-------
This comprehensive model consists, primarily, of the product of curve-fitting
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 x-rays and
low-intensity 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 x-rays 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
soil-surrogate 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 curve-fitting 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 |_ Curve-Fitting 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 X-rays - 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
uranium-bearing 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(d-z)] (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 tailings-cover 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 tailings-cover interface.
z = generalized distance, normal to tailings-cover interface
where z>0, above tailings-cover interface,
z = 0, at tailings-cover interface,
z<0, below tailings-cover 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 IQ-V1
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 one-tenth 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 source-free 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
"^jD-0,01
1D= 0.005
i i
(D= 0.002
D-OiOOl
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 curve-fitting 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 CONF-720805 (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," McGraw-Hill Book Company.
Copyright 1955. Fourteenth printing May 26, 1972.
Ford, Bacon & Davis, 1977 Phase II-Title I, Engineering Assessment of
Inactive Uranium Mill Tailings, for U.S. Energy
Research and Development Administration (Grand
Junction, Colorado) Contract No. E(05-l)-1658,
Salt Lake City, Utah, 1977
G.E.I.S. Uranium Milling, 1979 Generic Environmental Impact Statement on
Uranium Milling. NUREG-0511, Volume I,
Project M-25, 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, 1966-1967
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/TM-102, 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) 375-476 (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
Gamma-Ray Buildup Factors", ORNL-RSIC-10 Oak Ridge National
Laboratory, Radiation Shielding Information Center, February
1966
55
-------
Appendix A
Choice of Empirical Function to Represent Gamma-Ray 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 gamma-ray exposure rates from sources distributed in
absorbing media must include the effects of "secondary" radiation, consisting
mostly of Compton-scattered photons with the addition of annihilation
radiation from pair-production, and of X-rays 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 gamma-ray 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 energy-flux
buildup, energy-absorption 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 1-A. 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 spatially-dependent buildup factor
included in the integrand. This clearly necessitates expressing the buildup
factor as an explicit function of spatial coordinates.
57
-------
Table 1-A 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
energy-dependent coefficients. Two other polynomial forms, "Berger's" and
"Taylor's", include exponential terms with products of distance and
energy-dependent 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 energy-dependent 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]e-oa(E)p(E)r (i-A)
where Bf(E,iir) = energy and distance dependent buildup factor, dimensionless
A(E),a,(E),a2(E) = energy-dependent fitting parameters, dimensionless
u(E) = energy-dependent attenuation coefficient, cm'1
r = distance, cm
The energy-dependent 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 2-A
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 1-A) and considerably less so for
the various types of concrete, which are mixtures.
Table 2-A 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
2-5
2.1
1-3
1-7
2-3
1.6
3-2
3-2
2.9
2.6
Mean Percent
Range
Taylor
3-6
2.8
2-5
1.9
1.6
0.8
0.8
2.9
2.6
4.2
3-4
age Deviation
7 MFP
Berger**
1.2
0.7
0-5
0.2
0.3
0-7
0.4
2.0
1.4
0.9
0.6
Range
Taylor
3-7
2-5
2.5
1-7
1.2
0.5
0.5
4.0
3-3
4.8
3-7
*20-MFP parameters used.
**7~MFP parameters used.
(From ORNL-RSIC-10, "A Survey of Empirical Functions Used to Fit Gamma-Ray
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 uranium-238 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 air-tailings
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, tailings-ground, overburden-tailings, 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 air-tailings 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 gamma-ray 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 uranium-bearing 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 source-free 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
(1-0
air
where X(E) = exposure rate from photons of energy E, in R/s
FX = conversion constant
= 1.824401368 x 10-8 g . R/Mev
E = gamma energy, in MeV
(E) = "flux" of gammas of energy E, in gammas/(cm2-s)
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 source-free
overburden slab. With these basic premises and Figure 1-C, general equations
for the monoenergetic photon flux at any point "o" in the overburden, at a
distance "d" from the oberburden-tailings interface, are developed in the
following pages.
67
-------
Volume Element dV. with source strength Sv photons/cm3-sec
URANIUM
BEARING
SOIL OR
TAILINGS
SLAB
Point 0 in Cover Slab
I (Overburden), at a Distance d
I From Overburden-Tailings Interface
Figure 1-C. Geometry for flux calculations with a slab-distributed
source (uranium-bearing soil or uranium mill tailings) covered with
a source-free 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 e-r-sece-uc 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, cm-1
dV = volume element, cm3, equal to "r2sirie de d dr" (see Figure i-c)
To obtain the total flux <ฃ of photons of a given energy at 0, equation (2-C)
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 (1-A), Assumption 10 and Figure 1-C, the
spatial dependence of Taylor's Form of the buildup factor can be described,
* See Appendix B
69
-------
for the present case, by Equation (3-C)below.
BT(r,e) = Ag-auitfr-dsecej-ojucdsece +[i_A]e-a2lJt (r-dsece )-a2ycdsece
(3-C)
Equation (3-C) can now replace the generalized "B" in Equation (2-C) and the
resultant expression integrated. Prior to doing so, however, the integrand
can be simplified by multiplying Equation (3-C) by the exponential term in
Equation (2-C)
/ \ -wtCr-dseceJ-yrdsece ... , , / \ , /, *\f / \ / \ /A r\
BT(r,e)x e u ' u = Af^e) g^r) + (!-A)f2(e)g2(r) (4-C)
where f^e) = e Ut-uc)(l+ai)dsece
9i(r) = e-Vt(l+ci1)r
and f2(e) = e("t-Mc)(l+ซ2)dsece
g2(r) =e-ซd+ซ2)r
With these transformations, the integration of Eqn. (2-C) can be indicated
as
(l-A)F2(e)g2(r)] (5.c)
dV
v
where dV = r2 sineded(|>dr
Therefore,
2-n Tr/2 (t+d)sece
= d si
4ir J J
nede [Af^ejg^r) + (l-A)f2(e)g2(r)]dr (6-C)
J
oo d sece
The integration with respect to r produces the following two terms
70
-------
A e-Cisece(1.e-Tisece) (1.A)e-C2sece(1.e-T2sece)
Ti/t * T2/t
where (^ = ;ucd(l+cn) , C2 =
TI = yutt(l+cn) , T2 =
(7-C)
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
(8-C)
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 (9-C)
= y2 sine de
thus^= sinede
y2
As e varies from 0 to w/2 , y = sece varies from 1 to -
Equation (8-C)can now be rewritten
A
Ti/t
"Ciy
-(CiH-Ti)y
6 Ju
ฐy
y2
T2/t
[Kc*y
J ^
.1
dy-
I
i
-(C2+T2)y
-dy
(10-C)
1 1
The form of Equation (10-C)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 (6-C) which now becomes
Sv/
-E2(C2+T2)]j
Replacing Clf C2 , Tj and T2 with their equivalences, defined in Equation (7-C)
permits rewriting Eqn. (12-C) in a more meaningful form.
