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Numerical evaluation of the natural gamma radiation field at aerial survey heights
Løvborg, L.; Kirkegaard, P.
Publication date:1975
Document VersionPublisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):Løvborg, L., & Kirkegaard, P. (1975). Numerical evaluation of the natural gamma radiation field at aerial surveyheights. Risø National Laboratory. Denmark. Forskningscenter Risoe. Risoe-R, No. 317
O
Risø Report No. 317
hi
I Danish Atomic Energy Commission |
£ Research Establishment Risø
Numerical Evaluation of the
Natural Gamma Radiation Field
at Aerial Survey Heights
by Leif Løvborg and Peter Kirkegaard
February 1975
Sale* distributors; M. Gjellerup, ti, S«iv|Me, DK-1307 CepcnlMten K, Denmark
Available on exchange/rem: Library, DanMi Atomic Energy Contmiuion, Ru#, DK-4000 RMUMC, Denmark
/NIS Descriptors:
AERIAL MONITORINC ANGULAR DISTRIBUTION COUNTING RATES DOSE RATES ENERGY SPECTRA GAMMA RADIATION GAMMA TRANSPORT THEORY NATURAL RADrøACTIVrrY NUMERICAL SOLUTTON POTASSIUM PI-APPROXIMATION RADIATION FLUX THORIUM URANIUM
ULK 539.1ZM*3:550.g:553.495:6«1.3
February 1975 Risø Report No. 3? 7
Numerical Evaluation of the Natural Gamma Radiation Field at Aerial Survey Heights
by
Leif Løvborg
Electronics Department
and
Peter Kirkegaard
Computer Installation
Danish Atomic Energy Commission Research Establishment Risø
Abstract
In computational studies of the count rates to be expected in airborne, radiometric surveys of geological formations, knowledge is required of the aerial gamma radiation field produced by the radioactive minerals in the ground. The data presented in this work permits calculation of the energy and angular distribution of the gamma-ray flux at distances of between 0 and 200 m from a plane, homogeneous ground with known abundances of thorium, uranium, and potassium, A tabulation permitting calculation of the gamma dose rate in the air is also given. The data are applicable to any normal ground material (rock or soil) in which uranium and thorium are in secular equilibrium with their respective daughters. Besides, the air denaity may have an arbitrary variation with the distance to the ground.
In calculating the flux of gamma r a j s that s t r ike an a i rborne Nal(Tl) de
tector it is suggested to represent the bottom of the aircraft by an equiv
alent layer of a i r . F i r use in the numerical evaluation of aer ia l total count
ra tes the angular gamma-ray flux at the point of detection is approximated
in accordance with the double-P. polynomial expansion method. A detailed
tabulation of flux expansion coefficients, calculated for 0.05-MeV wide en
ergy intervals, is presented. To evaluate the differential count r a t e s of
high-energy Th-U-K gamma rays , it is convenient to make use of the well-
known formulas for a flux of unscattered gamma rays . With this application
in view data for insertion in these formulas a r e also given.
ISBN 82 550 0338 9
- 3 -
CONTENTS
Page
1. Introduction »
2. Model Assumptions and Sources of Basic Data ?
2 . ! . The Ter ra in Considered "
2 .2 . Natural Gamma Ray Emit ters 7
2 . 3 . Gamma Ray Interaction Cross Sections 9
3 . The Flux Concept and i ts Use in the Calculation ot Aerial
Detection Rates ' 5
3 . 1 . Angular Flux and Angular Counting Cross Section ' !
3 . 2 . Scalar Flux and Average Counting Cros s Section ! 3
4 . General Evaluation of the Aerial Gamma Radiation Field I 4
4 . 1 . Dcuble-P. Calculation of the Angular and the Scalar Flux '"*
4 . 2 . Absorbed Dose Rate in the Air ' '
4 . 3 . Numerical Example ' 6
4 . 4 . Extended Applications of the Flux and Dose Rate Data . . 2^
5. Evaluation of a Flux of Unscattered Gamma Rays 2~
5 . 1 . Formulas and Data -2
5 .2 . Numerical Example 24
6. Method Suggested for the Calculation of Aer ia l Count Rates . . . 25
7. Discussion 28
References 29
Appendix I: The Plane Gamma-Ray Transpor t Equation for Pure Compton Sca t te re rs 31
Appendix II: The Expansion Coefficients in the Double-P.
Representation of the Angular Channel Flux 33
Tables 36
1. INTRODl t 'TIoN
Aerial prospecting for radioactive minerals is based on the ust f iar^;e
scintillation c rys ta ls (Nal(Tl) crys ta ls) which a re car r ieo by aircraft at low
altitudes to detect the gamma radiation emitted from the tniu::;:. I he tola;
crystal volume required for an airborne radiometric survey is .JJ< tated by
counting s tat is t ics needs. Generally, the higher and faster it is necessar;,
to fly, the i a rper the crysta l volume must he. Besides a spectrometr ic
data acquisition system requires more crysta l volume than a system that
only reg i s te r s the g ross court ra te of gamma rays . Often the detector is
composed of severa l crystal-photomuitiplier assembl ies operated in par
al le l . In the design of such a detector the question a r i ses of how to choose
the dimensions (diameter and thickness) of the individual scintillation c ry s
t a l s . Clearly one wishes to obtain the greatest possible aer ial count r a t e s
per cubic centimetre of Nal(Tl) to be invested. For example, if a total of 3
about 7000 cm of Nal( II) is considered necessary for a particular survey,
the detector could be designed on the basis of ei ther four 6" x 4" crys ta ls 3 or three 8" x 3" c rys ta l s . In both cases the detector would contain 741 3 cm"
of Nal(Tl), but which of the two configurations woild produce the highest
count ra tes ?
A problem similar to that presented above was experienced by the
present authors severa l years ago when the Risø I'.lectrooics Department
became involved in airborne uranium prospecting .n Greenland. We there
fore began computational studies of the response of Nal(Tl) detectors to
natural, environmental gamma radiation. In these studies knowledge is
required of the energy and angular distribution of the gamma rays striking
the scintillation c rys ta l s . The part of our computational model describing
the aerial radiation field produced by the radioactive minerals in the ground
i s now complete. Generally speaking, we have studied a plane, two-media
gamma*ray t ransport problem, compiled basic data for use in the numerical
solution of the t ransport equation, and prepared tables for straightforward,
approximative determination of the angular gamma-ray flux in the a i r above
ground material with known abundances of thorium, uranium, and potassium.
Gamma rays penetrating rocks and air are multiple-scattered, for
which reason gamma-ray transport calculations are very complex even in
plana, semi-infinite geometry. In the computational method used by us the
flux of scattered gamma rays i s divided into one component directed away
from the ground and another component directed against the ground (the
"skyshine" flux). Each of these components is represented by the zero ' th
- 6 -
and th*> fir.si order terms of an expansion of the angular gamma-ray flux in
half-range Legendre polynomials. We have explained this so-called tiouble-
P. approximation in reference 1, in which we also descr ibe the implemen
tation of the flux calculation method in the form of a versat i le data p ro
cessing s y s l e n , consisting of programs and data files. In reference 2 it
was inferred, from spectrometr ic measurements in water on large, rad io
active concrete slabs, that the data processing system is free from e r r o r s
to within the limitations set by the quality of the basic data used and the
accuracy of the double-P. approximation.
In this work we introduce the application of the double-Pj approximation
to the determination of the combined flux of sca t tered and unscattered gamma
rays ; The resulting double- Pj equations a r e presented in Chapter 4 . To
gether with the flux expansion coefficients given in table 6 these equations
a re considered suitable for calculation of total gamma-ray count r a t e s . In
order to do so, the angular flux is multiplied by the angular counting c r o s s
section for the scintillation crys ta l s , see reference 2, after which the
product is integrated over ail angles of incidence and over ail energies
above the counting threshold. As it may be of interest to compare the
gross aer ia l count rate with the gamma dose ra te , at the survey altitude
or at ground level, we have included the dose rate values given in table 7 .
Gamma-spectrometr ic , aer ial surveys a r e normally based on a r e
cording of the signals from differential counting channels centered at the
energies 2. 61 5, 1.765, and 1.461 MeV, character is t ic respectively of
thorium, uranium, and potassium. It i s convenient to study these s ignals
numerically from a calculation of the ra te at which unscattered source
gamma rays interact with the scintillation crysta ls in the airplane. With
this application in view we present , in Chapter 5, data on the angular
distribution of the significant unscattered flux components at the point of
detection.
The information given in this report depends on several model assump
tions and sources of basic data. In Chapter 2 we discuss the assumptions
and make references to the data sources . The concepts of angular and
sca l a r gamma-ray flux and the use of these field quantities in the evaluation
of aer ial detection ra tes a re explained in Chapter 3 . In Chapter 6 we out
line our intended application of the data given in Chapters 4 and 5.
Calculation of the expansion coefficients is discussed in Appendix II,
- 1 -
2. MODEL ASSUMPTIONS AND SOLRCES OF BASIC DATA
2 .1 . The Terrain Considered
The terrain to be surveyed radiometric ally is supposed to have the
following characteristics:
1. The ground material is homogeneous and infinite in extension.
2. The radioactive minerals are evenly dispersed in the ground.
3 . The ground-air interface i s plane.
Moreover, we neglect variations in the air density with distance to the
ground (see, however, section 4 .4 ) .
In estimating the validity of these model assumptions we point out that
most gamma rays emitted from the ground come from, a shallow surface
layer. Even a one metre thick formavion can be regarded as infinite in the
vertical direction. For the same reason a possible gradient in the depth
distribution of the radioactive minerals just below ground level must be
expected to influence the aerial gamma radiation field quite significantly, 31 cf. the investigation by Beck and de Planque . Whether the ground-air
interface can be regarded as plane or not depends on the survey altitude.
In the first place it must be demanded that the terrain be viewed at a solid
angle that nearly equals 2 %. Besides, the distance to the ground must be
much greater than the magnitude of the natural terrain undulations.
Two common ground materials frequently encountered in aeroradio-
metric f urveya are considered in this work: Soil and granite. The densitites
and the chemical compositions ascribed to these materials are given in
table 1. The soil data are identical to those chosen in reference 3, while
the density and the composition of granite were taken from references 4 and
5 respectively.
2 .2 . Natural Gamma Ray Emitters
Table f shows that potassium oxide i s the third most common constituent
of a typical granite. This i s because potassium forms common minerals like
feldspar (orthoclase) ana mica (muscovite and biotite). The spontaneous
decay of the isotope K (isotopic abundance 0,0119%) into stable Ar by
orbital-electron capture i s followed by the emission of a gamma ray having
the energy 1.461 MeV '. The values found in the literature for the specific 71 gamma activity of potassium show a remarkable dispersion, Houtermans '
reports 24 determinations mads in the years V 947-1965. The values range
- 8 -
from 2. 6 to 3. S Y/(s • g). In this work we rely on the value of 3. 30 Y/(s . g 81
presently recommended in the Nuclear Data Tables \
The radioactivity of thorium and uranium, whose normal abundance? 9) in granite a r e of the o rder of 10-20 ppm and i -4 ppm respectively . i s
">3"> *»3g "*3S characterized by the decay chains formed by *" "Th, " U, and ~ U.
