Scattered Gamma 

Rays From Thick 

Uranium Sources

GEOLOGICAL SURREY BULLETIN 1052-A

Scattered Gamma 

Rays From Thick 

Uranium Sources

By A. Y. SAKAKURA

EXPERIMENTAL AND THEORETICAL GEOPHYSICS

GEOLOGICAL SURVEY BULLETIN 1052-A

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1957
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U. S. Government Printing Office 

Washington 25, D. C. - Price 25 cents (paper cover)
CONTENTS

Pag«

Letter symbols.___________________________________________________ vi

Abstract__ _ ________-____-_____________-___--_--_-__--_-____-___ 1

Introduction. _____________________________________________________ 2

Acknowledgments- --______--_____--__-_-_------_-__---_----____--_ 3

Gamma-ray intensity from elementary and broad sources.______________ 3

Determination of empirical relations- ____________________________ 3

Comparison with theory___________-________-___________________ 11

Effect of time constant and the cone of response_________________

Gamma-ray intensity from idealized sources and interpretation of radio- 15

activity anomalies_______________________________________________ 15

Selected idealized sources________-_-____________-_..______----___ 20

Relationship of observable data to source characteristics-_________ 28

Interpretation of anomalies-_____-__--__-_______-___-____-______ 33

Conclusions.-_-_-______--___________-__--_-_-____-___-_-_-_-__--__ 33

Computation of intensities_-__--_______---_-_--__-_-_----_-____.,-_-- 37

Calculation of detector response.____________________________________ 38

Intensity from selected sources-.____________________________________ 41

Area under counting-rate-meter curve._______________________________ 42

Literature cited.___-___________________-_-____-__---__-_-----_--_- 43

ILLUSTRATIONS

Page

FIGUEE 1. Elementary-source geometry...__________________________ 5

2. Broad-source geometry..._____________________________ _ 5

3. Broad-source data___._______--___-_--_-__-_____________ 7

4. Elementary-source data corrected for angle of view_________ 9

5. Elementary-source intensity as function of altitude.________ 11

6. Relative values of primary, scattered, and total intensities for

primary energies of 2.432 and 0.609 Mev from broad source, 13

7. Computed total and primary fluxes from broad source of

uranium emitting 1 photon per ft3 per second. __ _________ 14

8. Decomposition of total flux from broad source of uranium 

emitting 1 photon per ft3 per sec into contributions from 

several lines of spectrum._____________________________ 16

9. Relative values of primary, scattered, and total intensities for 

primary energies of 2.432 and 0.609 Mev from elementary 

source. ______________________________________________ 17

10. Computed total and primary fluxes from elementary source of

uranium emitting 1 photon per foot per second_________._ 18

11. Decomposition of total flux from elementary source of uranium emitting 1 photon per foot per second into contributions from several lines of spectrum_____________________ 19

12. Relative intensities at several distances from center of finite

sources of several radii._______________________________ 22

III
IV CONTENTS

Page 

FIGURE 13. Relative intensities at several distances from centerline of slab

sources of several widths.-_________________ 23

14. Relative intensities at several distances from centerline of a

slab source 400 feet wide at different angles of approach, __ _ _ 24

15. Relative intensities at several distances along flight lines at 

different distances from center of finite sources of 400- and 

500-foot radius_________________________ 25

16. Ratio of intensity at edge to intensity at center of source_ _ _ 26

17. Approach of intensity ratios to limiting values_____________ 27

18. Grade necessary to produce 50 counts per second for several

radii of finite sources on different flight lines_ ________ __ 29

19. Scaled area under curve ^(500, z,a) vs. scaled peak intensity

(?j(500,2,c) of finite sources of several radii (a) ___________ 30

20. Ratio of area under curve (Hj) to peak intensity (//) at z vs. 

ratio of area under curve (H/) to peak intensity (//) at 

1300-2 for finitesource of several radii-________________ 32

21. Ratio of area under curve (H,) to peak intensity (/«) vs. width

of a slab source at 0=0-______---------------_._______ 34

TABLES

Page

TABLE 1. Elementary-source data and associated computations __ 44

2. Primary gamma rays of uranium series________________^__ 45

3. Total flux and total intensities of various components from

uranium broad source emitting 1 photon per foot3 per second- 46

4. Primary flux and primary intensities of various components 

from uranium broad source emitting 1 photon per foot8 per 

second_____________________-_____-_-_-__- 46

5. Total flux and total intensities of components from uranium

elementary source emitting 1 photon per foot per second____ 47

6. Primary flux and primary intensities of various components 

from uranium elementary source emitting 1 photon per foot 

per second__________________________________________ 47

7. The function Gs (500,P,o)_________ 48

8. The function d (500,y,o)_______ 48

9. The function G4 (500,g,o)__________ ___- 49

10. Ratio of grade-area predicted from elementary source relation 

to actual grade-area as function of radius, a, and ground 

projection, 2, of closest distance of approach ______________ 49

11. Ratio of elementary source grade-area to actual grade-area 

averaged over two flight lines as function of radius, a, and 

ground projection, 2, of closest distance of approach._____ 49

12. Conversion factors to correct elementary source grade-area

prediction to true grade-area for finite sources.___________ 50

13. Ratio of total counts, Hj (500,2,0), to peak intensity, If 

(500,2,0), as funtion of radius, a, and ground projection, 2, 

of the closest distance of approach for finite source, _ _____ 50

14. Ratio of total counts, Hs (500,0,o) at normal incidence, to 

peak intensity, Is (500,0,a), as function of width, a, of a 

slab source__-----_--___----_----_------------------- 50

LETTER SYMBOLS

A=area of elementary source 

.Ao=area of standard elementary source=1,600 ft2

a=radius (or width) of source

.H/=area under curve recorded over finite source by counting-rate meter 

HI=area under curve recorded over line source by counting-rate meter 

Hp=area under curve recorded over elementary source by counting-rate meter 

HI =area under curve recorded over slab source by counting-rate meter 

7&=intensity from broad source 

//=intensity from finite source 

/i=intensity from line source 

IP=intensity from elementary source 

/«=intensity from slab source

B=response of counting rate meter

r=air distance between source and detector

/S=grade of source in equivalent uranium 

<So=grade of standard source in equivalent uranium=0.35 percent U

fo=time the source first falls within cone of response of detector

x=altitude of detector above source plane

j/=source plane projection of the air distance from the detector to source 

center line

2=ground projection of distance of closest approach of detector to source

a = l/t angle of cone of response of detector

/S=distance traveled in one time constant in mean free paths

6=angle determined by relative displacement of peaks on two adjacent lines

M=linear absorption coefficient of the most penetrating gamma ray of the 

U spectrum=1.46X10~3 ft at sea level

p=projection on the source plane of the air distance from detector to center 

of finite source

r=time constant of detector

VI
EXPERIMENTAL AND THEORETICAL GEOPHYSICS

SCATTERED GAMMA RAYS FROM THICK URANIUM

SOURCES

By A. Y. SAKAKURA

ABSTRACT

Semiquantitative interpretation of data from airborne radioactivity surveying 

requires detailed knowledge of scattered gamma-ray intensity at considerable 

air distances from natural thick uranium sources. Based on the concept of an 

elementary source of infinite thickness rather than a conventional point source, 

empirical expressions, which are of the form dictated by theory, are developed 

for measured gamma-ray intensities from two extreme types of natural uranium 

sources, the elementary and the broad source. Computation, based on published 

solutions of the Boltzmann equation for gamma-ray transport in one medium, is in 

agreement with experimental measurements. The calculated value for primary, 

scattered, and total intensities from thick uranium sources show that at considerable 

air distances the scattered intensities are more than half of the total intensity for 

energies above 0.4 Mev and become much more than half as the lower limit of 

detector energy response is decreased below 0.4 Mev.

From the expression for the gamma-ray intensity of elementary source, gammaray intensities from idealized sources, selected to approximate most nearly 

naturally occurring sources, have been computed. All computations are based 

on the assumptions of quarter-mile flight line spacing, and flight path at 500 feet 

above ground at atmospheric pressure. The computed data are strictly valid 

only for these conditions of airborne radioactivity surveying and for scintillation 

detectors of essentially identical spectral energy response.

The intensities and areas under the curve that would be recorded by a countingrate meter have been computed for selected source types, the elementary, the 

finite, the slab, the line, and the broad source. The peak intensity and the area 

under the curve are the two interrelated observable quantities most useful for 

interpretation. The shape of recorded anomalies as shown by analysis of experimental and computed data cannot be used for interpreting grade-area and (or) 

grade of natural sources although shape is useful in determining source type.

In interpreting anomalies, 4 of the 5 source types can be distinguished by the 

appearance of an anomaly, either on one or on two or more adjacent flight lines; 

one type, the broad source, can be readily distinguished by flattening of the 

anomaly on adjacent flight lines. With source type identified, the grade-area 

and (or) grade of a natural source can be determined from tabulated data for 

the two observable quantities, the peak intensity and the area under the curve. 

The accuracy of interpretation of grade of broad and slab sources has been confirmed by field investigations; the interpretation of grade-area and (or) grade of

1
2 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

finite and elementary sources, particularly those visible on only one flight line, 

is difficult and is satisfactory only within an order of magnitude.

The method of interpretation, based on a semiempirical approach, is applicable 

to measurements made with scintillation detectors of broad spectral energy 

response. For other than essentially identical radiation detection instruments, 

the necessary constants which specify the buildup factor for any specific detector 

can be established by empirical tests of the response to known thick uranium 

sources.

INTRODUCTION

Airborne radioactivity surveying is now widely and successfully 

used, primarily to prospect for uranium. In the majority of airborne 

surveys the sought-for target is a radioactivity anomaly, a sharp local 

increase in radiation intensity. Data on anomalies are usually 

reported on a qualitative basis (Stead, 1955).

To advance airborne surveying from the qualitative to the semiquantitative estimation of natural-source parameters,, a greater 

knowledge of air-scattered gamma-ray intensity at considerable air 

distances from naturally occurring thick uranium sources is necessary. 

The scope of the knowledge must be such that for any specific instrument of gamma-ray detection and measurement the observable data 

for a given anomaly, such as the peak intensity, the area under the 

curve, and the shape of the curve, can be correlated with the principal 

characteristics of a natural source, namely the strengh in terms of the 

equivalent uranium oxide content and the configuration in terms of 

the surface dimensions.