(13-C)
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 energy-dependence of the buildup factor,
attenuation, and source strength parameters. Nevertheless, the
energy-dependence 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 (13-C) 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 uranium-bearing soil or tailings
material for energy E, cm-1
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) l-A(E)
l+a2(E)J
(16-C)
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)[l-A(E)]
2pt(E)[l+a2(E)]
-\
|pc(E)d[Ha2(E)]|\
The energy-dependence of buildup, attenuation and source-strength
parameters has been repeatedly noted in Equations (14-C) through (17-C) to
stress the fact that their output is, in each case, a monoenergetic flux. By
direct application of Equation (C-l) to such resultant single-energy 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 (18-C) 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(r-dsece)+ydsece]
BB(r,0) = 1 + C [yt(r-dsece) + ycdsece]e (19-C)
Replacing the generalized "B" in Equation (2-C) with the above expression and
carrying out the multiplication produces
> + Cf(r,e)e(D-1)f(r'e)]dr (20-C)
where f(r,e) = Mtr-(u(.-yc)dsece
To integrate the bracketed expression with respect to r, note that
75
-------
dr
, thusdr=?l
(21-C)
which permits expressing the integrals as
(D-l)f
^- If e df
This produces the following two terms
e(ฐ-1)f [ (D-l)f -l]
liCTr L J
(22-C)
constant
(23-C)
With the limits of integration made explicit, the first term of (23-C) becomes
(d+t)sece|
i -ucdsece/, -Pttsece\
(24-C)
-e
|= J_e-ucdsece/ -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 ,
(25-C)
(26-C)
thus
dY = sec2e sine de
* Y2 sine de
dX = Sln9de
X2
with the corresponding change of limits, (25-C) can be rewritten as
dy . fe-(ytt^cd)y
dy
(27-C)
76
-------
As with Equation (10-C), the above integration results in two 2nd order
exponential integrals
(28-C)
Integrating this with respect to $ introduces a factor of 2n. Multiplying the
product of 2ir and (28-C) by the constant term of Equation(20-C) yields the first
term of the integration of(20-C)with respect to r, e, ^>.
(1st. Term) = 2v_ E (M) .
(29-C)
Now the process of evaluating the second term of Equation(23-C)is undertaken:
(d+t)sece
(D-l) utr-(up)dSece j (|M)
Expression (30-C)results in a 4-term 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
(30-C)
(31-C)
-------
For the terms including TI(B ) and 13(9), the following substitution is
useful
y = sece (32-C)
dy = sece(sinesece)do
y
For the terms including T2(e) and T4(e)> the corresponding substitution is
y = sece (33-C)
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,
(D-l)(ytt+ycd) E:[ -(D-l)(ytt+ycd)]
-E2 [-(D-l)(ytt+ycd)]
-(D-l)ycd
E2 [-(D-l)ycd]
(34-C)
The above expression can be simplified by making use of the following
relationships
-X
or
E,(X) = e
Ei(X) = -E2(X) + e'x
(35-C)
78
-------
With the transformations in (35-C) the first and third terms in brackets in
become, respectively
E2 [-(D-l)(ytt+ycd)J - e
(D-l)(ptt+pcd)
(36-C)
and -E2
-(D-l)ycdJ
(D-l)ycd
Cancelling like terms, this becomes
(D-l)wcd (D-l)(utt+ucd)
ut(D-l)2
- e
or C e
(D-l)ycd
(37-C)
Integrating with respect to ty, etc. results in the second term of the
integration of Equation (20-C)with respect to r, e,
Term) = ^ e^^fl-e^v]
SVC (D-l
(38-C)
2yt(D-l)2
Adding the 1st term from Equation (29-C) and expressing the energy-dependence
of relevant parameters produces
_ ME)
C(E)
- E2[yt(E)t
uc(E)dr CD(E)-1]
l-e
(39-C)
-2
79
-------
For the important case of an "infinitely thick" tailings slab (t = ป) without
cover material (d=0), Equation (39-C) reduces to
SV(E)
C(E)
(40-C)
With the values of "surface flux " obtained through Equations (14-C) ,
(15-C),(16-C),(17-C) or (40-C), applied in Equation (1-C), 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 (41-C)
where a(E) = energy - dependent fitting parameter, dimensionless
Replacing uc and d in (2-C) 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
(42-C)
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", (42-C) reduces to
(43-C)
The modifying factor is obtained by dividing (42-C) by(43-C),
ฐairuairh
(44-C)
The second term of (44-C) 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 (41-C)]. This reduces (44-C) to the following expression, with
energy dependences indicated,
(45-C)
81
-------
Although based on flux ratios, the modifying factor FM(E) is directly
applicable to exposure rates, as an examination of Equation (1-C) 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 (46-C) and (47-C), respectively.
XZE< -ฃ/xii?ii,) g . (46-c)
5
m
^d ;_ -E FxEi*(Ei)|^ni^-| .E2[pa1r(El)loOcm] (47-C)
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 (46-C) and (47-C) 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 1-D. 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. 1-D and the radionuclide decay data of
Kocher (1977), a complete spectrum of gamma emission energies present in a
uranium-bearing soil can be compiled. Postulating a "Base Case" of "1 pCi per
cubic cm", and making use of Kocher's intensities, an energy-dependent "source
term" SV(E) is found for each energy E, to implement Eqns. (14-C) through
(17-C).(40-C) and finally (43-C) and (44-C). In agreement to the form of these
last two equations, the SV(E) values are distributed between two tables.
Table 1-D contains SV(E) terms for energies up to 0.5 MeV, for a total of
n=105 values, while Table 2-D consists of the remaining m=177 values, for
energies over 0.5 MeV, where "m" and "n" refer to indices in (43-C) and (44-C)
83
-------
ATOMIC WGT.
ELEMENT
ATOMIC NO.
