.255 MeV < E ^ 0.5 MeV, the values obtained through Equation
(5F) should remain unaltered.
3. The resultant curve should lack discontinuities. Thus the maximum
2
buildup cannot occur at 0.255 MeV (d Bwc /^2 is negative at E = 0.255 MeV )
4. For lack of better information concerning soil crosssections, the
maximum buildup is assumed to appear at E 0.12 MeV, roughly midrange
of 0.01 MeV < E < 0.255 MeV.
An energydependent correction term CT(E), when subtracted from the
corresponding values of B (E), produces an "extrapolated" net effective
buildup factor BV11_(E) in agreement with the set constraints (Figure F).
AWL.
/ d3 \
where CT(E) = exp  + gJn + h, (7F)
\lnE+f3 V
and d3 = 1.757 679 538
f3 = 1.682 331 986
g3 =0.281 565 645
h3 = 2.116 732 933
The range of applicability of Equations (6F) and (7F) comprises energies
0.01 MeV < E < 0.185 MeV, thus complementing the range set for Equation (5F)
of 0.185 MeV < E l 0.5 MeV. The setting of E = 0.185 MeV as boundary between
the two ranges is based on the observation that C_(E) becomes negligibly small
as E increases to 0.185 MeV, but infinitely large when E = exp(f3) is exceeded.
105

B
wc
&
*
100
90
80
70
60
50
40
30
20
10
9
8
7
6
EFFECTIVE DOSE BUILDUP FACTORS,FOR A POINT AT THE SURFACE OF
AN INFINITELY THICK "WATER SLAB" WITH DISTRIBUTED SOURCES,
BASED ON BERGER'S COEFFICIENTS FOR THE 7MFP FIT,
As Function of Energy:
For 0.185 MeV i E < Q.5 MeV, B; JE) = 1.0 +
For 0.01
1.757679538
1.682331986
0.281565645
2.116732933
o.oi
0.12
ENERGY, MeV
0.185 0.255
Figure 5F. Effective Buildup Factors BWC(E) and BXWC(E) for a point at the surface
of an infinitely thick slab with distributed sources, based on Berger's
dose buildup coefficients for water (07 MFP fit), as function of energy.
106

Appendix G
Ancillary Curve  Fitting Equations
In addition to Taylor's and Berger's coefficients, the equations
introduced in preceding sections include other energydependent parameters,
namely
Pt(E) = energydependent attenuation coefficient of tailings, cm"1
yc(E) = energydependent attenuation coefficient of cover material,cm"1
uajr(E) = energydependent attenuation coefficient of air, cm"1
= energydependent massenergy absorption coefficient of air, cm2/g
P
Jair
In conjunction with several geometric parameters, most of those mentioned
above serve as input to the argument of yet another function,
E~ (argument) = Second order exponential integral
where argument = fra1(E),o2(E),ut(E),yc(E)ปVair(E)ปd and/or tl
All these quantities have been extensively tabulated in several publications
(e.g. Radiological Health Handbook), in which form they can be used directly
in any computer program possessing the necessary interpolating subroutines.
Nevertheless, since a simpler process was envisioned in developing the present
method, it was thought best to express them as explicit functions of energy,
or of the generalized argument in the case of the exponential integral.
Attenuation coefficients for "tailings" or "soil" (cover material) are not
available, which necessitates approximating these functions of energy on the
107

basis of coefficients obtained for other materials. The approach is suggested
by Figure 1G, in which massattenuation coefficients y/p are plotted as
functions of energy for various materials. These were chosen according to
their abundance in the earth crust and represent, in broad terms, the main
components of a generalized "soil". Quoting Hammond (1966), "oxygen accounts
for about 47% of the crust by weight, while silicon comprises about 28%, and
aluminum about 8%. These elements, plus iron, calcium, sodium, potassium, and
magnesium, account for about 99% of the earth's crust." Other materials were
added for diverse reasons. Carbon and water were included by at least one
researcher (Beck, 1972) among typical soil components. Since both water and
concrete were equally relevant in choosing parameters for Taylor's buildup
formula, the latter material was added for comparison purposes. Silica (SiO)
is the major component of tailings, with all the elements mentioned above,
plus uranium, being present as complex silicates (G.E.I.S. Uranium Milling,
1979).
One important observation can be made from Figure 1G, that for energies
0.25 MeV ฃ E ฃ 3.0 MeV, the u/p coefficients of the various materials lie
within a narrow band of values, with a maximum difference of about 15%
(between H20 and Fe, at E = 0.8, 1.0, and 1.5 MeV). This suggests, for these
energies, a generalized massattenuation coefficient approximately independent
of material, and depending solely on energy.
[lili] w[li(E)l where m = H20, Fe, Al etc. (I_G)
L p lm LP JG G = generalized "soil"
This generalization allows choosing the massattenuation coefficient of water
to represent the "generalized y/p" in (1G), without introducing gross error,
while retaining consistency with the choice of medium in selecting Taylor's
108

Energy, MeV
Figure 1G.Mass attenuation coefficients for various materials
109

and Berger's buildup parameters. Henceforth, by definition,
H20 v^u'
Consequently, the linear attenuation coefficient y(E)m of any of the soil
materials in Figure 1G, and thus that of soil itself, can be approximated by
multiplying the massattenuation coefficient of water, at the energy E, by the
density pm of the given material.
u(E)mป iii x Pm (3_G)
. JH20
Since the density of "tailings" is expected to be roughly that of "soil"
(cover material), it follows that
w . .n
x 1.6 g/cm3 /d_r\
where "1.6 g/cm " is the density of both "tailings" and "moist packed soil",
as per Schiager, 1974.
The selection of the massattenuation coefficient of water to represent
the "generalized massattenuation coefficient" in (1G) was influenced by the
choice of Taylor's and Berger's buildup coefficients for "water" in lieu of
the unavailable "soil" parameters, envisioned as a conservative alternative
that would increase, rather than reduce, the calculated values of "flux". The
wish for consistency discouraged other choices, although Equations (16C) and
(40C) indicate that a lower coefficient, such as that of C in Figure 1G,
would further increase calculated "flux", leading to an extremely conservative
model.
In that context, (P/P)H 0 is very conservative for E < OA MeV but, at
110