Inasmuch as naturally occurring sources are usually poorly defined 

in area and composition, and the radiation effects are obscured by 

surrounding sources, a purely empirical study would present great 

difficulties in determining the salient features of the radiation intensity 

from a given source type. It therefore seems more practicable to 

investigate numerically and analytically the features of certain simple 

source types, never found in nature, but nevertheless covering the 

entire range of variation in shape and size found in natural sources.

The following sequence of action presents itself:

Determination of the radiation intensity, theoretically or experimentally, from an elementary source from which other sources can 

be constructed.

Determination of the radiation intensity from simple sources by 

distribution of the elementary source over suitable geometrical 

configurations.

Investigation of the relationship between the source characteristics, such as size, grade, shape, and the observable data such as peak 

intensity reading and area under the counting-rate meter curve for 

several sources to establish criteria for distinguishing the different 

source types.
SCATTERED GAMMA RATS FROM THICK TJRANTUM SOURCES 3

Relation of source characteristic directly to the observable data 

without reference to (unknown) distance, with the criteria for distinguishing source types established.

Reduction of information gained in the previous steps to a simple 

recipe suitable for actual field applications.

This report is devoted primarily to the establishment of the expression for the elementary-source intensity, to a comparison of experimental results with existing theoretical knowledge, and to the effect 

on the apparent peak intensity of the finite cone of response and 

resolving time of a counting-rate meter. Characteristics of selected 

source types are established and a possible mode of interpreting airborne radioactivity measurements in terms of natural-source parameters is outlined.

ACKNOWLEDGMENTS

The writer is deeply grateful to L. V. Spencer, of the National 

Bureau of Standards, for a long enlightening discussion on the nature 

of gamma-ray scattering and on the format and contents of the paper. 

The section entitled "Comparison with theory" was inserted upon his 

suggestion. The writer is also grateful to F. J. Davis, of Oak Ridge 

National Laboratory, who compiled most of the data and to R. H. 

Ritchie, of Oak Ridge National Laboratory for many fruitful discussions. He is indebted to N. Lazar of Oak Ridge National Laboratory 

for determining the equilibrium spectrum of uranium.

The work herein reported is part of an investigation carried on by 

the U. S. Geological Survey on behalf of the Division of Research of 

the U. S. Atomic Energy Commission.

GAMMA-RAT INTENSITY FROM ELEMENTARY AND 

BROAD SOURCES

DETERMINATION OF EMPIRICAL RELATIONS

As the fundamental properties of gamma radiation are well known 

for the energy range observed in natural sources, the ideal method of 

interpretation would be based on theoretical computation of the radiation intensity emanating from several sources, by solving the Boltzmann equation for gamma-ray transport in two-media. This solution, 

in general, is a function of three space variables, two angular variables 

and an energy variable. The advantage of the computational approach is that the results do not have to be resolved in terms of the 

response characteristics of a particular detector, and that the response 

of any detector, if its spectral and angular characteristics are known, 

can be computed. No satisfactory solutions to the general two-media 

problems have yet been published, although the U. S. Geological 

Survey and the Atomic Energy Commission Computing Facility at 

New York University are now engaged in solving the two-media

428734 57 2
4 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

problems with plane symmetry. However, Fano (1953a, 1953b) and 

Spencer and Fano (1951) have made extensive analytical studies, and 

a vast quantity of numerical solutions to one-medium problems in 

plane and spherical geometry exists (Goldstein and Wilkins 1954), 

from which one may predict trends of solutions in two-media problems. 

Moreover, for moderate energy gamma rays, two-media problems with 

plane symmetry can be reduced to equivalent one-medium problems. 

The general procedure for solving the Boltzmann equation was developed by Spencer and Fano. A topical summary and pertinent bibliography can be found in recent articles by Fano (1953a, 1953b).

However, the simplest approach is a semiempirical one, wherein an 

experimentally determined or assumed form of radiation-intensity law 

for an elementary source is integrated over a suitable geometric configuration to obtain the intensities from various sources. Peirson and 

Franklin (1951) performed such an integration assuming an inversesquare exponential law which not only neglects the effect of scattering 

but also the multitudinous lines of the uranium gamma-ray spectrum. 

Carmichael and others (Steljes and others, 1952) used the inversesquare law but took air scattering into account by experimentally 

determining the "effective absorption coefficient" by measuring the 

radiation from radium sources. Cook (1952) assumed an "effective" 

energy and utilized "effective" computed cross-section and the inversesquare law. However, there is no priori reason why any "effective" 

quantity computed or measured in one source-to-medium orientation 

should be applicable to others. To avoid this difficulty measurements 

were made on infinitely thick sources, wherein the integration in depth 

is already performed in the data, and the different sources are synthesized through integration over a suitable surface.

Although the empirical approach was chosen for simplicity, it is 

useful to retain the form and the structure predicted by theory for 

the purpose of extrapolation into areas not covered by the data. 

Then, the radiation intensity will take the form of the product of the 

most penetrating primary component and a buildup factor, a polynomial in the distance variable for the case of moderate penetration 

(Fano, 1953a; Spencer, and Fano 1951).

There are two extremes in source types found in nature. One of 

them, which will be called an elementary source, from which every 

other source can be computed by integration, is depicted in figure 1, 

an infinite line embedded in the ground. The other extreme, which 

will be called a broad source, is depicted in figure 2, a half space uniformly filled with radioactive matter. It is obvious that the second
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 5

Air

Source

FIGURE 1. Elementary-source geometry.

FIGURE 2. Broad-source geometry.
6 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

source can be formed by infinite superposition of the former. Thus 

an empirical form chosen for the elementary source should be such 

that it not only fits the elementary-source data, but also, when integrated over an infinite area, should fit the broad-source data. In this 

manner, any intermediate form of sources should be well represented 

by the integration of the elementary source over a suitable geometrical 

configuration.

In this section, empirical expressions will be established for the 

elementary and the broad sources. It is known that the radiation 

intensity can be written as a product of two factors, one representing 

the primary contribution from the unscattered components, and the 

other, a buildup factor representing the contribution from scattered 

radiation (Fano, 1953a, 1953b; Spencer and Fano, 1951). The latter 

factor is generally a polynomial for moderate penetration.

An extensive series of experimental measurements to provide statistically sound data bearing an absorption and scattering of gamma 

radiation was planned and completed in cooperation with the Health 

Physics Division of the Oak Ridge National Laboratory. Measurements were made at different altitudes above a simulated elementary 

source and an infinite broad source with scintillation detection equipment designed and constructed by Oak Ridge National Laboratory 

and installed in a multiengine airplane.

The simulated elementary source is at Walker Airport, Grand 

Junction, Colo., and consists of a slab of carnotite ore 40 feet square 

and 6 inches thick sealed by a thin rubberized fabric to minimize 

escape of radon. The source contains 48 tons of ore at 0.35 percent 

U308 (1.31 X103 gU and 47.1 mg Ra).

In comparison with the infinitely thick source, the 6-inch thickness 

or ore yields 95 percent of the primary intensity of the most abundant 

gamma ray (0.609 Mev) and 75 percent of the intensity of the most 

penetrating gamma ray (2.432 Mev). The areal extent of the simulated source is small enough to be considered an elementary source 

at air distances several times greater than the effective diameter of the 

slab.

The experimental measurements over the source were made at 

100-foot spacings from 100 to 800 feet above the ground both directly 

over the source and at horizontal distances from the center-line to 

one side of the source, of 125, 250, 375, and 500 feet. The flight level 

of the aircraft was determined by a continuously recording radio 

altimeter. Flight-line markers for pilot guidance were set up across 

and parallel to the source with one line of grid markers through the 

source and perpendicular to the flight-line markers. A gyrostabilized 

continuous-strip-film camera recorded each flight; thus, from the known 

focal length of the lens, the distance from the ground and also the
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 7

horizontal distance offside the source could be calculated within about 

10 feet. The nominal flight levels as determined by the radio altimeter were often in error by as much as 80 feet when compared with the 

more accurate determinations of position based on the continuousstrip photograph.

In figure 3 the net counting rates from a broad source at minimum 

pulse height acceptance levels corresponding to 0.05, 0.1, 0.2, and 

0.4 Mev are plotted against altitude above the source. Each point 

corresponds to the average of five readings taken by three observers. 

The solid lines correspond to exponential curves fitted by least squares.

400 600 

ALTITUDE, IN FEET ABOVE SOURCE

FIGUKE 3. Broad-source data.

The slopes are constant, and consequently it is concluded that within 

the limits of experimental error, the spectral composition does not 

change. Consequently, all further measurements and considerations 

are based on the highest counting rate, corresponding to the lower 

energy acceptance level of 0.05 Mev.

The broad source is near Fruita, Colo., an area of Mancos shale 

several square miles and of farily uniform composition. Representative trench samples of the shale contained 1.6 percent K, 0.0017 

percent U (fluorimetric), and 0.0013 percent equivalent U. Experimental measurements over the source were made at 100-foot spacings 

from 100 to 1,000 feet above the ground.
8 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

The elementary-source intensity, Ip, is expressed as

where

x=altitude of detector above source plane

p=projection on the source plane of the air distance from source to

detector ___ 

r = air distance="

C, Oi02= constants to be determined 

.4 = area of an elementary source 

.A 0= area of standard elementary source=1600 ft2 

S=grade in eU of source

Se= grade in eU of standard source=0.35 percent U 

At 0=linear absorption coefficient of the most penetrating gamma ray of 

the U spectrum = 1.46 X 10~s ft"1 at sea level

The factor xfr is the effective area of the source as "seen" by the 

detector (fig. 1). Because of the strongly absorbing nature of ground 

only the gamma rays from a finite depth of an infinitely thick source 

contribute appreciably to the counting rate. The next factor,

e~V

 3 > is the primary intensity from the most penetrating component.

The square bracketed factor is the buildup factor which is not identical to the buildup factor of theory, as it also includes the effects of 

more rapidly decaying exponentials of the less penetrating primary 

components.

The intensity from a t broad source, /6 (x), is

li,(X) T (~\ = <>rC Airl a> p -j Ip(a, P) dp

As these equations are largely arbitrary, and for the sake of retaining 

simple equations, at is taken to be 1. Thus

CSA Xe~!*0r r-, , i / NOT / «> > 8] (3) 

(4)

These are the equations to be fitted to the data.