HALF-LIFE
U-238
Th-234
Pam-234 (99.8756)
Pa-234 (0.13SO
U-234
Th-230
Ra-226
Rn-222
Po-218
Pb-214
Bi-214
Po-214
Pb-210
Bi-210
Po-210
Pb-206
Figure 1-D. Uranium-238 Decay Series
84
-------
Table 1-D 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
.> 13000E-01
t49600E -():!.
,13300E-01
,63282E-01
,92367E- 01
,92792E-01
, .1 1.28.1
, 76 OOOE -0 1
, 1360 or;: --01
,43450E--01
,632001;;: -oi
,699ooi;;:-oi
,805ooE-oi
,94665E-0.i
t98439E"- 0.1.
,99700E--oi
, 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
.9QOOOE-0.1
,390001;;:-- o:i.
,2570oi:;:--oi
,3ooooi;;:--oi
* 24900 E --02
,200001;;: --02
, :!. n;:520E--02
,i5600i;:: --05
,403()0i;;:--04
,29900E--05
, 50 700 E --05
,20150E-03
,32630E-03
,61100E-04
- 15600E--05
* 15210E-03
, .' 1300E-04
.26000E-03
.2/300E.".- 05
, 19500E--05
,120901;;:-- 04
,49400E- 05
,26 OOOE -05
vV3459E--04
,.?4900E-05
,6240oi;;: --05
,247001-: -05
,24700E--05
,702ooE -05
, 9.1 OOOE -05
,14 040!.- -04
, 14300E-04
, 29900 E- 05
.84500E-04
,49400!-: --04
,1 1. 700E-04
,40300E-04
., 22100E-05
,15600E-04
,i820oi;::-- 05
,14040E-05
,41600E-04
,37700E-05
,15600E--05
,15600E-05
,3/.A)OE-04
,37700E-04
, 11440E-04
.74100E-05
,36400E-04
Volumetric
Source
Strength, in
"gammas of
energy E per
cm3 per sec"
SV(E).
,32190E-02
, 25900 E- ()4
,36260E-02
, 14430E-02
,95090E-03
,11100E-02
,92130E-04
, 74000E-04
,54334E-04
,57/20E -07
, !. 49 HE -05
, 1. 1063E-06
, i.8759E-.06
< /4555E-05
, L2073E-04
,22607E.-05
, 5 77 20 E- 07
.56277E-05
.52910E-06
,96200!;;: -05
,1.()101E-06
.72:! DOE --07
,44733!:---06
,'!8278E-06
, 96200E- 07
.34580E--0:vi
,3:51 13 E- 06
,23088E-06
,91390E-07
v91390E"-07
, 259 74 E- 06
33670E-06
< 51 94 8 c- 06
,52910E-06
, 1 1 063E-06
,31265E-05
, 18278E-05
,43290E-06
, :i.491:iE--05
,8:1. 770E-07
,577201;;: --06
,67340E-07
,51948E-07
.15392E-05
,13949E...-06
,57720E-07
,57720E-07
, 13949E-05
,13949E-05
,4232HF-06
,27417E-06
,13468E-05
NUCLIDE
238u
234
Th
234_
Pa
85
-------
Table 1-D (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
,40000E-03
,85()OOE-01
.38000E-02
,70000E-03
^SIOOOE -02
, 18000E-02
, 29900E-02
< i.3::>ooi;;:-02
,32800E-01
.. 1 3595
, 1 100 OF -01
,63300E-01
, 10700
,4 76 00 !> 0:1.
,. 7 4 7 () 0 1:;. () ;|
,55.100E -02
, 32000E-02
.> 1 9200
,37081
,33800E-02
,44000E ;)2
,50000E-03
,52000E -OP
.35800E-02
>60000E-02
,26900E-02
,18000E-02
,36000E-02
,41000E-02
, 16700E-02
,iioooi;;:-02
,31800E-02
,21800E-02
, 13300E-02
,1 1800E-02
,24300
,40500E-01
Volumetric
Source
Strength, in
"gammas of
energy E per
cm-* per sec"
SV(E).
!i1i?^l^
, 18759E-06
, 57720;:; -07
,72i50E -06
* 76960E-07
,18278E-06
, 11544E-06
,86580E-07
< 1 3949;;:;--o6
, 13949E-06
>. 1 6259E-03
, 4 2 7 5 3 ;;;: - o ^
, 69470E-04
"X o A i:i r\ <;. .... i"; '".'
f -..' '...' -v *.:> \.- i... \.- .*..
, 4 366 OE -04
, 1 4800E-04
,3:l.450E-02
, :!. 4060E-03
, 25900E-04
,29970E-03
,66600E-()4
,1 1063 E -03
v 50320E -04
,:l.2:i36E-02
.50300E-02
,40700E-03
,2342 IE -02
,395901;;. -02
,17612E-02
,2/639E-02
,20 38 7 [ 03
, 1 1840E-03
, 71040E-02
,13 720 E-01
- :l.2506E'-03
,16280E-03
, 18500E-04
,19240E-03
, 132 46 E -03
, 22200E-03
,99530E-04
,66600E-04
, 13320E-03
,15:l.70E-03
,6:!.790E-04
,40700E-04
,1 1766E-03
,80660E-04
,49210E-04
,43660E-04
.89910E-02
,14985E-02
NUCLIDE
234Pa(eont.)
234 Pa m
u
23ฐTh
226Ra
2MPb
Bi
J10Pb
86
-------
Table 2-D 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
, 15600E-04
< 136 OOE -04
, 156 OOE- 04
.50700E-03
, 260 00!!!.' -05
,20800E-05
> 3 2 5 0 0 E - 0 5
> 2 21. OOE -04
, 1 3520E-03
,40300E--04
.. 26000E-04
. 19500E-05
: 78000E-03
, 1 6900 E- 04
, 10400E-04
,260()OE--05
.13000E-05
3S400E !)5
'"i -\ * I ) l' i j- ( 1 !"i
,37700E--05
. 27300E-05
,286 OOE -05
,2 86 OOE -05
, 1 6900E--04
,79300E-05
.... ..., .., (..; ^j::- ....-,-s|
, :i 95 OOE- 04
, 19500E-04
, 19500E-04
31 ;:;OOE--05
, 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
, 39000E-04
, 1 1700 E -04
. 18.2 00!!" -04
.20800E-05
, 10400E-04
,26000E--05
. 7:i 500E-05
,26000E-05
,20800E-04
,63700E-05
,20800E-04
.. 19500E-04
Volumetric
Source
Strength, in
"gammas of
energy E per
cm-* per sec"
,67340!: -Of-
,57720i:- -..{
,5/720i;;-0,-.