•»35 Owing to its low isotopic abundance (0.71 % by weight), ** I contributes
s i l i t t le to the gamma-ray emission frem a radioactive mineral ' . The prin-232 238
cipal character is t ics of the Th and the L decay chains a r e shown in
tables 2 and 3. It is a basic assumption in this work that thorium and
uranium a r e in secular equilibrium with thei r daughters anywhere in the 23 ** 238
ground. The gamma-ray emission spectra of "Th and U with daughters-were re-examined by Beck in 1 972 . We have adopfcsd his data together
I 21 v i th the "best values" given by Hyde et a l . ' for the specific activities of
the parent isotopes, i . e»
2 3 2 T h : 4100 d is / ( s - g Th)
2 3 8 U : 1 2227 d i s / ( s - g U)""* The most intensive gamma rays emitted by the daughters of thorium
and uranium are listed in tables 4 and 5 in order of increasing energy. 232 Whereas there a r e four significant gamma-ray emit ters in the Th decay
chain, the great majority of uranium gamma rays comes from only two
isotopes ( Pb and Bi). There a r e d«scay products of radon ( Rn -218
Po, see table 3). One of the most important sources of disequilibrium 236 in t h e U decay chain i s the exhalation of radon from the ground in t . the
s i r . This phenomenon is part icularly pronounced for weathered, 222
loosely-bound mater ia l s like soi l . Assuming that one Rn atom is typically exhaled from the ground per second, Jacobi and Andre ' have derived
the vertical distribution oi radon and i ts daughters in the a i r under various 141 meteorological conditions. The resu l t s were used by Beck ' in a calcula-
tion of the gamma radiation field produced by airborne Pb - Bi . It
was concluded that for normal atmospheric turbulence 2-5% of the gamma -
ray flux above a thick layer of soil must be ascribed to radon emanating
at 235 ' The gamma-ray emission spectrum of U + daughters adopted by us
16) was derived from the mos t recent Atomic and Nuclear Data Tables ' , From reference 2 i t follows that more man 99% of the ae r i a l gamma
238 dose r a t e produced by uranium must be ascribed to daughters of U,
" * ' The specific activity of 2 3 5 U in natural uranium is 562 d i s / ( s , g tj).
- 9 -
from the soi l . In aeroradiometric prospecting radon in the air represents
a source of gamma-ray background that has no interest in the present con
text.
2 .3 . Gamma Ray Interaction Cross-Sections
The energies of the gamma rays that strike an airborne Nal(Tl) crystal
range from almost zero to 2.61 5 MeV. In recording the total gamma-ray
count rate the counting threshold i s normally set at a value of 0.1 MeV or
greater, for which reason the flux of gamma rays with energies below 0 . !
MeV i s disregarded in this work. This means that we can confine the
possible gamma-ray interactions in the ground and the air to the processes
of incoherent (Compton) scattering, photoelectric effect, coherent scattering,
and pair production. Our source of microscopic cross sections for these
processes i s the Livermore library DLC-7D ' which is part of the U.S*
evaluated nuclear data files. Of the elements determining the major com
position of common rocks, iron is the one having the highest atomic number
(Z « 26). The figures 1 and 2 show the relative contributions from the four
principal interaction processes to the cross sections of the elements from
hydrogen through iron for 0.1 -MeV and 3-MeV gamma rays.
Following common practice in gamma-ray shielding calculations we
neglec* the 0. 511 -MeV gamma radiation emitted when the positrons liberated
in the pair-production process are annihilated in the ground and the air.
Besides we consider coherent scattering a process that neither changes the
direction nor the energy of the interacting gamma ray. The total macro
scopic cross section »(E) (the linear attenuation coefficient) for the material
in question is consequently taken as the sum of the macroscopic cross sec
tions for incoherent scattering, photoelectric effect, and pair production.
The granite and the soil described in table 1 have average atomic numbers
of 10.1 and 8 .2 respectively, and the average atomic number for dry air i s
7.2 '. In the energy interval considered the photoelectric cross section i s
greatest for E • 0.1 MeV, while the pair-production cross section is greatest
for £ • 3 MeV. It follows from figures I and 2 that gamma rays predomi
nantly interact with granite, soil and air by incoherent scattering. There
fore our gamma-ray transport model implies, typically, that the source
gamma rays ar t deflected and degraded in energy by a series of scattering
*' In thia work air i s assumed to consist of 79% nitrogen and 21 % oxygen
by weight.
10
m%
to
so
40
20
coNTiteunoN re CROSS SECTION
E»0,tM»V
•COHERENT SCATTEWNG
PHOTOELECTRIC EFFECT
O S » * » 25
Fig. 1. Percentage contribution« from incoherent scattering, photoelectric effect, and coherent scattering to the elemental interaction cr.>ss sections up to Z • 26 for gamma rays with the energy 0.1 MeV.
w % ,____-_ , , , .—-r-
80
60
40
20
, C0NTRJ8UTI0H TO ! CROSS SECTION
•COHERENT SCATTERS«
E-3MtV
COHERENT SOVTEMNO
PHOTOELECTRIC EFFECT
PWR PR00UC7ON
0 S »3 % 20 If Fig. 2. Percentage contributions from incoherent scattering, pair production, coherent scattering, and photoelectric effect to the elemental interaction cross sections up to Z • 26 for gamma rays with the energy
- 1 1 -
events that terminates when the energy drops below 0.1 MeV. For each scattering event the relation between deflection angle and energy loss is described by Compton's well-known formula. The angular variation of the scattering cross section is given by the Klein-Nishina formula, see for example reference 16 or 1.
3. THE FLUX CONCEPT AND ITS USE IN THE CALCULATION
OF AERIAL DETECTION RATES
3.1 . Angular Flux and Angular Counting Cross Section
Consider a cylindrical Nal(Tl) crystal whose axis is oriented perpendicularly to the ground on board an aircraft flying at constant altitude z, see figure 3. It follows from the assumptions made in Chapter 2 that the gamma radiation field is constant along the flight line and axisymmetricai with respect to any vertical line. The latter implies that the angular properties of the field are fully described in terms of the polar angle 6 between the vertical and the unit vector Q.
////////////////////J////////////////////// Fif. 3. Definition of the parameter! z and « used in the calculation of aerial detection rates.
Neglecting the gamma-ray attenuation in the aircraft structure, the
differential rate of detection for the scintillation crystal can be written as
- 12 -
(1)
(2)
The function F(z, E,») describes the energy and angular distribution
of gamma rays at the altitude z and will here for brevity be called the
angular flux. F(z, E,*») gives the number of gamma rays with energy E,
travelling ac ross a unit a rea perpendicular to Q, per unit t ime, unit energy
interval, and unit solid-angle element. The other function, o(E,») , entering
the equation (I), is the angular interaction c ross section (counting c ross
section) of the scintillation crystal for a broad parallel beam of gamma rays
with energy E and direction fi, see references 2 and 1 7,1 8.
The angular flux i s of fundamental importance for a detailed description
of the radiation field. In o rder to calculate F(z, E,w) above ground mater ia l
with known abundances of thorium, uranium, and potassium, a t ime-inde
pendent gamma-ray t ransport equation (Boltzmann equation)* must be solved.
If we character ize the emission lines from the radioelements in the ground
by a sequence number (i), we can express F(z, £,*•) as the sum
F (z ,E .« ) = ^ F ^ z . E , « ) . (3)
i
where F-(z, E,w) is the angular flux resulting from the i ' th emission line.
In the t ransport equation for F,(z, E,») the gamma rays that a r r ive at the
point of detection without having been scattered in the ground material or
the a i r a r e considered separately, i. e, the solution of the equation is
written in the form
F . ( z , E , * ) » u i ( z , « ) * ( E - E i ) + • ^ z . E , « ) , (4)
where E i denotes the energy of the i 'th emission line, and where 6 stands
for Di rac ' s delta function. The two te rms on the right-hand side of formula
(4) a r e the contributions to F,(z, E, m) from unscattered and scat tered gamma
rays respectively. The energy distribution of *.(z, E ,# ) i s characterized by _-
' Cf. references t and ! 6 and Appendix I.
P ( z , E ) = 1 F ( z , E , « ) o ( K , - d Q , J4m
where
w = cos © >
dC = 2«d» = 2«sin©d6 > .
- 13 -
having discontinuities at the energies Ej and E . * E./(t + 2 E./O. 5 J I) (the Compton-edge energy). In the energy interval* to E. •.«, E,») is a continuous function of E. Compton-edge energy). In the energy intervals from 0 to E • and from E ,
3.2. Scalar Flu« and Average Counting Cross Section
It i s interesting to compare the differential detection rate P(z, E) for the actual Nal(Tl) crystal with the differential detection rate PQ(z, E) for a hypothetical, spherical gamma-ray detector. The latter is defined as a detector whose angular counting cross section o (E) is constant in aU directions, the value of e_(E) being given by
'° ( E ) = **L°iE'm)d-- (5)
The quantity on the right-hand side of formula (5) is the average counting cross section for the actual crystal, i. e. the angular counting cross section averaged over all angles of incidence. From formula (1) it follows that P z, E) can be written as
P0(2,E) - N(*,E)* 0 (E) , (6)
where
N(z,E) - | F(z,E,i*)dO. (7) z,E) - ( F ( z , E , - ) d y .
The function N(z, E) is the differential, scalar gamma-ray flux, in this work simply called the scalar flux, at the distance z from the ground.
The scalar flux is a useful quantity as it describes the overall energy-distribution of the aerial radiation field as a function of the survey height. In analogy to formula (4), the scalar flux N,(z, E) contributed by the i'th gamma-ray emission line from thorium, uranium, or potassium is given by an expression of the form
N^z, E) - UjUJME-Ep + »jCz, E) . (8)
where the two terms on the right-hand side are the scalar flux contributed
by unscattered and scattered gamma-rays respectively.
- »4 -
4. GENERAL EVALUATION OF THE
AERIAL GAMMA RADIATION FIELD
4 . 1 . Double-P. Calculation of the Angular and the Scalar Flux
In describing the energy dependence of the flux quantities F(z, E,«) and
Nz, E) we divide the energy range from 0.1 to 3 MeV into consecutive
intervals of the width AE = 0.05 MeV. Given the survey altitude z and the
abundances of thorium, uranium, and potassium in the ground, we shall
present data permitting calculation of the functions
e(z ,E,«) = I F ( z , E ' , - ) d E -•1(E)
(9)
and
*(z.E) = j N(z,E»)dE», (!0)
where the integration is extended over that of the consesecutive intervals
1(E) which contains E. In accordance with the similari ty between the flux
energy spectra thus defined and the spectra recorded with a multichannel
analyser we shall cal l f (z, E,») the angular channel flux and *(z, E) the
scalar channel flux. These field quantities vary stepwise with the energy 2 2
E, and they a re expressed in units of Y/(cm • s * sterad) and ¥ / (cm • s)
respectively.
In the double-P. representation, on which our flux calculation method
rel ies , the angular channel flux is given by the equations
f ( z . E , w ) = f + ( z , E , « ) + f " ( z , E , « )
f + (z, E ,») * fo
+ (z, E) • 3 »1+ (z, E) (2» -1)
t ~ ( z , E , « ) » f o - ( z , E ) + 3 , ] - ( z , E ) ( 2 « + 1) Of)
w s cos 0 (figure 3) .
The first equation expresses that the flux i s taken as a sum of two
components, ? (z, E,w) and ?"(z, £ , « ) . The former i s defined to be equal
to ?(z, E, ») for 0 ( » « l ( 0 « 8 ( » j ) and to be zero otherwise. Conversely
t"(z, E,«) equals e(z , E,») for -1 * w < 0 (w ( •*«) and z e r o otherwise.
This implies that • (z, E, w) i s the flux contributed by gamma rays
- 15 -
travelling away from the ground, while e"(z, tl.m) i s to be interpreted as a
flux of gamma rays that have been backscattered against the ground at
aerial heights greater Hum z. The next two equations show that the two flux
components are approximated by linear expressions in 2*» - 1 and 2«+ 1
respectively. Such an approximation corresponds to an expansion of
»z, E, w) in half-range, spherical harmonics ' where only the zero'th and
the first order expansion terms are retained. The polynomials in which
«(z, E,m) is expanded are orthogonal, from which it follows that tht scalar
channel flux only depends on the zero'th order terms in the expansion:
• (z.E) - 2« U o+ ( z , E ) + f o > , E ) . 0 2
+ -t-
Table 6 gives the flux expansion coefficients * (z, E), * (z, E),
e 0 (z, E), and ?j"(z. E) at aerial survey heights of z s 25, 50, 75, . . . , 2 0 0 m.