Simple exponential inverse-square expression for a point source 

such as those given in Peirson and Franklin (1951), will yield, with 

the use of a linear build-up factor, exponential broad-source expression 

upon integration over an infinite half space. Whereas such an expres-
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 9

sion can represent the broad-source data adequately, the inversesquare-exponential point-source expression, multiplied by a linear 

build-up factor and integrated over a semi-infinite line, fails to fit 

the data for the elementary source. Thus, equation (3) should be 

used in defining the expression for the elementary-source intensity. 

Moreover, as the two sources are the extremes in the geometrical 

configuration of the source rock, the, evaluation of the constants by 

using the data from both sources would yield an expression for the 

elementary-source intensity most suitable for superposition into 

arbitrary sources.

The dotted line curve (fig. 3) represents the expression

= (638 (5)

which was fitted by least squares to the broad source data, from which 

is obtained the value, 0.342 for a2 from the broad source data. 

With a2 established, it remains merely to determine C. In figure 4,

ouuu 

4000

2000

1000 

800

600 

400

200

100 

80

60 

,40,

\

V

___ \J

k

\

^

O ON.

0 ON^ °°: X 0 °

O

O

(9

°0° 0>J

0°

o

n

J.

O^Xxv

) 100 200 300 400 500 600 70

AIR DISTANCE, IN FEET FROM SOURCE 

FIGITRE 4, Elementary-source data corrected for angle of view.
10 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

the data for the elementary source, corrected by the factor r/x, are 

plotted against air distance r. The expression defined in equation 

(3) was fitted to all data taken more than 450 feet from the source 

because the time constant correction is small and because this is the 

region of greatest concern to this particular method of aerial surveying. In table 1 the following data are given: column 1, the air distance in feet; column 2, the altitude in feet; column 3, the experimental 

counting rate; column 4, the counting rate multiplied by r/x; column 5,

6 ~ltor

the data of column 4 divided by 2~[l+A*o?'-~0.3420u0r) 2] yielding

C; and column 6, the averages of column 5 with the nominal altitude 

and distance of the measurements offside.

The average values of C (table 1, column 6) are grouped so that 

any systematic deviation will point out an error made in the assumption of equation (3) for the elementary-source intensity. Within 

experimental error, the C's so obtained are equal. The observational 

error is calculated as the absolute mean deviation from the mean of 

data read by three observers from the same charts.

Using the values established for C and a2 , equations (3) and (4) 

become

0.342 Ur)*] (6) 

-0.342^) (7)

In figure 4, equation (6) at the point r x (directly overhead) is 

compared to experimental data corrected for the angle of view. The 

fit of the experimental data to the plot of equation (6) is particularly 

good for air distances less than 450 feet.

In figure 5, the intensity at different altitudes above an eleme^t^ry 

source, calculated from equation (6) , is plotted for flight lines directly 

over and at 125, 250, and 500 feet to the side of the source. It is clear 

that the maximum intensity increases with altitude as the distance 

of nearest approach to the flight lines is increased.

The accuracy of equation (7) in predicting the equivalent uranium 

content of broad sources can be determined by comparing it with the 

experimental equation (5) from which it follows that

o_ q

= 638, or 5=0.0018 percent eU308 (8)

The predicted value of 0.0018 percent equivalent U3O8 is considered 

a satisfactory fit with the analytical value of 0.0013 percent equivalent 

U3OS for a large representative sample of the broad source.
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 11

6000

4000

20.

200 300 400 500 600 700

ALTITUDE, IN FEET ABOVE SOURCE

FIGURE 5. Elementary-source intensity as function of altitude. 

COMPARISON WITH THEORY

From existing computations of solutions to the Boltzmann equation 

in one medium and the knowledge of the primary spectrum of the 

uranium series, the expected integrated intensity from 0.4 Mev to 

2.432 Mev was computed for the broad and elementary source of unit 

strength; that is, 1 photon per cubic foot per second and 1 photon per 

foot per second. The latter corresponds to a source 1 square foot in 

area emitting 1 photon per cubic foot per second. The details of the 

calculation are given on pages 37-38.

428734 57-
12 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

For the purpose of this section, the following arbitrary nomenclature 

is adopted:

Primary intensity: Number flux due to unscattered photons from one line of 

the uranium series.

Total intensity: Number flux originating from a given line of the uranium 

series and including all energies degraded from that line down to 0.4 Mev, and 

including the unscattered flux. This is the response of a non-energy-dependent 

detector per square foot of detector area.

Primary flux: Superposition of all the primary intensities of the uranium 

spectrum above 0.4 Mev energy in proper proportion and normalized to unit 

source strength.

Total flux: Superposition of all the total intensities. This is the response of a 

nonenergy-dependent detector with a lower energy acceptance at 0.4 Mev. The 

cutoff has been chosen at 0.4 Mev as the spectrum of the uranium series is unambiguously known down to 0.34 Mev and as the Compton effect predominates in 

this range so that published one-medium computation can be adapted to two 

media.

Table 2, constructed from the work of Mladjenovic and Hedgran 

(1954) and Mladjenovic and Slatis (1954) and confirmed by Lazar 

(written communication, 1955), gives the energy of the gamma rays 

above 0.34 Mev and their fractional abundances. These are the 

values used in computing the individual intensities and the fluxes. 

Tables 3 and 4 contain the computed total and primary intensities 

and fluxes which are normalized to a broad source emitting 1 photon 

per cubic foot per second and are plotted in figures 6, 7, and 8.

In figure 6 the primary, scattered, and total intensities are plotted 

to an arbitrary scale for initial energies of 0.609 and 2.432 Mev, the 

two extremes in energy found in the spectrum. The importance of 

the scattered components, particularly for deeper penetrations, can be 

readily seen from the graph. The futility of any analysis in which 

scattering is neglected is obvious.

In figure 7, the computed total and primary fluxes are compared 

with the experimental data normalized to the theoretical value at the 

500-foot altitude. The large deviation in slope between the curves 

labeled "data" and "primary" indicates the large contribution from 

degraded photons. The total flux and data agree well, even though 

the energy dependence of the detector has not been taken into account. 

The steeper slope of the computed curve relative to the experimental 

curve is to be expected because the Computation weights all energies 

equally, whereas scintillation detectors are more efficient in detecting 

the low energies and both primary and scattered and the fractional 

contribution of primary flux to the total flux decreases with increasing 

distance from the source. Absolute agreement of computed with 

experimental data cannot be demonstrated because the true spectral 

efficiency of the detector is not known. However, one can compute 

the overall efficiency at 500 feet. Under the assumptions of 2 gamma
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCE'S 13

?£fte

f-0.5

0.1

0.05

0.01

200 400 600 800 

ALTITUDE, IN FEET ABOVE SOURCF

1000

FIGUEE 6. Eelative values of primary, scattered, and total intensities for primary 

energies of 2.432 and 0.609 Mev from broad source.
14 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

0.5

o

R 0.1

0.

I

uj 0.05

0.01

0.005

0.001

200 400 600 800 

ALTITUDE, IN FEET ABOVE SOURCE

1000

PIGUEE 7. Computed total and primary fluxes from broad source of uranium emitting 

l photon per ft8 per second.
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 15

rays per disintegration of radium, a density of 2.3 for the source, a 

source concentration of 0.0013 percent eU3O8 , and a detector area of 

0.25 square feet, the expected counting rate is 597 counts per second 

for a 100 percent efficient detector. The experimental counting rate 

is 64 counts per second and the efficiency of the detector used is 

about 17 percent.

The decomposition of the flux and intensity from an elementary 

source of uranium are presented graphically in figures 8 through 11 

in order to show the relative contribution of both primary and scattered radiation. The total primary flux (fig. 8) from a broad source 

is decomposed into the contributions from each of the primary energies 

given in table 2. For an elementary source, the calculated flux and 

intensities, normalized to the emission of 1 photon per foot per second, 

are given in tables 5 and 6 and plotted in figures 9, 10, and 11. The 

relative primary, scattered, and total intensities from 0.609 and 2.432 

Mev primary gamma rays are presented in figure 9. The scattered 

intensity is again a significant proportion of the total intensity. A 

comparison of primary, total, and measured fluxes from an elementary 

source is given in figure 9, the source data being adjusted to agree 

with the computed total flux at 500 feet. The agreement in general 

behavior is excellent, the discrepancy being due to the equal weighting 

of energies in the computation. Finally, the decomposition of the 

total flux into contributions from each energy is shown in figure 11.

EFFECT OF TIME CONSTANT AND THE CONE OF RESPONSE

As computed on page 33-36, a continuously recording radiation 

detector in an aircraft flying at the 500-foot altitude at 220 feet per 

second directly over an elementary source will record 87 percent of 

the stationary value signal, whereas the same detector in an airciaft 

flying at the 100-foot altitude at 88 feet per second will record only 

76 percent of the stationary value. The lag in recording the peak 

response at the 500- and 100-foot altitudes will be 0.320 and 0.675 of 

the time constant, respectively. Also, it may be seen from equation 

(6), on page 4, that for low-flying aircraft, a decrease in the product 

of velocity and time constant and (or) an increase in the cone of response will give an increased response. For high-flying aircraft the 

product of velocity and time constant should be made as low as 

possible.

GAMMA-RAY INTENSITY FROM IDEALIZED SOURCES 

AND INTERPRETATION OF RADIOACTIVITY ANOMALIES

On pages 3-11 semiempirical expressions of general validity were 

developed for the gamma-radiation intensities at considerable air distances from the two extreme types of natural thick uranium sources,
16 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

0.5

Q

z

CO

cr

UJ 

Q.

o

O

o

CO

EC 

UJ 

OL

CO

z

UJ

0.1

0.05

0.01

0.005

- 0.001

0.0005

0.0001

200 400 600 800 

ALTITUDE, JN FEET ABOVE SOURCE

1000

FrouKE 8. Decomposition of total flux from broad source of uranium emitting 1 photon per ft* pet sec. 

into contributions from several lines of spectrum.
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 17

0.5

0.05

0.01

0.005

0.001

0.0005

0.0001

100 200 300 400 500 

ALTITUDE, IN FEET RELATIVE INTENSITY. ABOVE SOURCE

600 700

FIGURE 9. Kelative values of primary, scattered, and total intensities for primary energies of 2.432 and 

0.609 Mev from elementary source.

the elementary and broad souices. To interpret semiquantitatively 

the measurements made in airborne radioactivity surveying, quantitative relationships must be established between the characteristics 

of natural sources intermediate between the two extremes of natural 

source types, and the observable quantities recorded by a specific 

detector of the counting-rate-meter type, such as the peak intensity, 

the area under the curve, and the shape of the curve.