higher energies, it exceeds the coefficients of most soil materials generally by
some 10%, that of Al rather uniformly by 13%, and that of Fe by up to 15%
(Figure 1G). Correspondingly, the fluxes and exposure rates at these higher
energies would be unquestionably lower than those for true soil, were it not for
the compensatory effect of the conservatively chosen buildup factor (Appendix
E).
This effect is illustrated by a rough comparison of fluxes and exposure
rates calculated by using water to represent soil versus those obtained by using
aluminum "which is a fairly good approximation for soil" (Beck, 1981).
Referring to Equation (16C), the effect of using the H^O buildup factor
rather than that of Al is that of increasing flux and exposure rate, at any
given energy, by a factor "[Bl^O/ [B]AI", eclua1 to tne ratio of the
corresponding bracketed "buildup terms" in Figure 1E (indicated by "[B]" in
present notation). On the other hand, the use of the H20 massattenuation
coefficient (^//o)HpO instead of (^/O)AI in Equation (4G) increases pt by
the ratio "(^/p)HoO/^/p)Al"ป which amounts to reducing flux in (16C) by a
factor "(^/(o)Al/(^J/p)H9o"< Listing both increase and reduction factors in
Table 1G for the energy intervals used by Beck (1972) indicates that their net
effect, or product, is one of increasing low energy fluxes and reducing high
energy fluxes by up to 9%, respectively, assuming unit intensity for each energy
interval. Considering the actual tabulated intensities (also from Beck, 1972)
and average energies for each interval indicates an overall flux overestimation
of 0.5% and a total exposure rate underestimation of 4%, always assuming that Al
is the exact analog of soil. A discussion in Appendix I suggests some
liabilities of this assumption. In the interim, the above calculationsserve to
point out that the choice of H20 to represent soil will not result in gross
error.
Ill

Table 16. Effects on Flux and Exposure Rates of Using Water Buildup Factor and
Mass Attenuation Coefficient Instead of the Corresponding Parameters for Aluminum
Energy
Interval
(MeV)
Average
Energy
(MeV)
Intensity
(gammas per
disintgrtn)
Reduction Factor:
.05 .15
.15 .25
.25 .35
.35 .45
.45 .55
.55 .65
.65 .75
.75 .85
.85 .95
.951.05
1.051.35
1.351.65
1.651.95
1.952.55
2.55
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.2
1.5
1.8
2.25
2.55
.139
.104
.196
.361
.022
.436
.027
.084
.032
.014
.252
.137
.218
.081
.002
i.o
.8905
.8739
.8745
.8719
.8705
.8704
.8702
.8686
.8670
.8680
.8696
.8725
.8788
.8840
Increase Factor Exposure Rate
(Buildup "Term" and Flux Ratios
Ratios): per Unit Intsty.
1.0
1.089
1.069
1.07
1.066
1.045
1.014
.989
.966
.952
.926
.915
.914
.916
.915
0 *
223 *
223 *
223 *
1.223
1.2
165
136
112
098
067
052
1.047
1.042
1.035
Notes:
* From Figure 1E, the ratio of the "bracketted" buildup "term" of water
to that of aluminum is 1.223 at 0.5 Mev, and likely to increase for
energies E< 0.5 MeV. Thus, a minimum ratio of 1.223 was assumed to be
valid for energies 0.15 MeV*E<0.5 MeV.
(3) From Beck (1972)
(2) Midpoint of energy interval
(?) From Equations (15C),(16C),etc. and Equations (lG) through (4G):
for E>0.1 MeV, OVp)H 0> (*VP)Ar thus using (ปVp)H 0 in (4G) will
produce somewhat higher values of ME) which, in turn, will reduce
flux calculated through Equations (16C), etc. by the indicated ratio.
From Figure 1E: [B] =
,1.e. "bracketted buildup 'term'
Net effect of reduction and increase factors; product of
Overall
Effect
effect
on total
LL(3) x
E
exposure rate
' '
(ง)]
1.0]
[ฎ x
[CD x
1.005
ฉ x
1.0 x
,i.e. a 0.5%
 n ofi
u . 3D ,
ฉ]
increase
a 4% decrease
112

No similar complications attach to uair(E)ป which is simply
* 0.001203 g/cm3 (56)
f> air
where "0.001293 g/cm3" is the density of dry air at 760 mm Hg and 0ฐC.
Values obtained from the Radiological Health Handbook (1970) were used in
fitting curves to the mass attenuation coefficients for water and air, and to
the mass energyabsorption coefficient for air, as functions of energy:
Mass Attenuation coefficient of Water ป as function of energy
(Figure 2G) :
for .01 MeV ฃ E _< .08 MeV, (y/,o)H Q = F^E) + G4(E)
for .08 MeV 1 E 1 3.0 MeV, (y/

10
I
0.1
0.04
Mass Attenuation Coefficient for Water.
For 0.01 MeV * E =ฃ 0.08 MeV :
For 0.08MeV ^l ^ 3.00MeV:
, As Function of Energy
H2ฐ
where: F4(E) 6Xp ( p E ^fa + C4 InE
c, =
and : a4 = 0.5710089221
b,, = 2.485192485
3.082595417
12.34569243
53.13528831
= 10.621 80209
 2.35316425
9, ='
max.error = ฑ1.151
o.oi
o.i
Energy,MeV
Figure 2G.Mass attenuation coefficient for water, as function of energy

Mass Attenuation Coefficient of air U(E)1 , as function of energy (Figure
, n L<ฐ J .
3G): air
for .01 MeV < E <_ .015 MeV./y/p) = F5(E) = exp (a5lnE + b5) (SG)
V /air
where a5 = 2.883 555 097
bs = 11.671 826 05
for .015 MeV 1 E 1 .6 MeV,(y/p) . = F5(E) + 65(E)
\ /air
where G5(E) = exp ( + f 5lnE + g5]
\lnE+d5 /
and c5 = 1.028 577 166
d5 = 4.464 072 73
f5 = 0.451 578 597
g5 = 2.482 816 293
for .6 MeV < E < 3.0 MeVtfp/p) . = G,(E) + H,(E) (10'G)
where H5(E) =  + 15
ln 5
and h5 = 3.409 847 524 x 10~2
k5 = 2.730 717 269
15 = 1.536 800 255 x 10
2
115

o.i
0.03
Mass Attenuation Coefficient for Air, as Function of Energy
For 0.01 MeV < E < 0.015MeV,
For 0.015MeV ^ E < 0.6MeV,
F,(E)
air
= FC(E) + GC(E)
air
wher.
where H,(E)
5
InE + ke
and:
a5=2.883 555097
b5=11.671 82605
C5=1.028577166
(J5= 4.46407273
f5=0.451 578597
g5=2.482 816 293
h5= 3.409 847 524 x 10
K5= 2.730 717 269
I5=1.536 800 255 x 10
2
2
max. errors : 1.7% and 0.72%
0.01
0.1
Energy,MeV
Figure 3G. Mass attenuation coefficient for air, as function of energy