,57720E-06
, 18759E-06
,96200E-07
,76960E-07
* 1 2025 i::. -06
,81 770 iV- 06
,50024E-05
* :!. -191 1E-05
..96 20 0!'" -06
.. 721 5 OK:- 07
,2 886 OF- 06
,62530E-06
,384 HOE -06
,96200E-07
ป 481 OOE -07
,32708E-06
, :!. 9 7 2 1 E - 0 6
,:!.3949E-06
,10 10 IE -06
,1()582E-06
,10382E-06
, 6 2530 E -() 6
,29341 E-0 6
, 13949E-06
,72.I.50E-06
* 72:!. 30!:.'.'- 06
,72.I50E-06
, !. 1 54 4c -06
. 12987E-06
, 6 2 5 3 0 i'." - 0 6
.-22126h: -05
, 1 5 3 92 E- 05
,76960E-07
, 40885E- 05
, 3848 OE- 06
,14430E-05
,43290E-06
,67340E-06
,76960E-07
,38480E-06
,96 2 OOE -07
.26455E-06
,96200E-07
,769601- -06
.23569E-06
,/6960E-06
.72150E-06
NUCLIDE
234
Pa
87
-------
Table 2-D(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
f-k>
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
:!. ,07-44
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
,50700E-05
, 4 29 00 E- 04
, 63700 E --04
,63700E-OS
,286<)oi;::-- 04
, 46800 E -04
,41600E-04
,72800E-04'
,18200E-05
,76700E-05
, 15600E-05
,35100E--()4
,53300E-04
, 84 500 E- 04
, 1 5600E-03
,52000E-04
,63700E-05
,50700E-05
,364()()E-.04
, 14300E-03
,6 630 OF --04
, ;;!60ooi:-;:-o3
, 10140E-03
, 18200E--04
, :!.8200L":-04
,28600E-04
, 19500E-04
.31200E--04
,46800E-05
, :iOO:l.OE () 4
, 63 700 E- 05
,22100E-05
,9R80()E-05
,37700E-05
,63700E--05
, 10140E-04
,29900E-05
,32500E-05
. ,37700E--05
,49400E-05
,27300E-05
,37700E--05
,16900E-05
,89700E-05
, 221 00 E -04
,26000E-05
,50700E-04
,27300E-05
,27300E-05
Volumetric
Source
Strength, in
"gammas of
energy E per
cm3 per sec"
SV(E)ซ
.16.;;54E" 05
, :i.87!::;cn';:--06
, 15873E-05
,23569F.... ()5
, 23 56 9 E -06
>10582E-05
, 17316E-05
,15392E-05
,26936E-05
,67340E--07
, 28379E-06
,5//'20!-.-0/
, 12987E-05
,19 72 IE -05
,31265i:-05
, 577201 :-05
,1.9240!:-05
v23569E-06
, 18759E-06
,13468E-05
,52910E-05
,24531E-05
. 96200E-05
,37518E-05
,6734()E-06
,67340E-06
,10582E-05
,72150E-06
, 1 1544E-05
,17316E-06
,3703?E-()6
,23569!;;;. ..-06
,81770E-07
.36556E-06
,13949E..-06
.23569E-06
,3751 8E-06
,11063E-06
,12025E-06
, :l 3949E-06
,18278E-06
,10101E-06
,13949E-06
,62530E-07
,33189E-06
.81770E-06
,96200E-07
,18759E-05
,10101E-06
,101 OIF -06
NUCLIDE
234Pa
-------
Table 2-D (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
.72800E-05
, 1 5600E-04
*37700E-05
,27300E-05
,50700E-05
,20800E-05
.20800E-05
.50700E--05
,16900E-05
,32500E-05
, 195 OOE -05
*13780E-04
,50700E-05
, 1 6 900 E- 04
,19500E-05
* 3 .12 OOE -05
, 1950GE-05
,37/OOE-05
,24700E-05
,'>0800E-05
,351 OOE -05
,!;7200E-05
,19240E-04
* ?0673E-02
,:;8846E-02
,:<6952E-02
^'OOOOE-03
,. 9000E--02
> 36400E-02
,10 9 OOE. --Ol
, 59000E--02
,:i.4700E-02
,46.1.80
,15600E-01
,47222E~02
,4 03 OOE- 02
.13300E-02
,48800E-01
,310 OOE- 02
,12300E-01
,15000E-02
,10500E-02
,3. 1.6 OOE -01
,38300E-02
,31500E-02
, 2 85 OOE --02
,15000
* 255 OOE- 02
,16900E-02
.46000E-02
Volumetric
Source
Strength, in
"gammas of
energy E per
arj3 per sec"
*26936E-06
, 1 3949E-- 06
,10 10 IE () 6
,18759E--06
*76960E-07
,76960E--07
,18759E-06
.62530E-07
,12025E-06
,72150E-07
.50986E-06
,18759E-06
.62530E-06
,72150E-07
,1 1544E--06
,72150E-07
,13949E-06
,91390E-07
,76960E-07
,12987E-06
,21 164E-06
.7.1. 188 E- 06
..76490E-04
, 2 1 7 73 E -O 3
.13672E--03
,2 96 OOE- 04
,70300E-04
, 1 3 4 6 8 E - 0 3
,40330F -<> 3
v 2 1 830 E- 03
,54390E--04
*17087E-01
,57720E-03
,17472E-03
,149. HE -03
ป492.10E"-04
, 180561;:: -02
,11470E-03
. 45510 E -03
.55500E-04
,38850E-04
. 1169 2 E -02
,14 17 IE- 03
, 1 1655E-03
.10545E-03
,55500E-02
ป94350E-04
.62530E-04-
.17020E-03
NUCLIDE
234B
D^M /^*ซ ป |
rQ (Cont. ;
234pam
Rn-222
Pb
214Bi
89
-------
Table 2-D(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
, 59200E-01
, :i.4700i;::--o:i.
< 12J.OOE- 02
> 40200E.' -01
i 7 8 0 0 0 i::" - 0 ::.:'
, 13900E-0:i
,24800E-01
, 21 900 E -01
,41 OOOE--02
, 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 E-01
, 2 3 6 0 0 E - 0 2
, "? 0 5 0 0 !:." - 0 :i
, .'. !Vr 9 0 0
-. 'X8300!:: - 02
, 21 2 OOF -(> 1
, 22600E--O2
. 1 7700 E" 02
> 1 2100E' OJ
, 49900E'-0';
., 79400E--09
, 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()E--04
, :l 48 / 4 1:7- 02
.. 2 8 8 6 O i:;. () 3
,51430E--03
. 9 -j ;:',;;)!::.... (>;x
^ f-i 'j C> 30|:: () 3.