In this portion of the report, the radiation intensity and area under 

the counting rate meter curve for idealized sources are established, 

their relation to the characteristics of the source is discussed, and a 

method of interpreting anomalies is given.
18 EXPERIMENTAL AND THEORETICAL GEOPHYSICS 

10

5 x icr6

icr6

5 X 10" 7

10-'

5 X 10-«

i x icr8

100 200 300 400 500

AITITUDE, IN FEET ABOVE SOURCE

600 700

FIGTJBE 10. Computed total and primary fluxes from elementary source of uranium emitting 1 photon per

foot per second.

The unknown factors that enter into the interpretation of an 

anomaly are the source position relative to the detector, the surface 

dimension of the source, and the source strength in terms of equivalent uranium content. As the usual practice in airborne radioactivity 

surveying is to make continuously recorded measurements along 

equally spaced, constant-altitude flight lines, the location of an 

anomaly along the direction of flight can be determined with rela-
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 19

icr5

5 X 10"6

5 X 10-'°

10

100 200 300 400 500 

ALTITUDE, IN FEET ABOVE SOURCE

600 700

FIGTJKE 11. Decomposition of total flux from elementary source of uranium emitting 1 photon per foot per 

second into contributions from several lines of spectrum.

tively small error. The location of an anomalous source is thus reduced to one unknown, the distance along the line normal to the 

flight path which passes through the peak position of the anomaly. 

If either linear or circular symmetry is assumed, the surface dimensions of the source are reduced to one variable, either the breadth of 

a slab source or the radius of a finite (circular) source.

428734 57 4
20 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

The three unknowns for any given source must be related to the 

observable data. The most obvious is the peak intensity; another is 

the area under the curve. A third is the shape of the curve, characterized by the slope or by the width at half-maximum intensity.

Computation of the peak intensities and areas under the curves 

for selected sources are made and criteria for distinguishing the different source types are established. Then the observable data are 

expressed directly in terms of source characteristics, eliminating explicit reference to the (unknown) distance between the source and the 

detector. Moreover, as natural sources vary greatly in grade, some 

grade-independent quantities readily computable from observable 

data should be used to determine salient characteristics of the sources.

SELECTED IDEALIZED SOURCES

The general behavior of the radiation intensities from natural 

sources is best discussed in terms of a few sources selected to approximate most nearly natural configurations. These sources are all assumed to be infinite in depth. Because of the absorption of radiation within the source, a thickness of 1 foot may be considered 

infinite.

Elementary source. An elementary source is one whose areal extent is sufficiently small so that the radiation anomaly produced by 

it is identical to that produced by a theoretical point. The intensity, IP (x,p,a), irom an elementary source (fig. 1, sec. /), is a function of x, the altitude of the detector above the source plane; p, the 

source-plane projection of the air distance between the source center 

and the detector; and a, the radius of the source. The area under 

the curve recorded by a count-rate meter, Hp (x,z,a] is a function 

of previously defined x and a, and z, the source-plane projection of 

the air distance of nearest approach between the detector and source 

center.

Broad source. A broad source is so large in areal extent that the 

intensities over the central region remain constant no matter how 

much additional source material is added to its boundary. The intensity, /&(#), from a broad source (fig. 2), is a fuiction of x, the 

altitude above the source plane. The area under the curve is infinite.

Line source. A line source is very large in length yet so narrow in 

width that the radiation anomaly produced by it is identical to that 

of a theoretical "line" source, one whose width is zero. The intensity, Ii(x,y,a), from a line source which is infinite in length, is a 

function of x, the altitude of the detector above the source plane; y, 

the source-plane projection of the air distance from the detector to 

the source center line; and a, the width of the source. The area under
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCE'S 21

the curve recorded by a counting-rate meter, Hi(x,Q,a) is a function of 6, the angle between the flight line and the plane normal to 

the axis of the source, as well as of x and a. These variables are 

depicted in the insert to figure 14.

Slab source. A slab source is so large in length that the radiation 

anomaly produced by it is a function of its width and independent 

of the length. The intensity, Is (x,y,a), and the area under the 

curve recorded by a counting-rate meter, Hs (x,Q,a) are functions 

of the same variables as for the line source. The limiting value of 

the slab source as a >0 is the line source.

Finite source. A finite source has a circular area of radius, a, which 

is intermediate in value to the elementary and broad source. The 

intensity, If(x,p,a), and the area under the curve recorded by a 

counting-rate meter, Hf(x,z,a), from a finite (circular) source of radius, 

a, are functions of the same variables as for the elementary source. 

The limiting case of the finite source as a >0 is the elementary source.

The above enumerated quantities are derived on pages 41-43. 

All numerical computations were performed under the assumption 

of quarter-mile flight-line spacing and a 500-foot flight altitude.

It is shown in equation (9), page 59 that for a particular flight 

level, the intensity of a finite source, //, is equal to a numerical constant times the function Gs (x,p,a). The .quantity 6/3 (500,p,a) is 

tabulated in table 7 and plotted (fig. 12) for values of a against p. 

This plot of 6/3 thus yields the relative intensities at several distances 

along a flight line at 500 feet above a finite source of fixed grade. 

The relative intensities plotted are equivalent to intensities that would 

be measured by a scintillation detector of short time constant in a plane 

flying at 500 feet above the ground along the flight line as shown in 

the insert to figure 12. The striking features are the rapid rise of the 

peak intensity with increasing radius of the source and slow increase 

in width of the anomaly with increasing radius. The minor variations in the slopes of the curves with major changes in radius militate 

against the determination of source size by slope analysis. The 

source 40 feet square by Q inches thick at Grand Junction, Colo., 

used to establish the semiempirical equation for an elementary-source 

intensity, would give rise to too low an intensity to be depicted in 

this figure.

Similarly, it is shown in equation (8), page 42 that for a particular 

flight level, the intensity of a slab source, Is, is equal to-a numerical 

constant times the function 61 (x,y,a). The quantity Gi(500,y,a) is 

listed in table 8 and plotted in figure 13 for various values of a against 

y, yielding the relative intensities at various distances, y, along the 

flight line at 500 feet above slab sources of various sizes and of the
22 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

-4000 -3000 -2000 -1000 0 1000 

GROUND DISTANCE, p, IN FEET

2000 3000 4000

FIGURE 12. Relative intensities at several distances from center of finite sources of several radii.
RELATIVE INTENSITY 

b
24 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

same grade. The curves are equivalent to profiles of anomalies 

that would be obtained under normal airborne surveying conditions. 

Again, slope analysis to determine source size is not feasible.

In figure 14, the relative intensities at 500 feet above a slab source 

400 feet in width are plotted as a function of y'', the ground projection 

of the air distance along the flight line from the detector to the center 

line of the source, where the individual curves are plotted for different 

angles of approach between the flight line and the center line of the 

source. The effect of oblique angles of approach completely invalidates any attempts at slope analysis.

-4000 -3000 -2000 -1000 0 1000 

DISTANCE,y', IN FEET

2000 3000 4000

FIGURE 14. Eelative intensities at several distances from centerline of a slab source 400 feet wide at different

angles of approach.

The impossibility of slope analysis is further illustrated in figure 

15, where the relative intensities from finite sources of 400- and 500- 

foot radius are plotted against y, the distance along the flight line at 

500 feet above the source plane, where the closest distance from the 

flight line to the source center is given by its ground projection, z} as 

shown in the insert. These curves can also be considered as identical 

with anomalies that would be actually recorded on these flight lines. 

The shapes are strongly dependent on the distance of the flight line 

from the source center.

Examination of figure 16 reveals another difficulty in utilizing the 

shape of anomalies to determine their sizes. One cannot, in general, 

say that the point where the intensity drops to a certain fraction of
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 25

the peak intensity is the edge of the source, for the intensity ratio 

at the edge to that of the center is one for zero radius (width) and 

asymptotically approaches one-half for an infinite radius (width). 

The limiting values are easy to understand: for zero radius (width), 

the edge and the center are the same point; whereas for the infinite 

radius (width), the point at the edge corresponds to slicing away 

exactly one-half of a broad source. Because of the rapid variation

1.5

1.0

0.5

Flight line

Observation point

Plan view: detector at 500 ft 

above source plane

-2000 -1000 0 1000 

DISTANCE,y, IN FEET

2000 - 2000 - 1000 0 1000 

DISTANCE, y, IN FEET

2000

FIGURE 15. Relative intensities at several distances along flight lines at different distances from center of 

finite sources of 400- and 500-foot radius.

of the ratio at small source dimensions, boundary identification by 

this method is impossible, but for broad sources, the one-half intensity 

point yields the source boundary.

In figure 17, curves A and B answer the question, "How infinite is 

infinity?" or what is the upper limit of the size of a finite or slab source 

required so that for a point centrally located 500 feet above the source, 

the intensity is essentially equivalent to that from an infinite broad
26 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

1000 2000 

WIDTH (RADIUS), IN FEET

FIGUEE 16. Ratio of intensity at edge to intensity at center of source.

3000
SCATTERED GAMMA RAYS FROM THICK URAOTUM SOURCES 27

source. Curves A and B show (as a function of increasing radius or 

width) the ratio of the intensities from a finite and slab source respectively to the intensity of an infinite broad source. If it is assumed 

that for practical purposes 80 percent of the intensity from an infinite 

broad source would be indistinguishable from the intensity (or behavior) of an infinite broad source, then a finite or a slab source with 

a radius greater than 850 feet or a width greater than 1,200 feet respectively could be considered an infinite source. Curves A and B are

o.os

o.oi

100 500 1000 10,000

WIDTH (RADIUS), IN FEET

FIGURE 17. Approach of intensity ratios to limiting value.

plots of expressions in equations (9) and (8) at p and y set equal to 

zero, both divided by expression in equation (2), where x in all cases 

is 500 feet (see equations (1), (8), (9), pages 41-42); thus, the curves 

are applicable only to measurements made at 500 feet above the 

source plane.

Curve C in figure 17 shows the ratio of the intensity from a finite 

source to that from an elementary source and thus indicates the 

lower limit of size of a finite source where the finite source intensity
28 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

would be essentially identical to that from an elementary source. 