Since it i s of general interest to compare the radiation field at the actual
survey altitude with the radiation field near the ground level, table 6 also
includes the expansion coefficients for z - 0 and z = I m. The table was
prepared at the Risø Computer Installation using the multi-file tape GAMMA-
BANK and the program GAM PI described in reference 1. In constructing
table 6 it was decided to represent the ground material by granite (table 1)
and to ascribe a density of 0.001 293 g /cm to the air, corresponding to an
atmospheric pressure of i atm and a temperature of 0°C ". To permit
calculation of the gamma-ray flux produced by arbitrary contents of thprium,
uranium, and potassium in the ground, table 6 includes three sets of flux
expansion coefficients, giving, respectively, the contribution to the flux
from t ppm Th, 1 ppm U, and I % K. An example of the use of table 6 is
given in section 4. 3, The applicability of the flux expansion coefficients in
cases where the ground does not consist of granite, and where the density 3
of the air i s different from 0. 001293 g /cm , is discussed in section 4 . 4 ,
4. 2. Absorbed Dose Rate in the Air
One of the basic problems in aeroradiometric prospecting i s to correlate
the total count rate of gamma rays with the overall abundance of the radio
active minerals in the ground. The results reported in reference 2 indicate
that for a certain setting of the counting threshold the total gamma-ray
count rate will be almost proportional to the aerial gamma dose rate at the - ,
' Temperatures close to freezing point are characteristic of the conditions
for airborne uranium prospecting in Greenland.
- 16 -
survey altitude, independent of the ratio Th:U:K for the ground mater ia l .
If this assumption is valid, aerial gross count ra tes can be converted into
dose ra te units. Moreover, if the decrease in the dose ra te with the d is
tance to the ground is known, it should be possible to deduce the dose rate
at ground level, the quantity that appears to be the most suitable measure
of the radioactivity of the ground.
The absorbed gamma dose-ra te D(z) at the survey altitude z is given by
E " max
D(z) = j .N(z ,E)H e a (E)EdE (13) o
with E = 2.615 MeV. In this formula N(z, E) is the sca la r gamma-ray
flux, while |t (E) (cm /g) is the energy absorption coefficient ' for a i r .
Our program GFX permits calculation of D(z) expressed in units of
MeV • g" • s" . Using the same reference conditions as in the preceding
section, table 7 was constructed, in which the values of D(z) a re given in 7
units of ttrad/h (1 rad = 6.242 x 10 MeV/g). As is well known, gamma
doses (in rads) received by people a re numerically equal to the radiation
exposure measured in Rdntgen units (R), The absorbed dose in standard
a i r (0°C, 1 atm) is related to the radiation exposure by the conversion
equation
1 n • 0.869 rad, (14)
which rel ies on the assumption that 33. 7 eV i s required to crea te one ion 1 9) pair ' . Though radiometric instruments for geological field work a r e
often calibrated in exposure ra te units, we find it more natural to state
the resu l t s of total gamma-ray measurements in dose ra te units,
4 . 3 . Numerical Example
To evaluate an aer ial gamma radiation field, the respect ive flux and
dose ra te data in tables 6 and 7 are to be combined in accordance with the
actual abundances of thorium, uranium, and potassium in the ground. The
granite for which the tables were derived contains 4 .11% K , 0 (table 1),
corresponding to 3.41 2% K. In the following we assume that the abundances
of thorium and uranium in the granite a r e 12 ppm and 3 ppm respectively.
The energy distributions of the resulting flux expansion coefficients e (z,E)
and f "(z, E) for z * 1 m and z • 100 m a r e shown in figures 4 and 5. It
will be recalled that t (z, E) gives the flux of gamma rays travelling away
- M
10 PHI0- A PH[0
u n
u
0.5 1.0 ! .S 2 . 0 2 .5 ffc'U
Fig. 4. The energy distribution of the flux expansion coefficient »+z. Ej and *Q(«. E) «t the altitude i « 1 m »bove granite containing I ? ppm Th, 3 ppnt V, and 3.41 2% K.
from the ground, while » (2, E) represents the skyshine" contribution to
the flux. Since 9 z, E) includes the expansion coefficients for the flux
of unscattered gamma rays, the graphs of 9 (z, E) a re character ized by
peaks. The most prominent peaks in the two figures a re labelled according
to the radioelements by which they a r e produced.
- 18 -
* 0 ° ~ PHIO- 4 CHIO- 2 - !00 n
1 1 4
,<«a>
^ 0 " 2t
^ « I!
S/ w»
•C'^fc
!!S bJ
s p i
i!
I!
1
' . 5 2.0 2. S rev
Fig. 5. The energy distribution of the flux expansion coefficients • ( i , E)
and •" (z, E) at the altitude f l O O m above granite containing t 2 ppm Th,
3 ppm l . and 3.41 Z% K.
Fig. 6 shows the vari; tion of the angular channel flux f (z, E, «*) with
the polar angle d for E = 0.1 25 MeV at the altitudes z * 1 m and z » 100 m.
Since none of the emission lines of the natural radioelements have energies
in the interval from 0.10 to 0.15 MeV, the figure exemplifies the angular
properties of a flux of scattered low-energy gamma rays. The small
jumps seen in the graph of s(z, £,w) for z * 1 m in the transition from
positive to negative values of w * cose arise fromthe truncation error in
the double-Pj approximation. It is seen that close to the ground % • 1 m)
the gamma-ray flux per steradian varies l ess than a factor of two as ø i s
increased from 0 to s . At both altitudes considered the contribution from
"skyshin«" ( « ( 9 * s ) to *he scalar channel flux amounts to about 40%.
19 -
u*eo$ e
fim * * K » I * I
Fig. 6. The variation of the angular channel flux fyrens"- e • sterad) j with
the polar angle • in the energy interval from 0.10 to 0. I 5 MeV at the
altitude« t • I m and z * 100 m.
In fig. 7 we illustrate the relative diroinuation in the total gamma-ray flax * t o t(z) and the dose rate D(st) resulting from an increase in z from 0 to 200 m. Since gamma rays with energies of less than 0.1 MeV are disregarded in this work, the total gamma-ray flux is given by
W*>= f max
'E , mm N(z,E)dE « ?« L e*(z.E) + E f j z . E ) (15)
with E min 0.! MeV and E m a x = 2.6! 5 MeV. Values of £ • J (z, E) and £ t j (z , E) are derivable from the rows labelled "sum" in table 6. The figure shows that the total flux decreases less rapidly with altitude than the ••Jose rate. The reason is that the multiple scattering events cause the spectrum of gamma rays to be shifted towards lower energies as the aerial altitude increases.
- 20 -
Qtø i 1 i i i 1 i i 1 0 » SD 15 »O 125 M ffi 200 m
Fig. ?. The relative diminuaUon in the total gamma-ray flux and the gamma dose rate with the distance z to ground material (granite) containing thorium, uranium, and potassium in the proportions 12:3:341 20.
4.4. Extended Applications of the Flux and Dose Rate Data
After having presented data on the flux and the dose rate in standard
air above granite of the composition given in table 1, we shall now show
that tables 6 and 7 can be used for approximate calculation of the gamma
radiation field produced by all normal ground materials in air of arbitrary
density. Moreover, it will appear that the radiation field can be evaluated
even if the air density varies with altitude. Since the main purpose of this
work i s to establish a basis for the calculation of aerial count rates, we
shall also mention how the structural materials surrounding an airborne
Nal(Tl) detector can be taken into consideration. The following discussion
is based on the fact that soil and all common rocks, the atmosphere, and
most of the materials used in an aircraft and a detector housing consist of
elements with low atomic numbers (Z * 26). In section 2. 3 it was pointed
out that gamma rays with energies of between 0. f and 3 MeV interact with
low-Z elements almost exclusively by incoherent scattering. It appears
from Appendix I that the electron densities for the materials involved are
essential in describing the transport of gamma rays through pure Compton
scatterers. The electron density n for a homogeneous material i s given
by
- 21 -
ne'*No h r . < , 6 > i i
where P = bulk density of the material
23 N = Avogadro's number (6.o25 x 10 )
(w-, 2 . , A.) = (abundance by weight, atomic number, atomic weight)
for the i'th element in the compositional code for the
material.
It follows from the formulas (AS) and (A6) in Appendix I that the radi
ation field above s uniform mixture of Compton scatterers with electron
density n and gamma-ray emitters with source density q xs proportional
to the ratio q/n . This implies that the aerial radiation field produced by
unit weight concentrations of thorium, uranium, or potassium in the ground
is inversely proportional to the .quantity
<*> • I w i*7 (17)
for the ground material. The granite for which tables 6 and 7 were derived
has ( TT ) * 0.4965. Therefore, to determine the flux and the dose rate
above a ground having another value of ( -r) , the figures in the two tables
should in principle be multiplied by a factor of 0.4965/ ( T ) • Generally the
correction i s quite small and can often be omitted. This is because the
quantity r varies by as little as from 0.4551 (manganese) to 0. 5000
(oxygen) for the elements, with the exception of hydrogen, which enter the
common rock-forming minerals. For example, the value of ( jr ) for the
soil described in table 1 i s 0. 5026, so the field quantities per 1 ppm Th,
I ppm U, and 1 % K are only 1.2% smaller for soil than for granite. Since
hydrogen has j - • 0. 9921 it may be necessary to include the correction in
calculating the radiation field above material that is soaked with water.
The formulas (A4) and (A5) (Appendix I) show that the vertical distri
bution of the gamma-ray flux in an assembly of Compton scatterers, whose
electron density n J z) varies with the distance z to a plane, infinite source 2 medium, i s a function of the total number of electrons per cm in a vertical
column of the height z. If we let the Compton scatterers represent the
atmosphere, it follows that tables 6 and 7 permit calculation of the radi-
- 22 -
ation tieki in axr whose density v a n e s according to a function ø(t). Thus.
if z . denotes me >f the altitudes for which the flux expansion coefficients
and the dose rate a r e tabulated, these field quantities a r e the s ame a t the
altitude z.y in the a i r characterized by tha function P(z), where z 0 i s
derivable from the equation
, * • - >
0. Oil 293 - 2 = - ofz'Jdz' - '8) r»
In the normal case in which the a i r density can be regarded as constant,
p(s) -9 . i t altitudes of between 0 and - 2 0 0 m, z 2 i s simply given by
z.y = z, • 0. mi 293/o . (19)
If the a i r temperature t ( C) and the atmospheric p ressure p (atm) a r e
specified, p i s calculated by means of the well-known expression
p = 0 .001293 rr m - ( 2 0 >
In order to evaluate the flux of gamma rays that s t r ikes an airborne 2 Nal(Tl) detector one has to calculate the total number of electrons per cm
in all mater ia l present between the detector and the ground. In such a
calculation it i s convenient to express the thickness of the detector housing
and the bottom of the aircraf t as an equivalent layer of a i r which has to be
added to the survey altitude. For example, a 1 cm thick layer of an 3
aluminium alloy having a density of 2 g/cra is equivalent to a 1 5 m high
column of standard a i r . We point out that only the upwards-directed
gamma-ray flux, expressed by the expansion coefficients •_ (z. E) and
• . (z, E) in table 6, is influenced by the s t ructura l mater ials surrounding
the detector. The "skyshine" flux i s unchanged, because i ts energy and
angular distribution a re independent of the position of the scat ter ing
electrons above the point of detection.
