NBS MONOGRAPH 42

National Bi

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N.W. Bids

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Structure Shielding

Against Fallout Radiation

From Nuclear Weapons

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U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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Office, Washington 25, D.C.
UNITED STATES DEPARTMENT OF COMMERCE • Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Structure Shielding

Against Fallout Radiation

From Nuclear Weapons

L. V. Spencer

National Bureau of Standards Monograph 42

Issued June 1, 1962

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 75 cents
Foreword

The problem of providing protection against the radiations generated by

nuclear weapons is of much concern to the Federal Government and to the

people of the United States. A great deal of eflfort and money is being spent

currently on attempts to analyze existing structures for shielding properties,

and plans are being carried forward to design and build special shelters of

many kinds.

The National Bureau of Standards has maintained a continuing research

program directed to the development of engineering data and methods basic

to the solution of these radiation shielding problems. Most of the work of the

past few years on this project has been concerned with gamma rays from radioactive fallout. This Monograph summarizes both the methods and the data

which have been generated as a result of research on the penetration of fallout

radiations.

Financial support for this project has come primarily from the Office of

Civil and Defense Mobilization and from the Defense Atomic Support Agency

of the Department of Defense.

A. V. AsTiN, Director.

II
Contents

I. latroduction Page

1. Purpose 1 2. Comments about radiation shielding problems and publications 1 3. Choice and presentation of material 1 II. Description of fallout

4. Fallout vs initial effects ^ 2

5. Regional variations in fallout 2

6. Local variations in fallout 3

7. Types of fallout radiation 3

8. Characteristics of fallout gamma rays 3

9. Exposure to gamma rays 3

10. Decrease of fallout radioactivity with time 4

11. Time variation of the fallout spectrum 5

12. Magnitude of fallout gamma ray intensities and exposures 6

III. Description and measurement of the radiation

13. Introductory remarks 7

14. Physical properties 7

15. Radiation detectors 7

16. Measurement of the radiation flux from particular directions 8

17. Radiation sources 10

18. Protection and reduction factors 10

19. Attenuation of gamma rays 11

IV. Introduction to structure shielding analysis

20. Basic approach 12

21. Discussion of barrier and geometry reduction factors 14

22. Barrier shielding and effective mass thickness 14

23. Geometry reduction and solid angle fraction (co) 16

24. Brief summary of the analvsis procedure 18

V. Data

25. Introductory comments 18

26. Basic data 19

27. Catalogue of functions 24

28. Description of graphical data 29

VI. Elementary structure types

29. Comments 53

30. Density interface 53

31. Foxhole problems 56

32. Shelter covered with fallout 57

33. Vertical walls 59

34. Light superstructure with shielded basement 60

35. Apertures 62

36. Compartmentalization 64

37. CoUimation and divided barrier effects in compartmentalized

structures 64

38. Mazes 66

VII. Rectangular source shapes

39. Introduction 67

40. Solid angle calculations for circles and circular sectors 67

41. Solid angle fractions for rectangular surfaces 68

42. Approximation of rectangles by circular sectors.-. 75

43. Calculation of geometry factors for rectangles 77

44. Further discussion of geometry factors 78

45. Acknowledgments 79

Appendix A. Computations

46. Preparation of data: Use of digital computers 79

47. Preparation of data: Hand computations . 81

48. Comments on accuracy 81

49. Alternative data types and procedures 82

Appendix B. Data for C0«° and CS^"

50. Introductory comments 83

References 134

III

Structure Shielding Against Fallout Radiation From Nuclear

Weapons

L. V. Spencer

The theory of structure shielding from fallout gamma radiation is developed to the point

of applications to elementary structure types. Examples discussed in the text include the

density interface, foxhole, shielded foxhole or basement, light superstructure, vertical wall,

blockhouse, vents, compartmentalization effects, and mazes. A large number of engineering charts and graphs are presented for engineering calculations, including many obtained

from angular distributions of the exposure dose. Results are given for a fission spectrum,

and for Co^" and Cs'*^ sources. This information has been obtained almost completely

by machine calculations utilizing basic cross section data. A number of sources of experimental data are mentioned, but detailed comparisons with experiment are not included.

I. Introduction

1. Purpose

This Monograph is to assist scientists and engi- neers in the solution of problems of protection

from ionizing radiation, particularly radiation

from fallout. Important activities in which such

problems arise include (a) the design of new protective structures, (b) analysis of existing structures to determine the natuie of their shielding

capabilities, and (c) assessing the possibilities for

improvising shelter.

2. Comments about Radiation Shielding

Problems and Publications

The analysis of structures for their protection

against ionizing radiation represents a new field

of engineering which has many similarities to

illuminating engineering. Estimating radiation

levels at different locations in a complex structure

is comparable to the problem of determining the

illumination levels in a similar stiucture on a cloudy day, but with all partitions and walls

having varying degrees of transparency or translucence rather than being opaque. The complexity of the problems is such that, despite

intensive research, significant gaps in our knowledge still remain.

Reactor shielding problems are rather different

from the problems of shielding against nuclear

weapons. The former involve a localized source and shield whose various aspects are readily

available for study and possible modification.

Further, both source and shield differ in many

ways from the sources and the barrier configurations encountered in weapons shielding problems.

For these and other reasons the different reactor

shielding handbooks are not an adequate basis for

the study of weapons shielding problems, though

they do contain a great deal of useful information

[1-5].*

The joint AEC, DOD, and OCDM publication,

"The Effects of Nuclear Weapons," gives a general

description of all aspects of nuclear explosions,

•Figures in brackets indicate the literature references at the end of this Monograph.

together with many tables and graphs relevant to

radiation shielding [6]. However, the material

included in the book on the subject of protection

against fallout constitutes an introduction to the

problems rather than a detailed methodology for

their solution.

A number of reports have been published in the

past three years on shielding against fallout

radiation [7-11]. The problems and the different

approaches to their solution contained in these

reports tend to have close similarities to each

other and to the material in this monograph.

Nevertheless, there are differences worth noting.

The approach taken here is less empirical, relying

on basic data derived from first principles rather

than on field test experiments. This has the

advantage of providing moie diffeient types of

data, as well as a basis for different kinds of

estimates. Enough experimental confirmations of

the basic data exist to give confidence in its overall

accuracy [12] ; but further research can be expected

to complement and modify this information in

various ways.^

3. Choice and Presentation of IVlaterial

The contents and organization of this Monograph have been influenced by the fact that the

analysis of structure shielding may properly be

considered a branch of either nuclear or civil

engineering. It is, however, a subject with which

few people are familiar; hence it has seemed worth

while to include general information about fallout

radiation and its characteristic properties. Thus,

Part II describes fallout in an elementary way and

Part III gives properties of gamma radiation.

Part IV introduces the formal approach to

structure shielding analysis adopted in this

manuscript together with the basic parameters of

' Recently, OCDM has published a series of manuals, prepared by L. N.

FitzSimons of that agency, and based on the methods and data presented in

this monograph. See, e.g., "Fallout Shelter Surveys: Guide for Architects and Engineers," Report No. NP-10-2 (May 1960), and "OCDM Engineering

Manual," (Dec. 1, 1960).

1
description and measurement. Part V is a discussion of graphical data for use in applications.

Part VI gives an analysis of elementary structures.

Finally, Part VII presents a methodology for

generalizing the calculations to a wide group of

barrier shapes.

It was originally intended to incorporate into

this Monograph analyses of a number of complex

structures in terms of the data presented here.

However, this has been omitted for a number of reasons. Instead, we include an appendix which

gives two additional sets of graphical data, one

for Co^° and the other for Cs^", for use in analyzing

experiments. The data contained in the body of

the manuscript are for radiation from fission

products.

II. Description of Fallout

4. Fallout vs Initial Effects

Whenever a nuclear bomb is exploded, a tre- mendous pressure wave, large quantities of heat

and light, and a vast amount of radiation and

radioactive material are produced concurrently

[6, Chapters II and III]. Where these three

effects are all important, their interrelations complicate nearly all protection problems. For ex- ample, completely protective structures must be

blast resistant as well as radiation resistant, and

they must be built with due consideration for the

possibility of a fire storm. Furthermore, the

nature of blast and heat phenomena affect the

radiation problem, e.g., through vertical drafts

which alter the spatial distribution of radioactive

material.

Exposure to the initial radiations ceases about

a minute after the explosion, by which time the

radioactive material from the explosion is carried

out of range by the upward drafts. Large

amounts of earth and debris are also drawn upward, and the resulting mushroom-shaped cloud

may rise 80,000 feet or more. This cloud contains

particles ranging in size from submicroscopic

specks to grains as large as coarse sand. Radioactive residue from the bomb and its surroundings

adheres to these particles. The general term for

this particulate matter, as it is brought to earth

again, is "fallout."

5. Regional Variations in Fallout

Some of the fallout particles rise higher than

others; and some are more buoyant than others,

thus falling to earth less rapidly [6, Sections

10.1-10.24]. Close to the explosion there occurs

the "throwout"— rocks blown out by the blast;

while at the other extreme are particles so small

and raised so high above the earth that they are in

the stratosphere above the weather and are little

influenced by gravitational forces. It takes years

for these tiniest particles to settle to earth again,

and they may come down anywhere on the earth.

We shall not discuss this long-duration "worldwide" faUout, nor shall we be concerned with the

"throwout" [6]. A large fraction of the fallout material is big

enough to come to earth within hours or days, yet

small enough to be easily blown about by winds.

Correspondingly, the distribution of particle

200

190

180 -

170 -

160 -

150 -

140 -

130

120 -

(A

uj no -

s— lOOh

i 90h

o

(E

u. 80

UJo

Q 60 -

50

40 -

30 -

20 -

10 -

0 -

10 -

20

300 ROENTGENS

500 '

f l.000\

2,000

3,000

20 10 10 20

DISTANCE FROM G.Z. (MILES)

Figure 5.1. Idealized total (accumulated) dose contours from fallout in the first 36 hours after the high yield explosion at Bikini Atoll on March 1, 1954. (fief. 6.)

2
sizes and the regional wind structure determine

the spatial distribution of the deposit. Sustained

strong winds may carry fallout in lethal amounts

for hundreds of miles, whereas light winds may

leave most of it much closer to the explosion.

Figure 5.1 gives idealized contours of total fall- out exposure from a large (multi-megaton) explosion, indicating how the fallout was carried by

winds away from the point of explosion.

Attempts to calculate such gross variations have

had some success, at least to the point of being

useful in estimations for broad planning purposes

[6, Sections 9.55-9.138].

It is clear from figure 5.1 that intense fallout can occur in wide areas far beyond blast and heat

effects. This makes it reasonable to consider the

fallout as a problem separate from blast and heat

complications. In this Monograph, problems of

protection from fallout radiation are discussed

without reference to the effects of blast and heat;

it should be remembered, however, that intense

fallout also occurs in the proximity of the

explosion.

6. Local Variations in Fallout

Just as the gross features of fallout spatial

distributions are affected by the general wind

pattern over a large area, so the fine features

depend on the local winds. For example, fallout

may collect more on one side of a building than

on another, or blow down one set of streets more

than another, more or less in the manner that the

drifting of snow is determined by local variations

in the surface winds. We do not have explicit

data on the effects of such variations, and a thorough analysis of the problem is needed before

a reliable estimation of the magnitude of these

"micro-meteorological" effects can be established

[13]. In the absence of specific information we

assume for the present that fallout will be uniformly

distributed horizontally over exposed surfaces.

A question of particular importance relates to

the amount of fallout material which may remain

suspended above the earth in trees and shrubbery.

This has not been accurately determined. There

is evidence that not more than a few percent of

the material descending, e.g., upon a tree will

remain on the limbs and leaves of the tree if the

surfaces are dry [14-15]. But the possibility of

weather conditions which will cause fallout to

adhere even to vertical wall surfaces, and the

sensitivity of basement protection to the amount

of suspended source material, make this a major

problem for future research.

7. Types of Fallout Radiation

Two types of radiation are emitted by radioactive fallout materials,^ namely, (a) gamma rays

and (b) fast charged particles (alpha and beta

rays). Both types are biologically destructive;

however, the charged particle ladiations are not

' Neutrons are not emitted by fallout particles.

very penetrating. The fast beta-rays (electrons) can produce burns on unprotected skin, especially

when fallout material is in contact witli the skin

but very little shielding, such as is pi'ovided by

average to heavy clothing, is capable of providing

almost complete protection. The alpha rays are not even able to penetrate through the external

layer of skin. In problems of ingestion of con- taminated food, water, and air, however, both

alpha and beta rays must be considered, because

when taken internally their full energy is effective

in causing damage to the internal tissues [16].

The gamma rays present the primary shielding

problem because they are extremely penetrating

as well as biologically destructive. They may be

described as high energy X-rays, but for historical reasons the term "7-rays" is frequently used when

the radiation is produced by nuclear processes

and "X-rays" when it is produced by electron

bombardment of an anode.

8. Characteristics of Fallout Gamma Rays

Both the quality and the quantity of fallout

gamma rays have been investigated in many

research studies [17-23].

Gamma radiation may be visualized as a stream

of individual "energy packets," called "photons."

The energy content of each packet may be re- ferred to as the "photon energy." Quantitative

descriptions of fallout gamma rays may utilize

directly the total photon energy crossing unit area

of a surface per second; alternately they may

involve a standard detector whose response is in

some way descriptive of the strength of the beam,

as discussed in Sections 9 and 15.

Qualitative descriptions are nearly always in

terms of the "spectrum," which gives the relative

importance of the different gamma ray photon

energies present. A typical spectrum will be a

list of numbers or a graph giving, e.g., the fraction

of the total beam energy in the form of photon

energies between 1 and 2 Mev, between 2 and 3

Mev, etc. Radioactive material is a mixture of many types

of atoms, each emitting characteristic photon

energies. These radioactive atoms originated in

one of two ways: (1) Some were originally part

of an atom which fissioned in the nuclear explosion (mixed fission products); (2) others were part of

the bomb or were nearby and were made radioactive through capture of a neutron (neutron induced activities). The mixed fission products

have usually been considered the most important

radiation source, to the extent that the neutroninduced activities have often been neglected in

calculations pertaining to fallout. From both

sources a variety of photon enei'gies is generated over a range extending up to perhaps 3 Mev.

9. Exposure to Gamma Rays

Biological effects depend upon the energy absorbed from the gamma rays per unit mass of

3
biological material [24]. This absorbed energy is usually measured in "rads" (1 rad=100 ergs/

gram).

The energy absorbed in biological material depends in turn on the strength of the radiation

field and the duration of exposure. For protection

puiposes, this is measured in terms of the cumulative effect of the field on a standard material,

namely air, in units of roentgens (r) or milliroentgens (mr)}

Though the roentgen is a measure of exposure

to a radiation field, whereas the rad is a measure

of the result of this exposure, it has long been

common practice to blur this distinction by speaking of the biological effect of a "dose" of y roentgens, meaning the effect resulting from energy

absorbed during an exposure totalling y roentgens.

The International Commission on Radiological

Units has recommended that the term "exposure

dose" be used in this context to prevent confusion

with the energy absorbed in the biological (or

other) material.*

The strength of a gamma ray field at a given

time is frequently measured in terms of the "doserate," in units of roentgens per hour (r/hr) or, in

weak fields, in units of milliroentgens per hour

(mrfhx)

.

Perhaps the best way to gain a feel for the magnitude of different exposure dose rates and total

doses is by noting the effects upon human beings,

insofar as this is known or surmised. Specific

effects depend upon the health of the individual

the time-duration of exposure, etc., but some generalized statements about the effects caused by

short-duration exposures can be made, as in Table

9.1 below.

Table 9.1. Probable Effects of Fallout Gamma Radiation on Humans "

Short Term

Whole-Body

Exposure

(roentgens)

Probable Effect

0 to 100

100 to 200

200 to 300

300 to 600

Over 600

No obvious effects Some sickness

Sickness and some deaths

Severe sickness and many deaths

Few, if any, survivors

Note that the effects given all refer to the extent of general incapacitation following exposure. Effects on the

gonads, the eyes, etc., which may be extremely important

but not immediately obvious, have not been included.

10. Decrease of Fallout Radioactivity With

Time

When fallout material has stopped falling upon

a given location gamma ray intensities produced

' Exposure dose of X- or gamma radiation at a certain place Is a measure of the radiation that is based upon its ability to produce ionization. The unit

of exposure dose of X- or gamma radiation is the roentgen (r). One roentgen

is an exposure dose of (X or) gamma radiation such that the associated cor- puscular emission per 0.001293 g (ram) of air produces, in air, ions carrying 1 electrostatic unit of quantity of electricity of either sign [24].

' See footnote 3, above.

by the fallout can be observed to decrease with

time. If the fallout is not being spatially displaced

by winds, this decrease is described roughly by

the following formula:

D{U)^{t2lU)-'-'D{U), (10.1)

where U, t-i represent different elapsed times following the explosion and D{t) is the dose-rate at

time t [6, Sections 9.4-9.17]. Equation (10.1) can be roughly described in another way: For

every sevenfold increase in -time there is a tenfold

decrease in radiation intensity. For example,

if the radiation dose-rate in a given location is 1,000 r/hr four hours after the explosion, the

dose-rate twenty-eight hours after the explosion

would be about 100 r/hr.

Another convenient rule-of-thumb relates radiation dose-rate (r/hr) to total exposure (r): The

total exposure dose (r) over a very long time

interval beginning one hour after the explosion

is numerically about five times the dose-rate

(r/hr) at one hour in the same place, assuming

that no displacement or arrival of the fallout

material has taken place after one hour. Thus,

if the dose-rate one hour after the explosion were measured to be 30 r/hr, the total exposure

at the same location over a very long time interval

following this measurement would be about

150 r. A more general expression is

E{U,U) =^UD{U)[l - {Ulur % (10.2)

where E{ti,t2) is the total exposure dose accumulated

between times <i and ^2 following an explosion.

Note that if ti = l and ^2=°°, E(l,o=) =5D(l),

in agreement with the special rule-of-thumb

mentioned. These rules do not apply if the

shielding conditions are changed during the time

interval.

The functions of the ratio (^2/^1) appearing in

eqs (10.1) and (10.2) are given in figure 10.1.

It should be remembered that the time variation

expressed by the foregoing expressions is only approximate, that it depends upon the predominance

of fission product radiation over induced radiation,

and that it becomes a very poor approximation

when times after the explosion stretch out into many

months or years [25].^

EXAMPLE. The dose-rate of radiation in a given place 4 hours after an explosion is measured

to be 32 r/hr. (a) What will the dose-rate be

48 hours after the burst; and (b) what will be

the total accumulated exposure dose during the

time interval from 4 to 48 hours?

(a) Taking <i =4 hours, ^2 =48 hours, one obtains t2/ti = 12. According to figure 10.1,

{t./ti)-' -'={12)-' -'=0.053. Multiplying this by

D{ti) =32 r/hr, one obtains D(t2) = 1.7 r/hr at 48

hours. ANSWER.

5 For a recent investigation of this time variation see the report USNRDLTR-425, by Carl Miller (U.S. Naval Radiological Defense Laboratory,

May 27, 1960).

4
tz/t,

FiGUBE 10.1. Functions for obtaining the radiation dose rates and cumulative dose at different times after fission.

(b) To obtain the accumulated exposure dose,

one determines from figure 10.1 that [1 — (^2/^1)" -^] =

1- (12)- -2=0.39. The product (5)X(4 hr)X(32

r/hr)X(0.39) then yields the value 250 r. ANSWER.

11. Time Variation of the Fallout Spectrum

The fallout gamma ray spectrum changes with

time [17-23, 25-27]. This nnportant feature comes

about because many varieties of fission fragments

are contained in the fallout mateiral, and each

goes through a succession of transformations.

Thus the very nature of the radiating material

changes with time, and this affects the spectrum

emitted.

An example of these spectral changes is given

in figure 11.1. At one hour after fission (top

figure) the total energy is fairly evenly distributed

among gamma rays with photon energies from

about 0.5 Mev to 2.5 Mev. At the end of a day,

however, most of the radiation is made up of

photons below 1 Mev. With further increase in

elapsed time following the fission process, higher

energies again become very important. Finally,

although not shown, the lower portions of the

spectrum increase in importance.