, j. 5 j 7 0 E! - 0 3
, 1 2 '-'50 E- 03
,26.440E- 03
,98050E--04
i '"''x '''', o\'.:' ...,, '.i
,42550E--03
,87320E-04
'i '; '.' J { ! '! i ' -07
:. 58830E-02
- 1417 E-03
,78440!;;: ---03
,83620E--04
,.6549 (> |:; - 0 ^
,447V()E-03
v i 3 ^ /, 3 ;:;; .... (i ;;.;
, 1 1 988:;7-7 >...;;
.. 57330F--03
, ;[ 4 3 [ v r - 0 3
. 370001:; -05
NUCLIDE
21V
Dl (Cent.)
Po-214
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 2-A
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 2-A,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 (16-C) 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 1-D and 2-D,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 1-E for
each of the 11 media previously listed in Table 2-A, 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) + (l-A)/(l+a2)] has been plotted, for each material, over
the indicated energy range, in Figure 1-E.
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 1-E Buscaglione-Manzini* 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
0-5
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
13-218
11.1*1*0
11.1*26
8.783
5.^00
2.655
3.231*
3-50V
l*. 722
1.677
2.981*
5-1*21
5-580
1. 1*1*1*
2.081
3-287
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 1-E( 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
0-5
1
2
3
A
38.225
25.507
18.089
13-640
61.341
46.087
14.790
10.399
75.471
49.916
14.260
8.160
33-026
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 Gamma-Ray
Buildup Factors." By O.K. Trubey, ORNL-RSIC-10, Published
February 1966.
94
-------
As Function of Energy E and Transport
Medium. !
Energy, MeV
Figure 1-E Magnitude of [A/(!+ซ,) + U-A)/(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 1-E drawn using known values
of A, a1 , and c^ at energies of 0.5, 1, 2, and 3 MeV (Table 1-E). These four
values are obviously insufficient for meeting the requirements of Equations
(43-C) and (44-C) and equations leading thereto -Tables 1-D and 2-D 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 curve-fitting expressions.
I. Taylor's Coefficients
For Taylor's coefficients, required for 177 gamma energies from 0.5 to
2.45 MeV (Table 2-D), the following equations apply:
(Figure 1-F) 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 2-F) OI(E) = ax + bl InE
96
-------
where a} = -0.090 035
b: = 0.053 141 671 84
(Figure 3-F) a2(ฃ) = ^- + c2E + d2 (3-F)
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 2-A , 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 1-F. 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 2-F.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 3-F.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 1-D) 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
2-A & supra). This is an undesirable effect in calculating relaxation lengths
or any thickness-dependent 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
(40-C) 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 (16-C), with other results
adjusted accordingly when pertinent. Spatial dependence being absent from
Equation (40-C), 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) (4-F)
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
4-F, 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 4-F. Buildup
factors as functions of
distance, in mean-free-
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/(D-1) ], for
the bare, infinitely thick source slab case (see Table 1), is evaluated with
both sets of C and D parameters: the 20-MFP-fit values for C and D result in
a buildup term that is also higher (by 8% or 9%) than the value produced by
the 7-MFP-fit 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 7-MFP-fit buildup factor at 10 MFP's from the 0.255 MeV source
exceeds the 20-MFP-fit buildup factor by 87% (see Figure 4-F), 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 (4-F) may be gainfully evaluated with C(E) and D(E)
we
coefficients applicable to distances of less than 10 MFP's, with 7-MFP-fit
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 1-F.
103
-------
Table 1-F. 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 curve-fitting equation for Bwc as function of energy follows:
/a. \
(Figure 4-F) Bwr(E) = 1.0 + exp
- b,
(5-F)
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 (5-F) 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 (5-F) produces B.,.., = 17.6 . This is clearly
WLr
a fallacy, since at this energy the mass energy-absorption coefficient %?/
approaches the value of the mass-attenuation coefficient vyp , suggesting that
104
-------
B (0.01 MeV) =* 1.0 . Consequently, some correction is required if Equation
we
(5-F) 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
(5-F) 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 cross-sections, the
maximum buildup is assumed to appear at E 0.12 MeV, roughly mid-range
of 0.01 MeV < E < 0.255 MeV.
An energy-dependent 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, (7-F)
\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 (6-F) and (7-F) comprises energies
0.01 MeV < E < 0.185 MeV, thus complementing the range set for Equation (5-F)
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 7-MFP 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 5-F. 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 (0-7 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 energy-dependent parameters,
namely
Pt(E) = energy-dependent attenuation coefficient of tailings, cm"1
yc(E) = energy-dependent attenuation coefficient of cover material,cm"1
ua-jr(E) = energy-dependent attenuation coefficient of air, cm"1
= energy-dependent mass-energy 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 1-G, in which mass-attenuation 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 1-G, 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 mass-attenuation 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 mass-attenuation coefficient of water
to represent the "generalized y/p" in (1-G), without introducing gross error,
while retaining consistency with the choice of medium in selecting Taylor's
108
-------
Energy, MeV
Figure 1-G.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 1-G, and thus that of soil itself, can be approximated by
multiplying the mass-attenuation 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 mass-attenuation coefficient of water to represent
the "generalized mass-attenuation coefficient" in (1-G) 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 (16-C) and
(40-C) indicate that a lower coefficient, such as that of C in Figure 1-G,
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 1-G). 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 (16-C), 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 1-E (indicated by "[B]" in
present notation). On the other hand, the use of the H20 mass-attenuation
coefficient (^//o)HpO instead of (^/|O)AI in Equation (4-G) increases pt by
the ratio "(^/p)HoO/^/p)Al"ป which amounts to reducing flux in (16-C) by a
factor "(^/(o)Al/(^J/p)H9o"< Listing both increase and reduction factors in
Table 1-G 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 1-6. 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
.95-1.05
1.05-1.35
1.35-1.65
1.65-1.95
1.95-2.55
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 1-E, 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 (15-C),(16-C),etc. and Equations (l-G) through (4-G):
for E>0.1 MeV, OVp)H 0> (*VP)Ar thus using (ปVp)H 0 in (4-G) will
produce somewhat higher values of ME) which, in turn, will reduce
flux calculated through Equations (16-C), etc. by the indicated ratio.