5. EVALUATION OF A FLUX OF LNSCATTERED GAMMA RAYS
5 . 1 . Formulas and Data
In this chapter we shall present data for insertion in the well-known
formulas for a flux of unscattered gamma rays. We nave previously
(Chapter 3) introduced the quantities u.(z,») and \iAz\, describing the
- 23 -
thf angular and the scalar flux produced by source gamma rays of the
energy E-. The formulas 4) and (8) show that u(z,*») and U (z) are ex
pressed in units of '»/(cm" - s - sterad) and V/cm" • s) respectively. The
expressions for u-(z,w) and l',z) can be written as (cf. reference 2)
Q i u (z, *) = r fcxp(-x/*) 4 " * m * E i >
Ll<2> = TTT^T E2«"> '
(21)
m i
where
» » cos© » © (fig. 3)
x =* A (E )pz m v
9 - density of the air x
S* (E-) " total cross section without coherent scattering
for air, in cm"/g
ti ' (E.) * ditto for the ground material
Q. * emission rate of gamma rays with energy E. per gram of the ground material.
G A The cross sections p J* (E.) and » (Ej) are often referred to as the m * m i mass attenuation coefficients for the ground and the air. The function
K0(x) is a special case of the exponential integral of the order n:
CD !
En(x) * xn"' j exp<-t)/tndt * J * n " 2 exp(-*/ - )d« . (22)
X O
201 E„(x) has been tabulated, for example, by Trubey '. In fig. 8 we
show the graph of E2(x) for 0 * x * 3 .5 .
Table 8 shows the values of Q., normalized to abundances of T ppm Th
>r 1 ppm U in the ground material, for the thorium-uranium gamma rays
whose energies and intensities are listed in tables 4 and 5. The value of
Q| for potassium (1 % K) i s also given. Besides, table 8 gives the mass
attenuation coefficients of granite, soil (cf. table I), and air for the gamma-
ray energies in question. The mass attenuation coefficients were calculated
by means of the data file GAMMABANK and the program GBCR061 \ For
material that can be regarded as a pure Compton scatterer, the mass
- 24 ~
Q.001
Fig. 8. Graph of the second-order exponential integral E„(x) entering
formula (21).
attenuation coefficient is proportional to the quantity (, •*•) defined by
formula (1 7). The ratios between the values of ( -%•) for granite, soil, and
a i r equal 1:1. 01 2:1.007, and since the ratios between the three sets of
mass attenuation coefficients in table 6 a re in fair agreement with this
proportion, the table further i l lustrates the fact that incoherent scattering
is the predominant gamma-ray interaction process in typical ground ma
ter ials and air .
5. 2. Numerical Example
Fig. 9 exemplifies the variation of the angular flux of unscattered
2.61 5-MeV gamma rays from thorium with the polar angle ©. The plot
pertains to granite and standard a i r ( p * 0. 001 293 g/cm ). As one would
expect, the flux distribution is upward peaked at aer ia l survey altitudes
(z • 100 m) but nearly half-isotropic close to the ground (z « 1 m). It may
be useful to introduce the fraction a of gamma rays whose directions a re
confined to the interval from 8» 0 to 8 > 0 . Given a, the angle 6 • o • o
- 25 -
XalOOm i.lw
Fig. 9. The variation of the angular flux (V^cni"- s - s terad • MeV)) of
unacattered 2.615 - MeV gamma raya at the altitudes z « t m and z - ! 00 rn.
0 (z, E., o) is derivable from the equation
E2(x/cos8 J
— E 7 * r - c o * o * ' - • • <23>
where % is given by formula (21) ff. Since z and e define a cone whose
intersection with the air-ground interface is a circle with the radius
R * ztanft , the interface can be divided into concentric, circular areas,
each contributing a specified fraction of the scalar flux at the survey
altitude considered ("circles of investigation", cf. reference 21). In the
present example we find for a * 0, 5 thai (8 , R) = (45°, 101 m) at z = 100 m
and (*0. R) • (59°, 1.7m) at z = 1 m.
6. METHOD SUGGESTED FOR THE CALCULATION OF AERIAL COUNT RATES
We have now concluded the main objective of this report, namely to introduce formulas and tables for numerical evaluation of the aerial gamma-ray flux from the radioactive minerals in rock or soil making up a plane, uniform terrain. Since the data have been compiled with a view to their application in the optimum design of airborne Nal(Tl) detectors, cf. chapter 1, we shall round off the report by briefly mentioning the
- 26 -
method we intend to use for the calculation of aer ia l count ra tes .
The first of the simplifying assumptions on which the method rel ies is
that the pulse-height spectrum derived from a detector, which consists of
two or more crystal-photomultiplier units, is the sum of the pulse-height
spectra produced by the individual units. In other words, we neglect the
screening and the scattering effects in an a r r a y or a block of airborne
scintillation c rys ta l s . Let f(z, V) be the count ra te of pulses with ampli
tudes between V and V + dV obtained with a single Nal(Tl) crystal flown
in an aircraft at the altitude z fig. 3). The spectrum f(z; V) can b#> written
as
E 1 .- max r-
f ( z , V ) = 2 n j J F(z. E,»») o(E,«) G(E, V.« dwdE (24) o -1
with E =2 .615 MeV. F(z, £,*») denotes the angular gamma-ray flux
at the point of detection. In section 4. 2 we have explained how the gamma-
ray attenuation in the aircraft s t ruc ture and the detector housing can be
taken into consideration in the calculation of F(z, E,*»). The angular counting
c ross section a (E, ») for the Nal(Tl) c rys ta l has been defined in section 3 . 1 .
The third function, G(E, V,w), appearing in the formula is the response
function for the crystal-photomultiplier assembly. G(E, V,*») i s to be
interpreted a s a normalized pulse-height spectrum, generated by a parallel
beam of gamma rays having the energy E and forming an angle of 8 =
a rc cost* with the crys ta l axis . As is well known, monoenergetic gamma
rays interacting with a Nal(Tl) crysta l a re ei ther totally absorbed, in
which case they give r i se to a full-energy peak (and two annihilation escape
peaks if the energy is g rea te r than 1.022 MeV), o r they a r e scat tered out
of the c rys t a l whereby a Compton continuum i s formed.
As the next simplifying assumption we disregard the angular dependence
of the response function, i. e. we se t
G(E,V,») = G(E,V) . (25)
It i s common pract ice to write G(E, V) as the convolution of an energy
deposition spectrum D(E, E') and a Gaussian resolution function R(E», V), i, e.
E
G(E,V) * J D(E,E») R(E',V)dE» . (26)
o
- 27 -
Thus, to calculate an aer ial pulse-height spectrum knowledge is r e
quired of the following quantities describing the performance of the detector
unit:
1. The angular counting c ross section o(Ef»» )
2. The components of the function D(E, E')
(photofraction, escape-peak intensit ies,
and paramete rs characterizing the Compton continuum
3 t The parameters giving the width at half maximum
(FWHM) of the function R(E», V),
Whereas the calculation of angular counting cross sections is a com
paratively simple numerical problem, cf. reference 1 8, energy deposition
spectra must be evaluated from extensive Monte Carlo calculations, or
they must be determined experimentally. In reference 2 we have derived
the response function for a 3" x 3" Nal(Tl) detector using information on
D(E, £ ' ) available in the l i te ra ture . As far as we know, detailed response-
function data for la rge scintillation detectors have not been published so
far, and therefore further approximations have to be introduced.
Even the most advanced aeroradiometr ic recording systems rarely
sor t the detector pulses into more than four counting channels. Of these
one is used for measurement of the total gamma-ray count rate, while the
three others form a gamma spect rometr ic data acquisition system, by
means of which one endeavours to determine the abundances of thorium,
uranium, and potassium in the te r ra in beneath the aircraft . Normally the
spect rometer channels a r e about 0. 2-MeV wide and centered at the energies
2. 615, 1. 765, and 1.461 MeV respectively, cf. section 2. 2. In an attempt
to determine the optimum crys ta l dimensions for a recording system to be
flown at a given altitude, it is reasonable to confine the examination to the
integral counting channel and the differential counting channel with cent re
energy 2.61 5 MeV. Recalling that P(z, E) denotes the ra te of detection for
the scintillation crys ta l (section 3.1), the following approximate formulas
can be established for the total count ra te " . . ( z ) and the differential count
ra te n0 R 1 ( i(z):
- 28 -
Qo E r max n to t(z) - j f(z.V)dV * J P(z,E)dE
(27)
E Ert
o o
1 • — f*
* 2* ) j *(z. E .» )e (E .« )d«
i -1
i
n2 61 5(z) » 2« po(E.) J Ui(z.« ) o ^ . * )d « . (28)
In formula (27) E is the counting threshold expressed in MeV, while
• (z , E,») is the angular channel flux given by the double-P. equations (11).
In the last expression in (27), the symbol ) is understood as a summation
over the "energy channels" of width & E = 0. 05 MeV introduced in section
4 . 1 , beginning with the channel to which E belongs. Formula (28) ex
presses that n„ g . Az) i s taken as the count rate in the total absorption
peak produced by unscattered 2.61 5-MeV gamma rays from thorium (cf.
formula (21) and table 8). Therefore both the counting cross section
«(E., «) and the photofraction p (E,) must be known for E. - 2. 6f 5 MeV.
7. DISCUSSION
It may be questioned whether the double-P. representation of the
angular channel flux (formula (11)) is sufficiently accurate for determination
of aerial total gamma-ray count rates (formula (27)). To comment on this
problem we first observe that two types of error occur:
(i) No harmonics of order 1 ) 1 are retained in the expansion of the flux.
(ii) The calculation of the zero'th and the first harmonics is not exact,
cf. reference 1.
Errors of both types will affect the accuracy of response calculations
for anisotropic detectors, whereas only type (ii) errors will do so in the
case of an isotropic (spherical) detector*. Regarding the scattered flux,
' The accuracy of the dose rate values in table 7 also exclusively depends
on type (ii) errors.
*>o
which varies rather slowly in the intervals -1 * *» ( 0 and 0 « * 1, we 1 ?)
have strong evidence * " ' that the type (ii) e r r o r s a re small ; for the uu-
scat tered flux they a r e zero. Again, e r ror« of type (i) are expecte« to be
small for the scattered flux, but unfortunately this :s not true for the un-
sc arte red flux. In fact, we have to admit that the angular distributions -A
the unscattered flux (graphs like those in fi^. 9 are rather poorly r e
produced by retaining only the t e rms I = 0 and 1= ! in the double-P. ex
pansion cf. Appendix II). We believe, nevertheless, that the e r r o r intro
duced by the strong anisotropy of the unscattered flux can be tolerated,
because the numerical evaluation of formula (27) involves integration with
respect to directions and summation over energy intervals, by which the
e r r o r tends to be diluted.
REFERENCES
!) P . Kirkegaard and L. Løvborg, Computer Modelling -A Te r r e s t r i a l
Gamma-Radiation Fields . Risø Report No. 303 (1974) '6'a pp.
2) L. Løvborg and P. Kirkegaard, Response of 3" x 3" Nal(TI) Detectors
to T e r r e s t r i a l Gamma Radiation. Nucl. Inst. Meth. 1 2) (I 974) 23ti-25t.
3) H. Beck and G. de Planque, The Radiation Fie 'd in Air due to Distr i
buted Gamma-Ray Sources in the Ground. 11ASL-1 95 (! 968) 53 pp.
4) R.A. Daly, G. E. Manger, and S. P. Clark, J r . , in: Handbook of
Physical Constants. Revised edition. Edited by S. P. Clark, J r .
5) S. P. Clark, J r . , ibid 1-5.
40 6) J . D. King, N. Neff, and H. W. Taylor, The Energy of the K Gamma
Ray and its Use as a Calibration Standard. Nucl. Inst. Meth. 52
(1967)349-350.
7) F . G. Houtermans, in: Potassium Argon Dating. Edited by O. A.
Schaeffer and J, ZShringer (Springer Verlag, Berlin, 1966) 1-6.
8) M . J . Martin and P . H. Blichert-Toft, Radioactive Atoms. Nucl. Data
Tables A8 (1970) 1-198.
9) J . J . W. Rogers and J . A. S. Adams, in: Handbook of Geochemistry.
Edited by K. II. Wedepohl. Vol. 11/2 (Springer Veriag, Berlin, 1970).
Chapters 90 and 92.
- 30 -
10) W.W. Bowman and K. W. MacMurdo, Radioactive-Decay Gammas.