Mass EnergyAbsorption Coefficient of Air
(Figure 4G):
'ปen'
, as function of energy
air
for .01 MeV 1 E 1 .02 MeV,(wen/ \ = F (E) = exp
\ /ail
air
where ac = 3.157 083 5
D
b6 = 13.0
for .02 MeV E .5
.,
'"
\ = F6(E) +
/air
66(E)
where G6(E) = exp (^ + f6lnE + gA }
VlnE+dc 6/
(116)
(126)
and c, =
b
96 =
1.812 611 059
3.938 990 767
0.103 883 0383
3.030 852 910
for .5 MeV < 2.45 MeV,
lnE+ke
(136)
and h6 = 1781.994 330
k6 = 24.226 319 540
16 = 2.866 717 707
m6 = 69.980 580 070
117

Mass Energy Absorption Coefficient for Air, as Function of Energy
= F(E) =
For 0.01 MeV < E < 0.02 MeV,
_ . = FR(E) + G,(E)
p I 6 6
For 0.02MeV SEฃ 0.5MeV,
+ f, InE + g
For 05MeV ฃ E ฃ2.45 MeV,
 3.1570835
13.0
 1.812611059
3.938990767
01038830383
 3.030852 91
1781.99433
24.22631954
2.866 717 707
69.980580070
o.oi
0.01
Energy,MeV
Figure 4G. Mass energyabsorption coefficient for air, as function of energy.
118

The values used in curvefitting the 2nd order exponential integral as
function of the argument were taken from the Handbook of Mathematical
Functions AMS 55, National Bureau of Standards (1964).
ฃ2, 2nd Order Exponential Integral, as function of the argument X (Figure 5G)
for 0 <_ x 1 0.5,
, / a7 \
E2(x) = expl + C7] + xlnx (14G)
\x + b7 /
where a7 = 0.666 274 740 5
b7 = 1.200 944 510
C7 = 0.554 709 010 2
for 0.5 ฃ x 1 100.0
/ d7 \
1.0 + exp + g?lnx + h7 ) /15.6\
\lnx+f7 /
and E2(x) =
(2
where d7 = 282.378 704 2
f7 = 10.976 502 83
g7 = 3.179 407 102
h7 = 24.195 713 71
The maximum error observed in this curvefit was approximately 0.5%.
119

ro
o
19
05547090102 * xlnx
For 0 > x > 0.5
0.5547090102 + xlnx
For x >0.5
E,(x) =
!. 3787042
1.97650283
3.179407102Inx + 24.19571371
(2 + x) e*
Figure 5G. Graphical representation of the
Second Order Exponential Integral E2> as a
function of the generalized argument x,
obtained on the basis of values from the
Handbook of Mathematical Functions AMS 55,
flatl. Bureau of Standards (1964). The
curvefitting expressions, by the present
author, approximate these values with a
maximum observed error of about 0.5% (*$)
10 20 30 40 40 50 60 70 80
10
10
r'

Appendix H
Computer Implementation
One of the main objectives of the present report is to implement a method
for determining exposure rates over uranium bearing soils that not only would
be fairly reliable and well founded, but also be reasonably simple to apply.
An extreme case involving the use of a programmable desk calculator has been
envisioned. This would require partitioning the energy spectrum into several
ranges, in accordance with the range limits set for the various curvefitting
equations. The minimum number of ranges would thus be roughly a halfdozen,
with a maximum depending on the values of the argument for the second order
exponential integral E , Equations (14G) and (15G). The pertinent equations
would then be applied to each of the energies within a given range, exposures
summed, and the calculator reprogrammed for the next range.
Treating 282 gamma energies by the process described above is likely to be
tedious and time consuming. An alternate approach was followed by the present
author, involving the use of a computer. However, to test ease of
application, software development was abrogated in favor of implementation
through the ISIS program on a CDC 6400 computer. ISIS (1975) is an
interactive statistical package permitting the creation and manipulation of
data files through simple commands following the conventions of FORTRAN
EXTENDED. New files may be generated from previously created files and stored
by the computer. Naturally, no user commands are "stored" beyond the time at
which a new file has been created, i.e., no permanent new software is
maintained by ISIS.
121

A basic example of the operational scheme is provided by Figure 1H,
depicting the process whereby effects of varying thicknesses of the uranium
bearing soil slab on exposure rates are determined. As initial input, the two
files "BELOHAF" and "OVERHAF"* were created by using the ISIS "utility" TYPDAT
(TYPe DATa), each containing energies Ej and the corresponding source terms
Sv(Ei) for E < 0.5 MeV and E > 0.5 MeV, from Tables 1D and 2D,
respectively, and stored in memory. Subsequently, two other files, "BERGERS"
and TAYLORS", were generated through "utility" TRADAT (TRAnsform DATa),
listing buildup coefficients, attenuation parameters, etc. for each of the 105
and 177 energies in files "BELOHAF" and "OVERHAF", respectively. These two
new files were also stored.** Since the buildup coefficient in file "BERGERS"
includes an extrapolation of unverified validity for E < 0.255 MeV, and
Berger's coefficients are range dependent, no further use was made of this
file in the present case (see Table 2A, Equations (6F), (7F) and
accompanying discussions). File "TAYLORS", however, was transformed
repeatedly with TRADAT, using specific values of "t" (uranium bearing soil
slab thickness) to create successive files "XPOS1", "XPOS2", etc., containing
"fluxes" and exposure rates for each energy Ej in each of the given cases
t = 1 cm , t = 2cm, etc. Again, the various "XPOS..." files were stored.
Average exposure rates were obtained with ISIS utility MULDES (MULtivariate
DEScription) applied to each "XPOS..." file, and multiplied by "177" to
determine the total exposure rate X(t) for each specified t.
* ISIS data file names are restricted to seven alphabetic characters.
** "Storing" a file implies "making a file permanent", without curtailing the
user's facility for altering copies of this permanent file in the process of
generating new files.
122

Note that the several X(t) values are given in terms of R/sec per pCi/cm3
necessitating an increase by a factor of 3.6 x 10 to be expressed in pR/h per
pCi/cm3, and multiplication by 1.6 g/cm3 (soil density) to produce values in
yR/hr per pCi/g.
Four separate calculations were carried out using this and similar
schemes. A brief summary of Tables and equations relevant to each calculation
is given below.
Calculation 1) Maximum exposure rates at the ground surface.
This calculation was performed assuming a uraniumbearing soil slab of
infinite thickness without overburden, employing the files and equations
referenced below.
For E < 0.5 MeV:
Table 1D (File "BELOHAF"), 105 energies and source terms,
Equations (5F), (6F), (7F) for "Berger's effective buildup factor for'worst
case conditions'",
Equations (3G), (6G), (7G) for the linear attenuation coefficient of soil,
Equations (11G), (12G) for the mass energy absorption coefficient of air,
Equation (40C) for "flux",
Equation (1C) for exposure rate (File "LODOSEM" with 105 energies and
exposure rates).
For E > 0.5 MeV:
Table 2D (File "OVERHAF"), 177 energies and source terms,
Equations (1F), (2F), (3F) for Taylor's buildup coefficients,
123