From Figure 1-E: [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 (5-6)
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 energy-absorption coefficient for air, as functions of energy:
Mass Attenuation coefficient of Water ป as function of energy
(Figure 2-G) :
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 2-G.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 .
3-G): air
for .01 MeV < E <_ .015 MeV./y/p) = F5(E) = exp (a5lnE + b5) (S-G)
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 3-G. Mass attenuation coefficient for air, as function of energy
-------
Mass Energy-Absorption Coefficient of Air
(Figure 4-G):
'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/
(11-6)
(12-6)
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
(13-6)
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
-0-1038830383
- 3.030852 91
1781.99433
-24.22631954
2.866 717 707
69.980580070
o.oi
0.01
Energy,MeV
Figure 4-G. Mass energy-absorption coefficient for air, as function of energy.
118
-------
The values used in curve-fitting 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 5-G)
for 0 <_ x 1 0.5,
,- / a7 \
E2(x) = expl + C7] + xlnx (14-G)
\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 curve-fit was approximately 0.5%.
119
-------
ro
o
-19
0-5547090102 * xlnx
For 0 > x > 0.5
0.5547090102 + x-lnx
For x >0.5
E,(x) =
!. 3787042
1.97650283
3.179407102-Inx + 24.19571371
(2 + x) e*
Figure 5-G. 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
curve-fitting 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 curve-fitting
equations. The minimum number of ranges would thus be roughly a half-dozen,
with a maximum depending on the values of the argument for the second order
exponential integral E , Equations (14-G) and (15-G). 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 1-H,
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 E-j and the corresponding source terms
Sv(Ei) for E < 0.5 MeV and E > 0.5 MeV, from Tables 1-D and 2-D,
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 2-A, Equations (6-F), (7-F) 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 uranium-bearing soil slab of
infinite thickness without overburden, employing the files and equations
referenced below.
For E < 0.5 MeV:
Table 1-D (File "BELOHAF"), 105 energies and source terms,
Equations (5-F), (6-F), (7-F) for "Berger's effective buildup factor for'worst
case conditions'",
Equations (3-G), (6-G), (7-G) for the linear attenuation coefficient of soil,
Equations (11-G), (12-G) for the mass energy absorption coefficient of air,
Equation (40-C) for "flux",
Equation (1-C) for exposure rate (File "LODOSEM" with 105 energies and
exposure rates).
For E > 0.5 MeV:
Table 2-D (File "OVERHAF"), 177 energies and source terms,
Equations (1-F), (2-F), (3-F) for Taylor's buildup coefficients,
123
-------
Equations (3-G), (7-G) for the linear attenuation coefficient of soil,
Equation (13-G) for the mass energy absorption coefficient of air,
Equation (16-C) for "flux",
Equation (1-C) for exposure rate (File "HIDOSEM" with 177 energies and
exposure rates).
Equation (46-C) 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 (5-G), (8-G), (9-G) for linear attenuation coefficient of air,
Equations (14-G), (15-G) for 2nd order exponential integral values, modifying
factor,
Equation (47-C) for summation of exposure rates (1st term).
For E > 0.5 MeV:
File "HIDOSEM", with 177 energies and exposure rates,
Equations (5-G), (9-G), (10-G) for linear attenuation coefficient of air,
Equations (14-G), (15-G) for 2nd order exponential integral values, modifying
factor,
Equation (47-C) for summation of exposure rates (2nd term)
124
-------
Equation (47-C) 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 2-D (File "OVERHAF"), 177 energies and source terms,
Equations (1-F), (2-F), (3-F) for Taylor's buildup coefficients,
Equations (3-G), (7-G) for the linear attenuation coefficient of soil.
Equations (14-G), (15-G) with a specific value of t, for 2nd order exponential
values,
Equation (15-C) for flux,
Equation (1-C) for exposure rate
Equation (46-C), 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 (15-C) for "flux", here replaced by Equation (17-C), "d" becoming the
new input variable.
125
-------
ro
_
ISIS.TYPDAT
ฃ"^ " U2*
/
Equations (7-GU4-G) / .. /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 half-space geometry:
Basis: 1 pCi/cm3 E = 1.464 MeV
1 pCi/cm3 = > 3.7 x 10"2 decays
cm3-second
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 (1-F) A = 14.576 250 06
From Equation (2-F) c^ = -0.069 778 860 58
From Equation (3-F) az = 0.003 288 967 120
SOIL ATTENUATION COEFFICIENT FOR E = 1.464 MeV
From Equation (7-G) (p/P) = 0.058 688 605 16 cm2/g
2
From Equation (4-G) psoj-| = 0.093 901 768 26 cm"1
GROUND SURFACE FLUX FOR E = 1.464 MeV
From Equation (16-C) d> = 4.506 843 598 x 10"2 y's
r cm2-s
AIR MASS - ENERGY ABSORPTION COEFFICIENT FOR E = 1.464 MeV
From Equation(l3-G) (uen/p)air = 2-576 712 795 x 10"2 cm2/9
127
-------
EXPOSURE RATE PER pCi/g AT GROUND SURFACE FOR E = 1.464 MeV
From Equation (1-C) 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 (9-G),(10-G) (ji/p)a1r = 5.244 559 536 x 10'2 cm2/g
From Equation (5-G) 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 (14-G) 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 mass-attenuation parameters for
water [Equations (1-F) through (3-F) and (7-G)] with those for aluminum results
111 X, = 0.189 uR/h per pCi/g
im
The curve-fitting equations for the relevant aluminum coefficients appear in
Figure 1-1, including that for the buildup parameters in compound form [ see
bracketted "buildup term" in Equation (16-C)].
128
-------
(lJ/0)/n= O-0613 exp(-0.5C
v i\ \
at E - 1.464 MeV, (*Vp)A1 =
' ' i
' " ! i
3 H j.4l_If ...
-1 H- --l-f- - T-T+-
-r-t'-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 , 1-A ...Jtlttl
. + t-i- - -
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 i|r H"?! ijiljji
J -f+j-f- -']-; "i"!~yjTr -i-rti
- - -J-i ^rl* ,-^fi- fj-t-r -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, tt|t 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 _t-j
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
Illj
\b>
rm
r~
Urf
^
il
>,
\< i
f1!
i t'
! '
-ttrr
i'ii
hi
i!j:
|ll!