The intense, high energy component M hich dominates the spectrum about a week following the
PHOTON ENERGY (Mev)

FiGUBE 11.1. The relative intensities of different spectral components at several times after fission. The height of each box

is proportional to the (energy content) of gamma rays in the energy interval, and the central line indicates an energy assumed

for all photons in the interval for purposes of calculation. The lines do not correspond to spectral energies actually present.

The total intensity is unity for each spectrum. (Ref. 18, Table 11. Volatile components have been removed from the fission products.)

fission is due to an isotope of Lanthanum, which

is the fourth member of a radioactive family which

begins with Xenon fission fragment. The spectral

energy of these gamma rays is 1.6 Mev, and they

are important over a time interval of more than

two weeks.

Because of the tim.e variations of the gamma

ray spectrum, it is necessary to make some decision

regarding choice of a spectrum to be used as the

basis for shielding calculations'. It turns out that

penetration properties of fallout gamma rays are

less sensitive to spectral changes than might be

supposed, except for very large penetrations—this

will be discussed in the next Sections. The data in

s Note that Xenon is a rare gas, so that when this type of fission fragment

is initially formed, it will not enter into chemical combination. The Xenon

quickly (in a few seconds) disintegrates to produce Cesium, which is very

active chemically and which should become attached to fallout material. Thus Lanthanum and other daughter products of this chain should contri- bute to the fallout spectrum.

this monograph correspond to the spectrum existing 1.12 hours after the fission process. This spectrum is the one given in figure 11.1 (top curve),

with no additions for neutron capture components.

This choice results from, two considerations: (1) A

large part of the exposure to radiation is apt to occur during the first few hours; and (2) the spectrum one hour after the explosion is representative

of other early times in the sense of having about

the same penetrability.

12. Magnitude of Fallout Gamma Ray Intensities and Exposures

Precise information about likely fallout doserates is not available; but there is some information

about possible dose-rates produced by large (megaton) explosions. These vary up to thousands of

6
r/hr at early times after the explosion, and could

result in accumulated exposures up to tens of

thousands of roentgens for unprotected people,

even outside blast areas (see fig. 5.1). Furthermore, there exists strong likelihood of overlapping

fallout fields, increasing total intensities additivel}'.

Since a hmnan exposure to a few hundred roentgens is considered very severe, one attempts to

provide shielding sufficient to reduce radiation intensities by "protection" factors of the order of

1000 to insure in practically all cases reduction of

exposures to an order of magnitude below the danger levels. In different types of structures, different "protection factors" turn out to be feasible.

For exam.ple, in an underground shelter it may be

relativel}' easy to incorporate a protection factor

of 5000, whereas in an above-ground shelter, it ni.ay be difficult to incorporate more than a factor

of, say, 100. Figure 5.1 illustrates the value of

protection factors still lower than 100.

III. Description and Measurement of the Radiation

13. Introductory Remarks

This part contains a brief outline of terms and

ideas used in the latter parts of this Monograph.

Also included is a description of some of the characteristic physical properties of gamma radiation.

References are given to other publications containing a more complete discussion of some of the

topics.

14. Physical Properties

Gani.m.a ray photons act in many ways as if they

were tiny "bullets." They travel in a straight line

until they "collide" with an atom. When a "collision" takes place the photon may vanish altogether, or it m.ay "ricochet" and continue traveling

in a new direction with lower energy. The "collisions" are usually referred to as interactions.

Those in which the photon disappears altogether

are called absorptive interactions. The others in

which the photon "ricochets," are called scattering

interactions [12, Part A].

As a photon moves along its trajectory, there is always a possibility that in the next segment of

the path an interaction of some type may take

place. Although the nature and location of an

interaction is a matter of chance, the probability

of occurrence for each interaction type per unit

path length traveled depends upon the photon

energy and the type of material being traversed.

At very low photon energies an absorptive interaction has such a large probability that a long

path unimpeded by such an interaction is ex- tremely unlikely.

The average distance between interactions in a given type of material is called the "mean free

path." The mean free path is proportional to the

density of the material traversed, and this is the

main reason why the gamma rays travel about

700 times farther between interactions in air than

in water. Mean free paths in air for fallout

gamma rays are of the order of several hundred

feet, i.e., large compared with the dimensions of

most buildings. This has the consequence that

most gamma ray interactions within buildings do

not take place in the air, but in the walls, floors,

and objects of furniture.

When gamma rays "scatter" tlie>" continue

with lower energy in a new direction, so that they

may undergo one or more additional interactions.

There is a correlation between the angle of deflection and the energy loss which accompanies a

scattering interaction. On the basis of this cor- relation the following statements can be made:

Gamma rays deflected through 90° by a scattering

interaction cannot have a residual photon energy

above 0.511 Mev. Sunilarly, gamma rays which

have been deflected through 180° by a scattering

interaction cannot retain an energy above 0.256

Mev. Thus, gamma rays which have undergone

large changes in direction are apt to be much

lower in energy than unscattered gamma rays.

This is the case even when the direction change is the cumulative result of several interactions.

15. Radiation Detectors

An object which is affected by radiation to a measurable extent can be used as a radiation

detector [28]. Since the effect of radiation on

human beings is of primary interest, one might

wish to discuss all shielding problems in terms

of tissue damage to the exposed human body.

This is impractical because the body does not

make a very suitable detector for experimental or

theoretical investigations. Instead, detectors are used whose response has similarities to the response

of a human body, while being more easily measured

and interpreted. These detectors tell us about

the radiation field. The tissue damage caused by

exposure to a known radiation field is treated as a separate problem, and we do not concern ourselves with it further.

Every detector is characterized by its "response

function," which is the "efficiency" of gamma

rays of photon energy E, incident on the detector

from various directions, in producing the measured

effect. It is frequently advantageous to have

an "isotropic" detector, i.e., one equally efficient

regardless of the direction of incidence of the

photons.^

The most commonly used radiation detector

consists of an enclosed air space containing a

pair of oppositely charged electrodes. Wlien

exposed to radiation, the gamma rays interact

with the material surrounding the air space,

producing high-velocity recoil electrons, some of

' Note that the response function depends, in general, on both direction and photon energy and might be denoted by e{E,$,<t>). An isotropic detector would correspondingly have a response function «(£), i.e., independent of e, .

7
which traverse the cavity and ionize the air

within. A measurable electron current is produced as the ions travel to the electrodes and

deposit their charge. It is this current, divided

by the volume of the air space, which constitutes

the response of the instrument.

These so-called "cavity ionization chambers"

are usually constructed in such a way that their

response is nearly isotropic. But the current

per cm^ measured by the instruments is not proportional to the total energy incident upon the

detector. Instead it h nearly proportional to the

energy deposited per gram of detector material, as a result of gamma ray interactions occurring

within the detector. When such detectors have

cavity walls of a material similar to air in its

reaction to irradiation, the response can be

measured in roentgens and referred to as "dose,"

or more precisely as the "exposure dose," as ex- plained in Section 9. Reference to the "detector response" in this

monograph will always imply an isotropic detector of this type, i.e., of air-equivalent material,

whose response in roentgens is called the "exposure

dose."

16. Measurement of the Radiation Flux From

Particular Directions

To determine the flux of radiation traversing a given region one needs to introduce a suitable

detector as a probe. For this purpose one of the

isotropic detectors just described can be used,

with a response calibrated in roentgens. The

detector should be small enough so that all parts

of it are exposed to the same radiation flux, and

also so that its presence does not appreciably

disturb the flux distribution.

The response measured is due to photons which

traverse the detector material. These photons

are characterized by the direction of their trajectory at the time they enter the detector. We

identify direction by means of polar coordinates

0,0 which are measured with respect to an

arbitrary polar axis through the center of the

detector and a reference half-plane terminating

along the polar axis. Both "obliquity," d, and

"azimuth," 0, are determined from a line extending

out from the center of the detector parallel to the

trajectory but opposite the direction of travel of

the photon.* The angle which this line forms

with the polar axis is 6, while 0 is the angle between

the reference plane and the plane terminating

along the polar axis which contains the line.

(See fig. 16.1.) Any direction of incidence is uniquely specified by values for these two direction

coordinates.

Note that the coordinate pairs {9,4>) are in

one-to-one correspondence with points on a sphere

* It would seem more natural, perhaps, to identify (9,*) with the direction

of photon travel rather than its inverse. We would do this if the current of radiation were to be our focus of attention. But we will be more directly concerned with radiation sources than with radiation fields. It is therefore natural to refer directions to a polar axis extending from the detector outwards

towards the radiation source, since this is the reference system which is easy

to use in solid angle calculations of radiation sources.

of unit radius centered at the detector. We can

refer to (9,</)) either as a direction or as a point on such a unit sphere.

The total detector response D is made up of

contributions from all {9,<p): li D{d,4>)sineddd<i)

is the contribution to the detector response due

to gamma rays with incident trajectories between

6 and d+dd, and between 0 and 0 +c^<A, the total

response from all directions is expressed by the

integral

D-- sinddd d(pD{d,4>)

Jo Jo

(cos 61) J

d4>D{e,4>). (16.1)

We refer to 0(6,4)) f^s the detector dose angvlar

(or directional) distribution.

One reason for introducing D{6,4>) is simply

that the radiation aff'ecting a detector may be

analyzed into components coming, say, from a wall

or a ceiling. If there is negligible air scattering,

the photon trajectories of importance all intercept

both detector and wall or ceiling surface of interest.

We designate this surface by S, and define a function gs{S,<f) which is to be unity if a direction (0,0) can intercept both surface and detector, but zero otherwise. Then if Ds is the detector response to radiation from S,

Ds=j'_^d (cos d) j'J d<t>D(e,4>)gs{e,cf>). (16.2)

This expression reduces to (16.1) if contributions

from all (0,0) are included. If D{d,4>) does not

depend upon 0, we write D{d,<f)) = (2ir)~^D(d) , and

gs{9) ^1/2t£ d4>gs{d,<t>)

to obtain

D. cos e)D{d)gs{e) (16.3)

Note that gs[,d) is the fraction of the azimuth contributing, at obUquity d [29, 30].

The directions (0,0) which intercept both a given

surface {S) and a point at the center of a sphere

of unit radius form a kind of cone with apex at

the point. (See fig. 16.2.) The aperture of this cone is measured by the size of the area intercepted on the unit sphere; if unit area is intercepted, the

surface is said to "subtend a solid angle of one

steradian at the point." Since the area of the

unit sphere is 47r, the solid angle subtended at any

point cannot exceed 47r steradians. We shall find

it convenient to use a different measure of the solid

angle in this Monograph: Our unit will equal 2-k steradians. We shall designate the solid angle in

these units by w, and to avoid confusion will refer

to CO as the "solid angle fraction."

The solid angle fraction ws subtended by a surface S can be obtained from eq (16.2) by replacing

8
POLAR AXIS

Figure 16.1. A detector about to be traversed by a photon.

The photon path passes through the edge of the detector. The polar angles {d,4>) correspond to the line parallel to the photon trajectory which passes through the reference point

at the center of the detector.

POLAR AXIS

UNIT SPHERE

Figure 16.2. A surface subtending at the center of a detector a solid angle whose magnitude in steradians is equal to the shaded area on the unit sphere.

Figure 16.3. A circular aperture of radius 3 feet, through

which radiation passes towards a detector placed 4 feet from the aperture and on the perpendicular line through

its center.

9
D{e,ii>) by the constant (27r)~^ or from eq (16.3)

by replacing D{e) with 1.

=J'

(Z(cos 6) d4>{\l2iT)gs{9,<i>)

d{cosd)gs{e). (16.4)

For a plane surface extending in all directions to

infinity, with polar axis taken along the perpendicular to this surface, 6^s=l for O^cos e<l, so that both expressions in eq (16.4) give ws=l for

this case. Any single plane surface of finite

extent will subtend an co smaller than 1

.

EXAMPLE. Calculate the detector response D

at a position opposite the center of a circular

aperture of radius 3 ft through which radiation is entering a building, and distant 4 ft from the

aperture. Assume that the dose angular distribution from all points of the aperture has the

constant value Da, that only radiation entering

through the aperture contributes to the detector

response (see fig. 16.3), and neglect air scattering.

We choose as polar axis the direction from the

detector through the center of the aperture. Then

the angular distribution of the dose becomes Da for

(9<^max=tan-i 3/4 and zero for e>(?max=tan~i 3/4.

Since tan"' 3/4 =cos"' (4/5), we write

gs{e,4>)

_l,cose>4/5

"0, cos 6i<4/5.

Since both gs and D are independent of 0, eq

(16.2) reduces to eq (16.3), and

Ds= r d (cos e) Da=1/5 Da. answer

J 4/5 Note that for constant Da, as in this case, Ds is always given by Dacos, i-e., that cos=l/5 for the

aperture of this example.

17. Radiation Sources

Fallout material has already been referred to as the "source" of the fallout gamma rays. We

often refer to the gamma ray generating material

as the primary source, because it is a convenience

at times to treat any interface through which

radiation emerges as a "source." If the radiation

is not actually produced at such an interface, it is termed a secondary source. Note that the aperture

of the preceding example acts as a secondary

source. Fallout material can be spatially distributed

into almost any geometric configuration conceivable; but two configurations are of special

importance because they occur frequently and also

because many others of real importance can be

obtained from these two. One of these is called a

point source and is approximated by a local concentration of fallout material into a volume of

negligible dimensions. The other is called a plane

source and corresponds to a uniform spread of

fallout material in a plane. If the source is as- sumed to cover the whole plane, one speaks of the

source as an infinite plane source; but if the source has finite area it is called a finite plane source. The strength of a radiation source can be described in various ways. It is customary in physics

to measure the source strength in terms of the

number of nuclear disintegrations occurring every

second. A source is said to have a strength of one

curie if 3.7X10'° nuclear disintegrations occur per

second. On the other hand, in radiology one

speaks of a source which produces an exposure dose

of one roentgen per hour at a location one meter

from the source [see, e.g., 31]. Since protection is a relative matter, we discuss source strengths in

this Monograph in terms of the exposure dose they

produce at fixed positions ; and we do not attempt

to specify source strengths in curies even though

this is possible. Our practice here is thus akin

to that just attributed to radiologists. More

specifically, the real problem in radiation protection is to determine how much safer one location is than another. We therefore define a standard "unprotected" position and measure the

protection according to reduction of the exposure

dose below that experienced in the standard

position.

18. Protection and Reduction Factors

We wish to introduce a quantitative measure of

the protection afforded by structures, by one

structure relative to another, or one location in a

structure relative to another. Also, in order to

gauge the overall effectiveness of protective construction, a standard "unprotected" position is needed for comparison.

Fallout is assumed to cover uniformly all siu"-

faces according to their horizontal projection, as already mentioned. From this primary source

the radiation travels to the detector, wherever it may be. A fairly obvious choice for a completely

unprotected detector location would be above

ground in a large, open field. This choice has

been made frequently; but it has two difficulties:

One is that the detector response depends upon

the "roughness" of the ground surface; and the

other is that location of the source at the groundair interface makes it very difficult to calculate

the detector response accurately even if the spectrum and strength of the primary soiu'ce are completely known. The first difficulty makes it hard to "standardize the standard," so to speak,

while the second makes it hard to evaluate the

standard.

For these reasons we have chosen as a "standard

unprotected position" a detector location three

feet above a hypothetical source of the same

character as the fallout on the ground, but at a hypothetical smooth, infinite plane "interface"

with the ground replaced by compressed air of

10
the same density.' By making the source plane

ideally smooth we avoid the ground roughness

difficulty; and by replacing the ground by "compressed air" we make an accurate theoretical

analysis possible in a fairly realistic case and thus

avoid the second difficulty. To complete the

description of the "standard unprotected position"

it is necessary to add that the 3 ft separation is in

dry air at 76 cm Hg pressure and a temperature

of 20° centigrade.

There are at least three additional reasons for

this choice of reference detector location: (1) It gives an extreme, but not imrealistic estimate of

the dose to which the centroid of the body is exposed in an open contaminated field. (2)

Given the spectrum and strength of the gamma

rays emitted per unit area of the primary source,

the reference dose rate can be calculated to about

2-3 percent accuracy, which is an order of magnitude more accurate than necessary for most

shielding estimates. (3) It appears easier and

more natural to consider ground roughness as an

additional "bonus" shielding from fallout on the

ground, because the same "bonus" protection

does not exist against the fallout on top of the

protective structure.

We define the "protection factor," P, as the

ratio of the detector response A in the standard

improtected position to the detector response D

in a protected position, i.e., P=Do/D. (18.1)

The reciprocal quantity is what is actually determined in structure analysis, and for easy reference

it \vill be termed the "reduction factor" and

usually referred to simply as D/Dq. In this

monograph Do always refers to detector response

at the "standard unprotected position."

Note that the protection factor is not defined in

terms of a "standard" spectrum, so that it is a quantity which will vary, e.g., with time after

fission, weapon type, etc. This reflects accurately

the situation because the protection which a structure gives will itself vary with circumstances ; and

in all likelihood, the actual variation will cover a wider range than given by a variety of different

estimates. All fission data in this monograph

correspond to a spectrum calculated for 1.12 hour

old mixed fission products, as already indicated.

19. Attenuation of Gamma Rays

Before proceeding to the discussion of structure

analysis, we wish to examine briefly what happens

to gamma rays emitted by a point som"ce embedded in som.e material, in order to form, a picture of their "history" and an appreciation of the

different factors leading to their attenuation (reduction in strength). [See also 12, Section 5.]

Very near to the source most of the gamma rays

are diverging radially away from their point of

origin. As they diverge from the common source,

' Note that the density of the "compressed air" is immaterial except for purposes of visualization, so that the reference configuration is actually an

jnfinite plane source in an infinite homogeneous medium. (See Section 25.)

they also tend to spread apart from, each other.

The effect can be visualized by thinking of concentric spheres surrounding the radiation source,

with radii designated by r. As gamma rays from,

the source travel outwards they will intercept

spheres of increasing radii which have an ever

greater surface area. Even if all the gamma rays

pass unhindered through the spheres, they will be

spread ever n^ore thinly the farther out they travel,

simpl>" because they will be distributed over a

larger area. This has the consequence that the

exposure dose measured by a small detector of

cross sectional area a will decrease with distance r from the source in proportion to r~^, because the

detector can only respond to a fraction a/iTrr^ of

the total flux emerging from the sources. This is the inverse square law of attenuation with distance

from a source. Each gamma ray photon from the source penetrates unhindered through the surrounding material until an interaction occurs, but the interactions

are chance occurrences. There is a fixed probability determined by the photon energy and type

of material traversed that in a given segment of

the path an interaction will occur. This means

that in each succeeding layer of m.aterial a fraction

of the photons undergo their first interaction, and

the number of photons penetrating the layer without interaction is correspondingly reduced. We

say that the "unscattered beam." undergoes exponential attenuation. Successive layers of material

equally thick reduce the number of unscattered

photons by the same fraction, assuming the layers

to be homogeneous and equally dense. Tables of

attenuation coefficients are given in [32].

If the first interaction is a scattering, the gamma

ray photon may travel further, though with re- duced photon energy and changed direction.

Other interactions follow, separated by path segments which tend to become shorter as the photon

energy becomes smaller. Eventually an absorptive interaction terminates each photon "history."

Many of the gamma ray photons continue to

increase their distance from the source even after a number of scattering interactions. Thus, at

any fixed distance there will be gamma rays which

have been scattered once, twice, many times,

in addition to some which have not been scattered

at all. Because all "orders of scattering" must

be considered as giving a contribution to the

detector response, the exponential attenuation of

the unscattered gamma rays does not describe

accurately the law of attenuation due to the

combined effect of all interactions. It is customary

to modify the exponential law (accurate for

unscattered photons) by a factor called the

build-up factor to take into account the presence

of scattered gamma rays. (Tables of build-up

factors are given in [33].)