If it is assumed for practical purposes that 80 percent of the intensity 

from an elementary source would be indistinguishable from the 

intensity (or behavior) of an elementary source, then a finite source 

with a radius less than 275 feet can be considered an elementary 

source. Curve C is a plot of expression in equation (9) divided by 

expression in equation (1), at p set equal to zero and x set at 500 feet 

(see equation (1) and (9), p. 41-42).

Curve D in figure 17 shows the ratio of the intensity from a slab 

source to that from a line source and indicates the lower limit of size 

where the slab source intensity would be essentially identical to that 

from a line source. If it is assumed for practical purposes that 

80 percent of the intensity from a line source would be indistinguishable from the intensity (or behavior) of a line source, then a slab 

source with a width less than 800 feet can be considered a line source. 

Curve D is a plot of expression in equation (8) divided by expression 

in equation (4), where y is zero and x is 500 feet (see equations (4) 

and (8), p. 41-42).

For the specific radiation detection equipment currently used by 

the U, S. Geological Survey, the minimum detectable increase in 

counting rate is about 50 counts per second, equivalent to twice the 

standard deviation of measurement. Figure 18 shows the grade 

(ellsOs) of finite sources of radii that would be detectable at a 500- 

foot altitude of flight lines directly over, 600 feet away, and 1,300 

feet away from the source center. Curve A gives the most unfavorable case in which the flight line is at a horizontal distance of 1,300 feet 

from the source center. Even then, an ore-grade body with more 

than 0.1 percent equivalent uranium and a minimum radius of 200 

feet is detectable on two adjacent flight lines spaced at 1,300 feet 

(about %-mile) apart. Curves B and C are for the flight lines 600 

and 0 feet away, and give correspondingly lower radii. Taking 

the 600-foot distance to be the most likely case, an ore grade body 

with a minimum radius of 600 feet located midway between flight lines 

spaced at 1,300 feet can be detected.

RELATIONSHIP OF OBSERVABLE DATA TO SOURCE 

CHARACTERISTICS

The general behavior of the radiation intensities from natural 

sources discussed above indicates that the useful observable quantities 

are the peak intensity / and the area under the curve H. The 

third readily apparent observable quantity, the shape of an anomaly 

as characterized by the slope or by the width at half-maximum 

intensity, has been demonstrated to be of little value for interpretation 

of source characteristics.
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 29

RADIUS, a. IN FEET

FIGURE 18. Grade necessary to produce 50 counts per second for several radii of finite sources on different

flight lines.

A plot (fig. 19) of Gt(5QQ,2,a) of table 9 against G3 (5QQ,z,a) of table 

1 corresponds to a plot of arbitrarily scaled Hf(5QQ,z,aj against 

arbitrarily scaled If(5QQ,p,a) at the points where p equals z, the 

ground projection of the air distance of closest approach to the 

source center. This can be seen by dividing the expression in equation (9), (p. 42) by the expression in equation (7), (p. 43) to form 

Gz/G^=2lf/Hf. It is obvious that these plots are of little practical 

use as the scaling factors for real anomalies are unknown. However, 

by considering the graph, the following relationship is suggested.

(?4(500,2,o) = .F(a)[(?3(500,z,a) CD

where F(a) is a function of a. Now for the case of small a, the finite 

source reduces to an elementary source, and from equations (1) and 

(9), (p. 41-42) and equations (3) and (7) (p. 42-43), the functions 

#3 (500,2,a) and 6r4 (500,2,a) can be readily computed for small a.
30 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

100

10-*

10-3 10-2 10-1 

G 3 (500.z.a)

10

FIGTJEE 19. Scaled area under curve <?4(SOO,z,o) vs. scaled peak intensity 6»(500,z,a) of finite sources, of

several radii (a).

Then, assuming the form of equation (1), and fitting the equation 

to the computed values, by the method of least squares, it is found 

that

<W600,.,«> =2.18

/ a \ -210.

fiOCUa) (^) ] (2)

upon application of equations (9) (p. 42) and (7) (p. 43) ; the above 

equation becomes

4CS

g,(500.^) 7/(500,3,a) I0 - 788 (3)
SCATTERED GAMMA RAYS FROM THICK URAOTUM SOURCES 31

Thus, the grade-area, Sira2 SA,

CM _rfl>-40500ir"| r4.36{//(500,2,a) i»-788"|~a2i2 ,,. 

I 2CMo JL H/(500,«,o) J W

Thus, two transcendental equations, (9) (p. 42), and (7) (p. 43), 

in two unknowns, z and SA, were solved empirically to give explicit 

equation for grade-area for any value of 2, in terms of the two observed 

quantities, Hf(5QQ,z,a) and 7X500,2,0-). Equation (4) is true for 

elementary sources and approximately true for finite sources.

From tabulated values of ^(500, z, a) and 6^(500, z, a) and from 

equations (9) (p. 42), and (7) (p. 43), insertion into equation (4) 

yields incorrect grade-area for a finite source. The ratio of this 

incorrect value to the true value of grade-area is listed in table 10, 

as a function of 2 and a. One notes that the range in values is considerable. However, for any fixed a, the arithmetic mean of the gradearea ratios calculated at 2 and at .1300-2 (average of ratios over two 

flight lines straddling the source center) is reasonably constant as 

shown in table 11. The arithmetic mean at a fixed radius of the 

reciprocals of the values in table 6 is taken, and these average values 

are listed in table 12 along with maximum and minimum values of 

the reciprocals at a fixed a. Thus, if the approximate surface dimension of the source were known, application of equation (4) followed 

by multiplication by the proper correction factor in table 12 should 

yield the correct grade-area of finite sources. The tabulated extreme 

values in table 12 indicates that the error in source location entails 

only 15 percent error in the correction factor.

It remains now to determine the radius of the finite source. A

useful grade independent quantity is T f,Knn ' \ which is tabulated

in table 13. In figure 20, the data in table 13 at z is plotted against 

the quantity at 1300-s for values of a, a comparison of data from two 

adjacent flight lines straddling the source. The solid curve represents a given source radius; the dotted curves connect constant 

ground distances from the nearer flight line to the source z. To get 

grade-area, one can get the radius of the source and closest distance 

of approach from this plot and apply a correction from the proper 

entry in table 10 to the computed value from equation (4) . However, 

a 10 percent error in H and / would result in an error of the ratio 

K/I, represented by the grid length in figure 20. This could result 

in a possible error in 2 by as much as 600 feet and an error in a by 

200 feet. The variation with distance, z, of predicted to true gradearea ratios at a given radius is quite serious, as an examination of 

table 10 will show. Thus, it is better to estimate by figure 20 what
32 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

Hf' (500.1300 -z.a) 

I,1 (500,1300-z,a)

. Ratio of area under curve (Hi) to peak intensity (It) at z vs. ratio of area under curve (Hf to 

peak intensity (If) at 1300-z for finite source ot several radii.
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 33

the radius is, to compute the grade-area on two adjoining lines.through, 

equation (4), and to apply the proper correction factor in table 12. 

With grade-area and radius known, it would be easy to find the grade.

For slab sources table 14 furnishes the quantity, r g,:. nn ' \

H J ' Is (500,o,a)

which is plotted in figure 21 as function of the slab width, a. These 

grade-independent quantities can be used to estimate the width of 

the slab source. Once the width is determined, the grade can be 

determined from equation (8) (p. 42) with y set equal to zero.

INTERPRETATION OF ANOMALIES

Based on the foregoing discussions, the following method suggests 

itself. H and / represent respectively the measured area under a 

curve recorded on a counting rate-meter and the peak intensity reading. The procedure would be:

1. // the anomaly shows distinctive flattening at the peak. The 

source can then be considered a broad source, and the grade computed from equation (2) (p. 41). The boundary of the source is at 

the one-half peak intensity point.

2. // the anomaly is visible on one flight line. As an ore-grade body 

greater than 200 feet in radius would be visible on two lines and as 

any finite source of radius smaller than 275 feet can be considered 

an elementary source, application of equation (4) p. 41, yields the 

grade-area.

3. // the anomaly is visible on two lines. Two cases are possible, 

namely:

a. If the anomalies on two adjacent lines are approximately equal, 

it is a slab source. Compute .Hs(500,o, a) from the relation, Hs- 

(500,0,2) =H cos 0. The angle, 9, is determined from the relative 

displacement of the peaks on the two adjacent lines. Then from 

figure 21 and from the computed Hs (5QQ,o,a)/I, determine a. From 

equation (8), p. 42, with y set equal to zero and the measured /, determine grade-width and thence, the grade.

b. If the anomalies are not equal in amplitude, it is a finite source. 

Let subscripts 1 and 2 denote two adjacent flight lines with the highest 

peaks, and /i>/2 . Then with Hi/Ii as ordinate and Hz/Iz as abscissa, 

determine a from figure 20. Compute SA on lines 1 and 2 and average 

the results. Multiply the average by the factor hi table 12 corresponding to the proper a. The result is the grade-area from which

grade can be obtained.

CONCLUSIONS

Comparison of experimental data with theoretical calculation 

shows fair agreement for both the broad and the elementary sources, 

outside a multiplicative constant. Thus, the effect of the interface
34 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

12

10

500 1000 1500 

WIDTH, a, IN FEET

2000 2500

FIGUBE 21. Ratio of area under curve (H,) to peak intensity (/.) vs. width of a slab

source at 9=0.
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 35

and the effect of the spectral energy responses of different detectors 

can be approximately taken into account by merely changing a 

multiplicative constant. This result can be attributed to the fact 

that radiation emerging from thick sources possesses an equilibrium 

spectrum that does not change appreciably in traversing a mean free 

path of air.

It should be pointed out that the data of tables 3 and 4 can be 

adjusted for altitudes above sea level other than that at Grand Junction, Colo., by using the "appropriate /*, a function of air density, to 

correct tabulated values for different air distances from the source. 

The values in tables 5 and 6 can be similarly adjusted after multiplying by x2 because in this case r=x. The density of the source 

was taken to be 2.3; for other densities, corrections should be applied 

inversely as the densities. »s*fj

Certain general concepts underlying the semiempirical approach 

to airborne radioactivity surveying warrant attention. These are:

1. Use of an elementary source of finite thickness instead of 

the classical point source. In calculating the intensity of a theoretical 

point source at an interface, it is necessary to take into account 

scattering and absorption and the effect of the interface. This 

requirement is circumvented by observing the radiation intensity 

from an elementary source of infinite thickness, because the measurements themselves already contain the effects of both scattering and 

absorption and of the interface.