---f"-^-;
rZf:":
^?
:z&
--i
:^B
-
;
.._
--
.._.
. 1.
r
^__
1
-1
:--
~
- 1
:^
s^
HH
.rb:
i
....._..
-.. .
i
-
;.. .
'
^^^__^^
M
1
;
"
f- i -V
~J:- ": i. -
t
1
" r~
--i--
._ ....
;,. .
. 4 ...
.4....
-t-
I
-J
p
^t*^
...,_,
::'.:
--
>v
.J...
. .J. J
-j;.
-j-
^
i , ^
-1 "'
.: .
_
-
.. .i .
i
i
' 'i '
i
..... ....
i
. . -4 -:_
_..-.
_.:_) .-...
i
'
'
i
1
i
!
i
i
>
i
!
I
4_
!
i
i
1
i
i
j
...
-i-
-T-
::
...
: .:::
-r.tr:
r-j
1
i-
-~
j"
i
^*^. i
-
>s>
.
i
'-
-
v! '. '
^
i
V
"s
.
;
--
--
^
\-
1 i'.J
.c:
-"r-
r-
-
-
-i...
._:
--
N
-
_
..; -
-
' . "*
. . , ,-j~-
-
~T- , :
-..!*
1
-
I
!
i
i
i
i
i
-!-
i
"1
L.
1
I
_u4r.
-H-T-
4-j-
J_
--
-I-
...
--J--J--
s
-------
The exposure rate calculated using aluminum mass-attenuation 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
-rr-H-^t
-f-|-~u tr'T
: ! ' Ljjl
: T^ J
T TT" 4ปJ ^L.
' J I " i ~* j T < 1 ! 1* ^7.
i i | i -fj-Lj (->-- 4-P
| 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 . 1-A :::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+
ri-r-
i i | ' i : i j j ! ! i 1 i i i ; , : i ' i
ul- r*~ 'T^' 4f -fr ' i i | rf^- - --*t*.-
^~ i^ . i ! 1.1 ; 1 i 1; 1 , . , i :..
1 . i l|.!'' 'i;
r1 ~si^. rr-'-p-- ri-Hi-i-- -M-j'-:
"tr1^ TT*n^> "TT-Hri-1- -
' i i ; : ' ' ^ *^* >rm_ ,
_, .TTT rl,. yijt '7tr*r* -tf!
i ll ; II -,;! ,NI ' = ! "
! i ! ' * i ! |: M
.. lA^ _^. J: 111 ..L JL
; 1 ! 1 ' !, i - I *T jt!
I ' * ' 1 i ! i i [ i | 1 : j 1 1
T ' 1 1 ! ! 1 1 1 1 : ! i
I ' 1 1 j 1 ! j ; i "
IT-tir rfr"iT[:' ||j ^
T]i;:M::t;jmji J
, i I I ! "I Ell ill
::^:ฃ:::44iiti}5rฃ
Ii TI i uu ijOS
! 1 J 1 I ! ii:
_L i II | 1 1 |j
M ' iii I l:\
i ' i l!h ;!'
1 1 ; ; i '
TT MI TJ i M i'i
1 ill i :':
I T t TTTIn l| tTiT
I i ;i
i , ! i'!*
A i-QI Ilili
i-'-
1 :
' ii
ii
TT
i-rt
Tff
iit!
'!
I I '
i' .;
|-.
[
i!
!
'h
I!
1. -.
^ _
~~~_
I 11
^
.r^
i
"Jr."
~
t._..
ฑ
...
u_
....
..:. .
. i-
.._..
i
|
^|=
j
7Z
"N
1
-^
HE
zir
'-^-
... !_
"._
' - '- -
- -J~ i
._.
..__
;
]
1
^^^^_^
-I1
1
i
1
I
;
^f..:- -r.:
mi-iLr
i
0-
1
' t
.Tix.
;_..
]-'-
-f-
-
T"
I
. .[....
r
j
"^^^
-"-;--
v:'.::
"_
:up
....
.-4;-.:
Ssj^
. : :.i :
.. :
-
...:..
.1
i ' -
...
....
- : -j
^^
IS,
-H---
*
1
|
i
i
i
i
!
1
U- i
i
i
1
i
i
j
-;-
rz.v:
;.:ฃ
-
^
- r
~
;.
1
..;..
.-!._
~
.:ftja
:;r-
r.Lr:
-1 ",-i
rjr
T
ฃr
-
i ....i_
T-
,7-
t
- --V--
;.-
-
111
rr;
..,,_
_;...
'1
;. ...
;
-i-
i
i~
.: .j ..j .
4-
....
- - !
- - - i-
-
...
. . . . -.--)--
1
_._L..._|
-1-
...
1
-\
_J
I
1
1
i
1
i
-
.
_L
i
I
>
-H"T-
4-j-
"
rs-
'
-T-
-;-
|
i
1
1
1
j
]_
T-
-
hi-
"
i
-t-
1
i
;
j i
^
-
S
-
..:..
i
i
!
_J
-t-
1 ~
*T"
:-:
I
J-
"i
.1.
t
-T-
"T~
1
.1.
t
-r
1
'T
~r
IL'
~j-
T
-i.
T"
--
j
1:
i
f
-h
t
---
-~
4-
....
-4-.
I
-"
|
-|-
.5
.6
.7 .8 .9 1
Energy E, MeV
Figure 1-1. Mass-attenuation coefficient and buildup "term" for aluminum, for
the energy range 1.0 MeV
-------
The exposure rate calculated using aluminum mass-attenuation 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 Curve-Fitting Exposure Rate Models
None of the Equations (25), (26), (28) and (30) has been obtained,
independently, from theoretical considerations, but from curve-fitting
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 "pseudo-analytical"
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
uranium-bearing 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 uranium-bearing 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 (15-C), (16-C) and
(17-C), implemented by Equation (1-C) 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 (1-J)
X(co) X(0)
X(~) = X(0) as defined for Tables 2 and 3.
The applicability of Equation (1-J) is restricted by the requirements that the
uranium-bearing 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 (4-G) in the
present study, and lead to results supporting the validity of (1-J), as
comparison of Tables 2 and 3 may verify.