If the primary source is located in the air,

gamma rays from the source may strike a wall.

Because gamma rays can have scattering interactions which completely reverse their direction,

11
some of the radiation will be "backscattered"

from the wall, re-emerging again into the air. A great deal of interest attaches to the fraction

of the gamma ray energy entering the wall which

comes back out again in this way, i.e., the "energy

albedo." If the different spectral components

are weighted according to the efficiency of detection by an air-equivalent detector, an analogous

"dose albedo" may be determined. We view

this latter ratio as a type of reduction factor for

purpose of comparison with the other factors,

and refer to it simply as the albedo [34].^°

Other definitions might be useful, but for the

most part these four—the inverse square law,

the exponential factor, the build-up factor, and

the albedo—are sufficient for discussion purposes.

Figure 19.1 illustrates the first three factors in a particular case.

IV. Introduction to Structure Shielding Analysis

20. Basic Approach

The analysis of structure shielding can best be

described in connection with a typical, but elementary example. Figure 20.1 shows a "blockhouse," i.e., a building with four walls and a roof.

Fallout rests on the roof and on the ground around

the structure. We are primarily interested in the

dose measured at detector position A.

Radiation contributing to the detector response

comes from all directions. Because of the low density of air, most of the detected radiation moves in

straight lines to the detector from points of emergence at the walls. Thus it is natural to speak of

detector response due to radiation "from the

walls," "from the roof," "from the window," etc. We expect that radiation "from the walls" will

have little resemblance to radiation "from the

roof" in intensity or directional distribution. Radiation "from the roof" originates from the fallout

resting on top of the roof, while radiation "from

the walls" originates from fallout on the ground,

possibly a long distance from the structure. These

marked differences make it reasonable to differentiate between detector response "components"

from the different wall surfaces.

Radiation from a wall ordinarily has entered the

wall from the outside. Thus when one speaks of

radiation "from a wall," the reference may be

either to the radiation passing from wall to detector or to radiation initially incident upon the exterior surface of the wall in question. In other

words, one can classify the detector response components according to interior or exterior surface

crossed. For our purposes the latter is usually preferable and can be expressed as follows: If that

part of the detector response due to radiation entering the "^"-th distinct wall section from the

outside is designated Di, the total detector re- sponse wiU be

D=JlDi, (20.1)

matter to treat backscattering from interior surfaces by a simple corrective factor."

We next select a particular wall surface, namely

that adjacent to detector positions B and C in

figure 20.1; only radiation entering and leaving

this wall surface is to be considered for the

moment. To contribute to the detector response,

radiation must first pass through the wall and

then "locate" the detector. The detector re- sponse at B divided by the response at position C

is approximately the reduction factor due to

interactions within the wall material. If the

detector is moved away from B towards A, the

detector response will decrease further. This

additional "attenuation" is due to the "inverse

square" effect described in the preceding Section.

We distinguish between these two types of

attenuation and refer to them separately. The

first, attenuation by interactions with wall material, we shall call harrier reduction due to harrier

shielding. The second, accompanying a displacement of the detector farther from the wall, we

shall call geometry reduction and speak of it as due

to geometry shielding. The following subsections

describe these quantities in greater detail and give

reasoning which leads to their introduction. Here

it will suffice to say that geometry reduction can

often be considered as mainly a limited-source

effect. The wall acts as a secondary source; and

if this source were constant on a plane extending

laterally to infinity, there would be no appreciable

difference in intensity between detector positions

A and B. The geometry reduction comes about

primarily because the wall, being finite, subtends

a smaller sohd angle at A than at B.

If a reduction Bi of the detector response occiu-s

between C and B. and a further reduction G\ occurs when the detector is moved from B to A,

the detector response Di at A may be written

D^=6,BiDc (20.2)

where the sum extends over all distinct exterior

wall surfaces. Such a classification is useful for

many reasons. In addition to differences in incident radiation characteristics, the attenuation is apt to differ widely from one wall section to another. This has the consequence that the dominant

source of radiation can be a particular wall or wall

section. Further, it is usually a straightforward

where Dc is the detector response at a reference

i» still other types of albedo factor have been defined, using number of photons in place of energy or dose. Only the "dose albedo" appears in this manual, so that the use of "albedo" to refer to dose albedo should not be

confusing.

II By classifying according to exterior surfaces one estimates that a certain

fraction of the radiation entering the room from one wall will affect the detector after backscattering from other walls. An estimate of this corrective

factor is then made with little concern about the detailed history of the radia- tion or about effects due to radiation entering through the other walls.

12
Distance from Source, cm of H^O

a 1 Mev point source m H2O. irie aasneu y

give the solid curve.

13

622827 0—62 2
Figure 20.1. A blockhouse, with fallout on roof and ground.

position C, which is here shown as just outside

the wall.

The dose from a wall section nearly always will

be written in the form eq (20.2), so that calculation of the Dt involves determination of appropriate barrier and geometry reduction factors for

each type of wall surface.

Though Dc in this example is the dose just

outside the exterior wall, the comparison with the

standard value Do is of most fundamental interest.

This change requires definition of a modified

barrier factor B'i =BiDclDo, which expresses the

combined reduction due to attenuation in the wall

and exterior air. Henceforth barrier factors will

be of this type, i.e., ratios of the detector response

at position adjacent to a barrier to the standard

response Dq.

The approach described here depends for its success on the fact that only ~15 percent of the

radiation energy incident on a surface is backscattered. This relatively low albedo insures that

most of the detector response is due to radiation

which has had contact with only one wall.

21. Discussion of Barrier and Geometry Reduction Factors

In the preceding Section a distinction was made

between two types of reduction in the detector

response. The reasoning which leads to this

distinction is essentially as follows:

Most theoretical data derives from the study of

an especially simple case in which the medium has

no irregularities (infinite homogeneous medium).

The source may be concentrated on a plane in the

medium or localized at a point. Calculations of

the detector response as a function of distance

from the source in such an elementary case can be done to within a precision of a few percent.

Because these calculations have internal consistency and reliabihty, and also because they

usually represent a good approximation to the

reduction in intensity due to actual slab barriers,

we nearly always find it convenient to consider

ratios between actual detector response due to

radiation from a particular wall, say, and detector

response as estimated by such infinite medium

calculations. The resulting ratios express nearly

all the special characteristics of the configuration,

but they vary slowly as the position or the configuration is changed. It is possible to estimate

these ratios with calculations for an infinite

mediimi, since they express effects such as source

"finiteness."

Thus, the detector response is expressed as the

product of two factors: One gives the attenuation

in the wall material which would occur if the source were of a particularly simple type and the

medium everywhere uniform in density; and the

other is a ratio which expresses all the other

features of the actual configuration. For con- venience we give these two factors names. The

name "barrier factor" for the first is a fairly

obvious choice. The name "geometry factor" is chosen for lack of a more accurate term of reasonable length. (Note that the term "configuration

factor" would be more appropriate but is not used

here because of its length.)

One more aspect of this development deserves

comment. In practice, structure shielding analysis requires a great variety of standardized

functions expressible in the form of tables, graphs,

and possibly empirical formulas. Many of these

functions are rather naturally classified as geometry or barrier factors. We make the attempt to

carry this classification as far as possible, with the

consequence that standard functions may be

referred to as "geometry factors" which are not

obviously related to other functions also referred

to as geometry factors. The next two sections

describe some of the independent variables which

determine these factors both as to type and as to

nimierical value.

22. Barrier Shielding and Effective Mass

Thickness (X)

The shielding provided by a barrier depends

upon many variables, among which are

(1) the weight per unit area of the barrier,

(2) the type of barrier material,

(3) the gamma ray spectrum, and

(4) the directional distribution of the radiation

striking the barrier.

We discuss items (3) and (4) first: Data will be

given only for a single fixed spectrum, namely

that of figure 11.1, (top); variations of the fallout

gamma spectrum with time after the burst are not taken into account. The directional distributions of incident gamma rays vary widely,

so that results for each of the most important

cases will be identified with a special symbol.

Item.s (1) and (2), the type and thickness of the

barrier, can be treated together because of a fortunate circum.stance : Nearly all important

construction materials have atomic numbers low

enough so that the attenuation is due primarily to

scattering interactions, which are independent of

the energy states occupied by the electrons.

Thus, the attenuation produced by a barrier

depends almost com.pletely on how m.any electrons

it puts in the path of the gam.m.a rays, and this is

12 Basically, reduction due to attenuation in barriers is exponential, while

reduction due to configuration is a power function. We are in fact using

the same ideas that lead to the introduction of the Build-up Factor concept. Estimation of these new ratios requires a knowledge of dose angular

distributions.

14
simply the product of the number of electrons per

unit volume with the thickness of the barrier.

Therefore, to measure the effectiveness of barrier shielding a parameter X is used which is proportional to A (barrier thickness), and to p<^Z/A'^

where p is the density and <Z/A> is the ratio of

atomic charge to atomic mass number, averaged

over the constituent elements of the barrier.'^

(Note that p<Z/A> is proportional to electrons

per unit volume, the proportionality constant

being Avogadro's number.) In particular, we

define X by the expression

Z=2<ZM>pA. (22.1)

The factor of two is introduced because <Z/A>

is nearly 0.5 for such important construction

materials as brick and concrete, so that 2<^Z/A^ = 1 for those materials. If the factor 2<Z/A> ~

1 is treated as a dimensionless proportionality

constant, X can be measured in units Ib/ft^ (psf),

since the unit weight is usually given in lbs/ft^,

thickness in feet, and lb/ft" x ft =lb/ft^. Because

X is nearly always close to the weight per unit

area, or mass thickness, we call this quantity the

effective mass thickness}'^ Table 22.1 gives

2<Z/A> for a number of common materials,

together with the density of the solid material.

Table 22.1. Values of 2 <_ZIAy and p

Material 2 <ZIA> p, Density in pcf

Water 1. 11 62.4

Wood 1. 06 34.0 (average)

Air 1. 0 0.076

Brick 1. 0 115

Concrete 1. 0 144

Soil (depending on water

content) 1. 00-1. 02 100 (average)

Steel 0. 931 480

Lead" 0. 791 710

" Lead is included strictly for comparison. It should be

remembered that the data in this monograph do not apply

to lead because it absorbs rather than scatters the radiation. See the last paragraph of this section.

Table 22.2. Mass thickness of common building materials

used in various building components.

INDIVIDUAL BUILDING MATERIALS

Item Thickness Weight Psf Component

Inches Adobe.. _ 12 116 Wall

Asbestos board Me 1. 7 Do.

Asbestos, corrugated.. 4 Roof, wall Asbestos shingles 1.8 Do.

Asphalt roofing, 3-ply 1 Roof

ready.

Asphalt roofing, 4-ply 5.5 Do. & gravel.

13 In particular, <Z/A> = 2 f<, where /, is the fraction of the mass

due to the ith constituent.

1*" Effective mass thickness" and "equivalent thickness of concrete," a term used in the ODM report [7] are different names for essentially the same

quantity.

Table 22.2 Mass thickness of common building materials

used in various building components—Continued

INDIVIDUAL BUILDING MATERIALS

Item 1 nicKness vv tjigjit i SI \_' u111 puiie11

1

-

Inches Asphalt roofing, 5-ply 6. 2 Do. & gravel. Asphalt shingles 2.3 Do.

-DliCK » OA 1

, 1 Wall VV dll Q Do

T)rt Tin Tin UiJ 1 7 Do

Clay tile shingles. 15±5 Roof

^ld,y Lllc, Oil ULtUl d.l._ _ A

'± 1 fi VV dll

Concrete block, hol- 4 30±4 Do.

vin xJU^ Q 00±0

Do 12 85±10 Do.

Concrete, reinforced.

,

1 12^2 Do 12 144

Fiber board K> 0.8 Wall

T llJ\il oilcditlllllg K> u. y Do.

yj y |JDlliii UlUlyK. 2 y. 0 Do. Do 19 t=,-i-9 1^1. O^lZ. D Do.

\Ji y poUlJX UUdl U 1/. 2 1 u^ypbuiii, blieallllllg Mi 9 nu Wall

iV^ O I'V* Irfl lO nGT IVidl IJlV IdUillg o 9fi Do.

Plaster, directly ap- H 5 Wall, ceiling

5 0 Do

Plaster on gypsum Vi Do!

lath.

X lobier uii iiieiai laLii-- ¥i D. U

Tin Plaster on wood lath /* 5.

0

Do.

X^idrOlCl, OUilU. z\j Wall Do 4 30 Do.

x^iyWOOQ, llXlloli 11. nu Do. Do 1.5 Ceiling

Plywood, sheathing-

.

n 1.1 Wall, roof

Slate Me 7.3 Roof

Steel, corrugated, 20 2 Roof, wall

ga. Steel panel, 18 ga... 3. 26 Wall, roof

1 9 1 Wall

stone, cast, facing o

z Do.

3Ayi

9 0 Do.

stucco, wood lath Do.

11 Do. Wood block, flooring.. 3 10 Floor Wood finish flooring. . 2^2 2. 5 Do. Wood sheathing H 2.5 Floor, roof Wood shingles 2.5 Roof Wood shingles in. 1. 1 Wall

to weather. Wood siding, 8 in. 1.5 Do.

bevel. Wood siding, 6 in. 2.5 Do.

drop.

COMPOSITE MATERIALS

Brick 4 and structural clay 4 60 Wall

tile. Ceramic tile... Me

on mortar bed H 11 Do. Do

{ 23 Floor

Concrete, reinforced Do.

ribbed slabs (see section on components). Marble or terrazzo 4 50 Do. on concrete fill. Plaster, hollow wall 4 22 Wall

with steel studs. Plaster on suspended 10 Ceiling metal lath. Wood finish floor H

on wood sleepers 4 Lightweight concrete 4 23 Floor

fill.

15
Table 22.2 Mass thickness of common building materials

used in various building components—Continued

DOORS AND GLASS

Item Thickness Weight Psf Component

Inches Door, wood exterior, m 4. 5 Small buildings standard 3 ft-0 in. x

6 lt-8 in. solid core

flush panel. Door, wood interior. 1. 9 standard 2 ft-6 in. x

6 ft-8 in. hollow

core. solid core m 4.0 Do.

Door, glass, exterior. 5.1 Large buildings

d 1 li 111 1 11 U 111 cUgCf

standard 3 ft-0 in. x 7ft-0in. Door, glass, exterior H 11.0 Do.

standard, 3 ft-0 in. X 7 ft-0 in. Glass, double H 1.6 Small buildings

strength.

Glass, plate 3.5 Large buildings

• Glass.

Barrier shielding factors are considered functions of the effective mass thickness, i.e., B=B{X).

Use of the concept of effective mass thickness

makes it possible to perform calculations even when barriers are composed of several layers of

different materials. One merely sums the effective mass thickness (Xi) of the different layers to

obtain the total effective mass thickness X^ZXi.

i It should be remembered that the effective mass

thickness is not pA, the weight per unit area,

though the two can be numerically equal. Table

22.2 summarizes effective mass thickness data for a variety of common construction materials and

combinations. This data is taken from the compilation of [35]. It is important to realize also

that the mass thickness of a wall, floor, etc., can

ordinarily be obtained by use of the usual dead

load tables in engineering handbooks, though for

very thick walls (several hundred psf), one may

have to adjust for hollow spaces within the

construction (see Section 36).

EXAMPLE: A barrier is composed of wood and

concrete layers having weights per unit area 5

Ib/ft^ and 50 Ib/ft^, respectively. Find the

effective mass thickness for the barrier.

X={5 lb/ft2)X1.06+(50 lb/ft^)X 1.00 =55.3 psf.

ANSWER.

Figure 22.1 illustrates the dependence of barrier

attenuation on the mass thickness and atomic

number. Notice that even steel (atomic number

26) gives only slightly more protection than concrete of the same effective mass thickness. If

desired, the ratio between the curves in figure

22.1 can be applied as a correction to take into

account the minor differences between low atomic

number material; but with composite barriers this

must be done with care. Materials with high atomic numbers, such as Pb, give additional protection; but the difference

is not so great as is commonly supposed. To

reduce the intensity of 1 Mev radiation by a factor of 1,000 requires a concrete barrier having X=305 psf as compared with a Pb barrier having X=190 psf. But the weight per unit area

is Xi2<CZ/A»-\ which for Pb is 190/.791=240

Ib/ft^ while for concrete it is 305/1.0 =305 Ib/ftl

Thus nearly % as much Pb (by weight) is required

as concrete.

23. Geometry Reduction and Solid Angle

Fraction

To identify other important variables on which

the detector response depends, we consider again

the blockhouse illustration and in particular the

detector response at position A due to tlie wall

on the right (fig. 20.1).

(1) Solid angle fraction: It is clear that the

detector response will depend on the "apparent"

size of the wall, as seen from the detector.

Since "apparent size" is conveniently measured

in terms of solid angle or solid angle fraction, we

say that the detector response will depend upon

the solid angle fraction co subtended at the

detector by the radiation source.

(2) Barrier thickness: The thickness of the wall

(X) will have an effect upon the detector response over and above the attenuation which we describe

by a barrier factor. This is because the directional distribution of radiation emerging from the

wall affects the detector response but is in turn

affected by the wall thickness. In most cases and

up to a limit, thicker walls tend to produce directional distributions more and more concentrated

along the perpendicular to the wall surface.

(3) Wall shape and detector position: The

dependence on detector position is easy to demonstrate in the blockhouse illustration by the fact

that one can find detector positions at the ceiling

and floor in which the wall subtends the same co, but in which the detector response is expected to

differ considerably. Shape effects can also be

easily demonstrated: A detector 10 ft out from

the center of the wall will have one response if the

wall is long and narrow and another if the wall is square, though subtending the same solid angle.

(4) Type of source: The type of primary source,

as in the case of barrier factors, varies widely

enough so that we identify each with a special

symbol.

(5) Type of wall material: We treat for the

present only the case of materials of atomic number ~30 or below, this range including the most

important materials commonly employed in construction. As indicated in the preceding Section,

these low-Z materials have a certain equivalence

which we utilize.

In line with the discussion of Sections 20 and 21,

we consider all effects other than that due to

15 The detector response could equally well, be said to be a sum of contributions from each element of the source. Following this idea, the detector response could be written in terms of the radiation current flowing through

the differential area elements of the wall. This point of view is equivalent

to one we adopt, in which the emphasis is on flux directional distributions

at the detector, provided the radiation current emerging from the wall is independent of position on the wall. We make this assumption throughout

this manuscript.

16
X,psf

Figure 22.1. Attenuation of 1 Mev gamma rays in different materials (plane perpendicular source). Notice that Fe, concrete, and H2O are separated by small, nearly constant factors.

17
simple attenuation in the wall as expressed by a

single "geometry" factor G. Variations due to X

and CO are taken into account by writiag

G=G{X,w). (23.1)

To take shape dependence into account we represent actual source shapes as superpositions of

simpler, and different, source shapes. This makes

it possible to estimate geometry reduction factors

for a set of "standard shapes" which can be combined to approximate most of the important

shapes encounteied in practice (see Part VII).

Formally, this means that we write

G{X,o>) =J:G{X,o>,), (23.2)

where G{X,Wi) is the geometry factor for one of the

"standard shapes." Note that to insure the

correct solid angle fraction, the condition

03=^^03i must be imposed.'®

i Although the blockhouse illustration is useful

for purposes of identifying variables, it does not

show how one might refer to quite different

quantities as "geometry factors." For this piirpose we use a different example: Consider a heavy

multistory buUding with fallout on the (flat)

roof and detector in the basement. Radiation

can reach the detector only after passing through

several heavy floor slabs. To account for barrier

attenuation and remove it from consideration, we

may divide the detector response by the expected

response if the material between source and

detector were uniformly distributed, and if the

radiation source extended unchanged in the plane

of the roof to infinity in all directions (infinite

plane source). The resulting ratio, which we

consider a type of geometry factor, would depend

upon the following things: (1) The extent of the

roof, which can be expressed in terms of the

solid angle fraction subtended by the roof at the

detector; (2) the thickness {X) of the barrier

between roof and detector ; and (3) the shape and

position of the roof relative to the detector.