2. Use of the form and structure predicted by theory for the 

analytical expressions of the intensities to insure that correction 

for conditions differing from the ones prevailing during experimental 

measurements can be made unambiguously.

3. Use of elementary- and broad-source data to determine the 

constants in the elementary-source intensity expression.

4. The investigation, numerically and analytically, of the relationship between particular observable quantities and characteristics 

of sources selected to approximate natural sources.

The two observable quantities useful for determining grade-area 

relationship are the peak intensity and the area under the curve. 

The shape of the anomaly such as the breadth or slope at half-maximum intensity, cannot be used for interpreting grade-area and (or) 

grade of natural sources although shape is useful in determining 

source type.

This particular method of utilizing the peak intensity and the area 

under the curve, for the interpretation of natural source characteristics 

is not unique. For example, once the source types have been identified the grade might be computed on the assumptions that the source
-36 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

is either directly below the detector or at the point midway between 

two adjacent flight lines, yielding the two extreme values of grade. 

Based on investigation, numerical and analytical, of the relationship among the observable quantities and natural source characteristics, the following sequence is established for interpretation of actual 

anomalies:

1. Identification of one of five source types the elementary, 

the finite, the slab, the line, and the broad source by appearance 

of an anomaly, either on one or on two or more adjacent flight lines, 

by the relative magnitudes of the anomaly on adjacent flight lines, 

or by flattening of the peak of the anomaly.

2. Use of grade-independent observable quantities to determine 

source size.

3. Determination of grade (grade-area and grade-breadth) of 

sources without reference to the (unknown) distance from source to 

detector.

The following results have been established for the conditions of 

surveying used by the U. S. Geological Survey.

1. Broad sources are finite sources that are greater than 850 feet 

in radius and slab sources that are greater than 1,200 feet in width.

2. Elementary sources are finite sources less than 275 feet in 

radius.

3. Line sources are slab sources less than 800 feet in width.

4. Line and slab sources greater than 1,700 feet in length are 

infinite in length.

In application of equation (4), the total fractional error in the 

determination of the grade-area, 0 A , is related to the errors in

TT 1 T -L O-& H and / by

< 47o^4.o 7o^_o^ S4.7o-^FrTO./o F=o.O0 SA

where b is the fractional error in measurement assumed to be the 

same for both H and /. Thus, in order to obtain the grade-area within 

a factor of two, the error in measurement must be less than 12 percent. 

Moreover, to obtain the grade within a factor of two, the sum of the 

fractional error in grade-area and twice the fractional error in a 

must be less than 1. Particularly, in the case of overlapping anomalies, the uncertainty of resolving a compound anomaly into component anomalies will introduce very large errors in both H and / 

for the component anomalies.

The method of interpretation of the grade of broad sources is 

.highly accurate as shown by many field investigations (R. M. Mox-
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 37

ham, oral communication). Interpretation of the grade-width and 

grade of slab or line sources is also satisfactory. Finite and elementary sources, particularly those visible on only one flight line, 

are difficult to interpret; insufficient data are available to evaluate 

the accuracy of interpretation.

COMPUTATION OF INTENSITIES

The computed values (Goldstein and Wilkins, 1954) are the 

quantities 47rr2 /0 (ju/,£',-E'i) where /oG^-E^-) are the scattered 

energy flux at energy, E, distance, r, from the source, originating 

from a point source of energy E{ and strength 1 photon per second, 

for several media, ju* is the linear attenuation coefficient. Of the 

various media in Goldstein's tabulation, air is most nearly approximated by the Compton scatterer.

The quantity

)= fEi

t/0.4

(1)

was computed for energies .2^ 0.5, 1.0, 2.0, 3.0, and 4.0 Mev, and 

ju fr 1, 2, 7, 10, 15, and 20. The trapezoidal rule was used except 

at the end points where linear interpolation was followed by direct 

integration of the interpolant. Then quadratic interpolation was 

performed over these to find B(ntr,Et) for the 11 primary energies 

of the uranium series. The result was 77 values of B(^ tr,E^.

As the solution of multi-media Boltzmann equation of pure Compton scatterer in one distance variable can be cast into one-medium 

form, an exact solution can be obtained for the broad source by computing as though the media were homogeneous

(2) i)y ^fjr I

p pi*y (*

=V-r

2pi" Jo

»,

 

o

=e ^x r m /*

2Ui* JO M + Mi"'

For the case of an elementary source, such an equality does not 

exist, and we assume that the two-media elementary-source expression

is given as -^^(^(r x^-^-^x) wherein the scattered intensity at point a; 

directly over head is
38 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

=e~^ *< Jo" OH^ *(«+« *,*<) d« (3)

which is not exact. The integrations in either case are performed by 

the method of Gauss wherein

( <

f

JO

where LB (a;fc)=0, a^'s being then roots of Laguerre Polynomial of

order n.

Thus

(5)

*-- (6)

The B's were plotted on log-log papers and read off the graph for 5 

values of the argument for each x point and energy value. 

The primary intensity expressions can be readily found.

6 7*- dv < 7 >

Equations (7) and (8) multiplied by the relative abundance of table 2 

are listed in tables 4 and 6 under their respective energy headings 

for several values of x. The final column is the sum of all these 

components.

The sum of equations (7) and (5) and the sum of equations (6) 

and (8) multiplied by the relative abundances are found in tables 3 

and 5 under respective energy headings. The final column again 

represents the sum.

The superscripts a and s refer to air and source respectively.

CALCULATION OF DETECTOR RESPONSE

The relative response of a counting-rate meter at time t to a elementary source of intensity / is

«-ft~l/) I«/ )*' (l)
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 39

where time is expressed in units of the time constant, T, of the instrument, tQ is the time the source first falls within the cone of response 

of the detector, and the origin of time is chosen as the moment of 

closest approach. Expansion about t=Q exists for this integral, i. e.,

5 < 2 >

where the derivatives satisfy the relationship

Jt=0

(4)

where:

a altitude in mean free paths

z ground projection of distance of closest approach 

^=distance traveled in one time constant in mean free paths 

_x2 tan2 a z2

<*.=% angle of the cone of response 

By partial integration

d? \dtf ) \ dt'

(5)

Because the result of the partial integration is a semi-convergent 

series, within the range of parameters pertinent to airborne surveyhlg 

the minimum error is incurred by evaluating only the first term at 

the upper limit, and both terms at the lower limit*

(6)

where

ZQ =X sec a

Then the ratio of the response to the stationary value at the point of 

closest approach is

R(t) _A
40 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

where

(8)

(9)

(10)

(12)

The maximum is found from the equation

a - (13)

If we retain only two terms, we find

fe=i (14) 

the same result as found by Peirson (1951). If we retain three terms

= -2^i As 

13 2

If we retain four terms, we can expand the cubic about ia to a linear 

term and find

Then the insertion of £4 into equation (7) yields the ratio of peak 

response to the stationary value. This is carried out for

«=45°, z 0, and r=l second for two typical cases:

x= 0.730 (500 feet) *= 0.146 (100 feet)

0= 0.329 (220 fps) 0=0.128 (88 fps velocity)

<2=1 fc=l

is= 0.323 ts= 0.635

^=0.320 «4

It should be noted that due to the nature of the approximation the 

result can be considered the lower limit of the peak response. To 

recapitulate, the ratio of peak response to stationary value is
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 41

I M_ J__\_!_J_ \__ (17)

\ ' j\') /*>: \ r f(r) /4!

where

r /(r)

»

INTENSITY FROM SELECTED SOURCES

It was shown previously in equation (6), p. 10 that the intensity 

from an elementary source is

where

^ = source grade in percent ell

/So=source grade in percent eU standard 0.35 percent

A = area of source=7ra2

of standard source 1600 ft. 2

W=1.461X10-3 ft.-1 at sea level

7 = 0.342

C-3.19X107

and from a broad source is 

we rewrite

^ ^ /0 "~ ior) (3)

where K\I2 is modified Bessel function of the second kind, order 1/2. 

The line source intensity is

\ <k/ (#) =-0-7- as/2 (w) (4)

J co OQ.AO

where
42 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

It is known (Watson 1948) tl at

{ m Kv [pi/W+tf] [*2+z2]-f dz = J^- K,-H (pi)l^~x (5)

JO T *p

. , , , 2C/S , TXiOwo) i \~\ f7\ S.Ii (x,y,a)=-~-r- juo2 ax\ ^^-^- 7^0(^0") (7) OoAo L Mo" J

The slab source intensity is

(x,y,a) (8) -a/2

where

I (500,7/,a) is tabulated in table 8. 

The finite source intensity is

=--- x \ df \ydy=-x-r MxGz (x,P,a) (9) JO JO oo-d-o

where

2_2p2/ cos f +y2 

G3 (500,p,a) is tabulated in table 7.

AREA UNDER COUNTING-RATE-METER CURVE

The response, R(t), of a counting- rate meter of response time at 

time t to a time dependent signal I(t) is

i rt (*-<')

- e I(t'W (1)

f Tj-oo

Then the area under curve, H, is

H= (+t°R(t)dt= f" J(«')d«' (2)

J 00 J 00

For the elementary source

Hv(x,z,a}= f" I9 M*+iW, a)dt (3)

J-oo
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCE'S 43

where sz=x2 -\-z2 and v is the velocity of the detector. 

For the line source

Hi (z,M)= f " Ii (z, vt' cos e, a) d«'=4^r w>2 ax f " [Kl -yK0 ( wR) dt'

J-co 00A0 JO L M0# J

(4)

where 52=a;2+ w2^2 cos2 0. Thus,

H, (x, 6, a) = ~ -; e-V (Z-<yMo*) = M G, (*,*,«) (5) oo-Ao \t> cos 0/ oo-Ao

For the slab source

J

-f-eo 4 p.c

7S (z, »*' cos e, a) dt'=^f- (MOX) G2 (M,a) (6) -co OO.AO

identical to the line source. 

For the finite source

J

4-co / 4. r«s

If (x, ^z*+v*t' 2, a) dt' =!^ (MOX) G4 (z,«,o) (7)

-co 00-Ao

The function #4 (500,s,a) is tabulated in table 9.

LITERATURE CITED

Cook, J. C., 1952, An analysis of airborne surveying for surface radioactivity:

Geophysics, v. 17, no. 3, p. 607-706. 