The relationship in Equation (1-J) 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 (1-J) provides a valuable
criteria of accuracy in "curve-fitting", by implying that any model
successfully replicating the values of Table 3 must also, when transformed by
(1-J), closely reproduce the value of Table 2 to be considered valid. This
amounts to requiring that one-curve 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 "depth-dependent relaxation
length" L(d) and necessary introduction to the latter, which applies this
concept to summarize the results of Table 3 as a curve-fitting 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 curve-fitting 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-(e-l)(t/t())e-2] 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-(e-l)(d/do)e-2]
obtained by applying (1-0), 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 (2-J)
The simplified formulation implies negative values of L(d) for small values of
d, which makes it conceptually unattractive. It also increases curve-fitting
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
semi-logarithmic 2-point fit. By contrast, Equation (25) requires a 3-point
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 curve-fitting 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 (2-J). 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 exposure-reducing capabilities of
overburden, for d > 100 cm.
Having defended the advantages of the proposed models against tempting but
short-sighted 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 depth-dependent behavior of the slowly-varying 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, depth-independent L may provide a substantial "safety factor", if
used judiciously, through underestimation of the exposure-reducing 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 depth-dependent 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 (3-J)
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 1-J. 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
uranium-bearing slab of thickness t may be determined by combining Equations
(15-C) and (1-C),
X(E.t) = 6(E) {A1(E)[l-E2(t,E,alE)>A2(E)[l-E2(t,E,a2E)] } (l-K)
where G(E) = FftEp'entE)
L P -Mir
l
.Li,
Sv(E) A(E)
2w(E)[l+ai(E)]
A (E) Sv(E)[l-A(E)]
2 = 2u(E)[l+02(E)]
E2(t,E,alE) = E2{M(E)t[H-a1(E)]}
E2(t,E.o2 ) = E2
and u = yt = Vc
The expression (l-K) 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)](2-K)
For the special case t=ซ, this becomes
X(E,ซ) = 6(E)[Ai(E) + A2(E)] (3-K)
Dividing (2-K) by (3-K) 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 (17-C), (1-C), 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 (5-K) and 4-K) 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 (6-K) 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 uranium-bearing slab of thickness t,
covered with overburden to a depth d.
On that basis, the components of (6-K) 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, (7-K)
in both cases
i.e. thickness of uranium-bearing
slab t = ซ, in both cases
A comparison of above identities b) and d) serves to emphasize the fact that
the denominators in Equation (6-K) are equal. This permits rewritting (6-K)
in the following manner,
!
X(E,t) = X(E,ซ,o) - X(E,d) (8-K)
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) (9-K)
1 = 1 i = 1 i=1 N = #of 7 lines
Dividing both sides of (9-K) by the total exposure rate due to gamma? of all
energies from tailing slab infinitely thick with no cover, Ex(E,~,0), results
i-i
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 (6-K) 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 1-L.
z = d
z = 0
z = -<
f
Thickness of overburden, d
L
222Rn source, infinitely thick tailings slab
Figure 1-L. 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
(1-L)
dz
where J(z) = depth-dependent radon flux, in pCi/cm2.s
C(z) = depth-dependent "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
(2-L)
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 (2-L) 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)
(3-L)
(4-L)
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 tailings-cover interface.
The free radon concentration at the cover-atmosphere
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 (5-L)
or
From B.C.4: Cc(d) = 0
or Eead + Fe-ad - 0
so, E = -Fe'2ad
and Cc(z) = Fe-ad [ead ' aZ - e-(ad ' az)] for z > 0 (6-L)
From B.C.3: Jt(0) = Jc(0)
Equating the derivatives of (5-L) and (6-L) at z = 0 produces
A = -Fe-ad(ead + e-ad)
thus Ct(z) = i- - Fe"ad(ead + e'ad) eaz for z _<0 (7-L)
a*-
From B.C.2: Ct(0) = Cc(0)
Equating (6-L) and (7-L) at z = 0 results in
S r~~ad/ cxd . ^-ad\ _ i-^-otd/txd ซ-ad\
- - Fe (e +e )=Fe (e -e )
or F =-!_ (8-L)
Inserting (8-L) into (7-L) 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) (9-L)
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 (2-L) 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)
(3-L)
(4-L)
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 tailings-cover interface.
The free radon concentration at the cover-atmosphere
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 (5-L)r
From B.C.4: Cc(d) = 0
or Eead + Fe-ad = 0
so, E = -Fe-2ad
and Cc(z) = Fe"ad [ead ' az - e'(ad ' az)] for z > 0 (6-L)
From B.C.3: Jt(0) = Jc(0)
Equating the derivatives of (5-L) and (6-L) at z = 0 produces
A = -Fe-ad(ead + e"ad)
thus Ct(z) = \ - Fe"ad(ead + e"ad) eaz for z _< 0 (7-L)
From B.C.2: Ct(0) = Cc(0)
Equating (6-L) and (7-L) at z = 0 results in
J. . Fe"ad(ead + e-ad) = Fe-ad(ead - e-ad)
or F =_ (8-L)
2a2
Inserting (8-L) into (7-L) 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) (9-L)
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 formulas-from 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 1-M, 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 (11-L) and (14-L),
at regular intervals AZ. The average concentration between two successive
points is then calculated by
C^ _ cd-nAz + cd-(n-l)Az
2
where n = 1,2,3,..
and Cd_nAZ = concentration C(z) at location z = d-nAz
As n increases, Cn increases also, by an amount ACm = Cn - Cn_i (see
Figure 1-M), the increment becoming effective at a distance zm = d-(n-l)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 source-free cover of depth d - zm = (n-l)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 (n-1) 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 2-M, all for E = 0.2.
146
-------
ซ-0=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 2-M. 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.
EPA-520/6-82-014
2.
4. TITLE AND SUBTITLE
A Basic Technique and Models for Determini
Exposure Rates Over Uranium-Bearing Soils
7. AUTHOR(S)
George V. Oksza-Chocimowski
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 computer-implemented analytical procedures to predict
exposure rates over uranium-bearing soil deposits is demonstrated in this report.
The method is based, conceptually, on the energy-dependent point-source 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 uranium-238 decay chain, at equilibrium, and covered
with a source-free 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
uranium-238 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 curve-fitting equations included in the report. As direct application of the
method, maximum exposure rates over uranium-bearing soils are calculated, and the
dependence of exposure rates on the thickness of the uranium-bearing 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
Uranium-238 Decay Chain
Exposure Rates
Overburden
Radon Gas Exhalation
Buildup/Curve-fitting
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 2220-1 (FUv. 4-77) PREVIOUS EDITION is OBSOLETE
-------