Equations (23.1) and (23.2) are as applicable to

this case as to the blockhouse wall. The chief

difference is that w is determined for the faUout

on the roof, a primary radiation source, in contrast

to the blockhouse waU example in which co is subtended by a secondary radiation source.

V.

25. Introductory Comments

In Part IV a formal approach to shielding

calciilations has been described. This approach

developed out of a study of simple configurations

to be described in Part VI; but it has also been

designed to emphasize the use of quantities which

can be estimated theoretically.

'« For an alternative procedure, see Section 49 and references 29, 30, and 41.

The preceding illustration also makes it clear

that the estimation of actual barrier and geometry

factors consists mainly in finding a simpler configuration, resembling the actual one in important

respects, which is amenable to a detailed theoretical or experimental analysis. This point is elaborated in Parts V and VI.

24. Brief Summary of the Analysis Procedure

The preceding discussion may be summarized

by stating that the dose D measured in a structure

(such as a blockhouse) will be represented as the

sum of contributions Di entering through different

wall sections and ceiling. Each of the Di wiU be

the product of a barrier factor B(Xt) and a geometry factor G(Xi,o}i). Each geometry factor

consists of one or more terms corresponding to

"partial surfaces" chosen to represent approximately the actual source surface whether primary

or secondary. If the reference detector response

for the barrier factors is taken to be the standard

value Dq three feet above an infinite, smooth,

air-air interface having uniform contamination

equal to that on and aroimd the structure, the

complete expression for the detector response

takes the form

D=DoJ:B(X,)j^^G(Xi,o:ij)^ (24.1)

Structure configurations exist which call for

more complicated expressions than this. For

example, in maze-type geometries the radiation

may have to turn corners to reach the detector.

In basements below grade the detector response

may consist largely of contributions due to gamma

rays which have scattered downwards from the

exposed superstructure through intervening floor

slabs. Such "higher order" contributions give

terms involving additional factors, but otherwise

of the same type as the terms in eq (24.1).

To differentiate between different types of

data, we will replace "B" and "G" by a variety

of other symbols, each referring to a single type

of data, as indicated in the preceding Sections.

We discuss next the theoretical methods for

estimating B's and G's, i.e., the derivation of

data for applications.

Data

In Part V we turn to the presentation and

discussion of data for applications. First we

give a description of the "basic" data used to

obtain the fvmctions for applications, so that an

appreciation of the limitations of the graphs of

barrier and geometry factors is possible and also so that one can get an idea how additional information might be obtained. This discussion can probably be skipped or skimmed by readers

interested in a particular application, though the

18
"basic" data have some fairly direct applications

to shielding problems.

Not all the factors which one might desire can

be easily obtained theoretically, or even experimentally. In applications, the graphs of the

various factors will no doubt be used in cases beyond their range of applicability. This makes

it especially important to give a clear statement

of their origin and interpretation. Because this

type of background information is least ambiguous

when stated mathematically, no attempt is made

to avoid mathem.atical terminology.

26. Basic Data

To produce varied types of basic data, digital

computers have proved essential. They are used

to generate solutions of the integral equations

describing the transport, diffusion, and energy

loss of gamma radiation. These "transport

equations" have been solved by "moment

methods" and by "Monte Carlo methods." The

former yield solutions for a source in a medium

without boundaries, while the latter permit calculation of boundary effects [12, Section 7].

All tables and graphs presented here have been

obtained from four types of basic data: (1) Plane

isotropic source case; (2) point isotropic source

case; (3) plane oblique source case; and (4) albedo

results. The first three were obtained using

moment methods; while the last required Monte

Carlo calculations. We discuss each type briefly

here. Appendix A contains an outline of the

actual moment calculations of the first three data

types.

(1) Plane isotropic source case: Consider an

infinite, plane source of fallout radiation imbedded

in an infinite, homogeneous medium. Distance of

the detector from the source plane is represented

by d. Alternately, distances may be referred to

in terms of X, Ib/ft^ effective mass thickness

between source and detector. If the medium is air (at 20°, 76 cm Hg), the relation between the

two is d/X=n.S ft of air/psf. (26.1)

At a distance d from the source plane, the radiation produces a dose angular distribution which

we refer to as l{d,cosd), where d is the obliquity

measured from the perpendicular from the detector to the source plane.'' We fix the scale of

l(d,cosd) so that an isotropic detector which

registers Dq roentgens three feet above the source

will register dD=Dol(d,cosd)sinddd roentgens due

to gamma rays striking the detector between

obliquities 6 and d+dd. This implies that'*

" See Section 16, including the footnote. " We assume here that the function l(d, cos 9) corresponds to penetration

in air. Analogous functions corresponding to penetration in other materials such as concrete, will also be referred to as, e.g., l(X, cos $). They will cor- respond to the same source strength as in (26.2) (air penetration easel, but this implies a difTerent value for the integral over all angles.

j dicos 6)1(3 ft, COS 9) = 1. (26.2)

Computations of these angular distributions

have usually involved their representation as a sum of Legendre polynomials,

l(d, cos e)=f: in+l/2)l,{d)P, (cosd). (26.3)

71 =0

In one type of calculation the first seven or eight

functions l^id) were determined. An extrapolation n-^co was then performed using the fact

that l{d, cos d) resembles (cos d)~^ exp (—o/cos 6) for cos 0>O when the unscattered component is important. In another type of calculation the

unscattered component was calculated exactly and

the scattered component determined by summing a series of the type eq (26.3) up to seven or

eight terms. (See Appendix A.)

Figure 26.1 illustrates the appearance and

behavior of l{d,cosd).^^ The various features of

these curves have fairly simple interpretations:

Near the source plane (small d), l{d,cosd) is proportional to (cos 6)"^ for angles approaching but

less than r/2 (i.e., cos 0^0+). This results in

large values for directions nearly parallel to the

source plane, an important feature of the radiation

from this type of source. With increasing penetration the directional distributions become nearly

isotropic in the forward direction (for d^l50 ft,

corresponding to about 0.3 mean free paths for a photon of average energy). For still greater

penetrations the distributions become increasingly

peaked at cos 0= 1, as the initially dominant oblique

components are preferentially removed.

(2) Point isotropic som"ce case: A source concentrated in a region of negligible dimensions

generates gamma rays which penetrate outwards

into an infinite, homogeneous medium. At a penetration distance d from, the source, the dose

angular distribution is identified by p(d,cosd),

where 6 is measured relative to the axis from

detector to source.^" The scale of p(d, cos 6) can be

fixed conveniently relative to l{d,cos6) by considering a small contaminated area ^ of a plane

surface located a radial distance r from an isotropic

detector, with A/r^^l. We identify Dop{r,cose) X

A/4:Trr^ with the dose angular distribution at the

detector due to this source; but if the whole plane

surface were similarly contaminated, we would

want the total detector response to be Do at a

distance of 3 ft from the plane. To write this

condition in the form of an integral we take A to

be an annular sector of width dp and length pd(l>, as shown in figure 26.2. The detector response Do

at height 3 ft is then given by the integral over

19 These curves represent actual results for penetration by a 1.12 hr fission spectrum in an infinite medium of water, which is so similar to air in regards

to interaction probabilities that these curves have been scaled to correspond

to feet of air.

2» See Section 16, including the footnote.

19
COS 9

Figure 26.1. Dose angular d-islributions l{d,cosd) for an idealized plane fallout source, at different heightsin air (d) above the source. (H2O, 1.12 hr fission. See also Jigs. Rl and B2.)

20
Figure 26.2. A small annular sector of a source plane located a distance

r= VS^ +p^ ft from a detector 3 ft above the plane.

obliquities d and the total contribution from all area elements

Z?o=J d<p j

d (cos e)Dop{r, cos 6)

where P{r) = fLxd (cos d)p{r, cos 6). Cancelling Dq

from eq (26.4) we obtain the normalizing condition

on p(r,cos0) in terms of its integral over all obliquities.^^

The scattered and unscattered components of

p(r, cos 6) have quite different characteristic features. The unscattered component, p^^\r, cos d) is concentrated at the angle 0=0, while the scattered

component, designated p''\(r, cos0), is distributed over all obliquities. As in the case of the plane

isotropic source, p{r, cos 6) has been calculated by

using its Legendre polynomial representation.^^

The calculation proceeds more or less as follows:

A digital computer determines the total strength

of the unscattered component and also the first seven or eight functions p„'*'(r) by standard

techniques [12, Part C]. An extrapolation in n

is then performed, which uses a resemblance of

21 As in the case of l(d, cos 9), other functions, corresponding to penetration

in other materials than air (or water), will be called p(A', cos B) if the source

strength is the same as that implied by eq (26.4). " Do not confuse the Legendre polynomial P„(cos 9) with P(r) of eq (26.4). Note that P(r)=pci(r).

^'"'(rjcos^) to the function (a+cos d)~^''^. (See

Appendix A.)

Figure 26.3 shows curves of p{r, cosd) for several

values of r, but rescaled so that the integral of

p'-^^ (r,cos6) over all obliquities is unity. All

the distributions are strongly peaked at cos0=l;

but the truly remarkable feature is the similarity

of trends. About the only change worth noting

from one value of r to the next is in the relative

strengths of scattered and unscattered components, which is indicated by the separation of the curves vertically and (qualitatively) by the lengths

of the arrows at cos0=l. With increasing penetration the unscattered copmonent steadily decreases in importance.

(3) Plane oblique source case: Radiation from

a plane source goes in all directions; but for some

purposes it is advantageous to consider only those

gamma rays which go initially at an obliquity do relative to the normal to the source plane. The

detector response at different distances from a plane source originating gamma rays with the

fixed obliquity will be designated s{X, cos^o),

with X measuring distance from, the source in

terms of the effective mass thickness (psf). For

our purpose it is convenient to fix the scale of this

function so that cos^os'"' (-^'jcos^o) is unity at

X=0, where s^°'' (X,c.osdo) is the detector re- sponse to the unscattered component.

'3 Note that Lim cosSm"') (,X,cosei) is independent of cosSo, for cose(i>0. X-)0

21
COS 6

Figure 26.3 Dose angular distributions p{r,cos6) for a point fallout source, normalized by dividing by the total intensity of

scattered gamma rays. The arrows at cos6= 1 indicate the unscattered component. The arrow lengths have no quantitative

significance. {H2O, 1.12 hr fission. See also figs. BS and B4.)

Figure 26.4 gives this function for different

values of cos^o, corresponding to penetration by

a 1.12 hr spectrum of fission gamma rays in H2O

(or air). Special features exhibited by these curves have interpretations as follows: The curves correspounding to cos&o~l describe gamma rays

initially traveling directly away from the source

plane. These curves lie above the others for large

X because a large part of the "path length"

traversed by each gamma ray contributes to distance from the source plane. Conversely, when

gamma rays initially move almost parallel to the source plane (cos6o<C<^l)) the initial layers of

barrier material greatly reduce the radiation

intensity.

It is possible for radiation from, a plane oblique

source to be scattered back into the region behind

the source plane. The characteristics of this

back-penetrating radiation are of considerable

interest and have applications to shielding problems. It is possible to view such cases either as corresponding to obliquity angles greater than

Tr/2 (cosS<0) or as corresponding to negative

values of X, with cosSo>0. The terminology is a little easier to manage in the former case and

so we adopt that point of view. Figure 26.5 gives

22
X.psf

FiGtFEE 26.4. Gamma ray attenuation curves for monodirectional gamma rays from a plane source, for incident obliquities

6o<.90° relative to the normal to the source plane. (H2O, 1.12 hr fission. See also figs. B5 and B6.)

23
curves of s(X,cosdo) for negative cos^o- Only

scattered radiation contributes to the detector

response. The curves are continued to X=0,

although strictly speaking the detector response

changes discontinuously at this point because the

contribution due to unscattered gamma rays

begins here. Data for grazing incidence (cos^o) =0

is included. The various curves are seen to be

rather rough. High accuracy is difficult to obtain

in this type of calculation because a differencing

of large numbers is involved, and the irregularities

reflect the inaccuracy of the calculation.

It is by means of the plane oblique data that

we mqst easily make comparisons of the penetrability and scattering properties of fallout gamma

rays at different times after fission, together with

similar comparisons involving Co^° and other

radioactive materials. Figures 26.6 and 26.7 give curves proportional to s{X,l) and s{X,0) for different fission sources, with data also for Co^". In

figure 26.6, marked differences appear only for

rather large penetrations (X>144 psf). Differ- ences in scattering properties are indicated in fig- ure 26.7. The calculations are less reliable, and

these differences may be exaggerated by the inaccuracy of the data for X near 0. All curves correspond to penetration in water-equivalent

materials.

(4) Albedo or backscattering data [34]. The

gamma ray albedo has been mentioned in Section

19. If a plane gamma ray source is incident on a thick slab, a detector at the interface will observe

a dose angular distribution D{cosd). The albedo

(a) to which we refer is defined by

a- j d (cos e) cos d Z>(cos d)

d (cos e) cos dDicos 6)

(26.6)

This ratio depends on the incident source, and for

monodirectional sources is a function of the angle

of incidence (^o) relative to the normal to the slab

face.

Berger and Raso have made machine Monte

Carlo calculations, for monoenergetic sources, of

this function [34]. Figure 26.8 gives data for a

1.12 hr fission spectrum incident on a concrete

slab, as calculated from their data.

27. Catalogue of Functions

The detector response to radiation from an

arbitrary souice is represented by an expression

like eqs (16.2) or (16.3), which sums contributions

from different parts of the source. Source shapes can vary enormously, and to cut down the number

of variables, we do calculations for circular sources,

illustrated by figure 16.3, with the detector directly

opposite the center of the circle. Applying eqs

(16.2) and (16.3) to this simple case, we find that

d{cosd)\ d<pDs(d,,p),

1- u Jo

, Dsio:)=( d (cos 6) Ds(e), (27.1)

«/ 1— fc)

where w=l —cos 6^^^ as discussed in Section 40.

We approximate both barrier and geometry factors by use of integrals of this type with the

integrand taken as one or another of the functions

described in Section 26. Interpretation of these

integrals is given mostly in the next section; here

we simply list them for intercomparison and

reference.

(1) Barrier Factors: The following expressions

define functions which we use to approximate

different types of barrier factors: 2*

X (rf) =J

d (cos e) I (d, cos 6) , (27 . 2)

S {d) =J

d (cos e) I {d, cos 6) , (27.3)

r d (cos do) s {X, cos do) S'{X)=^ , (27.4)

J

rf(cos ^o)s(0, cos ^o)

P(.d)=j c?(cos e)pid, cos e), (27.5)

P«(cO=J'^(^ (cos e)p^'^(d, cos d), (27.6)

W(X,d)=j^ d (cos d) cos d s (X, cos 6)

X (l/27r)

J

d^ I (d, sin d cos ^) . (27.7)

Most of these quantities have fairly obvious

designations and interpretations. "L" stands for

"layer," "S" stands for "skyshine," "P" stands

for "point," and "W" stands foi "wall." P^'^

and p'-^^ refer to the scattered component only

from a point source.

(2) Geometry Factors: The ratios to be used

in estimating geometry factors are listed below:

£.U.„)=^[i(X)-i(j^)}

(27.8)

(27.9)

2< Notes: P<»'(d) is the same as pSf'W in eq (26.5). Pand should not be confused with the Legendre polynomials P„. We consider these defini- tions to hold even when the /, p, and s data corresponds to materials other than air (or H2O).

25 It is assumed that both denominator and numerator in these ratios correspond to penetration in the same material, by gamma rays from the same

source. Note that in the alternative approach mentioned in Section 49 and described in references 29, 30, and 41, it would be logical to use Legendre coeffi- cient ratios in a manner closely analogous to the use of the ratios defined here.

24
25
X,psf

Figure 26.6. Penetration of fallout gamma rays at different times after fission. Also included for comparison is a curve

for Co^". The curves are normalized to equal total energy dissipation, integrated over all penetrations. (H2O, plane

perpendicular source.)

26
X,psf

Figure 26.7. Penetration of gamma rays from a plane, monodirectional source generating gamma rays only in directions

along the source plane. Substantial penetrations away from the source plane presuppose large scattering angles. {H2O.)
28
(27.10)

Xj dtp lid, sine con <p), (27.11)

W^, (c?,co) J' ^ (cos 0) (l/2x)

/"2!r X J

c?<p ^(c?, sin 6/ cos ^), (27.12)

1 r-i+'^

A^a (C^,C0) J

(cos e) ^ (d, COS 0) ,

(27.13)

Pa id,w) =pj^ d (cos e) p (d, cos (?)

,

(27.14)

P'-' ((/,a;) =p(S^ ^

0? (cos 0)p<»' («5, cos 6)

,

(27.15)

{d,o^) =pij^ £ ^ d (cos e) (l/2,r)

Xj dtp p{d, sm e COS ip) . (27.16)

All these definitions are complete except for eq

(27.10). The factor v multiplying the right side

of eq (27.10) is chosen so that i6(A"2) = l, i.e., so that the integral over all directions gives the plane

isotropic source result, L{X). Since L{X), P{X),

and s{X, cos 6) are connected by the relations

L(X) = l/2\^~r P(X')=u\ (/(cos ^)s(^,cos e),

(27.17)

it turns out that f=(l/2)P(0).2^

The names assigned to the ratios eqs (27.8) to

(27.16) are less obvious than in the case of barrier

factors. In general the letter used, "L", "P",

etc., is determined by the type of barrier factor

appearing in the denominator. A more detailed

description of these quantities is reserved for the

next Section.

2' This is easily seen for monoenergetic sources, for wliicli

P<«)(AD = P(0)e-f^; si«i{X, cos e) = (cos 9)-ie-fJr/oo. », cos 9>0.

Inserting these functions into eq (27.17), and remembering that the unscattered component is completely dominant in the limit the value for u

is immediately obtained. But eq (27.17) holds for all X and for scattered as well as unscattered radiation, since both integrals represent the plane iso- tropic dose rigorously. Extension of the argument to include polychromatic

sources is therefore straightforward.

Notice that in sorne cases d and in some cases X is the penetration variable. The two are interchangeable, provided the proper conversion is used (eq (26.1)).

28. Description of Graphical Data

We next take each of the functions given in the

preceding Section, describe the physical situation

for which it applies most accurately, present one

or more graphs, and discuss important features

exhibited by these graphs.

(1) L{d), L{X): In eq _(27_.2), L(d) is defined

as the integral over all obliquities of the quantity

l(d, cos 6). Therefore Z(c?) is simply the total detector response at a distance d (in air) from an

infinite, plane, isotropic source, divided by the

total detector response at 3 ft in air from the same

source. L{X) is the same function except that

"distance from the source" is expressed in terms

of the effective mass thickness. Figure 28.1a is a sketch of the simple configuration for which this

quantity is directly applicable. The absorbing

medium is homogeneous, and the detector response

is isotropic. Figures 28.2a and 28.2b give typical curves of this quantity. Both figures show that near the source plane the radiation intensity falls

rapidly with increasing distance from the source. This is characteristic of gamma rays having initial

directions nearly parallel to the source plane (see

fig. 26.1). After considerable penetration, the

trend of the curves is nearly that for gamma rays

which leave the source plane perpendicularly (see

fig. 26.3).

(2) S{d): The definition of this quantity given

in eq (27.3) differs from the definition of L{d) in

regards to the limits of integration. Only obliquity angles greater than 90° are included. Since

gamma rays with these obliquities are traveling

towards the source plane, they have been backscattered. Thus this function gives the total detector response to skyshine radiation at a distance

d in air from an infinite, plane, isotropic source in a homogeneous medium. Figure 28.1b sketches

the physical situation. Figure 28.3 gives a curve

of S(d), obtained from the data of figure 26.1.

The intensity of the backscattered radiation can

be seen to change rather slowly with distance from

the source out to perhaps a mean free path of the source radiation. As indicated on the graph,

.S(O) =0.098.