At. Data Nucl. Data Tables ]_3 (1974) 89-292.
11) H. L. Beck, The Absolute Intensities of Gamma Rays from the Decay
of 2 3 8 U and 2 3 2 T h . HASL-262 (1 972) 1 4 pp.
12) E. K. Hyde, I. Per lman, and G. T. Seaborg, The Nuclear Propert ies
of the Heavy Elements. Vol. 2 (Prentice-Hall , Engle wood Cliffs, N. J . ,
1964) 1107 pp.
13) \v. Jacobi anrl K. Ar.dré, The Vertical Distribution r»f Radon 222,
Radon 220 and Their Decay Products in the Atmosphere. J . Geophys.
Res. 68 (1963) 3799-3814.
14) H. L. Beck, Gamma Radiation from Radon Daughters in the Atmosphere.
J. Geophys. Res. 79 (1974) 2215-2221.
1 5) W. H. McMaster, N. Keer Del Grande. J. H. Mallett, and J. H. Hubbel,
Compilation of X-Ray Cross Sections. UCRL-501 74 (Sec. 1 -4) (1 969-70).
1 6) H. Goldstein, Fundamental Aspects of Reactor Shielding (Addison-
Wesley, Reading, M a s s . , J 959) 41 6 pp.
1 7) A. Schaarschmidt and H. -J . Keller, Calculation of Efficiency and
Cross Section of Cylindrical Scintillators in Axisymmetrical Gamma-
Ray Fields. Nucl. Inst. Meth. 72 (1969) 82-92.
18) H . - J . Keller and A. Schaarschmidt, Empfindlichkeit und Wirkungs-
querschnitt zylindrischer Szintillatoren in axial-symmetr ischen
Gamma-Strahlungsfeldern verschiedener Winkelverteilungen, BMwF-FB
K-69-20 (1969) 125 pp.
19) F .H. At t ixandW. C. Roesch, Radiation Dosimetry. 2nd edition. Vol. 1
(Academic P r e s s , New York, 1 968) 405 pp.
20) D. K. Trubey, A Table of Three Exponential Integrals . ORNL-2750
(1 959) 63 pp.
21) J . S . Duval J r . , B. Cook, and J. A. S. Adams, Circle of Investigation
of an Air-borne Gamma-Ray Spectrometer. J . G sophys. R e s . , 7_£
(1971) 8466-8470.
- 31 -
APPENDIX I
THI CLANK GAMMA-HAY TRANSPORT EQUATION
KOK PIKE COMPTON SCATTERERS
The plane, one-dimensional Boltzmann equation goverrung the transport
of gamma rays through matter can be formulated as
w %-; F(z, E,*) + it (z, E) K(z, E,w ) =
(AI)
n (z) j K(z, E>, u*) o(E' - E. u' - i ) d fcfiK' + 'fy L ) (- a> < z < u>)
where
- ' -! - i
E(z, E,w) = angular gam ma-ray flux (V • > !i, " • s . s terad )
7. = distance along the z-axis (cm)
E = gamma-ray t-uer^y (MeV)
a = unit vector in the direction of gamma-ray
movement
<* - j_ • C, where _i is a unit vector parallel to the z-axis
iiz, E) = total macroscopic c r o t s section without coherent scattering (cm" ' )
n j z ) = position distribution of electron density (electrons • c n r ^ )
o(lZ'- K, U' — i*) = microscopic transference cross section for ~ ~ incoherent scattering from (E' ( Q*) into (E, 0)
(cm*-- MeV"' . s terad" ' per electron) q(z, E) = position and energy distribution of isotropically
radiating source (V - c m ' ^ • MeV"' • s " ' )
The double integral on the right-hand side of the formula is extended
over all previous energies and directions.
We now suppose that the gamma-ray interactions a re confined to the
process of incoherent scattering, the macroscopic c ross section t»(z, E)
being taken as
K The transference cross section for incoherent scattering is derivable
from Compton's and Klein-Nishina's formulas, cf. reference 1.
- 32 -
»«.£) = ne(z) t,JE) . <A2)
where
o £) = . o(E - E \ w - ^ M ^ ' d r " . * <A3)
i s the incoherent-scattering cross section (cm" per electron) for a free
electron. This implies that the mater ia l in the interval - o> z ( oo is
regarded r.s a cloud of electrons whose density varies according to n (z).
It i s convenient to replace the distance z bv the number of electrons per 2
cm , C» in a column of the height z, i. e. we define
z
C = n e(z ' )dz ' . (A4)
o
With this new position variable the t ransport equation reads
- ^ - F(C. E. -) t ce<E) F(C, E.u) =
tA5)
i Fil,E',»>)°(E' - E. &'- 6 ) d C ' d E ' -*• 3i^£_Ei (- æ < C <oo)
where Q(C. E). the source density per electron, i s given by
Q(C.E) - ffiffi . (A6)
. 33 -
APPENDIX II
THE EXPANSION COEFFICIENTS IN THE DOUBLE-P. i
REPRESENTATION OF THE ANGULAR CHANNEL FLUX
In section 4. 1 equations (II we considered the doubie-P, representation
of the angular channel flux e(z, E,%»):
, ( z . E.*») « «*(z. £. . . ) + jiz.E.m) . (A7)
and j
• ?z.E.t») * (21+1) e * ( z , E ) P .C2.+ I) . (A8)
l«o
P, being the Legendre polynomial uf the order i. The expansion coefficients
*.* (z, E) a r e given by '
+
»j (z. E) = j 9 (z. E.*) P.(2 • + ! ) d *
• I - o
(I a 0 , 1 ; s : J and = I ) .
o -!
(Ay)
and it is the objective of this appendix to discuss the. evaluation of the ri»,rht-
hand side of (A9). In doing so, we first partition #" (z, E.w) in*;.- :n un-
collided and a scat tered part:
9 ( i , E , n ) * f u ( z , E . - ) + » " ( I . E . - ) (Ain)
and make a s imi la r partitioning of the expansion coefficients;
«- • • t j (z. E) * f j"u (z. E) + ^ m (z. E) (A! ?
• + f" (z, E) are closely related to the expansion coefficients +'(z ,K) for
the angular energy flux (L e, energy x angular flux) discussed in I) and
available in the GAMMABANK data-file system (>. denotes the wavelength
in Compton units, that is X. » 0.511 / E L In /act , •". resu l t s from a
"channel integration", like that in equation (10), of +. (z, k).
- 34 -
+ + t~ g (z.E) = j *~(z. Vj/E'dE' . (Al 2)
1(E)
Here, the integration range 1(E) is the channel interval corresponding
to E (see section 4. 1 ), and V' = 0. 51 l / E ' . The numerical integration
technique applied in evaluating (A1 2) is the same as used in the program
GFX1.
Next we turn to discuss the "uncollided" part of (A? I), viz.
+
?• (z.E) = | 9 ( z , E , » ) P ( 2 « - 1 ) d « . (A13) 1, u j u l
As the "skyshine" does not include uncollided gamma rays, we have
f" u (z.E) = 0, (A14)
and we need only consider the plus term
1
^ u ( z , E ) = J f * ( z , E 1 # ) P 1 ( 2 « . | ) d « . (A15)
o
qp (z, E,w) = t (z, E,w) is the angular channel flux of uncollided gamma
rays; it is expressible as the following channel integral (cf. equation (4)
in section 3.1):
? u(z .E, w) = J ^u^z,*) bE< - EjdE' , (A! 6)
KE) i
or
• (z .E,«) = u(z,w) . (A17) u U 1
E.€I(E)
As indicated, the summation is restricted to source line energies
belonging to the channel interval 1(E); from (A15) we get
t[ f U(z ,E)» £ J Ujtz.tOPj^n- 1)d« (A18)
E^KE) o
- J3 -
Now, for z * 0 and w ) 0 we have '
Uj(z.») * 4 „ y (g ) exp( j — 2) . (AI9) G i
q. is the source strength of line no i (V/cm s)); M,.(E ) and y ,(E ) 1 . Cj 1 A 1
are the linear attenuation coefficients (cm" ) in the ground and the air ,
respectively, taken at the line energy E . Restricting 1 to 1 = 0 and i - '
and inserting P (2w - 1) = ! , P.(2w - 1 ) * 2 w- ?.. we finally a r r ive at the
expressions
and
f ^ u ( z . K ) = c t [2 E3xt) - K^xfl ( A 2 I )
K^TtE)
q i with the abbreviations c ; -% re«-r and x ; u ,(E )z. L (x) stands l 4* j*r(E7) i F A' i n' ' for the nth-order exponential integral Qefined in equation (22), section ">. I .
The term in brackets in (A2I ) can be equivalently expressed as
[ ] « exp(- xj - (1 + x.) E2(xx) , (A22)
so wp shall only require an algorithm for the E,,-function.
- 36 -
Table i
Densities and compositions of the ground mater ials considers«: ::. tr,.s W-TR
Per cent abundance
Granite Soil
3 I p = 2. 67 g/cm ! p = 1 . 6 g /
N a 9 0
H.,0
('(^
70.18
0.39
14.47
!. 57
J .78
0. 1 2
0.88
1.99
3.48
4 .11
0.84
0.19
61. =>
1 3 . 5
4 . =>
i 0 . 0
a
Tabi«? 2
23. Principal charact»»ristacs of the "Th deciv chain
Isotope
232, "Th
228 Ra
228 Ac
228 Th
224 Ra
Radiation
«
§
8 . T
« . t
Half-
1.39 x
life
I01 Ov
i
1 64%
2 1 2 -Po i
2 2 0 R n !
2j6Po 2]»Fb
2 1 2 B i
1 36%
208 T i
2 0 8 p b
1
P .T
• K64%)
« (36%)
•
8.T
stable
55.3 s J
0 .15 s |
10.64 h
60.6 m i
!
3 x 10* 7 s
3.1 m
- 38 -
Table 3
Principal character is t ics of the U decay chain
Isotope
OTA
1 Th 2
Pa
2(
3 4 u
2 3 0 T h
226_
i Ra
222Rn
2(
, 8 P o
2 | 4 P b
2 | 4 B i
2j4Po 2]°Pb
2 , 0 B i
1 210„
1 P ° 2 0 6 p b
Radia t ion
«
P
P
« . T
C
« . T
8
C
P . T
P , T
a
P , T
P
a
s t ab l e
Half- l i fe
4 .51 x 109y
24.1 d
1.18 m
2 . 4 8 x 105y
8 x 10 4 y
1600 y
3 .82 d
3 .05 m
2 6 . 8 m
19 .8 m
1.6 x I 0 " 4 s
2 1 . 3 y
5.01 d
138 .4 d
-
- 39 -
T a b l e 4
Dominant g a m m a - r a y s emi t t ed by Th d a u g h t e r s
I so tope
21 2
2 2 8 A c
2 0 8 T l
t t
2 1 2Bx
2 2 8 A c
!•
M
2 0 8 T 1
G a m m a - r a y energy
(MeV)
0 .2386
0 .3385
0. 5107
0.5831
0 .7272
0 . 9 1 1 !
0 .9667
1. 5881
2. 6147 — —
Intensi ty
<%>
4 5 . 0
12. 3
9 .0
30. 0
7 .0
29 .0
2 3 . 0
4 . 6
3 5 . 9
Tab le 5
Dominant g a m m a - r a y s emi t t ed by U d a u g h t e r s
Isotope
2 , 4 P b
i r
2 , 4 B i
i t
i i
i i
n
i i
t i
Gamma-ray energy
(MeV)
0.2952
0.3520
0.6094
1. 1 204
1.2382
1.3778
1.7647
2. 2045
2.4480 —
Intensity
(%)
17.9
35.0
43.0
14.5
5.6
4.6
14.7
4.7
1.5
- 40 -
Table 6
Expansion coefficients 9 (z, E), *. (z, E), » (z, E), and t"(z .E) in
c m ' • s~ to be used with the equations (11) for calculation of the energy
and angular distribution of the natural gamma-ray flux above a ground
(granite) with known abundances of thorium, uranium, and potassium. The
expansion coefficients a re given in 0. 05 - MeV wide energy intervals for 3
altitudes z of between 0 and 200 ni in air having a density of 0.001 293 g/cm' .