Equations (3G), (7G) for the linear attenuation coefficient of soil,
Equation (13G) for the mass energy absorption coefficient of air,
Equation (16C) for "flux",
Equation (1C) for exposure rate (File "HIDOSEM" with 177 energies and
exposure rates).
Equation (46C) for summation of exposure rates both for E < 0.5 MeV and E >
0.5 MeV.
Calculation 2) Maximum exposure rates at one meter above ground surface.
This calculation reduces the exposure rates due to each of the 282 energies in
1) corresponding to the effects of air attenuation.
For E < 0.5 MeV:
File "LODOSEM", with 105 energies and exposure rates,
Equations (5G), (8G), (9G) for linear attenuation coefficient of air,
Equations (14G), (15G) for 2nd order exponential integral values, modifying
factor,
Equation (47C) for summation of exposure rates (1st term).
For E > 0.5 MeV:
File "HIDOSEM", with 177 energies and exposure rates,
Equations (5G), (9G), (10G) for linear attenuation coefficient of air,
Equations (14G), (15G) for 2nd order exponential integral values, modifying
factor,
Equation (47C) for summation of exposure rates (2nd term)
124

Equation (47C) for summation of exposure rates (1st and 2nd term)
Calculation 3) Dependence of exposure rate on thickness of uranium bearing
soil slab.
This calculation determines the effect of varying the thickness "t" of a
uranium bearing soil slab without cover material on exposure rates due to
gamma energies higher than 0.5 MeV (85% of total energy emitted).
Table 2D (File "OVERHAF"), 177 energies and source terms,
Equations (1F), (2F), (3F) for Taylor's buildup coefficients,
Equations (3G), (7G) for the linear attenuation coefficient of soil.
Equations (14G), (15G) with a specific value of t, for 2nd order exponential
values,
Equation (15C) for flux,
Equation (1C) for exposure rate
Equation (46C), 2nd term, for summation of monoenergetic exposure rates
resultant from a slab of thickness "t".
The process is then repeated for the next chosen value of t, etc.
Calculation 4) Dependence of exposure rate on depth of cover slab.
This calculation determines the effect of varying the depth "d" of overburden
material covering an infinitely thick uranium bearing soil slab on exposure
rates due to energies greater than 0.5 MeV.
The tables and equations of 3) are used in 4) with the sole exception of
Equation (15C) for "flux", here replaced by Equation (17C), "d" becoming the
new input variable.
125

ro
_
ISIS.TYPDAT
ฃ"^ " U2*
/
Equations (7GU4G) / .. /r ป
"ISIS.TRADAT S *A*soirEi'
Direct transfer / S (E. ) \
ISIS.TRADAT / v/ 1 \ /
^f *s ** ^f
ISIS.TRADAT >^ _ t

APPENDIX I
Sample Calculations for a Monoenergetic Case
Application of Basic Computational Scheme to K uniformly distributed in
soil with infinite halfspace geometry:
Basis: 1 pCi/cm3 E = 1.464 MeV
1 pCi/cm3 = > 3.7 x 10"2 decays
cm3second
SV(E) = 3.959 x 10 7's
cm3.$
Intensity = 10.7% = > .107 Y'S
Decay
TAYLOR'S BUILDUP FACTOR COEFFICIENTS FOR E = 1.464 MeV
From Equation (1F) A = 14.576 250 06
From Equation (2F) c^ = 0.069 778 860 58
From Equation (3F) az = 0.003 288 967 120
SOIL ATTENUATION COEFFICIENT FOR E = 1.464 MeV
From Equation (7G) (p/P) = 0.058 688 605 16 cm2/g
2
From Equation (4G) psoj = 0.093 901 768 26 cm"1
GROUND SURFACE FLUX FOR E = 1.464 MeV
From Equation (16C) d> = 4.506 843 598 x 10"2 y's
r cm2s
AIR MASS  ENERGY ABSORPTION COEFFICIENT FOR E = 1.464 MeV
From Equation(l3G) (uen/p)air = 2576 712 795 x 10"2 cm2/9
127

EXPOSURE RATE PER pCi/g AT GROUND SURFACE FOR E = 1.464 MeV
From Equation (1C) Xs = 3.101 701 261 x 10" R/S per pCi/cm3
Conversion Xs = 1.786 579 926 x 10'1 yR/h per pCi/g
AIR ATTENUATION COEFFICIENTS FOR E = 1.464 MeV
From Equations (9G),(10G) (ji/p)a1r = 5.244 559 536 x 10'2 cm2/g
From Equation (5G) yair = 6.781 215 480 x lO'5 cm'1
ARGUMENT FOR 2nd ORDER EXPONENTIAL INTEGRAL
yair x 100 cm = 6.781 215 480 x 10'3
VALUE OF 2nd ORDER EXPONENTIAL INTEGRAL = Modifying Factor for Xlm
From Equation (14G) E2[yair x 100 cm] = 0.962 992 296 8
Exposure Rate at 1 meter Above Ground Level For E = 1.464 MeV
Xlm = 1.786 579 926 x 10"' yR/h per pCi/g x E2[yair x 100 cm]
.= 1.720 462 706 x 10"1 yR/h per pCi/g
= 0.172 yR/h per pCi/g
Beck (1972) result: Xlm = 0.179 yR/h per pCi/g
Replacing the buildup factor coefficients and massattenuation parameters for
water [Equations (1F) through (3F) and (7G)] with those for aluminum results
111 X, = 0.189 uR/h per pCi/g
im
The curvefitting equations for the relevant aluminum coefficients appear in
Figure 11, including that for the buildup parameters in compound form [ see
bracketted "buildup term" in Equation (16C)].
128

