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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

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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 « .

O H - O J SOC-0* O ' f O * U f O « •ic-o« MC-O« «ot-o« i«t*oo O H - 0 0 0 0 1 * 0 0 J « * 0 0 aai*oo a«c«oo au*oo o*c*os aac*os «*c«« I 0 C 0 4

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 .

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• • • • *. *• 1 . 4 .

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»»CO« I ' C O « » O f 0*

»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 .

«• *• ». «• 1 . 2 . 1 .

»• t •

»• *• • < I . 1 . 1 . 1 .

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*• *. 4 . 1 . 1 .

«. J . 1 . 1. 1 .

1 . 1 . 1 •

*. •»• •«. •*. *». •!• •*. •»• • ' . •!•

" H i I *

« ) 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

| . » « l - 0 2 J . ' I d J * . » » i " i ) J » . • » I ' i . i ' l '

- 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