1 * " » T H O H I W "
l < » l » >
. 1 0 * 0 . 1 5 • I S « « . * « . 2 » * » « » 5
.»•«.)• . 1 0 * 0 . 1 9
. » • • • • a ••••«•« . 4 9 * 4 . 9 « . 5 0 * 0 . 9 9 . » • • • • O . » 0 * 0 . 4 S .«9*a.-a . * « . * - • . > 9 .rs*«.«o .ao*o.is . • 9 - 0 . » 0 . » 0 * 0 . » 9 . *9* i .aa
. • 0 - 1 . 0 9
. 0 9 * 1 . 1 0
. j a n . n
. 1 9 * 1 . 2 0
. 2 0 * 1 . 2 9
.29* i . ia
. H M ) H
. 1 9 * 1 . 4 0
. 4 8 * 1 . 4 9
.<s-i.se
. 9 0 * 1 . 9 9
. 9 9 * 1 . * 0
. •an>«9
.os'-i.ro • r a n i*9 . M - 1 . 0 0 . 0 0 * 1 . 0 9 . • 9 » l . » 0 • • a n >t9 .»9>a*ao . 0 0 - 2 . O S .es*2iio . 1 0 * 2 . 1 9 . 1 9 * 2 . 2 0 . 2 0 * 2 . 2 9 . 2 9 * 1 : 1 4 . 1 0 * 2 . 2 9 . 1 9 * 2 . 4 0 . 4 0 * 1 . 4 9 . 4 9 * 2 . 9 0 . 9 0 * 2 ; 9 9 •99'i.oa . 4 0 * 2 . 4 9
»«*>«
*wia«
4 . 9 « C * « 1 4 . 1 W 0 1 4 . 4 4 C * « !
\.*.»*;*•» l . * « C * « l I . O l f O l • . » « - 0 4 • • • » 1 * 0 4 i . aacn i I . 4 K - 0 1 «.4rc*o« 4.2ac*04 r . » * [ * « 4 0 . 1 * 1 - 0 4 o . t t f a o 9 . m - a 4 i . r u - a i i . i 9 t * a i
I . 1 4 C - 0 4 l . 4 * C - 0 4 ) . 0 * C * 0 4 » . • 2 C - 0 9 1 . 2 4 C - 0 4 «.oac*09 0 . I 4 C - 0 9 o.oic-os r.«4C*as 2 . 4 « C * 0 4 * . I « C * 0 9 4 . 4 « c * a 4 4 . 0 4 C 0 4 r . ) 4 C * 0 9 4 . 9 - C 0 9 4 . S 0 C - 0 S 9 . 4 9 C 0 S 9 . 1 2 C - 0 S • • l a c a s 4 . 1 9 1 - 0 9
4 . 1 « f O S 4 . 1 U - 0 1 4 . W 0 S 4 . 2 * C * 0 1 4.are-as 4 . 1 * f - 0 9 4 . 2 * 1 - 0 9 4 . 2 * 1 - 0 9 4 . 2 U - 0 9 4 . W 0 1 4 . 1 2 f * 0 9 4 . l 4 t * a 9 1 . 0 1 C 0 1
l . * 9 C - 0 2
" M I I *
1 • 9 4 1 * 0 4 2 . 0 K - 0 4 l . 9 4 t * > 4 1 . 2 9 t * 0 * I . I K - 0 4 4 . 4 2 1 * 0 9 4 . 4 0 C - 0 5 ' . I O C - 0 9 9 . a i f « 9 4 . 4 K - 0 9 l . « 0 f * 0 9 ) . 4 0 t * 0 5 2 . O H - 0 9 2 . i r i * 0 9 l . * » £ « 0 9 1 . 9 « f * 0 9 l . l » ( - O S t . t l f O t
o.i*c-o» ' . 9 l f - 0 » * . « O C * 0 * 4 . 4 O C - 0 4 s.*oc-o* S . 4 5 £ * 0 4 S . O H - 0 4 4 . 4 2 C - 0 4 4 . 2 1 1 - 0 4 i . a i c - 0 4 ] . * U * 0 » M ' f O t 2 . * 0 t - 0 4 2 . 9 U - 0 4 2 . 1 0 1 - 0 4 2 . 2 ) 1 - 0 4 2 . 0 * C * 0 4 i.*rc-o» l . * 4 [ - 0 4 t . r 2 t - 0 4
1 . 4 0 1 - 0 * 1 . 4 0 C - 0 4 1 . IOC-04 l . 2 » C * 0 4 1 . 2 0 C - 0 4 I . I O C ' 0 4 ».r»t*or a.rac-ar r.4»c*or s.iac-or i .aic-ar i.ioc-or r.*rc-o»
1 . 4 0 C - 0 )
»mo*
» . • 0 1 - 0 1 a.sn-o) l . S U - O J o.oic-o« 4 . 1 9 C - 0 * i . i r i - 0 4 2 . I O C - 0 4 i . r o e - 0 4 I . 1 9 C - 0 4 • 1 . 0 2 C - 0 4 a . i r c - 0 9 4.ru-os 9 . 9 3 C - 0 S i 4 . 4 4 C - 0 S . 1 . 4 2 C 0 9 2 . * O C * 0 9 2 . I U - 0 3 i . ru -os i
1.9OC-0S 1 1 . 1 9 C - 0 S 1 1 . 1 2 C 0 9 1 1 . 1 1 1 * 0 9 < 1 . O K - O S 3 « . ) 4 ( * 0 4 ! 0 . 4 1 C - 0 4 ! M K ' t l < 4.**c*eo < 4 . 2 * 1 * 0 * i S . S S f O * i 4 . 0 4 C - 0 4 1 * . ) W * 0 » i 4.»oc-o4 a 1 . 0 0 C - 0 4 i 1 . 9 9 C 0 0 i 1 . 1 1 C Q 4 i ] . 0 t * * 0 4 1 i . iac-04 i 1 . 4 M - 0 4 I
2 . 4 0 1 - 0 4 1 1 . 1 0 C - 0 4 1 2 . I 1 C 0 4 1 1 . » 0 1 * 0 4 1 I . 0 4 C - 0 4 l . » 0 C * 0 4 1 | . 4 » C - 0 4 1 t . l l C - 0 4 t l . U C - 0 4 1 o.t9C-or i 4.*at*ar i i . i w i r 1 . I U - 0 0 1
1 . 1 0 1 * 0 2 1
- H j l .
> . 0 1 1 * 0 4 I . I 1 C * 0 * 1 . 4 0 1 * 0 4 l . 2 * 1 * 0 4 l . 1 2 1 - 0 4 > . » ) C - 0 3 l . * 2 t * 0 S ' . 1 1 1 * 0 9 > .»4C-0S 1 . 4 4 1 - 0 3 I . * 1 C * 0 9 > .40C*0S t . o r c - o s i . i r c » 0 9 .•rc*«9
l . S U - O S . I U - 0 S
) . 2 * C * 0 4
1 . 2 2 1 - 0 4 ' . 9 4 C - 0 4 > . » H * 0 » t . 4 ) 1 - 0 4 > . * ) C - 0 » k . 4 t £ * 0 4 > . 0 » l - 0 » I . 4 S C 4 4 I . 2 4 C - 0 4 l . 0 4 C * 0 4 I . 4 J C - 0 4 l . * » I - 0 » . r j c - o * . 9 9 1 - 0 4 . 4 0 1 - 0 4
I . 2 9 C - 0 4 U I l t - 0 4 . 4 * 1 - 0 4 . 0 4 1 - 0 4 . ' 4 t * 0 »
. 4 2 1 * 0 *
. S O t - 0 4
. 4 0 t - 0 »
. I O C - 0 4 1 . 2 t t * 0 4 • l l t - 0 4
>.»2t-or )«a*c*ar '.9*1-or >.i*c*ar i . 2 r t * o r . ) * t -or . 9 H - 0 *
. 9 4 1 * 0 1
1 • D »
1
• r » i
s i i
i i 4
» * 1
0 9
» 9
S S
2 4 1 9
2 1 4 1
r 4
i 4
4
) 9 1
1 1
1 1 1 2 1 0 9 9
)
» » l a «
. » H - 0 2
. 4 4 C - 0 1
. I S i - O )
. r u - o )
. 4 4 C - 0 )
. ' 2 C - 0 1
. »n -o )
. 0 4 1 - 0 )
. 4 1 1 - 0 )
. S 4 C - 0 )
. 2 ) 1 * 0 )
. 2 * 1 * 0 4
. 1 * 1 - 0 4
. * r t * o )
. 9 ) 1 * 0 4
.2SC-04
. 4 1 1 - 0 4
. 2 « t * 0 4
« » U - 0 » . S H - O *
» » f O l OOt-04 1 1 1 - 0 1
. 1 0 1 - 0 4 $ 9 1 - 0 4 2 ) 1 - 0 )
. » » C - 0 *
. r ) i - o » • U - 0 4 art-o* 4 U - 0 4 ou-o* !«•(! I 2 t - 0 1 ' 2 1 * 0 4 * * t - 0 4 1 4 1 - 0 9
. I l t - 0 5
1 * 1 - 0 5 t 9 t - 0 3 0 0 1 - 0 4 4 4 1 - 0 9
. U C - O l rrt-os
. ' 2 1 - 0 * r 9 t - o * r o t - o *
s 4
) ) 2
2 l l 1 1 r
4
) 4 4
) ) 2
2 2
l 1 1 1 1 0
r • 9
* 4 2 1 1 1
( * 7 4
9 1 2 1 r
1 1
1 * • « u
* » n -
. 4 1 C - 0 4
. 2 4 1 - 0 4
. » 9 C - 0 4 • 1 4 1 - 0 4 . ' 0 1 - 0 4 . 0 4 C - 0 4 .OOC-04 . s r t - 0 4 . 1 ) 1 - 0 4 . p r c - 0 4 • S S t - 0 5 . 4 0 C 0 S . • » ( - O S .»Of -DS . J U - O S . • I t - a s .irc-os . » » £ • 0 9
. 4 0 1 * 0 3 21C-0S
. 0 0 1 * 0 3
.src-os > ) 3 t - 0 S • l O C - 0 9 . 0 1 C - 0 9 H C I t
. 4 1 1 - 0 * 3 4 C - 0 * » I t * 0 4 0 1 1 * 0 * o r i - 0 4 2 4 1 - 0 * *oc-o» • I C - O *
. 2 ' C - 0 » a r t - o * i*i*or ) » i - o r
) t l - o r 2 ' t - o r 2rc-or r»c-er
.loc-or l o t - o r i o f - o a « r t - o o 2 0 t - 0 0
« « H l u »
* « i a -
1 . 1 0 1 - 0 2 • • 0 2 1 * 0 ) ) . * 4 t * 0 ) l . O » C * 0 ) 1 . 1 0 1 * 0 ) » • « * £ * 0 * 3 . 0 ' l - 0 4 ) . * » ( - 0 4 2 . * » C * 0 4 2 . 2 0 C - 0 4 1 . 4 0 1 * 0 4 1 . I U - 0 4 1 . l O t - 0 « * . 2 3 t - 0 3 1,»01-03 • . O l - 0 3 s.on-os s.ort-os
4 . ) ' l - 0 S ) . » ' t - o s a . m - a s 2 . 3 ' i - O S 2 . l » t - 0 3 i«»u-os 1 .431-OS t . m - o s i.irc-os 1 . 0 2 1 - 0 3 * . " t - 0 4 r . 4 « t - 0 4 » . 2 4 1 * 0 * 3 . 0 ) 1 - 0 4 ) . * « l - 0 * 2 . 4 * 1 - 0 * l . « » t * 0 » l . * * l * 0 * 1 . 4 ) 1 - 0 * 1 . 2 H - 0 *
1 . I U - 0 * »,«»t-or r . » ' t - 0 ' S . ' i t - O ' i.s*t-or t.rot-or i . o r t - o ' s . m - o o i.aat-oo
• » i i -
* * J
) i 2 1
l 1 t t
• 3
* 4 1
) 2
2 2
1 1 1 1 1
«. r * s. 4 . 4 .