(lJ/0)/n= O0613 exp(0.5C
v i\ \
at E  1.464 MeV, (*Vp)A1 =
' ' i
' " ! i
3 H j.4l_If ...
1 H lf  TT+
rt'r ^SsLt {{
1 [ i H"M ?>?*
_t_ ! ' ! ; M '
i ' '
1 1 1
\ \
\ J ! t
0 t
L. I ' r;
L A 4 A ~\ X
A , 1A ...Jtlttl
. + ti  
1+04 1+a0 .. TTT. t
Al i
i
i
i
.j. _p
i. .. L.L...
14855762 InE) <07
0.0505692 cm2/g
.06
\f) .05
Al
.04
i i l[ '
~"""TMtjtfjJ];
lit ir H"?! ijiljji
J f+jf ']; "i"!~yjTr irti
  Ji ^rl* ,^fi fjtr 4*7
i 4] j i.l ^4 t J^
v in+ t"" rftr t"^' "rl
i ' l ^^*i i '
1 1 : i : .' i 1 . *^ป
! , : l ' ' ! ! 1 M
! ' . ' )' "jit
T i ; 1 . i : l : l
i '' i i HI
l!i i i : i !'
T ' il' i ii '
r J r "rH" "jT" *tjT tr"
' 1 ' 1 ' I'l
i i ' !  j I ] : J
ltt^4+'^M:4:M
J . !
.i...Jlt.n..i.E3
f t n . t.. t s
t t Tt ft Jttt ftr
i 1! j J
ii 111
t lit T in i It
ii! Hi
J.LIlL.'''.^ 1"
r tr i, ttt i it
r IT t rntTT
it ' ni
ii
n * i ^i
4;
TTT "i
!'^ ir
tr~
Iir :]  "
n; i "
"T" "p'
rl 'fr1
?i 4i:rj
'
rwr _tj
in in
T i Ii!
;  1 '  ! !
T^ ^1'
fHl Tl
Hr! l!
Til t!
! *" 1 ,
i.':
i ij
T'! i!
TTHF
I ; 1 '
Jjil ^
I ,ij
. t j
* !:ii
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\b>
rm
r~
Urf
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ll!
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. .J. J
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s

The exposure rate calculated using aluminum massattenuation and buildup
factor coefficients is clearly more conservative  roughly 10 % greater
than the result based on the corresponding water parameters, at 1.464 MeV.
However, the latter result shows better agreement with the value published
by Beck in 1972. This suggests that aluminum is not necessarily a better
analog for "soil" than water, in applications of the present method.
130

(P/p)A1= 0.0613 exp(0.5048i
at E = 1.464 MeV, (^/p)A1= 0
r T i J
i
' ~ i ' '
1 H i l!r
rrH^t
f~u tr'T
: ! ' Ljjl
: T^ J
T TT" 4ปJ ^L.
' J I " i ~* j T < 1 ! 1* ^7.
i i  i fjLj (> 4P
 i ! t i
1
rr T~r~t1~T j
_l {I 1 i
i i ; ' i
1 i
2 ! M
{  j i .
fA t A ~\
A . 1A :::iiฑ ฑฃฃi
+ . T jrr
i+a, l+a, I r"TTTT
"JAI "lit"!"!". :i
Al TT T T;
"I
i TT t
i T
I I i
i. ...ฑ:i it
55762 InE) <07
.0505692 cm2/g
.06
ViP/ .05
x 'AI
.04
_ 11 Li U III 'I! ]:'
iMitiituia ฃ ~3
T rrrfttTTjTit p r+
rir
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ul r*~ 'T^' 4f fr ' i i  rf^  *t*.
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.6
.7 .8 .9 1
Energy E, MeV
Figure 11. Massattenuation coefficient and buildup "term" for aluminum, for
the energy range 1.0 MeV

The exposure rate calculated using aluminum massattenuation and buildup
factor coefficients is clearly more conservative  roughly 10 % greater
than the result based on the corresponding water parameters, at 1.464 MeV.
However, the latter result shows better agreement with the value published
by Beck in 1972. This suggests that aluminum is not necessarily a better
analog for "soil" than water, in applications of the present method.
130

Appendix J
Comments on CurveFitting Exposure Rate Models
None of the Equations (25), (26), (28) and (30) has been obtained,
independently, from theoretical considerations, but from curvefitting
techniques ultimately devolving to an iterative process for determining
coefficients of optimum fit. Fortuitously, these coefficients were found to
be simple powers of the natural logarithm base "e", leading to convenient,
concise expressions with a misleading resemblance to analytically derived
functions. Nevertheless, the interrelationship of these "pseudoanalytical"
expressions may be shown to be consistent with the theoretical bases of the
present work.
The obvious contribution of Equations (26) and (28) to the model of Equation
(30) may be reviewed in summarizing (30) as the product of two ratios, each of
them expressed as an independent function of a single variable, either
uraniumbearing soil slab thickness "t" or depth of overburden "d". Since
both variables are totally independent of each other, the model represents
each ratio to be independently valid, a validity that extends to their product.
Although the effects of uraniumbearing soil slabs of varying thickness, in
Equation (28), are conceptually independent from the consequences of varying
depth of overburden, in Equation (26), the two equations embody similarities
of form that indicate an interconnection. This interconnection may be
supported on analytical grounds. Comparisons of Equations (15C), (16C) and
(17C), implemented by Equation (1C) and the treatment of Appendix K
ultimately yield, for the special cases t = d, the formal relationship
131

Ml = 1  x(d) , for values t = d (1J)
X(co) X(0)
X(~) = X(0) as defined for Tables 2 and 3.
The applicability of Equation (1J) is restricted by the requirements that the
uraniumbearing soil and overburden have the same attenuation coefficient and
that the same building factor be applicable to both materials. These
conditions are fulfilled through Assumption (10) and Equation (4G) in the
present study, and lead to results supporting the validity of (1J), as
comparison of Tables 2 and 3 may verify.
The relationship in Equation (1J) suggests that any expression describing
accurately the behavior of X(d)/X(0) could be used to generate a reliable
model of X(t)/X(ป)t and vice versa, with little more than a change of
independent variable. Such procedure was applied to Equation (26) to generate
Equation (28).
In addition to being analytically useful, Equation (1J) provides a valuable
criteria of accuracy in "curvefitting", by implying that any model
successfully replicating the values of Table 3 must also, when transformed by
(1J), closely reproduce the value of Table 2 to be considered valid. This
amounts to requiring that onecurve fitting equation satisfy two sets of
tabulated values, independently calculated. This criterion is met by Equation
(26), and therefore by Equation (28) as well, enhancing the credibility of
132