(3) S'(X) : This quantity (eq 27.4) is a ratio of

integrals and is unity for A''=0. Both integrals

sum over negative cosd, i.e., over gamma ray slant sources of radiation which travels initially away

from the detector. The configuration is sketched

in figure 28.1c, which shows a plane source emitting

gamma rays isotropically in an upward direction

into an infinite homogeneous medium. The detector is below the source plane, separated from it by an effective mass thickness X. Radiation can

be backscattered to the detector, giving a response

measured b}' the function S'(X). The normalization to unity at X—0 is for convenient use as

622827 0--62——3
a reduction factor in the applications. Data are given in figure 28.4. The curve bears a considerable similarity to s{X,0) as given in figure 26.5;

but this is not sm^prising, since the grazing incidence com.ponent of the source contributes strongly

to this function.

(4) P(d), P{X)-. The integral in eq (27.5) includes all obliquities, so that P{d) relates to the

total detector response at a distance d from, a point

isotropic source in an infinite homogeneous medium (see fig. 28. Id). Results obtained from, the

data of figure 26.2 are given in figures 28.5a and

28.5b. The scale has been chosen to simplify the use of the curve in determining detector response

due to a small plane area (A) of contamination, as discussed in Section 26

D/Do^P(d)A/4wd'. (28.1)

(5) The only difference between P'"(X) and

P{X) is omission of unscattered gamm.a ray contributions to the integrand. Thus P^^''iX) re- lates to the total detector response due to scattered gamma rays. Graphs of this function and

also of P<"' (^)=F(Z)-P<«(X) are given in

figure 28.6, as obtained from, the data which also

yielded figure 26.2.

(6) W{X,d) : This is a more complicated quantity than those preceding. Figure 28. le is a sketch of the relevant physical situation. The

angular distribution of radiation incident on a

vertical wall is assumed to be the same as if the wall

were not present. This distribution, Z(d,sin6cos^),

corresponds to a distance d in air from, an infinite plane source in a homogeneous m.edium.,

but with directions referred to a polar axis perpendicular to the vertical wall (i.e., parallel to the source plane)." Assuming the spectrum, incident

on the wall to be that generated by the prim.ary source on the ground, an integral over all oblique

angles of incidence is made, each weighted according to our information about oblique penetration

(fig. 26.4). The result approximates the total detector response at a height d above the source plane and with a thickness X of barrier material

between detector and external barrier surface.

Errors in this calculation arise from, inaccuracies

in the angular distribution, changes in the incident

spectrum, as the scattered radiation com.ponent becomes m.ore important, and incorrect correlation

of spectrum, with direction. In general, this function is expected to represent the vertical wall barrier factor accurately for d sm.all compared with a mean free path of the source radiation, i.e., (Z^IOOO

ft. Data are given in figure 28.7. The curves have a m.aximum. value of W{0,d) ~{ll2)L{d),

corresponding to radiation froni. halj an infinite

plane, namely the half beyond the wall. Notice

that the bottom curves are not expected to be

reliable.

" Notice that l(d,cosQ) becomes Z(d,cosecosai+sinesinacos<^>) relative to a second polar a.xis inclined at an angle a to the original axis, with it> measured

relative to the plane of the two reference axes. When a= jr/2, this reduces to

Ud,sinflcos0).

(7) Lc{d,(ji),Lc{X,o3): The significance of Zc(X,w)

is illustrated in figure 28.8a: A circular area of

fallout has its center directly opposite a detector

and separated by a layer of air of thickness d and

equivalent mass thickness X. Noting eq (26.4),

together with the relation cosSinax=l — w, we

represent the detector response due to the circular area in the form

fx/a-o.) ^X' / X \ D/Do=l/2j^ ^PiX')=L {X)-l(j^J •

(28.2)

Equation (27.8) is the ratio of the response due to

the circular area to the response due to fallout on

the whole plane. Data corresponding to penetration in water-equivalent materials are given in fig- ure 28.9. While this quantity is easily obtained

from figures 28.2a and 28.2b by differencing, and

is included in the data of figure 28.18, the curves

of figure 28.9 are convenient for comparison

purposes.

(8) Laid,u), La{X,o>): The integral in eq (27.9) is over a cone of obliquities of incidence on the detector.

Thus, this ratio expresses the reduction in detector

response which occurs if an isotropic detector

(separated from an infinite plane source of fallout

gamma rays by a distance d in air, or a corresponding barrier thickness X) is replaced by a detector

responding only to gamma rays incident within a

particular cone of directions, as indicated in

figure 28.8b. The response cone has aperture

^max and subtends solid angle fraction w at the

detector, with w=l — cos^max- Curves of this

ratio are given in figure 28.10, as obtained

from the data of figure 26.1. Notice that these curves go to unity only for i.e., including

backscattered contributions for which l^u<2,

and amounting to ~15 percent.

(9) L,{d,w), U{X,oi)-. The integral of eq (27.10)

is over a cone of initial source obliquities. The

function Lb{d,u)) therefore expresses the fractional

reduction in detector response occurring if an

(infinite plane) isotropic source is suddenly

allowed to emit radiation only into a limited cone

of directions about the perpendicular toward the

detector (see fig. 28.8c). The cone of emission

has an aperture ^max defined by cosdmax — 1 —

where co is the solid angle fraction of the cone of

emission, subtended at the detector. Graphs of

Li,, obtained from the data of figure 26.4, are given in figure 28.11. These bear a noteworthy

resemblance to the curves for Lc and La. As in

the case of figure 28.10, the curves go to unity only

as w^2, although figure 28.9 doesn't give data

for the region of emission in directions away

from the detector (i.e., w>l).

(10) Equation (27.11) defines Wa{d,u>) as a ratio

whose numerator is an integral over a cone of

obliquities of incidence on the detector. The cone

axis is parallel to the source plane and the cone

angle (^max) is related to the solid angle fraction

30
subtended by the aperture by the usual relation

cos0max= (1 — w), as in figure 28. 8d. The denominator of eq (27.11) has the same interpretation but with cone angle d^^-^=Tr/2, so that' all radiation incident on one side of the detector is included in the response. Curves of Waid,ui) are presented in figure 28.12, as obtained from the

data of figure 26.1 for penetration in waterequivalent materials.

(11) In some cases one wants to know the

amount of skyshine radiation passing through a

vertical aperture. The function Wi,{d,ui), defined

by eq (27.12) , is presented here because of possible

applications of this type. The definition of is like the definition of Wa except for integration

limits which include only skyshine contributions.

Thus Wi refers to a half-cone of skyshine radiation of aperture ^max about an axis parallel to the

source plane, with cos0inax= 1 — where co is the

solid angle fraction subtended by the whole cone

(see fig. 28.13a). The denominator is the total

(a) (b)

^< )» -K » >» -X -X ^ H H

(d) (e)

Figure 28.1. Simple detector-source-medium arrangements:

(a) Isotropic detector, plane isotropic source; (6) inotropic detector shielded on the sxde toionrd the source, plave isotropic source; (c) isotropic detector, source "isotropic"

only in directions pointing away from detector; (d) isotropic detector, point isotropic source; (e) isotropic detector and plane isotropic source: The radiation field at heiqht d above the primary source is taiten as a new source at a penetration distance X to the right of the detector. In all cases the calculations are Jor infinite homogeneous media, and all cases but (.d) also correspond to media with plane density variations.

31
d,ft.

0 1000 2000 3000 4000 5000 6000

d.ft.

FiGTJRE 28.2a. Plane source of fallout radiation: The (isotropic) detector response ratio D/Do as a function of height d

air above the source plane. {H2O, 1.12 hr fission. See also Jigs. Bll and B12.)

32
X

X.psf

Figure 28.2b. Plane source of fallout radiation: The (isotropic) detector response ratio DI as a function of effective mass

thickness separating the detector and the source plane. {H2O, 1.12 hr fission. See also figs. BIS and BI4.)

33
34
X,psf

Figure 28.4. Attenuation curve for radiation backscattered from a plane source isotropic over one hemisphere only, corre- sponding to skyshine radiation incident on the ground. {H2O, 1.12 hr fission. See also figs. B17 and B18.)

35
d,ft.

FiGrRE 28.5a. Point source oj fallout radiation: P(d) as a function of distance d in air between detector and source. (H2O,

1.12 hr fission. See also figs. B19 and B20.)

36
X,psf

Figure 28 5b Point source oj fallout radiation: F(X) as a function of effective mass thickness between source and detector. {H2O, 1.12 hr fission. See also figs. B21 and B22.)

37
38
00 200 300 400

X.psf

500 600

Figure 28.7. Detector response ratio for effective mass thickness X separating the detector from a plane fallout source

1 angular distribution — I d<t>lid,sindcos<t>). {H2O, 1.12 hr fission. See also figs. B25 and B26.)

39
(a) (b)

(c)

Figure 28.8. Simple detector-source-medium arrangements:

(a) Isotropic detector, isotropic circular source; (6) conical detector pointed towards the source plane, plane isotropic source; (c) isotropic detector, conical source pointed towards the detector; (d) conical detector pointed parallel to the source plane, plane isotropic source. All configurations correspond to an infinite homogeneous medium

and the latter three to an infinite medium loith plane density variations.

40
Figure 28 9 Geometry factor describing detector response due to circular plane sources of fallout radiation, as shown in

figure 28.8a. {H2O, 1.12 hr fission. See also figs. B27 and B28.)

41
42
Hi

Figure 28.11. Geometry factor for response of an isotropic detector adjacent to a plane source emitting radiation in a limited cone of directions, as sketched in figure 28.8c. {H2O, 1.12 hr fission. See also figs. B31 and BS2.)

43
Figure 28.12. Geometry factor describing detector response to radiation incident in a limited cone of directions about an axis

parallel to the primary source plane, at height d, as shown in figure 28.8d. {H2O, 1.12 hr fission. See also figs. BSS

and B34.)

44
(a)

(c)

FiGUBE 28.13. Simple detector-source-medium arrangements:

(o) Detector which responds to the upper halj of a cone of directions with axis parallel to a plane isotropic source: (b) conical detector pointed away from a plane

isotropic source; (c1 conical detector pointed towards a point isotropic source; (d) conical source pointed 90° awayfrom the source-detector line, point isotropic source. All media may be considered infinite and homogeneous; and the first two may be considered infinite with plane density variations.

622,827 0—^62—4

45
Figure 28.14. Geometry factor describing detector response to skyshine radiation incident in a limited cone of directions about an axis parallel to the primary source plane, as sketched in figure 28.13a. This graph is accurate for d 200 ft of air. {H2O, 1.12 hr fission. See also figs. B35 and BS6.)

46
w

Figure 28.15. Geometry factor describing detector response to skyshine radiation incident in a limited cone of directions about an axis perpendicular to the primary plane, at height d above ground, as sketched in figure 28.13b. (H^O, 1.12 hr fission. See also figs. BS7 and B38.)

47
w

Figure 28.16. Geometry factor describing detector response to scattered gamma rays from a point source of radiation, striking

the detector in a limited cone of directions about the line from source to detector, as sketched in figure 28.13c. {H2O, 1.12

hr fission. See also figs. B39 and B40.)

48
Figure 28.17. Geometry factor describing detector response to fallout ganmia rays from a point source, striking the detector in a limited cone of directions about an axis perpendicular to the line from source to detector, as sketched in figure 28.13d.

(H2O, 1.12 hr fission. See also figs. B4I and B42.)

49
3

50
lUUU

51
52
skyshine intensity. Data for this function obtained from the data of figure 26.1 are given in

figure 28.14. Notice that co=l corresponds to an

aperture passing skyshine from one side only, i.e.,

only halj the total skyshine; this is reason for

the factor 2 in eq (27.12). The variations with d were too small to warrant separate curves for

different d values.

(12) Sa(d,oo) : This function is defined by an

integral over a cone of obliquities of incidence

pointed away from the source plane (see eq

(27.13)). Thus only skyshine contributes to the

detector response. The relevant configuration is sketched in figure 28.13b. The allowed cone has

aperture angle ^max and subtends solid angle fraction a)=l —cos^max- The denominator is the

response to all backscattered gamma rays incident on the detector. Data, given in figure 28.15,

have been obtained from the data of figure 26.1.

Note the steepness of the curves near co=l, ex- pressing the strong component of the gamma ray

directional distribution in directions parallel to

the source plane.

(13) Paid,u) and P^''(rf,w): All preceding ratios are for plane sources. The Pa ratios defined by

eqs (27.14) and (27.15) are like the La ratios, but

with reference to a point source. The numerator

is the response of a detector to scattered radiation

incident within a cone of directions about the

radial axis from detector to source (see fig. 28.13c).

The denominator is the total response of an

isotropic detector to the scattered radiation.

Data obtained from figure 26.3 is given in figure

28.16. Note that the curves go to unity only

for w^2, so that the backscattered contributions

are not included in figure 28.16. The following

expression relates Pa and ' •

Pa(X,o:) [P«> iX) -f P<^' (X)Pi" {X,c.)].

(28.4)

This merely expresses total detector response as the sum of responses due to scattered gamma rays

and to unscattered gamma rays. The latter are always concentrated along the radial line connecting source and detector.

(14) P^''(X,co): For completeness we include

the point source analogue to the Wa (plane source)

function. The function is defined by eq (27.16)

and illustrated by figure 28.13d. It has in the

numerator the response of a detector to scattered

gamma radiation incident within a cone of

directions perpendicular to the radial line from

detector to source. The aperture angle ^^ax

is related to the solid angle fraction w by

cos9max=l — Thc denominator represents the

total response of an isotropic detector to scattered

radiation, while the numerator never represents

more than half, hence the factor of 2. Data is presented in figure 28.17, as calculated from the

basic data of figure 26.3.

(15) L(X)L,{X,w), L{X)La{X,o>), L{X)L,{X,o^y.

These combinations of barrier and geometry

factors come up so frequently in applications that

contour diagrams have been prepared to permit

direct determination of the product. Figures

28.18, 28.19, and 28.20 present the diagrams.

Radial lines have been superimposed and calibrated to permit accurate interpolation. Use of

French curves facilitates this interpolation.

VI. Elementary Structure Types

29. Comments

A number of elementary barrier arrangements

have been the subject of special studies. Because

of their simplicity they permit a fairly complete

analysis, both experimentally and theoretically;

and they appear over and over again in various

combinations in nearly all structures. One might

say that complicated structures are a composite

of these simpler types. As an example, the blockhouse described already in Section 20 combines

vertical walls (Section 33) with a fallout-covered

shelter (Section 32).

In this part we discuss these configurations with

the help of the data and approach already presented. Our intention is to describe them schematically in a manner which leads to numeiical

values for the detector response. At the same

time we hope to give an appreciation for the nature

for the approximations involved, so that a feeling

for size of possible errors can be achieved.

Because this Part is intended to be illustrative

rather than exhaustive, we limit the examples to

circular source and barrier shapes. The general'

ization to arbitrary detector position relative to

arbitrary rectangular shapes is given in Part VII

.

30. Density Interface

The simplest configurations of real importance

comprise a radiation source at a density interface,

with the detector placed some distance above in

air (see fig. 30.1a). One may visualize a vast,

smooth, level field supporting the radiating

material.

(1) Air-air, infinite plane source case: In the

simplest of these cases, the interface is smooth

and separates two semi-infinite regions, one of

air at standard pressure and the other of compressed air of the same density as earth. We

have already commented that reference to "compressed" air has importance only as aid to visualization, if the source covers the plane uniformly.

This is because of a scaling principle which

guarantees that compression of the material into

plane layers of different density does not alter

53
®

Figure 30.1. (a) Plane, isotropic fallout source at a simple

density interface; (b) fallout source on rough ground.

Figure 30.2. (a) Detector above an area cleared of fallout;

(b) detector below a finite source on the ground.

the radiation flux anywhere, in intensity, spectrum,

or angular distribution, if "distance from the

source" is measured in mean free paths, lbs/ft^

gm/cm^, effective mass thickness, or other indicators of the amount of material between source and detector.^** Thus, this case is just the one

to which calculations of L(X) and L(d) apply

precisely, and the detector response as a function

of distance from the source is simply

D/Do=L{d). (30.1)

28 Note that this scaling principle does not hold for non-uniform or finite source spatial distributions.

Figure 30.3. Geometry of a defector above a circular fallout- free area, when the fallout elsewhere is considered to be buried beneath a hypothetical layer of thickriess A.

(2) Air-earth or air-concrete case, infinite plane

source: If instead of the location at an interface

between air and compressed air the source is at an interface between air and earth, or air and

concrete, a modification of the detector response

is expected. This is because the interaction

probabilities for gamma rays in earth and concrete differ from those of air, for low energy gamma

rays. This has the consequence that near an

air-earth interface the flux of low energy photons

falls a little below the flux near the source in an

air-air case. The effect is small, a few percent,

and is due to photons which in earth or concrete

undergo absorptive interactions at higher energies

than in air.

This same effect is expected for the flux penetrating downward into earth or concrete. By

and large, earth and concrete can be treated as

if they were equivalent to water in penetration

properties, because the differences in interaction

probabilities give differences in detector response

which are significant only for very large penetrations, or for very low energy photons. For our

purposes it is sufficient to write

Z)/Z)o«0.9Z(JO, (30.2)

within or at the surface of the semi-infinite earth

or concrete region, if the L{X) data is for penetration in H2O, as in figure 28.2b. The factor

0.9 allows for the lower flux of low energy photons

in concrete, and should be omitted if the data

is for penetration in concrete. For an experiment

of this type see [36].

(3) Ground roughness : If the interface between

earth and air is rough, rather than smooth, a

reduction in intensity is to be expected. The

reason for this can be appreciated with the aid of

figure 30.1b. That part of the radiation source which falls into "pockets" cannot contribute as much to the radiation component nearly parallel

to the source plane. The gamma rays emitted

nearly parallel to the source plane must penetrate

a substantial amount of dense material before

emerging into the air, and are correspondingly

reduced in intensity.

54
One expects all earth surfaces to be rough to

some extent; but this effect will be larger in the

case of a ploughed field than a carefully smoothed

baseball diamond. Investigation of ground

roughness effects is still in a fairly rudimentary

state. We note, however, that raising the detector height above a smooth interface has a similar

effect of reducing the contribution of the unscattered "grazing" component through attenuation

along the elongated path from the point of generation. Because of the importance of some type of

rule for obtaining intensities and angular distributions corrected somehow for ground roughness,

we tentatively adopt the simple procedure, suggested by Ksanda, Moshkin, Shapiro, and others,

of writing

DIDo=L{d+T), (30.3)

where r is a constant which is characteristic of the

local terrain [37]. Values of r are not available

for widely different terrains. At least one case has been investigated experimentally, however,

and for this case it has been calculated that ts=40

ft. 2^ The experiments were performed in an open

Nevada field without special treatment either to

increase or decrease roughness. Equation (30.3)

has no real justification beyond the comment that

introduction of the parameter r affects both intensity and angular distributions the way ground

roughness affects them. The argument can be

made to sound shghtly more convincing by describing the fallout as effectively buried beneath

a thin layer of earth.

(4) Finite plane sources: The preceding discussions assume that the source covers the plane of

the interface uniformly. If the region of contamination does not cover the source plane, new and

complicating aspects to the problem appear.

Consider, for example, a circular clearing in a

fallout field, with the detector above the center of

the field (fig. 30.2a). Due to ground roughness,

the detector response will be far from the value

expected for a smooth source plane. Hardly any

unscattered radiation will reach the detector

because it will be blocked on its path along the

ground by irregularities.

Another complicating feature of this type of

problem involves the scattered component of the

radiation. Gamma rays which travel long distances on a path which is always not far above

ground have a large probability of being deflected

into the ground. When this happens, distances

between successive interactions shrink from hundreds of feet to inches. If the photon is to travel

any significant distance farther, it must re-emerge

into the air. This effect reduces the detector

response in just the sort of configuration illustrated in figure 30.2a by an amount which is not

well known but which depends on the radius of

the clearing. Studies of this "path foreshortening" effect have been made, both theoretically

and experimentally; but results for clearings are not available.

2' C. M. Eisenhauer, private communication.