)• 2« 1 • I . I < * •
* ' . »• s< )• 2 l i Ti 4 .
1 .
. » 4 1 - 0 « • 4 * ( - 0 « , » ) l - 0 « > s r t - u »
r » i - a * 091 -U« » U - J 4 »n-u* 1 ) 1 - 0 4
. o » t - » » • s w - s * * L ' O S 4 » t * a 3 » t l - a s ) 2 ( . - 0 9 • 44.-U3 j r t - u 5 »rt-us
» 0 f 0 3 2 2 1 - 0 3 »Ol-OS 3»tC-03 å * t - o s 101-OS 0 * t * 0 $ r 2 i - o * • l t ' 3 « 3 » t - J * » $ t - o * »st-a» OOt 'O* J « f O * « 2 t - 0 * • 2 t - 0 * l » l * o » o r i - » « »2t-or »u-or i r t - u ; WQ> 1 2 1 - 0 7 « 2 t - o r i a t - o » i * i - o r i a t - o a O U - 0 4 2 2 1 - 0 0
I I » J ' « b l ! u «
' U -
l . » ' t - v > 2 1 . 1 4 C - 0 2 4 . 4 . 1 . - 0 4 • • • J t - O J 3 . 2 w t - U i • • 2 * t - w ) J . » l t - U J J . 2 J t * u J i.t'l'U* 2 . * 2 t - a i 2 . I 2 C - 0 J 2 . 2 ' t - 0 i 2 . 1 4 l * d j 2 . 0 4 1 - 0 j 1 . 0 * C - 0 2 l . K C - O i 1 . 1 4 L - 0 ) 1 . t O C O )
i,in.'o* l . ? 4 t - U J ur2C-o) > . M l - U J i . ? a t » 0 ) l . l i l ' « ) I . U l V ) l . » » t - 0 i i . r i t - j j i . 0 / i * U 2
* ) 2
) * ) 1
) i
2 2 2 2
1 l
I l l
1
» * r
» • 3
* 1 . 1 3 .
S H - 0 4 1 4 2 1 - 0 4 * 101*04 *
- • I C - O * 4 . 0 ) 1 - 0 4 2 > » ) t * 0 4 t 4 3 t - 0 4 1 J H - 0 4 » s t t - 0 4 r ' 2 C - 0 4 4 • l t t - 0 4 S 2 1 C 0 4 4 O l f O « ) » ) t - 0 4 ) 4 4 1 * 0 4 1 3 0 1 - 0 4 2 J41 -04 2 2 H - 0 * 2
H l - 0 4 1 •oe-os i »ot-os i f 4 t - 0 S l »ot-os i » • 1 - 0 3 * S t t - O S * 3 2 1 - 0 5 3 • U - 0 5 2 1 * 1 - 0 ' r
. 5 * t - J 2 r
. H f * i J
. • • t - U ) 2
. O ' t - O i )
. J J t - O ) «
. ' J t - O J i
. 2 ' £ - 0 J t , ' » £ - U « J •42C-U« 1 • 4 1 t - 0 4 2 . J i f j i 2 • 3 » t - U 4 2 . » S t - J « 2 . 4 4 ( - 0 4 I . M 2 1 - U * 1 . * > £ - M « 1 . j « c - o « i ><>*C»<J4 1
. » J £ - U « l . » U t - 0 4 * . • a c - 0 4 * . 2 2 C - 0 « r . 0 4 t - t t * » , 4 3 f < J 3 3 . » l £ - 0 > 4 . 2 * t - U l J . 3 J C - 0 3 1 . ' I C - O ' 3
• t U ' i i . * * t - ^ 4 . l « f j * • * 4 t * U 4 • O l t ' O * . * 4 C - » 4 • * * ( - > i 4
. )«'«»
. O l f j 4
. ' 2 f U 4
. 4 4 1 - U « • 2 U - J 4 . J I C J I . # J £ * J 4
.**(•»• • 30C-V4
. J » t - U *
.Ul'»*
• I U - D 4 . » l f » i 5 . O l t - 0 3 . Tii-n .»n-as . » » l - » ' 5 • 3 ' £ - a 3 . 3 * t - J 3 . » * t - U 5 . i * t - j '
o . i r i - 0 2 ) . j 4 t * o ) 2 . *» t *02 i . t i i ' O i 1 . 5 a t - 0 1 3 . * 4 l - 0 ) 4 , ' J t - 0 2 3 . 0 U - D )
I • I M
I M i l THORIUM I PfN UMNIUM t l fUt»SSlUM
t (KCV3
, 1 0 * 0 . I S • . 1 1 * 0 . 1 0 « . 2 0 - 0 . 2 3 « , 2 S * 0 . 0 1 . 3 0 * 0 . 1 * 1 . 3 3 * 0 . « 0 1 . 4 0 - 0 . 4 5 * .43-o.so * .30*0.55 1 .51*0i»0 t . 4 0 - 0 . 4 3 4 .4S-o.ro 4 .FQ-O.rS 1 •rs-o.«« i . » O - o U j 4 .»s-o.to s . 4 0 - 0 . » S I , « S * l i « 0 1
. 0 0 - l . O S 1
. O S - I . I O 1
. 1 0 * 1 . 1 3 1
. 1 1 * 1 . 2 0 t
. 2 0 * 1 . 1 5 t
. 2 3 * 1 . 1 « 1
. 3 « * 1 , 1 3 •
. I S - 1 . 4 0 •
.40-1.«s r
. • S - t . S O 2
. S O - l . S S »
. S S - 1 . 0 0 4
. *«•! . *S 1
. * s - i . ' o r
. 'O- l . 'S «
. f S - 1 . 0 0 4
.««*i.«s s
. • » • I . * « >
. 4 0 - 1 . t S 4 ,«s*a.c« o
. 0 0 - 2 . O S 4
. O S - t . l O «
. 1 0 - 1 . I S 4
. 1 3 * 2 . 2 0 •
. 1 0 * 2 . 2 3 4
. 2 S * 2 . 1 « «
. 3 0 * 1 . » 4
. 3 S - 1 . 4 0 4
. 4 0 - 2 . 4 S 4
. 4 5 - 1 . S O •
. 3 0 - 2 . 3 3 4 . S S - 2 . 4 0 4 . 0 0 * 2 . O S 2
*U"> ]
• M O «
, 4 « C * * 1 .« •£- • * • J 2 E - 0 3 . » 3 I * « 1 .««t*«l . 0 1 1 * 0 1 . * M - 0 4 • 0 4 C - 0 4 ,03t*«l . ' t t ' l l . 4 4 C - 0 4 . 2 * 1 - • « . 4 5 E - 0 4 . » » 1 * 0 4 . » • C * 0 4 . I O C * « . 4 4 C - 0 3 . 2 * 1 - 0 3
•ISC««« , * ? C * « * . 0 * E * 0 4 ••3C-0S . 2 4 1 - 0 4 . • 4 C - 0 S .wt -os •ou-os . • 2 1 - 0 5 .rrc-o« .o«c-os . 3 3 1 - 0 4 .»tc-»« . 2 3 1 - 0 5 •S«C-«J . « * E * O S .««t*os •ou*«» . I M - t S . J 4 f O S
• 3 3 C - « S
.!««•» , 2 « t * « S
.»•e*os ,2'c-os . 2 0 C - 0 J • 2 F I - 0 S • 1 F C - 0 S . 2 * 1 - 0 5 . I 0 I * « 5 • 1 2 t * « S • l * l * « S . • S C O 3
. M C ' « 1
• M i l «
2 . 5 3 1 * 0 4 2;«*c«o« 2 . 2 U - 0 4 1 . 3 4 1 - 0 4 Lire-04 « . m - « s 8 . 8 0 1 * 0 3 ' . » 0 1 - 0 3 •.2«C*«5 «.'*E»03 3 . * * C - 0 3 ..ore-os 4 . 0 2 C - 0 3 1 . W 0 3 2 .S3C-OS 2 . 3 W - 0 S S . 3 4 C - 0 S 4 . 2 2 C - 0 S
«.r«c*o* t . 0 4 C - 0 * 7 . 4 4 1 - 0 4 * . a u * o * 7 . 0 1 C - 0 * S . ' I E * « * S . I S C - 0 0 4 . 7 4 f 0 4 4 . 3 2 C - 0 4 8 . ' * E * 0 » 3 . * « ( * 0 » i.irc«os i.ooc-os I . H E - O » 2 . 4 4 C - 0 4 2 . 2 H - 0 4 2 . 4 S C - 0 4 a.i»e-«» l . * * C * 0 * ».rrc-o»
l . * 3 E - 0 » I . S l f O « I . « 1 E * 0 * 1 . 3 3 1 - 0 4 1 . 2 4 1 - 0 4 1 . 1 4 1 - 0 4 I.«1C«0* «,«*c«<* r.roi-or S.SM*«' 1 . 1 U - « ' i.«*f*or 5.431-03
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i . i t f u « i l.o«t-o« i .141-0« 0 .»•1*0« " . t i t - O « 4
l .JSt-81 • t . 7 * t *0» J • * . t * 0 » i . j «c -a» i . J ' t - O * I . *u t *o» 1 . 0 . 1 * 0 ) ' , " ' l - 0 » 5
• l » l - o * l . « ' £ • » • 1 . l»C-of \ • VVJ - 0 ^ « .«»t-l>? 1 .»»C-oii » . - • l - 0 » 1 ,*ffV "1 . 1•!-0 ' -7 . » - t * u * -»
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I P M THORIUM
i (< • ( •> M I « * M i l « M i l * M | J «
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OS 1 . 10 1 . I S 1 . ao t . as i . a« i . as i . « 0 1 .
>os t . SO 1 .
• SS 1 . 0« 1 . OS « .