these equations* plus, by implication, that of (30).
The origins of Equations (30) and (28) may be traced beyond Equation (26),
which has sources of its own. These are to be found in Equations (24) and
(25), the former being primarily a definition of "depthdependent relaxation
length" L(d) and necessary introduction to the latter, which applies this
concept to summarize the results of Table 3 as a curvefitting equation.
The logarithmic form of the resultant expression for L(d), in (25), was
suggested directly by Figure 2, a continuous graph based on Table 3 values of
"relaxation length".
The accuracy of Equations (30), (28) and (26) may be seen to depend on an
accurate fit of L(d), such as, presumably, that of Equation (25). In that
regard, the graph in Figure 2 invites tempting simplifications of the form
L(d) = a + b ln(d/d ) which must be discarded as undesirable. The various
*Author's note. It does not necessarily follow that any curvefitting equation
reproducing the values of one of the Tables 2, 3 will lead to a successful
model for the other. Applying the observation that powers of "e" appeared to
be particularly useful in obtaining such equations, the present author tested
an alternative fit X(t)/X(ซ) = tanh [e(el)(t/t())e2] to the
values of Table 2, with a maximum error at any point of 1.6%. Unfortunately,
the corresponding expression X(d)/X(0) = 1  tanh [e(el)(d/do)e2]
obtained by applying (10), yielded errors of up to 30%, at d = 100 cm.
133

shortcomings of this formulation may be examined quantitatively by comparing
Equation (25) with an example of the simpler form, also by the present author,
L(d) = 5 + 1.23 In(d/d0) , in cm (2J)
The simplified formulation implies negative values of L(d) for small values of
d, which makes it conceptually unattractive. It also increases curvefitting
error in the range 1 cm <_ d ฃ 100 cm, as compared to the results of (25).
This increase is inherent in the simpler formulation, corresponding to a
semilogarithmic 2point fit. By contrast, Equation (25) requires a 3point
fit.
Of greater significance are the consequences of the simplified formulation for
the region d > 100 cm. It was pointed out, elsewhere in this report, that
computer round off precluded obtaining reliable results for t or d greater
than 100 cm. Whereas for X(t)/X(ฐฐ) this is largely inconsequential, such
handicap in determining X(d)/X(0) for d > 100 cm is of greater importance.
Consequently, the need for accuracy in curvefitting equations for L(d) and
X(d)/X(0) in the region d _< 100 cm increases proportionately to the degree of
generality such expressions may be required to have; specifically for applying
them to the region d > 100 cm.
Any simple equation of the form L(d) = a + b ln(d/dQ) providing a reasonable
fit to the values in Table 3 may be expected to underestimate both L(d) and
X(d)/X(0) for d > 100 cm. This is due to the fact that the rate of increase
of L(d) with respect to Ln(d/dQ) in the above expression is a constant, "b",
whereas the graph in Figure 2 shows a slowly but steadily increasing slope.
134

The consequences of the simpler formulation may be tested by comparing L(d)
and X(d)/X(0) resultant from Equations (25) and (26) with the corresponding
values produced by (2J). The latter generates values which are consistently
and progressively lower than those produced by (25) and (26) as d increases
past d = 100 cm. This indicates that (25) and (26) are more conservative in
gaging the effectiveness of the cover slab in reducing exposures, i.e.,
theyare less apt to overestimate the exposurereducing capabilities of
overburden, for d > 100 cm.
Having defended the advantages of the proposed models against tempting but
shortsighted approximations, it becomes necessary to address the more
fundamental problem of a "depth dependent relaxation length", L(d). Without
this concept, Equation (25) and, by implication, Equations (26) and (30), lack
foundation.
The analytical bases of the present technique may be advanced in support of
this concept. On the other hand, the more traditional notion of a relaxation
constant appears to be supported by empirical data, in treating which,
however, the depthdependent behavior of the slowlyvarying function L(d) may
be all too easily neglected. Note, for instance, that Equation (25) predicts
a change in L of some 10 mm between depths of 1 foot and 2 feet, of another 5
mm between 2 feet and 3 feet, etc. Such differences may be easily attributed
to other factors, or ignored altogether in developing simpler models for which
a "safety margin" would be eminently desirable.
A constant, depthindependent L may provide a substantial "safety factor", if
used judiciously, through underestimation of the exposurereducing capabilities
135

of overburden. Such judicious use entails setting limits on the thickness of
the cover slab for which a given constant L may be used. Exceeding these
limits will produce the opposite effect, i.e., the exposure rates will be
underestimated.
With the aid of Equation (25), specific limits may be determined for each
given L. In the author's experience, proposed values of L vary between 10 and
14 cm. Replacing the depthdependent L(d) in (25) with a generalized constant
L representing these values, and solving for d produces
dL = exp[Le"1/4]/2e3  1/e (3J)
where L = generalized constant relaxation length
= 10, 11, 12, 13, 14 cm
d. = limit depth, in cm, which must not be exceeded if
a constant L is used in Equations (26), (30)
The results are summarized below.
Table 1J. Thickness of the Overburden Slab Which Must Not Be Exceeded
With the Use of a Constant L
Depth_limit d_ which must not be exceeded if equation
Constant L X(d)/X(0) = exp(d/L) is.to produce conservative results
(cm) i.e., overestimation of X(d)/X(0), thus a "safety factor".
10 59.7 cm or approximately 2 feet
11 130 cm or approximately 4 feet
12 285 cm or approximately 9 feet
13 620 cm or approximately 20 feet
14 1350 cm or approximately 44 feet
136

Appendix K
Interrelationship of Exposure Ratios
The relationship of X(t)/X(ป) in Equation (28) to X(d)/X(o) in Equation (26)
is based on the following analysis:
The exposure rate to gammas of energy E from sources distributed throughout a
uraniumbearing slab of thickness t may be determined by combining Equations
(15C) and (1C),
X(E.t) = 6(E) {A1(E)[lE2(t,E,alE)>A2(E)[lE2(t,E,a2E)] } (lK)
where G(E) = FftEp'entE)
L P Mir
l
.Li,
Sv(E) A(E)
2w(E)[l+ai(E)]
A (E) Sv(E)[lA(E)]
2 = 2u(E)[l+02(E)]
E2(t,E,alE) = E2{M(E)t[Ha1(E)]}
E2(t,E.o2 ) = E2
and u = yt = Vc
The expression (lK) may be rewritten, for convenience,as
X(E,t) = G(E)[A1(E) + A2(E)]  6(E)CA1(E)E2(tiE.aiE) + A2(E)E2ftfE,a2E)](2K)
For the special case t=ซ, this becomes
X(E,ซ) = 6(E)[Ai(E) + A2(E)] (3K)
Dividing (2K) by (3K) results in the following ratio
X(E,t) _ Ai(E)E2(ttE.alE) f A2(E)E2(t>Eta2:) (4_K)
= 1 f A2(E)
137