In spite of these difficulties it is possible to

obtain estimates of a sort of the detector response

to radiation above a clearing, taking ground

roughness into account. To accomplish this we

follow consistently the schematization in which the

effect of ground roughness is approximated by

performing a calculation at an increased height

above ground. This implies that ground roughness has effects similar to those due to "burying"

the source under a thin layer of earth, as already

mentioned. In figure 30.3 a detector is pictured

at a height d above ground. The source is buried

under a layer of earth which has effective mass

thickness A, equivalent to a layer of air of thickness T. The clearing is circular, with the edge of

the source at a radius d/tsnxd out from the point

below the detector. The distance in air from the

detector to the edge of the contamination is d/cosd, and the additional penetration through the

covering layer is A/cos0, which is equivalent to a distance in air of t/cosO. The penetration from

the edge of the source to the detector is thus

d-\-T equivalent to ^^^^

feet of air. If we use this penetration distance in the expression for the detector

response above a cleared area, we obtain the

desired approximation:

D/Do^L(^^y (30.4)

where co=l —cos0. No allowance is made in this

expression for path foreshortening.

One advantage of this simple approximation is the possibility of using it to calculate anisotropic

detector responses above clearings; but we don't

pursue this here.

The accuracy of our predictions of detector

response above a clearing is limited by lack of

knowledge of ground roughness and path foreshortening; but in a related problem which is perhaps of greater practical importance these

complicating effects are not operative. This

problem is the calculation of the detector response

in earth (or concrete) below the source plane, when

the source is of limited extent. Figure 30.2b

illustrates this case. Radiation from the source

traveling initially upward goes such long distances

througli tlie air that it doesn't come back to the

same locality. Thus the detector response will

be lower than in cases which are similar but

without the density interface. But this is a small

effect even when the barrier thickness between

source and detector is not very great. It is possible

to determine the detector response fairly accurately'

from the expression

D/Do=0.9LiX)Lc{X,o:), (30.5)

where w is tlie solid angle fraction of the source as seen from the detector. (The factor .9 is a rough

correction when L{X) and Lc{X,oi) represent H2O

penetration data.) This expression can be seen to

55
include a contribution due to radiation just mentioned as being, in fact, absent. Thus it will be

an overestimate, particularly for X~0.

EXAMPLE 1: Calculate the reduction factor

due to ground roughness, for a detector 3 ft above

a contaminated area, if the "roughness" parameter T has a value of 40 ft for this type of surface.

From eq (30.3) and figure 28.2a we obtain

L(3+40)=0.52. ANSWER.

EXAMPLE 2: How high above the ground must

a detector be raised to reduce its response by a

factor of 1000?

Applying eq (30.2), but omitting the factor 0.9

because of the height above the interface, we see from figure 28.2a that Z(rf) =0.001 for rf =2950

ft. ANSWER.

EXAMPLE 3: Assuming a ground surface such

that T =40 ft, calculate the radius of a clearing

sufficient to reduce the detector response to

Do/1000, at a height of 3 ft above ground.

Applying eq (30.4), using figure 28.2a, we find

that L f7^V0.001 for -^=2940 ft. From

\1 — CO/ 1—0)

this we see that 1 —co~3/p=43/2940, giving p =

3(2940)/43^200 ft. ANSWER.

This last should be considered only the crudest

sort of estimate, in view of the state of knowledge

of ground roughness effects.

31. Foxhole Problems

A typical foxhole configuration is shown in

figure 31.1. Radiation generated at the groundair interface can enter the aperture even though

it faces away from the primary source, by backscattering from the air. This backscattered component of the radiation is also called the "skyshine". Normally the aperture will permit

sk\ shine radiation to strike the detector only if it travels in an allowed cone of directions as indicated in figure 31.1 by the dashed line on the right.

The configuration bears a close correspondence

to that of figure 28.13b, in which the detector is

/

/

/

/

Figure 31.1. Detector on the centerline oj a cylindrical

foxhole.

in an infinite medium but is sensitive only to radiation incident within an allowed cone. We therefore make estimates of detector response in foxholes by means of the functions S{d) and Sa{d,u),

which represent infinite medium data relevant to

the case sketched in figure 28.13b.

It should be noted that in the foxhole configuration there will be contributions to the detector

response due to radiation backscattered from the

bottom and sides of the foxhole. This introduces

a correction factor of magnitude ~ 1.2. Taking

this into account, our exj)ression for the skyshine

detector response becomes

D/Do=l.2S{d-\-r)Sa{d+T,co), (31.1)

where d refers to height above ground, r is the

ground roughness coefficient and w is the solid

angle fraction of the aperture as seen from the

detector.

Note that foxhole-type problems can occur at

great heights as, for example in the case of a skylight in the roof of a tall building. (See Section

35.)

One complicating feature of the foxhole geometry is the possibility that radiation may go directly

from fallout near the "lip" of the hole to the detector. Strictly speaking, eq (31.1) applies only

if the radiation source has been removed to such a distance from the edge that it cannot penetrate

through the earth to the detector.

Attempts have been made to ascertain whether

direct radiation from fallout near the lip is significant in comparison with the skyshine radiation.

This will certainly depend upon the size of the foxhole. The coQtribution "through the lip" should

increase as foxhole size decreases, with the ratio of

lip contribution to total response approaching

unity as the foxhole size approaches zero. There

should also be effects due to foxhole shape: An

elongated foxhole, or one with the edges of the lip

rounded off should differ from a cylindrical foxhole, or one with a sharply defined lip. It appears

that for the ideal case of a foxhole large enough to

hold a man, with vertical walls and a sharp lip, the

direct radiation penetrating through the lip is comparable in importance with the skyshine.^°

Calculations of the lip contribution involve integrals of the type (see fig. 31.1)

JCOS^min

d{cosd)l[Xid),cose], (31.2)

which sums the contributions from radiation

emerging into the foxhole at various depths (X)

below the ground siu-face, assuming that the directional distribution as a function of depth is not

affected by the interface at the foxhole wall. Eq

(31.2) appHes only to a detector on the centerline

of a cyhndrical foxhole; but the generalization to

other "cases is fairly obvious and, in fact, great

variations are not expected (a) if the detector is

3» C. M. Eisenhauer, private communication .

56
I moved off-center or (b) if the foxhole is not cylindrical, so long as the solid angle subtended at the

II detector by the aperture is held constant.

[ For most purposes the integral of eq (31.2) can

, . , , . • 7/ «N d(cosd) j„

t be approxunated by writmg d{cosd)= — dX

and assigning ^^^j^^^ ^ constant value corresponding to the edge of the lip. This leads to an expression for Dc, dose through the lip:

Z^^/A ^

" ^ ^ ~"^^^

" (X, 1 - CO)

, (31.3)

in which p is the density of the ground, co =

1 —cos^min is the solid angle fraction of the aperture

and d is the depth below the ground surface.

Values of the integral in eq (31.3) can be obtained

from figure 32.2. Clifford, Carruthers, and Cunningham have very recently reported foxhole ex- periments [38].

EXAMPLE: Calculate the protection factor provided by a cylindrical foxhole 3 ft in diameter and

4 ft deep at a point on the centerline and 1 ft above

the bottom of the foxhole.

The solid angle fraction w= 1 -3/(3^+ (3/2)^)^

=0.106. We obtain the skyshine contribution

using this value and X=0 in eq (31.1). From

figure 28.3 we obtain »S'(0) =0.098, and from figure

28.15 we obtain <S„(0,0. 106) -0.044. Thus DjDo = 1.2<S(0)^a(0,0.106) =0.0052.

We calculate the lip contribution assuming that

the densitv of earth is p=100 pcf. From figure

32.2 we obtain using cose= 1 — 0.106 =0.894 the

value 13.1 for the integral in eq (31.3). Thus

O,/Z,,=i«:lM«L^)(:3.1)«0.008.

Thus the two contributions are of the same order

of magnitude and the total response is given bv

Z>/7)o«0.013. ANSWER.

32. Shelter Covered with Fallout

Figure 32.1 is a sketch of a typical shelter covered with fallout. A different example of a

similar configuration is the blockhouse illustration

in figure 20.1, which has fallout on the roof. In

both cases the radiation penetrates through the

roof slab almost as if the source were imbedded

in an infinite medium. The emerging flux and

directional distribution both can be well represented by infinite medium calculation, and we

make use of the analogy between figure 32.1 and

the directional detector illustrated in figure 28.8b.

Data for the latter case are employed.

The detector in figure 32.1 is exposed to backscattered radiation and to a total spectrum determined for concrete ratlicr than for air. As in the

foxhole configuration one sliould presumably inFiGUHE 32.1. Detector on the centerline of a cylindrical

fallout shelter.

elude a backscattering correction (1.15) and a

correction (0.9) which takes into account the lack

of low energy photons in concrete. These two

corrections almost neutralize one another; and

since our schematization is not expected to be

accurate to 5 percent, we omit them both and write

D/Do=L(X)La{X,w), (32.1)

where X is the effective mass thickness of the

roof slab and w is the solid angle fraction of the

roof as subtended at the detector.

It can be seen in figure 32.1 that radiation can

contribute to the detector response by entering

through the basement walls rather than through

the ceiling. To take account of this one should

include a correction very similar to the "foxhole

lip" correction discussed in the preceding Section.

In fact, almost identically the same argument can

be made, and it leads to an expression more general than eq (31.3), namely

Z>,/Z}o= "^^^~"^^i^~"^ rdX'l(X',l-<.), (32.2)

pd Jx

where w = l — cos0 and d is distance below the roof

slab. Values for this integral are to be found in

figure 32.2. Fortunately, this correction tends to

be relatively small if the slab is moderately thick.

Experiments on structures equivalent to source- covered fallout shelters have been performed by

Clarke, Batter, and Kaplan [39].

EXAMPLE 1: Calculate the reduction factor in a

cylindrical basement 10 ft in diameter, covered by

a roof slab of concrete 1 ft thick, and with the

detector positioned on the centerline 9 ft below

the roof slab.

For this case, w= 1 -9/(92 +5^)^^ =0.126. This

value of o), and a value J^=144 psf can be inserted

into eq (32.1). From figures 28.2b and 28.10, we

then obtain L(144) = 0.0067 and i:a(144, 0.126) =

0.257, so that Z)/Do= (0.0067) (0.257) =0.00172.

ANSWER

Tt is of interest to calculate the correction due

to radiation entering the compartnu>nt through

57
COS 6

Figure 32.2. Data for calculating "lip penetration" in foxholes and the wall contribution in shelters. {H2O, 1.12 hr fission. See also figs. B49 and B50.)

58
the walls instead of the ceiling. From figure

32.2 we obtain, using cos0=l —0.126 = 0.874, the

value (0.69) (0.73) - .5 for the integral in eq (32.2).

Thus,

0./Z,.„ 0^«1KO||(1^) (0.6) »0.00008.

The correction is about 5 percent.

33. Vertical Walls

The case of a vertical barrier between a detector

and a fallout source on the ground is well illustrated by the vertical wall in figure 20.1. Radiation from the source on the ground penetrates

upward into the air, so that at different heights

about the ground there are directional distributions

similar to those given in figure 26.1 for an infinite

homogeneous medium. The actual distributions are only "similar" because of modifications due

to ground roughness and the perturbing effect

of air-wall and air-ground interfaces.

Radiation from fallout penetrates vertical walls

with surprising ease. This is because penetration

into the wall depends on incident obliquity, i.e.,

the angle of incidence relative to an axis perpendicular to the wall (and therefore parallel to the

source plane). The main feature of the radiation

incident on the wall can be appreciated by making

a simple approximation. Referring to figure 26.1,

note a high concentration of radiation with cosffs^O,

thus travelling nearly parallel to the source plane.

Suppose all radiation should travel parallel to the

source plane, with directions uniformly distributed

otherwise. If obliquity angles 9' are measured

relative to an axis which is itself parallel to the

source plane, the corresponding distribution,

l{d,cosd'), would be uniform in 6' . In other words,

so that

l{d,cosd')dicosd') xde',

l{d,cos6') oc sine'

(33.1)

This illustrates that near the ground (small d)

the actual function l{d,cosd') should have a peak

value for 6' near zero. Since decreasing 6' corresponds to increasing penetrability, the con- sequence is that fallout radiation from the ground

and incident upon vertical walls is very

penetrating.

For our purposes it seems reasonable to neglect

the perturbation in the radiation field due to

air-wall and air-ground interfaces, and to take

ground roughness into account in the manner

indicated by eq (30.3). Thus we apply to this

problem the data represented by the function

W{X,d) ; and we write for the detector response

at height d above ground and behind a thickness

X of wall,

DIDo=0.9W{X,d+T). (33.2)

The factor 0.9 corrects for the deficiency of low

energy photons when the material adjacent to the

detector is concrete rather than air, water, or wood. It should be omitted if the calculated

W{X,d-^T) already assumes a concrete wall.

If backseat tered photons should make no contribution, the factor should probably be closer to

0.85; but in general backscattering will occur, even for a configuration like the blockhouse of

figure 20.1, in which there is no material immediately behind the detector (Position B).

Thus the corrective factor 0.9 is to be preferred.

Equation (33.2) does not apply to the detector

response some distance behind a wall of limited

size. For example, the detector response at B

in figure 20.1 due to radiation from the adjacent

wall is given by eq (33.2) but the detector response

at A due to radiation from the same wall contains

a further reduction, as discussed in Section 20.

At present we do not have directional distributions

for radiation emerging from such a vertical wall, so that at best we can only prescribe rough guesses

for this reduction factor.

Various methods can be used to construct

functions for estimating the detector response

behind a vertical wall. We can, for example,

interpolate between Waid,u), which is reasonably

accurate in the limit of zero wall thickness, and

f*a^'H°° ! '^)) which represents an upper limit to

the geometry factor in the limit of very thick

walls. To accomplish this we assume that

radiation not scattered in the walls contributes

according to the thin-wall function, while radiation

scattered in the walls contributes according to the

thick-wall function. Assuming the proportion of

unscattered gamma rays to be given roughly by

the ratio P'°'(Z)/P(X), which we designate

b{X), we arrive at a function Wai defined by

Wai(XM=b(X)Wa{d,o^)

+ 1.15[l-6(Z)]P<.<*>(co,co). (33.3)

The factor (1.15) in this expression is to normalize

Pa^" to unity at co=l, to make it comparable

with Wa.

According to figure 28.16, the function Pa^'^ id,w)

varies slowly with d (or X) ; and we use the

bottom curve (for c^=1000 ft) to represent

Pa'-^\<^,u) in calculations of Wai without ex- pecting to introduce errors more serious than

those inherent in eq (33.3). In general, eq

(33.3) is expected to give conservative estimates

of the geometry reduction factor because Pa'"

overestimates the factors expected for thick walls.

Using eq (33.3) in lieu of a better estimate of

the wall geometry factors, we represent the

detector response in the form

D/Do=0.9WiX,d+T)Wai{X,d+T,^), (33.4)

where d and X are height above the ground and

wall thickness, respectively, and co is the solid

angle fraction subtended at the detector by the

wall, and the factor 0.9 estimates the correction

59
of H2O wall data to concrete wall data. Equation (33.4) is intended to apply only to walls

nearly circular (or square), with the detector opposite the center of such a wall. It may certainly

be greatly in error if applied indiscriminately to

eccentric wall shapes and to off-center detector

locations. (Note that a correction for backscattering has been included implicitly.)

EXAMPLE 1: How thick must a vertical wall

be to reduce the detector response immediately

behind the wall to 0.00 IDq, if the front of the

wall faces a fallout field 6 ft below the detector

and having a ground roughness coefficient r =40 ft?

Making use of eq (33.2), we examine figure

28.7 to find that value oiXfor which W{X,4:Q ft) =

0.001/0.9. This value turns out to be approximately X=23S psf, which is just under 20 in.

of concrete. ANSWER.

EXAMPLE 2 : If the wall of the preceding example is square, 12 ft on a side, and if the detector

is 12 ft behind the wall, opposite its center, what

wall thickness would be required to reduce the

detector response to 0.00 IDq?

This calculation is considerably more complicated. To begin with, it is necessary to calculate

the solid angle fraction subtended by the wall at

the detector. This can be most easily done by

the method to be outlined in Section 41 ; but it could also be done by replacing the square with

a circle having the same area. Using the method

of Section 41, we note that for squares, e = l, and

for a distance from the square equal to the length

of a side, 77 =2. From figure 41.2 we then read

off the value co=.13. Inserting this value, together with the value (d +T)=4:Q into eq (33.4),

we select several values . of X which we expect to

bracket the desired value, namely X=144, 192,

and 240. For these values we obtain from

figures 28.6, 28.7, 28.12, and 28.16 the following

information:

X b(X) Tr„(46',.13) Pa'">(",.13) Ty„i(JSf,46',.13) WiX.iG') D/Do

144 0.182 0.18 0.46 0.47 0.0085 0.0040

192 .152 .18 .46 .48 .003 .0014

240 .127 .18 .46 .49 .00105 .00051

Plotting these values for D/Dq on double-log paper

we find by interpolation that D/Dq=0.001 for

^5=^207 psf, corresponding to about 17 in. of

concrete. ANSWER

34. Light Superstructure with Shielded

Basement

Figure 34.1 is a sketch of a structure which is similar to the blockhouse of figure 20.1 except

that walls and roof are thin and the configuration

includes a shielded basement. Our primary interest is in the detector response in the basement ; and

radiation penetrating into the basement may have

originated either at the roof or on the surrounding

contaminated ground.

Because the roof is thin it is no obstacle to

radiation. Therefore the floor slab protecting

the basement is subjected to a beam of gamma

rays from the roof ; and this radiation is incident

on the floor slab along lines intercepting some

part of the roof. Penetration through the floor

slab proceeds as if there were a source located at

the slab which could emit gamma rays only in

directions corresponding to the directions of travel

of radiation from the roof. Moreover, the cone

of incident directions doesn't change significantly

within a few feet of any point on the floor slab.

For these reasons we consider the actual configuration to be in close correspondence with the infinite

medium arrangement of figure 28.8c, and we estimate the detector response by means of data from

figure 28.11. If we include a factor 0.9 to account

for suppression of low energy components in

concrete as compared with water or air when the

L(X) data is for penetration in water, the detector

response (roof contribution only) is obtained from

D/Do=0.9LiX)L^(X,c^). (34.1)

Here co is the solid angle fraction of the roof as seen from the opposite point on the other side of

the barrier from the detector, as shown in figure

34.1.

In figure 34.1 it is clear that lowering the

detector from its position adjacent to the slab

will not change much the response to radiation

from the roof because the floor slab will still subtend a large solid angle fraction. But it is easy to visualize configurations of this general

type in which the detector response is sensitive

to the distance from the barrier. (As a simple

example one need only reduce the length and

width of the structure pictured in figure 34.1

while keeping the height constant.) One may

wish to include another factor in the expressions

for D/Dq to account for this variation:

D/Do=L(X)L^(X,o,)L^(X,o:'), or (34.2a)

^L(X)L^{X,c^)P,(X,o,'), (34.2b)

depending on whether w is nearly unity eq (34.2a) or much smaller eq (34.2b). Here, co' is the solid

angle fraction of the protecting floor slab as subtended at the detector. The factor 0.9, which

corrects the data to concrete, has been omitted

from expressions eq (34.2) because backscattered

radiation has been omitted altogether through

the special normalization used for La and Pa',

this backscattered radiation gives an increase of

about a factor 1.15, thus slightly more than

canceling the 0.9 factor. If data for concrete is used, a backscattering factor is appropriate.

The second part of this problem has to do with

radiation which originates at the ground and penetrates "in and down." This is much more difficult

to analyze. We proceed by trying to account for

major factors one by one: If the heavy protecting

floor slab were absent the problem would reduce

60
Figure 34.1. Fallout on and around a light cylindrical superstructure covering a shelter. The radiation incident on the barrier slab is largely confined to a

cone of aperture 6.

to a fox-hole problem and the detector response

would be given by an expression like eq (31.1).