Fano, U., 1953a, Penetration of X- and gamma-rays to extremely great depth:

U. S. Nat. Bur. Standards Jour. Research, v. 51, no. 2, p. 95-121. 

 1953b, Gamma-ray attenuation: Nucleonics, v. 11, no. 8, p. 8-12; no. 9,

p. 55-61. 

Goldstein, Herbert and Wilkins, J. Ernest, 1954, Calculations of the penetration

of gamma-rays, final report: U. S. Atomic Energy Comm. NYO-3075,

issued by U. S. Atomic Energy Comm. Tech. Inf. Service, Oak Ridge, Tenn. 

Mladjenovic, M. and Hedgran, Arne, 1954, An investigation of the Compton

secondaries from RaC gamma rays: Arkiv for Fysik, Band. 8, Hafte 4,

p. 49-63. 

Mladjenovic, M. and Slatis, H., 1954, Internal conversion spectrum of the active

deposit of radon: Arkiv for Fysik, Band 8, Hafte 4, p. 65-82. 

Peirson, D. H. and Franklin, E., 1951, Aerial prospecting for radioactive minerals:

British Jour. Applied Physics, v. 2, p. 281-291. 

Spencer, L. V. and Fano, U., 1951, Penetration and diffusion of X-rays. Calculation of spatial distribution by polynomial expansion: U. S. Natl. Bur.

Standards Jour. Research, v. 46, no. 2, p. 446-456. 

Stead, F. W., 1956, Instruments and techniques for measuring radioactivity in

the field, in Page, L. R, Stocking, H. E., Smith, H. B., (compilers), Contributions to the geology of uranium and thorium by the United States Geological Survey and Atomic Energy Commission * * *, p. 705-713. 

Steljes, F., Cowper, G., and Carmichael, H., 1952, Aerial prospecting for radioactive material: Canada Natl. Research Council, Rept. no. CRR-495. 

Watson, G. N., 1948, A treatise on the theory of Bessel functions: 2d ed., Cambridge University Press, 417 p.
44 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

TABLE 1. Elementary-source data and associated computations

Air distance 

r (feet)

Altitude 

x (feet)

560

683

601.3

526

605

566

552

581

485

462

480

455

471

622

619

577

586

543

485

516

472

364

402

364

387

389

472

476

449

442

506

490

405

416

417

453

430

Experimental 

intensity 

(counts per 

second)

Experimental 

intensity O (average)

600 feet nominal altitude, 375 feet offside

470

530

510

520

45±5

42.5±2.5

40

57.5±12.5

53.6±6.0 

50.2±3.0 

45.8 

66.5±14.5

2.21X10? 

2.74 

2.09 

3.27

(2.57±0.42)X10?

500 feet nominal altitude, 375 feet offside

60 77.9 2.748 (3.36±0.70) X10'

440 55±5 75.6±6.9 3.768

440 55±5 70.8±6.4 2.996

420 90 118 4.696

455 45±5 57.4±6.4 2.584

400 feet nominal altitude, 375 feet offside

315 45±5 69.3±7.7

300 82.5±2.5 127 ±4

315 65±5 99 ±7.6

320 70 99.6

285 87.5±2.5 145 ±4

600 feet nominal altitude, 250 feet offside

540 70 ±10 75.2 3.38

570 30 32.8 1.76 (3.40±0.70)X10'

580 77.5±7.5 82.6±10.6 4.35

520 67.5±2.5 74.9± 2.8 3.32

530 82.5±2.5 91.2±2.8 4.20

500 feet nominal altitude, 250 feet offside

490 62.5±2.5 69.3±2.8 2.650

425 60 68.4 1.987 (2.59±0.34)X10'

480 65±5 69.8±5.4 2.34

415 103±3 117 ±3 3.18

415 95±5 106 ±6 2.80

400 feet nominal altitude, 250 feet offside

285 148±3 189 ±3

315 125±5 160 ±6

285 106±5 134 ±6

320 125±5 151 ±6

315 103±3 127 ±9

600 feet nominal altitude, 125 feet offside

465 120 122 3.32

470 130 132 3.66 (3.09±0.10)X107

445 120 121 2.94

440 143 143 

500 83±3 83.4±2.4

490 87.5±2.5 87.5±2.5 2.61

500 feet nominal altitude, 125 feet offside

395 173±3 177 ±3

410 123±18 124 ±18

415 145±5 146 ±5

445 100 102

425 130 132
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 45 

TABLE 1. Elementary-source data and associated computations Continued

Air distance Altitude 

r (feet) x (feet)

Experimental 

intensity 

(counts per 

second)

Experimental 

intensity

"CO

O C (average)

400 feet nominal altitude, 125 feet offside

327 

311 1 

332] 

285. 

336J

320 

300 

325 

275 

325

143±43 

273±3 

180 

350 

203±8

146±43 

283 

184 

363 

209±8

600 feet nominal altitude, C feet offside

473J 

486 

515 

550 

536

470 

485 

515 

550 

535

155±6 

175±20 

100±7 

110±10 

96. 7±8. 9

156±6 

175±20 

100±7 

110±10 

96. 9±8. 9

4.27 

5.135 

3.37 

4.34 

3.58

(4.14±0.53)X10'

500 feet nominal altitude, 0 feet offside

470 

461 

460 

460 

446

470 

460 

460 

460 

445

133±9 

140±7 

148±2 

118±8 

163±2

133±9 

140±7 

148±2 

118±8 

164±2

400 feet nominal altitude, C feet offside

325 

310 

325 

320 

290

325 

310 

325 

320 

290

272±6 

310±7 

282±6 

255±3 

332±8

272±6 

310±7 

282±6 

255±3 

332±8

300 feet nominal altitude, 0 feet offside

260 

260 

260 

260 

245

260 

260 

260 

260 

245

427±6 

437±6 

433±10 

460±7 

490±7

427±6 

437±6 

433±10

460±7 

491±7

200 feet nominal altitude, 0 feet offside

140 

140 

160 

145 

145

140 

140 

160 

145 

145

1400 

1480 

1350 

1620 

1550

1400 

1480 

1350 

1620 

1550

100 feet nominal altitude. 0 feet offside

1111 

861

87 j 

85! 

85J

110

85 

85 

85 

85

2810±260 

3930±10 

3650 

4000 

3940±10

2840±260 

3990±10 

3730 

4020 

3960±10

TABLE 2. Primary gamma rays of uranium series

Energy (Mev)

0. 609

. 769

. 934

1. 120

1. 238

1. 378

Abundance

0.268

.0579

.0282

. 204

. 0741

.0869

Energy (Mev)

1. 509

1. 764

1. 848

2. 204

2. 432

Abundance

0. 0292

. 164

.0172

. 0522

. 0188
46 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

 

OcO

cooo

5J§06N OOgJjHOOtD rH

588 §8888 g

CO i l t*- 00 <£ D»C

ioas i-H 

»-HI it irno o

._,^ ^P«S»

jtDC<3<_ _. _ .

3SSS8 8J

ea -rh

OCOCDT-HM r-1 00 OO CO

i§ ill

0§I

'C

i l<P<£rt~ ti-C«OO cowooOi-f
TABLE 5. Total flux and total intensities of various components from uranium eleme

Altitude (feet)

100 . .

200 . 

300 

400 

500 

600 _ .

700 

Energy (Mev)

0.609 0.769 0.934 1.120 1.238 1.378 1.509 

Total intensity (photons per square foot pe

6.58x10-' 

1.35 

.503 

.144 

.113 

.0683 

'.0401

1.37x10-' 

.294 

.114 

.0553 

.0304 

.0186 

.0114

8. 05x10-8 

1.76 

.714 

.357 

.202 

.124 

.0796

6. 80x10-' 

1.54 

.617 

.352 

.177 

.110 

.0717

2. 62x10-' 

.597 

.241 

.122 

.0697 

.0441 

.0286

3.31x10-' 

.756 

.306 

.157 

.0911 

.0584 

.0373

1. 19x10

.271 

.113 

.0567 

.0327 

.0207 

.0139

TABLE 6. Primary flux and primary intensities of various components from uranium el

Altitude (feet)

100 __ .. . .

200 . ____ . 

300 .. __ 

400- __ _ _ 

500 .

600 ... . _ . __ .

700 . _

Energy (Mev)

0.609 0.769 0.934 1.120 1.238 1.378 1.509 

Primary intensity (photons per square foot p

2. 96x10-' 

.573 

.198 

.0731 

.0402 

.0231 

.0131

7. 25x10-' 

1.44 

.510 

.228 

.116 

.0641 

.0375

3. 98x10-8 

.807 

.294 

.135 

.0706 

.0400 

.0240

3. 19x10-' 

.660 

.243 

.128 

.0597 

.0343 

.0208

1. 23x10-' 

.255 

.0946 

.0444 

.0237 

.0137 

.00839

1. 53x10-' 

.323 

.121 

.0574 

.0310 

.0181 

.0112

5. 49x10-

1.16 

.440 

.210 

.114 

.0672 

.0419
48 EXPERIMENTAL AND THEOEETICAL GEOPHYSICS

TABLE 7. The function Gs (500,p, a)

p (feet)

0 ... 

100.-..-. -.

200 

300

400 

500 

600 

TOO 

800 

900 

1,000... 