A similar process, applied to Equations (17C), (1C), produces
X(E,d) = AjtE.jE^d.E.aiE) + A2(E)E2(d.E,a2E) (5_K)
X(E,0) ~ Ai(E) + A2(E)
where E2(dfE,aiE) = E2 {u(E)d[l+a1(E)]}
= E {y(E)d[l+a2(E)]
2
A comparison of (5K) and 4K) indicates that, for the special cases of t = d,
X(E,t) _ j X(E.d) (6_K)
) " ~X(E,0)
In keeping to the simplified notation used throughout the report, each of the
ratios of Equation (6K) is expressed in terms of either of two geometric
variables, t or d, while omitting any mention of the second geometric
parameter, which is held constant. A more complete rendition of the
dependence of exposure rate on energy and geometric variables would be
X(E,t,d) = exposure rate due to gammas of energy E,
from a uraniumbearing slab of thickness t,
covered with overburden to a depth d.
On that basis, the components of (6K) could be rewritten as follows
a) X(E,t) = X(E,t,o)
b) X(E,) = X(E,ซ,o)
and
c) X(E,d) H
d) X(E,o) E
i.e. cover thickness d = o, (7K)
in both cases
i.e. thickness of uraniumbearing
slab t = ซ, in both cases
A comparison of above identities b) and d) serves to emphasize the fact that
the denominators in Equation (6K) are equal. This permits rewritting (6K)
in the following manner,
!
X(E,t) = X(E,ซ,o)  X(E,d) (8K)
138

Consequently, a summation of exposure rates over all energies E^ may be
indicated as
N N N
2 X(Ei,t) = X) X(E1f.o)  2 X(Ei,d) (9K)
1 = 1 i = 1 i=1 N = #of 7 lines
Dividing both sides of (9K) by the total exposure rate due to gamma? of all
energies from tailing slab infinitely thick with no cover, Ex(E,~,0), results
ii
in
N N
EX(E,,t) EXUi.d)
i = l n = , 1 = 1 _ (10 K)
~ "
The above expression is exactly equivalent to that of Equation (6K) which, in
simplified notation, is
X(t) X(d)
X(ป) X(0)
139

Appendix L
Radon Distribution Through Overburden
Diffusion theory and Pick's law were used to model the 222Rn concentration
in an infinitely thick tailings pile covered with a finite thickness d of
overburden, both of infinite area! extent, as shown in Figure 1L.
z = d
z = 0
z = <
f
Thickness of overburden, d
L
222Rn source, infinitely thick tailings slab
Figure 1L. Tailings and cover configuration.
The fraction of 222Rn which emanates from the source material in the
tailings represents a flux which is proportional to the concentration
gradient, as per Fick's law
J(z) = D
(1L)
dz
where J(z) = depthdependent radon flux, in pCi/cm2.s
C(z) = depthdependent "free" radon concentration, in pCi/cnv*
D = diffusion coefficient of "free" radon in soil, in cm2/s
Applying Fick's law to the general diffusion equation produces, at steady
state,
 a2C + S = 0
dz2
where a2 = ^n
D
and S = cRa xRnE
D
(2L)
140

and S
= CRa
with ARp = 222pn decay constant, ins"1
CRa = ^Ra concentrations in tailings, in pCi/g
E = emanating power of 222pn jn tailings, dimensionless
The general solutions of (2L) for the concentration of 222pn as function of
depth, C(z), are
Ct(z) = Aeaz + Be"aZ + i , for z <. 0 (tailings)
or
aZ
and Cc(z) = Eeฐ" + Fe"0" , for z >. 0 (cover)
(3L)
(4L)
Four boundary conditions are required to determine the values of A, B, E and
F. They are
B.C.I Ot(ป) =0 At ป, the concentration of free radon is assumed to
be at an absolute maximum, thus dC(ซ)/dz = 0 and
Jt() = 0.
B.C.2 Ct(0) = Cc(0)
B.C.3 Jt(0) = Jc(0)
B.C.4 Cc(d) = 0
Both the free radon concentration and flux are
continuous at the tailingscover interface.
The free radon concentration at the coveratmosphere
interface is assumed to be very small, i.e., approach
ing "zero". In reality, Fick's law does not apply to
such interface.
Solving for A. B. E. F:
From B.C.I: Jt(ป) = 0
or D
dCt(z)
dz
= D(oAeaZ
= 0
141

this means that B = 0 , otherwise 0 would be infinitely large
thus, Ct(z) = AeaZ + for z _< 0 (5L)
or
From B.C.4: Cc(d) = 0
or Eead + Fead  0
so, E = Fe'2ad
and Cc(z) = Fead [ead ' aZ  e(ad ' az)] for z > 0 (6L)
From B.C.3: Jt(0) = Jc(0)
Equating the derivatives of (5L) and (6L) at z = 0 produces
A = Fead(ead + ead)
thus Ct(z) = i  Fe"ad(ead + e'ad) eaz for z _<0 (7L)
a*
From B.C.2: Ct(0) = Cc(0)
Equating (6L) and (7L) at z = 0 results in
S r~~ad/ cxd . ^ad\ _ i^otd/txd ซad\
  Fe (e +e )=Fe (e e )
or F =!_ (8L)
Inserting (8L) into (7L) produces an equation describing the free radon
concentration as function of depth in the tailings, i.e., for z ฃ 0
Ct(z) = ^ [1  ea(z"d)cosh (ad)] for z 1 0 (tailings) (9L)
142

and S =
with ARn
E =
222Rn decay constant, in s'1
^Ra concentrations in tailings, in pCi/g
emanating power of 222Rn in tailings, dimensionless
The general solutions of (2L) for the concentration of 222Rn as function of
depth, C(z), are
az
"az
Ct(z) = Aeaฃ + Be~ฐ" + ฑ , for z <. 0 (tailings)
o2
and Cc(z) = Eeaz + Fe"aZ , for z >. 0 (cover)
(3L)
(4L)
Four boundary conditions are required to determine the values of A, B, E and
F. They are
B.C.I Jt(ซ) =0 At ซ, the concentration of free radon is assumed to
be at an absolute maximum, thus dC(ป)/dz = 0 and
Jt(ป) = 0.
B.C.2 Ct(0) = Cc(0)
B.C.3 Jt(0) = Jc(0)
B.C.4 Cc(d) = 0
Both the free radon concentration and flux are
continuous at the tailingscover interface.
The free radon concentration at the coveratmosphere
interface is assumed to be very small, i.e., approach
ing "zero". In reality, Pick's law does not apply to
such interface.
Solving for A. B. E. F:
From B.C.I: Jt(ซ0 = 0
or 
dz
OtZ
Z=oo
= 0
141

this means that B = 0 , otherwise J would be infinitely Ta'r'g'e :
thus, Ct(z) = Aeaz + for z < 0 (5L)r
From B.C.4: Cc(d) = 0
or Eead + Fead = 0
so, E = Fe2ad
and Cc(z) = Fe"ad [ead ' az  e'(ad ' az)] for z > 0 (6L)
From B.C.3: Jt(0) = Jc(0)
Equating the derivatives of (5L) and (6L) at z = 0 produces
A = Fead(ead + e"ad)
thus Ct(z) = \  Fe"ad(ead + e"ad) eaz for z _< 0 (7L)
From B.C.2: Ct(0) = Cc(0)
Equating (6L) and (7L) at z = 0 results in
J. . Fe"ad(ead + ead) = Fead(ead  ead)
or F =_ (8L)
2a2
Inserting (8L) into (7L) produces an equation describing the free radon
concentration as function of depth in the tailings, i.e., for z _< 0
Ct(z) = L [1  ea^z"d^cosh (ad)] for z C 0 (tailings) (9L)
142