The floor slab introduces an extra attenuation,

and the real problem is to estimate this attenuation

as a function of slab thickness X. To this end

we note that radiation from ground contamination

must be scattered downwards to be incident on

the floor slab, and that such incident radiation is not greatly affected by the clear area covered by

the superstructure, for structures of average size.

Therefore, we expect this radiation to resemble

the skyshine both in intensity and directional

distribution. This suggests use of the function

S'{X) to estimate the reduction of the detector

response below that expected for a foxhole

geometry with no floor slab (see Section 28,

paragraph (3), and fig. 28.1c):

D/Do=S{d+r)S,id+T,c,)S'iX), (34.3)

where d, r, and w all have the interpretations of the

foxhole configuration.

Use of the function Sa in eq (34.3) is justifiable

if the fioor slab does not greatly modify the skyshine directional distribution, i.e., if the floor

slab is thin. If X is substantial and w isn't small,

it would be better to omit this factor.

EXAMPLE 1 : Consider a light, cylindrical superstructure rising 25 ft above ground level, and

having a diameter of 35 ft. Underneath this

superstructure there is a basement of depth 10 ft, as shown in figure 34.2. Calculate the protection

factor at the center of the basement floor with

and without a 6 in. first-floor slab {X=72 psf).

Assume a ground roughness coefficient t =40 ft.

Figure 34.2. Light superstructure geometry of Example 1.

First we calculate the roof contribution for the

two cases, using a,= 1-25/ V(25)^+ (17.5)^=0.181

and co' = 1-10/ V(10)'+ (17.5)2 =0.504, together

with A'=72 for the case with floor slab, and using

co=l-35/> (35)2+ (17.5)2=0. 106, a)' = l, and X=Q

for the case without a floor slab. From figure

28.20 we obtain L(0)Z6(0,0. 106) =0.023 and

L(72)L6(72,0.181)=0.0125. Using eq (34.2b) for

both cases we must next determine appropriate

values of Pa(A» : We note that P„(0,1) = 1, and that according to eq (28.4), Pa(72,0.504) =

(0.166)-^ [0.049 +0.1 18(0. 71)] = . 8. Therefore, the

roof contribution turns out to be

61

622)82,7 O—62
No floor slab : Z)/I>o= (0.023) 1 =0.023,

6 in. floor slab: D/Z)o= (0.0125) (0.8) =0.010.

Next we perform the "foxhole" type calculation of eq (34.3). The values S{46 ft) =0.068,

Sai^O ft, 0.504) =0.29, and -S' (72) =0.027 give

No floor slab : D/Do= (0.068) (0.29) =0.020,

6 in. floor slab : D/Do= (0.068)

(0.29) (0.027) =0.00054.

Combining the two contributions, we obtain

No floor slab: P=Z>o/i>= (0.023 +0.020) "' =23,

6 in. floor slab: P=Z>o/-D= (0.010 +0.00054) =95. ANSWER.

Note (a) that the floor slab is most effective

against radiation from the ground, and (b) that in

the absence of the floor slab approximately half

the detector response is from the ground

component.

35. Apertures

The analogy between gamma rays and visible

light may be used to clarify the effect of openings

upon the detector response within a structure.

Just as doors and windows permit light to enter

a room from outside, according to their size and

location, so also do doors and windows permit

radiation to enter when fallout covers the sur- rounding ground. In one respect this analogy

breaks down, however: Radiation from fallout can penetrate through opaque walls, so that if thin

they may be almost as transparent to radiation

as windows. In this case the size and location

of windows makes very little difference.

This illustrates that there is a connection

between effective mass thickness of a barrier and

the permissible design of openings in the barrier

from the point of view of radiation protection.

When radiation enters a structure mostly through

windows, the advantages of heavy walls are negated; and conversely, if the walls are very

thin the window design matters little. Openings of many different types occur normally

in construction, and there doesn't exist much

information, either experimental or theoretical,

about aperture effects. Nevertheless, some general rules can be given for estimating their contribution to the detector response. In this section

we discuss briefly openings in' walls and roof; and

we distinguish between two different cases: Apertures between source and detector, and

apertures not between source and detector.

Figure 35.1 is a sketch of circular windows in a blockhouse wall and ceiling, with the detector

placed opposite the center of each. In botli cases the window represents a type of surface which

differs from the main wall or ceiling in that the

effective mass thickness is nearly zero and the

solid angle fraction subtended at the detector is apt to be small.

In the skylight case the detector will receive a contribution due to skyshine in addition to a contribution from fallout on the skylight. The

relative sizes of these two contributions is of some

interest. For purposes of determining the skyshine contribution the skylight can be viewed as a

special type of foxhole, so that the detector

response is given essentially by eq (31.1):

D/Do=S{d)Sa{d,o,), (35.1)

where d is the height above the surrounding ground

and CO is the solid angle fraction of the skylight

subtended at the detector. Equation (35.1)

applies only for very small oi zero effective mass

thickness of window. If there is a covering of

fallout on the skylight, its contributipn plus the

skyshine contribution can be estimated by viewing

the configuration as an example of a falloutcovered shelter and applying tjie arguments

leading to eq (32.1)

D/Do=L(X)La(X,o:). (35.2)

Here co is again the solid angle fraction subtended

by the skylight at the detector, and X, the effective mass thickness of the window, is very small.^^

The case of a window in a vertical wall bears a

close relation to the directionally dependent

detector pictured in figure 28. 8d. Radiation at

the window entrance (fig. 35.1) should have a

directional distribution much like that above an

infinite plane source in an infinite medium, if we

consider only gamma rays traveling towards the

window. The detector behind the window in

figure 35.1 "sees" only that part of the radiation

field at the opening which travels in an "allowed"

cone of directions, very much in the manner of the

detector in figure 28. 8d. We therefore apply the

data obtained using eq (27.11) to obtain the

detector response to radiation passing through the

window, just as in the case of a thin vertical wall

(eq (33.4)):

D/Do-^0.9W(X, d)Waid,w) (35.3)

where d is the height of the window above ground,

0) is the solid angle fraction of the window subtended at the detector, and X is the window

thickness, which is presumed to be small or zero. Note that it may be desirable to take account of

ground roughness by increasing the value of d, and

that the factor 0.9 is to be omitted if the W{X,d)

data is for a concrete wall.

Figure 35.2 shows openings in partitions which

lie between the source and the detector. Radiation

entering through these openings can reach the

31 Backscattering has been omitted in both eqs 35.1 and 35.2. Strictly spealiing a factor 1,15 should be incorporated in eq 35.2 and a factor perhaps

as big as 1 3 in eq 35.1. The latter is larger because the radiation is lower m

energy. Xote, however, that if H2O data is used a factor of 0.9 or so is also needed.

62
Figure 35.2. Apertures directly between source and detector.

FiGtTKE 35.1. Apertures in a blockhouse wall and roof.

detector without penetrating as niucli material as radiation which passes through the partitions to

the detector. If the partitions are very hght, one

can probably take the openings into account by

assigning an average value to the effective mass

thickness between source and detector. On the

other hand, the partitions may be thick, so that

radiation penetrating through the partitions is significantly reduced in comparison with that

passing through an opening. In this case the

opening can be considered as giving a separate

term in eq (20.1), in which term the effective

mass thickness between source and detector is given a value equal to the combined thickness of

partitions between source and detector along lines

through the opening, and the solid angle fraction

is that subtended at the detector by the opening.

If more than one opening lies between source and

detector, additional terms may be required, each

nivolving effective mass thickness and solid angle

fraction determined by overlapping or nonoverlapping parts of these openings.

Figure 35.3 shows an example of openings

which are not between source and detector.

If the partitions are heavy, the presence of these

openings can be expected to modify the detector

response significantly. For example, trie floor

opening permits a substantial increase hi the

strength of the radiation field directly below,

while the wall opening gives ready access lateially

to the detector.

Notice that the radiation must turn the corner

in order to take advantage of these openhigs,

so that a scattering must occur at some position

which is favored in the sense of being below the

floor openings and (or) hi front of the wall opening.

This is actually a "maze" problem, and is quite

different from the case of openings which permit

radiation to travel directly from source to detector.

'2 We say here that the openings "overlap" when a straight line from source

to detector can pass through more than one opening.

Figure 35.3. Apertures not directly between source and

detector.

Methods for treatuig this case are essentially

those used in maze problems, which are discussed

hi Section 38.

EXAMPLE: A radiation detector is at the center

of a cubical blockhouse of side length 10 ft, and

with very heavy walls. Small, circular wnidows

of equal radius are placed hi the center of the

roof and one wall. Compare the contributions

to the detector response from skyshine and direct

radiation through the roof window and from

radiation through the whidow in the wall.

For definiteness assume X~0 and aj = 0.05.

We then obtain the following:

Figures

Roof, skvshine: D/Do=S ilOh)Sa (lOft, 0.05) = (0.084) (0.021) =0.0018 28.3,28.15

Roof, direct: D/Z)o=Lim L (X) La (X, 0.05)

=0.014 28. 19

Wall: DID,= W{0, 5ft) Wa (5ft, 0.05) = (0.45)(0.125) =0.056. 28.7,28.12

63
From this we see that 0.056/ (0.056 +0.014 +0.002) =0.78 of the detector response is from the window

in the wall, and that of the remainder only about

1 1 percent is due to skyshine. ANSWER.

Note that backscattering corrections would

change these figures somewhat.

36. Gompartmentalization

From the point of view of radiation shielding,

the single most prominent characteristic of

ordinary buildings might be termed compartmentalization. Most of the mass of the building

which may be expected to act as a radiation

shield is concentrated in the floors and walls,

which divide the building into compartments.

Consider a multi-story building, as schematized

in figure 36.1, with fallout on the roof and on the

surrounding ground. Radiation from the roof

fallout penetrates the building traveling downwards. As this radiation penetrates deeper into

the building, it must pass through successive

layers of material, namely the floors. On the

other hand, depending on the trajectory, the

radiation may or may not pass through the walls.

Thus the floors represent a sort of divided shield,

while the walls act as a collimator which favors

some trajectories over others.

Notice that the same effects are present also

in the case of radiation originating on the ground

around the building. As this radiation penetrates

into the building it must pass through a set of

walls. But the floors and the other set of walls

Figure 36.1. Multistory, compartmentalized structure with

fallout on roof and (/round.

are so oriented that the radiation may miss them

altogether if it follows an appropriate trajectory.

Thus one set of walls forms a divided barrier

while the other partitions lie more or less parallel

to the direction of penetration and give advantage

to radiation traveling directly inwards.

In addition to the compartmentalization of

buildings into rooms and corridors, maay or even most types of wall and floor construction involve

compartmentalization on a smaller scale, because

of voids introduced to reduce weight. The most

obvious examples are hollow concrete blocks,

hollow tile, and hollow bricks. Another example

of slightly different type is that of floor joists,

which have the effect of collimation on the downward-penetrating radiation. Such compartmentalization can cause overestimation of the

protection afforded by a wall.

The simplest way to take account of compartmentalization in building materials is that of using

approximations based on a more uniform distribution of material. For example, in the case of

compartmentalized walls one might estimate the

effective mass thickness by using weight/volume

ratios, thus assuming the material in the wall to be

spread uniformly through its volume. Experiments tend to confirm that such estimates are accurate enough for most purposes, largely because

the barriers so constructed are usually rather

light-weight. The effective mass thicknesses

given in Table 22.2 were obtained from weightvolume ratios.

Approximations in which the material is as- sumed to be uniformly distributed can also be

used in large, multi-room structures, depending on

partition thicknesses. The method is essentially

the same as for a single compartmentalized wall:

One calculates the total mass of material between

detector and roof, say, and assumes it to be distributed uniformly between. The accuracy of

such a simple approximation depends primarily on the partition thicknesses. In general, one

expects it to be accurate when the partitions do

not individually reduce the radiation intensity by

as much as a factor of e^2.7. From figure 28.2b

we find that this means partitions of less than

about 40 psf effective mass thickness. It should

be remembered, however, that such a limit is, for

the present, more of a guess than an accurately

known figure; and that it depends very much on

the configuration, as will be indicated more

clearly in the next Section in the discussion of

other simple types of approximate calculations

for compartmentalized structures.

37. Collimation and Divided Barrier Effects in

Compartmentalized Structures

In a more detailed analysis of compartmentalization, it is advantageous to consider collimation

and divided barrier effects separately, since they

are quite different, even though they depend on

tlie existence of particular directions of interest,

such as "downward" and "inward." In this

64
Section we make the distinction by giving separate

consideration to partitions placed "parallel" and

"perpendicular" to the direction of interest. (We

include floors when we speak of "partitions.")

(1) Partitions perpendicular to the direction of

penetration: When a barrier is divided into layers

through which radiation must penetrate successively, the mass distribution can usually be ex- pected to affect the detector response. But there

is one case in which this is not so. When layers

of material and a plane source do not terminate,

but extend unchanged over the whole plane, as indicated in Section 30, the dose measured by a detector is independent of the particular manner

in which the division into layers takes place, or even the existence of a layered structure, so long

as the total effective mass thickness between

source and detector remains fixed.

When the radiation source is finite in extent, or non-uniformly distributed, the nature of the

layering must be taken into account. We consider three extreme cases, namely (a) the total

barrier mass confined to a layer adjacent to the

source; (b) the total barrier mass confined to a layer adjacent to the detector, and (c) the barrier

material spread uniformly between source and detector. Other situations can arise which are combinations, or which can be expected to lie somewhere between the extremes just indicated. For

example, the mass may be concentrated in a layer

half-way between source and detector, or it may

be divided into two layers, one adjacent to the source and one adjacent to the detector.

When the total barrier mass is confined to a layer adjacent to the source, one has essentially

the fallout-covered shelter discussed in Section 32;

and the analysis utilizes data for La(X,u).

Cases in which the total barrier mass is confined

to a layer adjacent to the detector have already

been discussed in Section 34. They correspond to a radiation source emitting only within a restricted cone of directions, so that calculations for this

configuration utilize data for L,,{X,a}).

Distributed barriers have been discussed in Section 30. The finite source cases of interest utilize

data for the Lc{X,ui).

Very general arguments indicate that the detector readings in these three cases shouldn't differ

greatly, a factor of two usually being sufficient to

onconipass the spread of values. Tliis is essentially because the barrier and geometry reduction

factors account for the large part of the barrier

effectiveness, and these do not change significantly

from one mass distribution to another. Therefore, these tliree extreme cases can be used m

representing the different cases of layering in

practical problems. We do not expect a slight

mismatch of schematization and configuration to

result in significant errors. Notice that if the barrier is finite, as in figure

37.1, radiation can escape out the sides, lowering

the observed dose and giving a slight increase to

tlie barrier effectiveness.

(2) Partitions parallel to the direction of peneFiGURE 37.1. Fallout on a finite divided harrier.

,\ «

*

V

! T

Figure 37.2. Fallout on a simple structure with parallel

partitions.

tration: Colli mation effects are more difficult to

estimate than divided barrier effects. At present

neitlier experimental nor theoretical studies are

available. The discussion given here is mainly

confined to limiting cases and is therefore incomplete even though indicating the nature of some

of the main effects.

In the compartmentalized structure of figure

36.1, consider penetration by radiation from fall- out on the roof. It is clear that collimation effects

will depend on wall tliickness, floor thickness, wall

spacing and tlie distance of the detector from the

source in terms of the number of compartments.

Various limiting cases can be considered:

(1) The walls may be very thick, so that radiation can hardly penetrate into adjacent compartments. In this case the detector sees mainly

radiation which has not penetrated througli the

walls, and the problem reduces to one of the

divided barrier problems with a finite source.

(2) The fioors may be very thick: In tliis case the radiation only penetrates tln-ough the floors

if it travels always in a nearly vertical direction.

This is because radiation traveling at a substantial

obliquity has slight chance of getting througli the

65
barrier, except when the obUque portion of the

trajectory occurs near the exit surface of the barrier. The very thickness of the floors provides a

sort of colhmation in this way. It is possible,

though very unhkely, for this type of colhmation

to be stronger than that provided by the walls, so that it doesn't matter when the walls are present.

(3) The walls may be thin, so that radiation

readily travels from one "channel" to the next.

In this case the collimation effect is slight unless

the radiation can penetrate to distances large

compared with the "channel width."

The most difficult cases concern walls and floors

which are neither thick nor thin. It may be

possible in such cases to justify the approximation in which all material between source and

detector is distributed uniformly throughout the

intervening space. But this approximation can

lead to an overestimate of the shielding provided

by the compartment partitions. A safer estimate can be obtained as follows: If the radiation cannot

travel from source to detector without passing

through parallel partitions, the total effective

mass thickness of these parallel partitions is added

to that of the perpendicular partitions; and the

detector response is determined with this augmented barrier thickness.

To apply this rule the primary radiation source

is divided into components according to the number and type of parallel partitions which must be

penetrated, as shown in figure 37.2. Radiation

from that part of the source directly above the

detector is not shielded by parallel partitions;

whereas radiation originating far to one side of

the detector may have to go through several

partitions to contribute to the detector response.

The detector response for the simple case pictured

in figure 37.2 would thus be estimated by

D=UX+i)La{X^M + L{X)L,{X,o>2)

+L{X+t)La{X+t,o:,), (37.1)

wliere coi, W2, ^3 are the solid angle fractions subtended by Su Si, S3 at the detector.

Notice that even when the radiation might be

expected to penetrate parallel partitions at a slant angle we do not use the slant penetration

thickness, but rather the normal thickness of these

partitions in equation (37.1). There are several reasons for this, but the weightiest is our desire not

to over-estimate the protection factor seriously.

We are treating parallel portions as if they were

part of the divided barrier, and this procedure

under-estimates their efl'ect somewhat.

This approach to the parallel partition shielding

increases the amount of calculation necessary

because each component requires a separate

calculation.

38. Mazes

P"'igui'e 38.1 shows an elementary maze. To

pass from source to detector the racHation must

follow openings, because tlie barrier is nuich too

Figure 38.1. A simple maze with two 90° bends. The

solid angle fractions u>' and u are subtended at the detector by the near and far sides of the bend.

thick to permit "shortcuts." At least two

changes of direction are necessary for the gamma

rays contributing to the detector response in this

example.

Radiation readily passes down the initial passageway leading from the source. The intensity

of the radiation diminishes steadily with distance

from the source, maintaining a rough proportionality with the solid angle fraction which the source subtends at the detector.^^ At the end of the

first passageway, as at positions all along, the

detector response is partially due to scattered

gamma rays, though mostly to gamma rays coming

directly from the source. Once the corner has

been turned, direct radiation can no longer reach

the detector in significant amounts, and the

detector response is due to scattered radiation

almost entirely.

It is possible to examine this scattered radiation

to determine where it "turned the corner."

This leads to the conclusion that the wall surface

at the corner which can "see" both source and

detector contributes a substantial fraction of the

scattered gamma rays which go down the second

corridor. It follows that maze shielding can be

enhanced somewhat by removing or recessing

these scattering areas so that they no longer have

access to both source and detector, or by covering

the walls at the corner with a layer of tin or lead

sufficiently thick to reduce the albedo considerably.

Experiments now in progress are leading to a further analysis of radiation "turning the corner"

into contributions from individual wall surfaces,

and a report on this subject by J. C Le Doux

M This dcpciidi'iicc upon solid ansjlo would not hold for an anisotropic

source. In an cxlrcme case, for a jilanc-parallcl source, the detector response

is independent of the length of the first corridor.

66
and A. Chilton is to be published.^* The results are by no means simple. Apparently three different components are distinguishable: (1) Radiation

scattering from ceiling, floor, and the end wall of

the first corridor; (2) radiation scattering from

the end wall of the second corridor, i.e. the side

wall of the first corridor; and (3) radiation "cutting the corner." Wall positions contributing to

each component can be seen using figure 38.1.

As the detector is moved down the second corridor, each component is expected to be reduced in

strength according to the solid angle fraction subtended by the scattering surface at the detector.