1,100.... _

1,200 

1,300 

1,500 

2,000- 

2,500 

3,000 

3,500- 

4,000...- ....-

400

0.8390

.T856

.T152

ft97Q

.5236

.4023

.2962

.2122

.1500

.1055

.0743

.0582

.0442

.03207

.0138

.0027

600

1.0962

1. 0558

.9883

.8937

.7720

.6232

.4785

.3533

.2543

.1806

.1277

.1001

.0759

.0551

.0237

.0047

600

1. 3147

1. 2899

1. 2325

1.1427

1.0203

.8654

.6979

.5368

.3977

.2875

.2052

.1608

.1219

.0885

.0380

.0075

700

1.4931

1. 4797

1.4340

1 ^fifl

1. 2457

1. 1031

.9344

.7543

.5816

.4324

.3138

.2461

.1867

.1357

.0586

.0116

a(fe

800

1.6351

1.6286

1. 5935

1. 5297

1.4373

1. 3162

1.1637

.9860

.7972

.6164

.4597

.3614

.2750

.2007

.0882

.0173

.0032

et)

900

1. 7475

1.7445

1. 7177

1. 6673

1. 5932

1. 4954

1. 3677

1.2090

1.0254

.8306

.6438

.5084

.3894

.2867

.1303

.0255

.0049

1,000

1.8354

1.8336

1. 8132

1.7742

1. 7165

1.6402

1. 5379

1.4061

1.2437

1.0561

.8570

.6825

.5285

.3950

.1899

.0369

.0072

1,500

2. 0557

2. 0553

2.0500

2.0398

2.0248

2.0049

1.9783

1. 9416

1. 8937

1.8312

1. 7482

1. 5770

1.4098

1.2466

.9320

.2150

.0426

.0080

2,000

2. 1170

2. 1167

2. 1153

2.1126

2. 1087

2. 1036

2. 0964

2. 0870

2.0750

2.0588

2. 0374

2. 0318

2. 0036

1. 9528

1.7835

.9649

.2282

.0457

.0085

2,500

2. 1321

2. 1318

2. 1312

2. 1304

2. 1294

2. 1281

2. 1274

2.1246

2. 1191

2.1160

2.1138

2. 1010

2.0881

2.0755

2.0496

1.7962

.8827

.2362

.0478-

.008&

TABLE 8. The function GI (500,y,a)

y (feet) a (feet)

0 

400 

500 

600 

TOO... 

800 .

oon

1,000.__ _

1,500.. 

2,000 _ .. _ .....

2,500. 

3,000. 

400

0.8086

.4569

.3496

.2621

.1946

.1439

.1063

.0787

.0177

.0036

500

0.9780

.5784

.4470

.3372

.2511

.1859

.1375

.1017

.0228

.0047

600

1. 1303

.7027

.5500

.4183

.3129

.2322

.1718

.1270

.0284

.0059

700

1. 2655

.8287

.6584

.5061

.3809

.2835

.1553

.0347

f¥V79

800

1.3843

.9543

.7715

.6007

.4559

.3408

.1871

.0417

.0087

.0010

900

1. 4879

1. 0773

.8877

.7021

.5384

.4049

.3013

.2232

.0496

.0104

.0013

1,000

1. 5778

1. 1955

1. 0051

.8091

.6286

.4766

.3562

.2643

.0587

.0124

.0017

1,500

1.8720

1.6602

1.5267

1. 3569

1.1583

.9478

.7466

.5714

.1291

.0279

.0048

2,000

2.0098

1.9095

1.8422

1.7489

1.6246

1.4657

1.2748

1.0638

.2767

.0604

.0118

2,500

2.0727

2.026*

1.996&

.1.9532

1.890*

1.8076

1.6976

1. 5543.

.5765

.1292

.0271

.0041
SCATTERED GAMMA RAYS FROM THICK URANIUM SOURCES 49

TABLE 9. The function G± (500,z,a)

z (feet)

0 

100. ...........

200 

300. -

400 

500 

600 

700- 

800 

900. 

1,000...... ... _

1,100.. 

1,200.. .........._

1,300. 

1,500.-. 

2,000... 

2,500. 

3,000-. 

3,500- 

4,000- 

o (feet)

400

2. 0395 

1. 9134 

1. 7513 

1. 5530 

1. 3186

1. 0481 

.8049 

.6063 

.4509 

.3343

. 2475 

.1972 

.1529 

.1144 

.0554

.0016

500

2. 9588 

2. 8337 

2. 6455 

2. 3941 

2. 0796

1. 7019 

1. 3398 

1. 0241 

.7709 

.5745

.4261 

.3393 

.2627 

.1964 

.0946

.0198

600

3. 9346 

3. 8268 

3.6313 

3. 3480 

2. 9770

2. 5182 

2. 0450 

1. 6010 

1. 2224 

.9211

.6857 

.5461 

.4230 

.3164 

.1527

.0323

700

4. 9385 

4. 8469 

4. 6534 

4. 3579 

3.9605

3. 4611 

2. 9096 

2. 3549 

1. 8406 

1. 4029

1. 0496 

.8364 

.6484 

.4856 

. 2354

.0505

800

5. 9824 

5. 9096 

5. 7248 

5. 4279 

5. 0191

4. 4983 

3. 9037 

3. 2663 

2. 6318 

2. 0571

1. 5667 

1. 2496 

.9699 

.7275 

.3547

.0764 

.0123

900

7. 0540 

6. 9872 

6. 8081 

6. 5168 

6. 1133

5. 5975 

4. 9849 

4. 3063 

3. 5872 

2. 8886

2. 2501 

1. 7994 

1. 4014 

1. 0561 

.5234

.1133

.0202

1,000

8. 1419 

8. 0807 

7. 9094 

7. 6279 

7. 2363

6. 7345 

6. 1324 

5. 4361 

4. 6735 

3. 8868

3. 1170 

2. 5058 

1. 9648 

1. 4940 

.7632

.1658 

.0308

1,500

13. 1905 

13. 1576 

13. 0481 

12. 8622 

12. 5997

12. 2607 

11. 8389 

11. 3264 

10. 7206 

10. 0076

9. 1951 

8.0093 

6. 9047 

5. 8811 

4. 0773

.9870 

.2091 

. 0348

2,000

18. 1905 

18. 3635 

18. 2865 

18. 1511 

17. 9573

17. 7051 

17. 3894 

17.0114 

16. 5624 

16. 0254

15. 4124 

14. 6867 

13. 8871 

13, 0136 

11.0450

4.8304 

1. 1628 

.2425 

.0382

2,500

23.0945 

23. 0731 

23.0079 

22. 8988 

22. 7459-

22. 5491 

22. 3036 

22. 0047 

21. 6649 

21. 2993

20.8783 

20. 4541 

19. 9409 

19. 3386 

17. 8670

12. 6298 

5. 4713 

1. 3130

.2728 

.0420

TABLE 10. Ratio of grade-area predicted from elementary source relation to actual 

grade-area as function of radius, a, and ground projection, z, of closest distance of 

approach

z (feet)

a (feet)

0 .. . 

100

200 

300 

400 

500 

600-. .

700 

800

900.

1,000 

1,100 

1,200-.. ...... _. _________

1,300- 

400

2.32

2.04

1.91

1.76

1.61

1.55

1.29

1.18

1.06

.945

.877

.672

.627

.494

500

2.98

2.82

2.61

2.36

2.10

1.80

1.55

1.37

1.21

1.07

994

.808

.685

.559

600

4.08

3.85

3.56

3.19

2.82

2.36

1.95

1.65

1.39

1.22

1.08

.904

.788

.634

700

5.46

5.50

4.80

4.34

3.78

3.15

2.58

2.11

1.73

1.45

1.01

.867

.724

800

7.34

7.07

6.56

5.98

5.20

4.26

3.48

2.79

2.22

1.81

1.48

1.25

1.04

.852

900

9.78

9.42

8.89

8.09

7.01

5.62

4.78

3.80

2.82

2.31

1.84

1.54

1.29

1.05

1,000

13.10

12.60

12.00

11.00

9.65

8.13

6.73

5.25

4.10

3.13

2.40

2.00

1.84

1.34

TABLE 11. Ratio of elementary source grade-area to actual grade-area averaged over 

two flight lines as function of radius, a, and ground projection, z, of closest distance 

of approach

z (feet)

a (feet)

0 

100 

200 

300 

400 

500 

600 

400

1.40

1.38

1.29

1.32

1.24

1.30

1.23

500

1.76

1.75

1.71

1.72

1.58

1.50

1.46

600

2.35

2.32

2.23

2.13

2.02

1.88

1.85

700

3.09

3.01

2.91

2.78

2.61

2.44

2.35

800

4.09

4.05

3.91

3.73

3.50

3.24

3.13

900

5.41

5.35

5.16

4.96

4.66

4.22

4.29

1,000

7.20

7.20

7.00

6.70

6.39

6.11

5.99
50 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

TABLE 12. Conversion factors to correct elementary source grade-area prediction to 

true grade-area for finite sources. Minimum and maximum columns show extreme 

spread in factors

a (feet)

0__T __ _____

400_________

500__-______

600__ __ _._

Minimum

1.000

.714

.568

.426

Average

1.000

. 765

.613

.477

Maximum

1.000

.813

.684

.540

a (feet)

700________.

800______.__

900_________

l,000_---___

Minimum

0.324

. 244

. 185

. 139

Average

0.368

.276

.207

.151

Maximum

0.425

.319

.233

. 167

TABLE 13. Ratio of total counts, Hf(500,z,a), to peak intensity, If(500,z,a), as 

function of radius, a, and ground projection, z, of the closest distance of approach 

for finite source

z (feet)

a (feet)

0 .. 

100

200 ____ ...........

300 

400 _________ -

600 

600 

700 

800... __ .

900 

1,000 ___

1,100 

1,200 

1,300 

0

3.74

3.76

3.93

4.16

4.51

4.89

5.22

400

4.86

4.70

4.90

4.95

5.04

5.21

5.43

5.71

6.01

6.34

6.66

6.78

6.92

7.13

500

5.40

5.37

5.35

5.36

5.39

5.46

5.60

5.80

6.06

6.36

6.67

6.78

6.92

7.13

600

5.99

5.93

5.89

5.86

5.84

5.82

5.86

5.96

6.15

6.41

6.68

6.79

6.94

7.15

700

6.62

6.55

6.49

6.43

6.36

6.28

6.23

6.24

6.33

6.49

6.69

6.80

6.95

7.16

800

7.32

7.26

7.19

7.10

6.98

6.84

6.71

6.63

6.60

6.67

6.82

6.92

7.05

7.25

900

8.07

8.01

7.93

7.80

7.67

7.49

7.29

7.12

7.00

6.96

6.99

7.08

7.20

7.37

1,000

8.87

8.81

8.72

8.60

8.43

8.21

7.98

7.73

7.52

7.36

7.27

7.34

7.44

7.56

TABLE 14. Ratio of total counts, Hs(500,0,a) at normal incidence, to peak intensity, 

It(500,0,a), as function of width, a, of a slab source

a (feet)

0 _______________ 

400 ____ - ____ _______

500 ____ _ ______ ---_

600______________-______

700____-_-______ ________

800___ 

H.II,

4.36

4.68

4 82

5.02

5.24

5.47

a (feet)

900__ _________ ___ _

1,000 ___ _ __ ______ _

1,500___._ _ __ _____ __

2,000____ __ . ____ ____

2,500_-.____- ______ _

H./I.

5.73

6.00

7.58

9.38

11.30

o