Appendix M
Effects of Radon Diffusion on Exposure Rates
The effects of radon diffusion through cover material on exposure rates was
estimated by numerical integration techniques, employing the models developed
in the study and the radon concentration formulasfrom Appendix L. The method
assumes the typical "infinitely thick" tailings slab covered with overburden
of depth d to be equivalent to a large number of infinitely thick slabs,
occupying simultaneously the same space but with varying radon concentrations
and depths of cover. The fundamental concept is partially illustrated in
Figure 1M, for the specific case d = 100 cm and D = 0.02 cm2/s, and further
amplified by the following description.
Infinitely thick slab, C =
98)+C(98.5)
9B~.5HC(99)
Infiri.tely ; thick
concentration =
7r_ C(99)+C(99.5)
Infinitely
thick slab
of concentre
3 _ "C(i9.'5lfC(10pl
" "
of corcentra,tion (i,0 cqver
Infinitely thick
C(Z),pCi/g
Figure 1  M. Schematic representation of numerical integration method, applied to the
top two cm of a cover of thickness d = 100 cm, on top of an infinitely thick tailings
slab, with a radon diffusion coefficient of D = 0.02 cm^/s, for E = 0.2
145

Having decided on a specific set of values E, d, D, the distribution of radon
C(z) in the cover and tailings is determined by Equations (11L) and (14L),
at regular intervals AZ. The average concentration between two successive
points is then calculated by
C^ _ cdnAz + cd(nl)Az
2
where n = 1,2,3,..
and Cd_nAZ = concentration C(z) at location z = dnAz
As n increases, Cn increases also, by an amount ACm = Cn  Cn_i (see
Figure 1M), the increment becoming effective at a distance zm = d(nl)Az
Setting m = n, and adopting the convention that (T0 = 0, the above may
be restated as ACm not being present for all z > zm, appearing as a step
function at z = zm, and continuing to exist for all z < zm. This is
tantamount to assuming the existence of an infinitely thick slab of
concentration ACm, with a sourcefree cover of depth d  zm = (nl)Az.
Such configuration is ideally suited for the calculation of exposure rates
through application of Equation (30) to the various slabs of incremental
concentration ACm and depth of cover (n1) AZ. Adding the increments AX
resultant from each of these calculations produces the total exposure rate due
to an infinitely thick tailings slab with cover d and diffusion coefficient D.
Repeating this procedure for various d and D values leads to the exposure
rates depicted in Figure 2M, all for E = 0.2.
146

ซ0=0.02 cm Vs
0=0.01
D= 0.005
D= 0.002 ซ
10
40 50 60 70 80
Cover Slab Depth d, 1n cm
90 100
Figure 2M. Relative decrease in exposure rates, with respect to
maximum exposure rate possible, as function of increasing thickness
d of the overburden slab, for emanating power E = 20% and different
value of radon diffusion coefficient in soil, D, in the range 0.02
cm2/s > D > 0.0002 cm2/s.
147

TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA520/682014
2.
4. TITLE AND SUBTITLE
A Basic Technique and Models for Determini
Exposure Rates Over UraniumBearing Soils
7. AUTHOR(S)
George V. OkszaChocimowski
9. PERFORMING ORGANIZATION NAME AND ADDRESS
U.S. Environmental Protection Agency
Office of Radiation Programs, Las Vegas Fa
P.O. Box 18416
Las Vegas, Nevada 89114
12. SPONSORING AGENCY NAME AND ADDRESS
Same as above
3. RECIPIENT'S ACCESSION NO.
5. REPORT DATE
nq August 1982
6. PERFORMING ORGANIZATION CODE
8. PERFORMING ORGANIZATION REPORT NO.
10. PROGRAM ELEMENT NO.
Cl llty 11. CONTRACT/GRANT NO.
13. TYPE OF REPORT AND PERIOD COVERED
Technical Note
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
16 ABSTRACT
The application of simple computerimplemented analytical procedures to predict
exposure rates over uraniumbearing soil deposits is demonstrated in this report.
The method is based, conceptually, on the energydependent pointsource buildup
factor and, operationally, on two consecutive integrations. The dependence of
photon fluxes on spatial variables is simplified by an analytical integration over
the physical dimensions of the deposit, represented as a slab bearing homogeneously
distributed nuclides of the uranium238 decay chain, at equilibrium, and covered
with a sourcefree overburden slab; both slabs being of variable thickness but of
infinite areal extent. Elementary computer techniques are then employed to
integrate numerically the exposure rates corresponding to the specific energies of
uranium238 decay chain, for chosen thicknesses of the overburden and uranium
bearing slabs. The numerical integration requires the use of buildup factors,
attenuation and absorption coefficients expressed as continuous functions of energy
by curvefitting equations included in the report. As direct application of the
method, maximum exposure rates over uraniumbearing soils are calculated, and the
dependence of exposure rates on the thickness of the uraniumbearing slab and depth
of overburden is reduced to a simple model. These results, valid for uranium mill
tailings piles, are compared to those of other authors, and applied to determine
changes in exposure rates due to radon gas emanation from source materials.
17.
KEY WORDS AND DOCUMENT ANALYSIS
a. DESCRIPTORS
Radioactive Wastes
Uranium Ore Deposits
Gamma Irradiation/X Ray Irradiation
Radiation Shielding
Radon
Mathematical Models
18. DISTRIBUTION STATEMENT
Release Unlimited
b.lDENTIFIERS/OPEN ENDED TERMS
Uranium Mill Tailings
Uranium238 Decay Chain
Exposure Rates
Overburden
Radon Gas Exhalation
Buildup/Curvefitting
models.
19. SECURITY CLASS (Tills Report)
Unclassified
20. S EC U R 1 T.Y_ C LASS (This page)
TjncTassifiecr
c. COS ATI Field/Croup
1807
0807
1808
1806
0702
1201
21. NO. OF PAGES
159
22. PRICE
EPA Form 22201 (FUv. 477) PREVIOUS EDITION is OBSOLETE
 