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0 0 1 * 0 8 O i l - O S «ai*os 40C*0S S « C « S au*«s ooc-os O l f O S 0 « f * O S rocos oot-os | J C « S » I ' M I 0 1 * 0 J OSI-OS O l f O S M C t l 0 « € * 0 S F4C-0S • a t * * s
» O f OS OOI-OS » H H > 0 0 l * I S o«c*os O M - O S «tt*«s O I I * 0 S OOI-OS » 0 1 * 0 5 S O f O S » • c o s S O f • •
s»t-o« • a t - « s I S f «S 4 1 C 0 S H I - O S •o i -os O f f «S o*t*os 0 ? l * O S isc-os I 4 I - 0 S iae*os oac*os oac-os ou-os ««c**s I K ' H * l f 05
aoc*os • O f OS 1 0 f SS aoi-os I f f OS aoi-os i r t - os I O f OS U f O S iai-os i » f » s 0 « f OS SOI-OS •a f»o 1 1 1 * 0 0 141*00 ooc* »o «r i -oo OM-00 Oi l*0«
• i f oo O O f 0« O O f 00 tot*oo O O f 0« O O f 00 O O f 00 O O f 00 t s c * • • I t f O O 1 0 1 * 0 0 1 0 1 * 0 0 l * f 0«
O . O O f 0« 0 s . a i f o * a I . O ' f O « l i
».»«••« »< o.otfos a S . O O f OS I S . O O f O S I a.o«t*o» i i .aofos • I . I M ' O S *< o.orc*o* * . • • i i f o « «. s.aafo« a *.ooi-o« a l . O O f O O 1 a.iaco« i a . tofoo i l . ' U - O t 1
i . o r fo« »• I . I S f O * 0 o. io for • ' . » • C O * S. S . - M - O * «. 0 . l t f * 0 ' ». i . a ' f o r a. a.ooc*or a. i.oot-or i< l . « S f O * l< l . l t t * « * 1 ) . 0 » C 0 > 1 O . l l f O O • » . O ' f O t ' s.ascot s i .orcoo « J . O t f t t 1 a . i t c o t a i .rscoo • i .aocot t
o.«sc o* • o.ooco« s « . * « ( • « « 1 . « » ( • • « > i.ooco« i i .o' i*«« * i i . iacoo *• i.oocoo •> | .«M*ttt * l l . f O C O « * l I . O O C O O * » o.srco« *» a . s i f io •»
• S I * O S « l f OS » 1 1 * 0 8 • » C O S sai-os o«cos soc*os au*os aicoo 0 0 1 * 0 0 • o c o o 0 0 f * 0 0 0 0 t * 0 « 1 0 1 - 0 « • • C O « S I C O « I H * « t I 0 C O *
»ai*o» aoco* » O f « * «ot* • ' • O f « * s i c o ' Tfl-tT I M - O ' * i c « * » » C O * m * « * o r e o? 1 0 1 * 0 1 | t t * O t 8 0 1 * 0 1 aocot 101*01 J i f 0 1 •acot 1 0 1 - 0 t
• u*o» • I C 0 1 » " C « t 0 8 1 * 0 « loco« 0 * 1 • • « lot*«« a * c o«
. • 0 1 - 0 1 U C O I
. » » C O « O I C « « O O C K
»u«< r . i o c o i i . * « c e i i . i o c o i i .s tco«
1 • 100 »
k f<PM uOtNIUM I t » O ' M i i O N
•MIO«
l . O S C O ) i .ooco) I . 1 U - 0 1 1 . 0 1 1 - 0 1 r .oico« » • • I f 04 1 . 0 t t - 0 4 » . » I C O * 1 . 0 * 1 - 0 4 1 . 0 » f 04 I . 8 1 C 04 I . 1 8 C 0 4 I . U C 0 4 I . 1 0 C O * 1 . 0 0 1 - 0 4 i.o*t-o« J . O O f 0« i . * r t , -o«
l . * * C O * | . « 0 C 0 «
!•'»•(« i . i n - o « l . ' O f 0 * I . O O C O * r . 'u -os I . S O C O « l . l * C O « i .o icos I . 0 4 C 0 « r.oicos • • ' 1 C 0 S • . u c o s • •*>co» 1 . S 4 C 0 « • .» •cos ' • • • C O S 1 . 1 4 1 - 0 8 I . I 2 C 0 S
I . I I C 0 « i I . I O C O * i>*acos • • i l i -oo | . » 0 f 0« I . S 4 C 0 S ! > 0 * C 0 0 l . i l H ' O I l . l l f d t
1 . 2 . I . 1 . 1 . I . 1 . 1 . 1 .
1 . 1 .
r. r. ». •. » • t.
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•• 8 . 1 . 8 . J . 1 .
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»oco« * 1 C 0 « » I f 0« ast*o« . O C O « l « t * 0 4 1 4 C 04 »OCO« " S C O S 401-US
oocas IOC-OS l O f 08 4 1 1 - 0 8 • » C O S » S t * 0 8 » O f 04 l O f 04 i n * os » 0 1 - 0 8 » 0 1 - 0 8 J « C O S 0 1 1 - 0 8 • O f 08 • " I ' M O O C 0 8 • 1 C 0 8 » » C O S • 8 C 0 8 » • C O S • I C o o ' O f 05 i » c c s » I C O « » I f 00
* S C u * - 1 » » C O * •» locos - i l t l - 0 « • • 4 H - . 8 • » « i t * o o * i ( » C O * - 1 m - o « - i » 0 C 0 8 'i
" M O *
• 1 I C 0 J • I 5 C 0 1 . » I C O « . 5 0 C 0 « . » 0 C 0 4 • 1 » C 0 4 .rocos • 1 0 C 0 8 . 8 8 C 0 8 . • » C O S • oocos • 4 0 C 0 8 • ou-os . 1 1 C 0 * • au-o* . » • C O * . 1 S C 0 * . 4 * C 0 *
• 1 1 1 * « * • loco« .»oco* .««co* . » S C O ' . 0 0 1 - 0 ' . 1 S C 0 « • » I C O «
. )«•« . 1 * 1 - 0 0 • 1 0 C S 0 . I 1 C 0 » . I I I « " « . O I C O » . i H ' O t . » i c o » • 0 0 l - 0 » . » » C O * . » 1 C 0 » . 101*141
W t l ' H • O ' C O « • 0 * 1 * 0 1 .»•co» • • • C O « . « u * o » • l * C O » • 1 I C 0 « . » ' C I O
" 1 . 8 .
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1 . 1 . 1 •
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« ) C 0 « • 1 C 0 S • » 1 - 0 5 « 1 C 0 »
*»c«> * « C 0 8 « I C O » S * C O * 0 0 C U 8 JOCOJ 0 * 1 - 0 1 I O C S * I I C O * * 1 1 * 0 * * S C 0 * «oco» ' » C O « 1 I C U *
l H * o / » I C O ' t i l ' « ' iocs' l O l - B F »•» •o ' « 1 1 * 0 « » O C O t • u-ot • O l - U « • « C w ( l H - 4 1 • *l*u» 101-y* i l e a * « » l - o » » H - g * OU-M* 1 * C 0 » » • f i t
* I C 0 » 401*4* •OCw« * l l - u » »oc«» • H-K* ( » I ' l l . I t • . »
•"MiO' " n i l *
O t - U i 8 U-»4 1 » C - l 1 » C O i 1 J C O J 1 O C U J 2 i c o i i » C O J 2 2 C « « 2 » C O « 1 ' 1 * 0 * 2 oco* i » C O « 2 • C O « 2 OCO« 2 • C D « 2 " C » 4 2 J l ' V t 2
» C O * 2 • C O * 1 l l - o * i » C O « 2 » C O « 2 * C « * 2 I C O * 2 H'V* 2 I C O * 2 « t * v i 2
• J 2 C 0 « . * * C » * • 0 « f 0 * • o*co« • O I C O « . » 8 1 - 0 * • 2 8 C 0 « • J I C O * • 1 4 C 0 « . 1 * 1 - 0 « . J U - O « O J C O « . J I C O « • 2 * 1 - 0 « . 2 * 1 - 0 * • 1 « C 0 « . 2 * 1 - 0 * . H t - O «
• 2 1 C 0 * • 2 2 C O « • 1 2 C 0 « • 2 1 C 0 * . 2 ) 1 - 0 « • 1 « C 0 * . 2 8 C O « . 1 * C » « • m - o « • i i i ' i i
* . * U C O J i.'ufui i . o i c u i l . U a C - U i t . U J C - u a l . i i C i i i l . l i c « . I . ' C C U * » . 2 * 1 - 0 «
• .!«••> «.'»!"«» J . u i f i l ) J . l l [ ' t » l . » < ( ' < ) 1 . ' 2 C - J J » . o » C « j t . * 2 C 0 * J . I ' C O *
> . » 2 ( - 0 « • • » ' C O / I . . M C O / » . * » c o * ) . » t f t - 0 « l . « ' C u *
* » . » 1 C » » • l . l o f O ' • » . » » € • « ' • 1 , ' < C « »
2
/ 2 1 I
» 1
• t J 4 2 1 1 1
r » i
i i
» 1 t 1 i
• 1
• i • i
• 2 C - « U l ' M i »•l-vl «Ut"w* » » t - u « J l C u * o'Cu« » O C J » * » C u 8 l » C J l » 0 1 - " 1 » » • J * 2 2 C u » * « l - u " i I / L ' J I 1 2 1 - J *
. » 2 1 ' U t i l l f a t
. l ' [ « « l 0 » C « * l i t ' « ' 4 « C u '
, r » i - u * 181-UO 2 l t * u » l * C - ' 2 J C U ' • * l * < »
J ' o
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- 5t -
Table 7
Absorbed tiose r a t e s Dis) irs .sir of the dent.-!;. 0. 00 ; l'.ii g / u : .
r t s a l ' i n j ; fr>tm unit abundances of t h o j i u m , uramun;. ,
and p o t a s s i u m in the ground n :a le r i . i i (j»r.truu)
3
z ^ m )
'»
1
~* 1 ! ">0 ! i
\ p p m T h
. . . 277
0 . 21!
0 . J04
0 . ! f,l
0 . I 30
l)(7 ) U r a d / ;,)
? ppn , L
0 . 570
(). .-» J O
0 . 4 1 5
0. 324
o . -*»a i
K
i . 3 7
i . »1 T
I . 0 J
0. Mi',
0. b~>3
*
100
V2r,
! 50
1 7">
J00
O. 1 06
O.OR77
0.073?
O.Ofe! 2
0 .0516
0. 209
0.17!
0. 14!
0 . • ? 7
0. 0')74
0. 53K
0 .447
0. 374
0. i\ 4
0. 2b4
S p e c i f i c t r a m m a - r a s e m i s s i o n s Q f\.r t h e m . i u f a i fad ; , . e l e m e n t s
unci m a s s a t t e n u a t i o n coeffit. l e n t s 15 • f u s e .a m e c a leu a t i. •- • r i >! a
t l . ; \ o! u n s c a ' t e f e d ^ a m m a r a \ . s
Gamma- r;i\
source
1 p p m Tli
1 ppm U
"
1% K
K
t.MeV)
f 0 . 2386
(). 33H5
.). 5! 07
0 . 5 8 3 1
0 . 7 2 7 2
0 . 91 i 1
0 . 9667
1 .5881
2 . 6147
0 . 2 9 5 2
0 . 3 5 2 0
0 . GO94
1 . 1204
1 . 2 3 8 2
1 . 3 7 7 8
1 . 7 6 4 7
2 . 2 0 4 5
2 . 4 4 8 0
1 . 4 6 0 8
1 . 8 5
5. 04
3. (•'.'
1 .23
. . . O t
, i b
9 .4 3
1 . 8 9
| 1 .47
] 2 . 1 9
4 . 2 8
5. J.h
1. 77
6. 85
5. 02
1 . 80
5. 7 5
1 . 83
3. 30
t; )
u f 3
M l - 4
H . - 4
i o " 3
-'A 1 0 '
1 0 '
Mf4
ur4
u f 3
10" 3
'
! 0 " 3
u f 3
. O " 4
ur4
M , " 3
ur4
ur4
i o""
\ I <ss a t t e r . - ;
G r a n i t e
O. 1 1 -.3
(1. 1 01 •
li. 0 8 5 7
0 . 0 8 ! 0
U. 07 3,8
0. OoOO
0. 004 2
0. 0 5 0 0
0. 0 388
U. 1 0 0 0
0 . 0 9 90
o. 078-4
0. 05 90
0 . 0 5 6 7
0 . 0 5 3 7
0. 04 7 3
0 . 0 4 2 3
(]. 0401
0 . 0 5 2 2
i tl- TS I', e ' ' i t
Soi l
0 . 1 : 6 5
0 . 1 0 2 3
0 . 0 8 6 7
0 . 0 8 20
0 . 0 7 4 2
0 . 0t .68
0 . 0 ' .50
0 . 0 506
0. 0391
0. 1078
0. 1 (JOH
0 .Oh04
0 . Oo'i4
0 . 0 5 74
0 . 0 5 4 4
0. 04 7 9
0 . 0 4 27
0. 0 4 0 5
0. 52 8, ._
- e l . t ( . v..'- i f )
Alt '
0 . 1 i 4 9 i
0. i 0 ! 4 0. 08t .2
0. 081 4
0 . 0 7 3 8
0 . 0 6 6 4
0. 064 6
0 . 0 5 0 2
0 . 0 3 8 5
0. 1 0 6 8
0 . 0 9 9 9
0 . 0 7 9 9
0 . 0 6 0 0
0 . 0 5 7 0
0 . 0 5 4 0
0. 04 75
0. 0 4 2 2
0 . 0 3 9 9
0 . 0 5 2 4