This means that components (1) and (3) vary

inversely with the cube of the distance and component (2) with the square of the distance of the

detector from the corner. On the other hand,

components (1) and (2) do not depend on the

absolute dimensions of the maze, while component

(3) decreases in inverse proportion to the magnitude of an increase of scale. That is, if the maze

size is doubled without change in shape, and with

the detector in the same relative position, the

third component will contribute only half as much

to the detector response at the corresponding

position in the larger maze, but the other two

components are unaffected. The third component

is very important for small mazes and ducts, but

relatively unimportant for large mazes. This type of analysis suggests a simple expression for representing the detector response in

mazes, namely

VII. Rectangular

39. Introduction

Most of the surfaces of interest in radiation

shielding are rectangular or representable by

rectangles. On the other hand, it is much easier

to generate data for circular surface sliapes, with

the detector opposite the center of the circle; and

many, even most, calculations can be carried out

with acceptable accuracy by applying circularshape-data to rectangular-shape-walls. Therefore, in this and tlie following sections we discuss

methods for doing this. In tlie last section we

consider the problem of rectangular-source-data

more generally.

The use of circular-source-data to obtain tfic detector response to a rectangular-shape-sourcc

involves several steps and a number of simple

types of calculations, which we list below:

(1) Calculation of tlie solid angle fraction of

circular sectors;

(2) determination of the solid angle fraction

subtended by any rectangle at a point on tlie

central axis of symmetry ("centered rectangle"

case;)

"Private commuiiiciitioris from .1. C. I.f Doux to C, M. Eiscnhaucr,

Added Note: Very recent puhliciitions on this subject are: C. M. Eisenhauer,

Scattering of Co-CiO (iamiiia Itadiation in Air Ducts, MiS Tech. Note 74

(I'liKilf)-.';), Oct. lyiid (Olliceof 'I'echnical Services, U.S. Dept. of Commerce)

,

C". \V. Terrell, A. .). .leris, R. O. Lyday, D. Sperher, Radiation Streaming

in Shelter f:ntraMcesvays, AR1''-11S8-12, Oct. (Armour Research

KoundatioM, Chicago, 111. J.

Table 38.1. Albedo coefficients Ai, Bi for mazes with a

long first corridor and large, square cross sec'ion. wg is the solid angle of the entrance as subtended at the first bend.

0)0 A, B,

0. 007 0. 056 0. 08

. 063 . 04 . 19

. 165 . 054 . 27

jy^j^ n [(^,C0,+5,(C0',-C.,)] (38.1)

1-^0 i^O i=l

in a maze with N bends, where coj,co'i are the solid

angle fractions subtended by the far and near

ends, respectively, of the i'th corner at the l)'st corner on the detector; and Di is the detector

response at the midpoint of the first corner. (See fig. 38.1. Note that coq is taken to be 0.)

The two terms represent the and components mentioned in the preceding paragraph,

and Ai, Bt can be considered effective albedos for

these two components. Experiments with single

bend mazes, using Co-60 and a long first corridor,

give estimates of Ai, 5, as listed in Table 38.1.^^

Corresponding estimates for later bends are not

available, but for a safe overestimate the values

A{=1, Bi=0, i^l, can be used. A closer estimate

A B

for i>l might be Ai=-^— Bi—-:— which

relies on the assumption that {Ai/B{) = {AijBi).

Source Shapes

(3) determination of the solid angle fraction

subtended by any rectangle at any point, in

terms of a linear combination of the simpler

"centered rectangle" cases;

(4) approximation of a "centered rectangle" by

circles or circular sectors, so that the approximation iias the same solid angle fraction and perhaps

the same "elongation" as the rectangle;

(5) determination of geometry factors for

circular sectors. It is not difficult to see how these

different t>pcs of calculations can be combined

into a fairly accurate procedure for utilizing the

circular data: We represent the general rectangle

case first as a combination of "centered rec- tangles" and then, by representing each "centered

rectangle" as a combination of circular sectors,

we obtain the general case as a combination of

circular sectors. Then we calculate the geometry

factor for each circular sector independently and

combine these results to obtain a value for the

rectangle.

40. Solid Angle Calculations for Circles and

Annular Sectors

Tlu' solid angle fraction w subtended by a circular or annular sector is easy to calculate if the

Private communication from C. .VI. Eisenhauer, based on analysis of the data of Terrell, et al., of the [irecedinn footnote and as yet unpuhli.shed data duo to F. X. Riz/.o and .\. Ciuadrado, lirookhavon National Laboratory.

67
Figure 40.1. Coordinates describing circles and circular

sectors for calculations of the solid angle fraction subtended

at detector D.

reference point is on the axis (see fig. 40.1). This

is the only case we consider here.

If p is the radius of a circular area, z the distance

of the reference point from the sufrace plane,

and cos a = z/-^2^-\- p^, then

Ci)-- 1 — cos a. (40.1)

If two concentric circles have radii pi and P2 and

subtend solid angle fractions coi and uo, then the

solid angle fraction of the annulus between the

circles is (for 0)2^0^1)

--(1)2 —a)i = cos «! — cos a2. (40.2)

Similarly, the solid angle fraction of the

annular sector between circles of radii pi and p2,

and between azimuths (pi and v'2, is (p2^Pi)

W =7n(c02 —Wi), (40.3)

where m is the fraction of the azimuth occupied

by the sector, i.e..

-<P\)- (40.4)

EXAMPLE: Calculate the solid angle fraction

subtended by a 30° sector of the annulus between

two concentric disks of radii 5 ft and 7 ft, as subtended at a point on the axis 10 ft from the plane

of the disks.

Since cos a2 = 10/V7'+10^=0.820, and cos ai =

10^/125 =0.894, while v52-v5i = 30°=7r/6, we have

ir/6

and w =0.08333X (0.894 -0.820) =0.0062.

ANSWER.

41. Solid Angle Fractions for Rectangular

Surfaces

It is particularly fortunate that the solid angle

fraction for a rectangular surface can always be

calculated analytically. The calculation is simplified if the conventions of Cartesian geometry are observed. The notation in the following paragraphs is that of figure 41.1, with the origin of

coordinates in the plane of the rectangle and

opposite the reference point (usually the detector

location). The perpendicular distance from the

plane of the rectangle to the reference point is z; and Xi, yt {1=1, 2, 3, 4) are the coordinates of

the four corners of the rectangle. The coordinate

s.ystem is chosen so that the x and y axes are

parallel to the sides of the rectangle; and the corners are numbered in counter-clockwise succession.

Having specified a coordinate system, we next

list several important parameters, namely.

ti=\yilxi\,

rii = \z/Xi\,

qi=^yihi3!i-

(41.1)

The first two of these are referred to as "eccentricity ratios," while the last is ±1, depending on

the quadrant in which the corner is located.

Finallv, we define the function

r(e,?7)=-tan ^ — , (41.2)

which has the synmietry property

r(e,,)=TQ,j). (41.3)

In terms of these quantities, the following general rules hold precisely:

(1) Reference point opposite the center of the

rectangle: For this case ei =ei and ni='ni- The

solid angle fraction w subtended by the rectangle

is given by

o^ =r{e„m). (41.4)

Extensive tabulations of T{e,r]) have been made

and are presented in figures 41.2a and 41.2b.

(2) Reference point opposite one corner: The

solid angle fraction w is given by

(41.5)

This result is easily obtained from symmetry by

combining the rectangle of interest with three

31" For a discussion of comparable problems in illumination see reference 40. Note, however, that in illumination the integrals are somewhat modified

because the detectors are usually plane surfaces, having a response function which is not isotropic but is proportional to the cosine of the angle of inci- dence relative to the normal to the detector surface.

68
Figure 41.1. Cartesian coordinate system used to describe a rectangle and a

reference point {or detector), for calculations of the solid angle fraction.

other identically shaped rectangles in the other

quadrants (see fig. 41.3). The value of t in eq

(41.5) pertains to the larger rectangle comprising

the four parts together.

(3) Reference point opposite the center of one

edge: This arrangement is shown in figure 41.4.

The detector can be considered opposite the corner

of two similar, adjoining rectangles, each with

eccentricity ratios ei, tji, so that

(41.6)

(4) Reference point on a plane of symmetry:

An example of this case is pictured in figure 41.5.

The detector can be considered at the corner of

two similar, adjoining pairs of overlapping rec- tangles. The combination yields the result

<^=\ l2i^(ei)'?i) +23r(€3,'73)|. (41.7)

Notice that the terms add when the origin of co- ordinates is inside the rectangle and subtract when

it is outside.

(5) General case: Figure 41.6 illustrates the important fact that a symmetric combination of

rectangles can always be constructed which permits determination of the solid angle fraction subtended by an\' rectangular surface at any reference

point whatsoever. One obtains therefrom the

general expression

^=i\giTiei,Vi)—g2r{€2,V2) + qzT(e3,V3)—gir{€i,rn)\.

(41.8)

Each term on the right corresponds to the solid

angle fraction for one of the rectangles with a corner at the point opposite the detector. The

terms all add if the origin of coordinates falls

inside the rectangle; while two terms add and two

subtract if the origin is outside the rectangle.

Since the T(ei,r?,) correspond to solid angle fractions for "centered" rectangles, they are referred

to frequently. We shall therefore make use of the

shorter notation

ri = r{et,vt) (41.9)

for these "partial solid angle fractions."

EXAMPLE. Find the solid angle fraction subtended by a rectangular roof with dimensions 25

ft by 35 ft at points (a) 24 ft below the center,

(b) 24 ft below one corner, (c) 24 ft below the

middle of one of the short sides, (d) 24 ft below

the roof and 10 ft out from one of the short sides,

on the plane of symmetry, and (e) 24 ft below the

roof and half-way between the center and one

corner. All five positions are shown in figure 41.7.

Calculation of the solid angle fractions can proceed in two, sometimes three ways. For many, or most, cases it is sufficient to determine the parameters €i, til and then refer to figure 41.2. Thus,

for the point designated (a) one has ei=(25/2)/

(35/2) =0.7143, r7i=(24)/(35/2) = 1.371. One then

locates the point having these coordinates on fig- ure 41.2, places alongside this point a French curve which approximately matches the neighboring contours, and reads from a nearby radial line the

value 0.173. ANSWER.

By comparison, for point (b) one first can determine the values ei= |25/( —35)| =0.7143, =

l(24)/(— 35)1 =0.6857, and then calculate numerically the result CO =0 .2 5 (2 /tt) tan -

' (0 . 7402 ) =0 . 1 0 1 4

.

ANSWER.

Notice that in all these calculations the co- ordinate svstem is oriented in the same way. For

point (c),' usiiig €i = 12.5/35 =0.357, »;i =24/35 =

0.686, one obtains from figure 41.2 the value w =

0.124. ANSWER.

69
70

72
Figure 1:1.2b. Conlours of constant T{e,ri),e'> 1. This function is the solid angle fraction subtended by "centered rectangles."—

Continued

73
Figure 41.3. Reference point {or detector) opposite one corner of a rectangle.

Figure 41.4. Reference point {or detector) opposite the center of one

edge of a rectangle.

41.5. Reference point {or detector) on a plane of symmelnj of a rectangle.

To calculate the solid angle fraction for point

(d), one first lists

ei = 12.5/(35-10) =0.5

€3=l(-12.5)/(-10)| = 1.25

T?i =24/(35-10) =0.96; 2i = l

rj3=i24/(-10)|=2.4;g3=l.

Figure 41.2 then provides the results r(0.5, 0.96) —

0.216, t(1.25, 2.4) =0.111, so that a;=0. 5(0.216 +

0.111) =0.164. More precise numerical evaluation gives the result w =0.1059. ANSWER.

Finally, for point (e), one has

6:=|(3/4)(25)/(3/4)(-35)| =0.714

62=|(-l/4)(25)/(3/4)(-35)| =0.238

e3=|(-l/4)(25)/(l/4)(35)| =0.714

74
e4=l(3/4)(25)/(l/4)(35)| =2.143

T,i-|24/(3/4)(-35)| =0.914;2i = -l

772= '?i; 92=1

„3=l24/(l/4)(35)| =2.743;g3=-l

Numerical evaluation of the r(e,,i7,), followed by

combination as indicated in eq (41.8), yields the

result w =0.1 524. A trial calculation using figure

41.2 as a shortcut gave the less precise value 0.153.

ANSWER.

X

(a)

42. Approximation of Rectangles by Circular

Sectors

We next consider simple methods for representing rectangles as combinations of annular and

circular sectors. If this can be accomplished for

"centered rectangles" the extension to the general

case is straightforward. Of course, it is in principle possible to make such representations with

any accuracy desired, by using many small

annular sectors. But this amounts to a form of

numerical integration, which we seek to avoid.

To be acceptable, a representation must use a minimum number of sectors.

Three methods will be discussed, each illustrated by one of the sketches on the left in figure

42.1. In the first, the elongation of the rectangle

is neglected altogether. In the second, the elongation is accounted for by the use of two circular

sectors. In the third, mainly for use with rec- tangles far off center, annular sectors are used.

(1) Approximation by circles: In some cases, eccentricity effects can be shown to be weak, so that two surfaces can be considered equivalent if they subtend the same solid angle fraction. One

can then calculate the solid angle fraction T{e,r]) of a "centered rectangle," and replace the rectangle

with a circle subtending u =T(e,rj) (see fig. 42.1a).

In the general case of an off-center rectangle, each

of the four "centered rectangles" is replaced in

this way by a circle, and the corresponding approximation has the form

i=l (42.1)

Ficii RE 41. G. Reference point {or deleclor) opposite arbitrary

localions: (a) inside and (h) outside a rectangle.

in analogy with eq (41.8), with the Wj representing

circles. The combination can be viewed as a pair

of annular sectors whose combined solid angle

fraction is equal to that subtended by the offcenter rectangle.

(2) Approximation by two circular sectors:

This method attempts to obtain greater accuracy

through an attempt to take eccentricity effects

into account. In figure 42.1b the rectangle is indicated by a solid line. The dashed lines indicate two squares, one having its sides equal to the

length of the rectangle, while the other has its

sides equal to the width of the rectangle. The

circular arcs represent parts of circles which, if complete, would subtend solid angle fractions

equal to values for the squares. The representation consists of the complete smaller circle plus

enough of the annular ring between the circles to

give the combination a total solid angle fraction

equal to that of the rectangle.

IF tlio solid angle fraction of the rectangle is r(e,77), the solid angle fractions subtended by the

two scjuares are T{\,ri), and t (1,17/6). It is easy to show that the api)roxiniation takes the form

FifM HE 41.7. Detector positions for the Example, relative

to a 2-') ft 1)1/ !') ft rectanr/ular roof. co = aw' + (1 —<7)c (42.2)

V

75
j COS d

\

Figure 42.1. Approximation of a "centered rectangle" by circles and annual sectors:

(a) Neglecting elongation (Method 1): (b) taking elongation into account in a simple two-term approximation {yiethod 2); (c) using an annular sector to represent an

off-center case (Methods). The graphs onthe right show qualitatively the circular sector approximations (dashed lines) totheg,(B) functions (solid lines) which correspond

to the sketches on the left.

76
where w is the soUd angle fraction of the ("centered") rectangle, w' and w" represent circles

having solid angle fractions 7(1,77) and T{l,r]/e),

respectively, and

a=

T(e,T?) — T(l,T?/e)

7(1,77) —t(1,77/€)

(42.3)

Extension to the general case of off-center rec- tangle is accomplished by the use of eq (42.1),

with each of the aj< referring to a sector-represented

"centered rectangle."

(3) Approximation by annular sectors: This

method is for use when the four terms in the

general case cancel strongly against one another,

as when the rectangle is far to one side of the origin

of coordinates.

Concentric circles passing through the farthest

and nearest corners of the rectangle subtend

solid angle fractions w 1- 1} and

V3

V'73^ +e3^ +

V'7i=' +6i' +

respectively. A sector

of the annulus between these circles can represent the

rectangle if it has the same solid angle fraction.

Our representation thus takes the form

CO =m(co'- (42.4)

where w' , w" represent the two circles, with solid

angle fractions as indicated, and

1=1

/ ^3 _ \

(42.5)

Since this last method is applied to off-center

rectangles primarily, there is no reason to consider

combinations of such representations.

Similarly, in method (2)

sectors, eq (42.2) becomes

utilizing circular

6?(Z,co) =a G{X,co') + (1 -o) G(X,c^")

,

(43.2)

where the value of a is given by eq (42.3). Combination of the four results of the type eq (43.2)

into an expression of the form eq (43.1) gives the

circular sector approximation for the general case. Finally, the annular sector approximation (3)

for a geometry factor is obtained by translating

eq (42.4) into

GiX,o^)=m[G{X,c^')-G(X,^")], (43.3)

where m is given by eq (42.5).

These methods can be applied to any of the geometry factors represented generically by G{X,o)),

with greater accuracy in some cases than in others.

No attempt has been made to compare systematically the accuracy of the different approximation methods in different types of calculations.

EXAMPLE 1 : A rectangular underground shelter

40 ft X 20 ft is covered with a concrete slab of

effective mass thickness X=150 psf. Fallout on

the slab emits gamma radiation of intensity sufficient to give a response Do to a detector 3 ft above

an idealized plane similarly contaminated. Calculate the detector response 10 ft below the slab,

half-way between the center and one corner, using

the circular approximation. Figure 43.1 illustrates the configuration.

To determine the detector response one first

calculates the parameters €«, 77^, and t; for each of

the corners of the roof slab, using figure 41.2.

I

12

3

4

30/15

|30/(-5)|

|(-10)/(-5)

l(-10)/15|

7?.- T, Qi

10/15 0. 58 1 10/(-5) . 278 -1

10/(-5) . 205 1 10/15 . 40 -1

43. Calculation of Geometry Factors for

Rectangles

As a last step in the calculation of geometry

factors for rectangles, we replace w in the preceding

section by G{X,w), a geometry factor. We can do this because the (j's combine linearly just like

the co's. Thus, (42.1) takes the form

Using figure 28.10 we next calculate La for circles

having the same solid angle fraction

12

3

4

0. 58

. 278

. 205

. 40

L„(150,a),)

0. 73

. 49

. 39

. 61

G{X,<^) =

i=l (43.1)

where the term on the left is the approximate

geometry factor for the rectangle, and the terms

on the right are geometry factors for circles subtending solid angle fractions co<."

" Note that eq (43.1) is exact when the G(.Y,u,) correspond to the centered

rectangles determined by the four corners of the ofl-center rectangle, rather than to circular sector approximations to these centered rectangles. Figure 43.1. Detector below the roof slab of an underground

shelter, as in Examples 1, 2, and 3.

11

622827 0—62 6
Finally, we make the combination for the general

case:

!=1

=Z>o(-0057) (.555)=0.0032Z?o. ANSWER.

The circular approximation has fairly obvious

limitations. It cannot be applied accurately for

length/width ratios which differ by more than

perhaps a factor of three from unity, unless the

directional distribution involved is either nearly

isotropic or nearly monodirectional. In the

example just worked, the term Zia( 150,£02) is expected to be rather inaccurate because €2 is 6; but this term contributes only about 10 percent

of the final result.

Clearly this method can be expected to become

more inaccurate when the terms cancel rather

than add, i.e., if the detector is to one side of the

roof slab. Also, values obtained will represent an overestimate or underestimate depending on

whether radiation contributes more at angles near perpendicularity or at angles nearly grazing

the source plane. Thus the value obtained in

this example is an overestimate because radiation

emerges from the roof slab with greater intensity

at angles near the perpendicular.

EXAMPLE 2 : Re-calculate the detector response

for the preceding example, using a circular sector

approximation.

Using the e*, and Xj values already determined, the following table is easy to calculate:

1 u' w" a l-a £„(150,u') i.(150,u")

1 0.48 0.71 0. 569 0.431 0.68 0. 78

2 .128 .71 .743 .257 .26 .78

3 .128 .336 .630 .370 .26 .55

4 .336 .48 .656 .444 .55 .68 From these values, using eq (43.2), we obtain

i o>i La{150,u>i)

12

3

4

0. 58

. 278

. 205

. 40

0. 723

. 394

. 367

. 608

The final combination then yields

Z>=I>oX(150)(l/4) 2 (-)*+'2.L„(150,co,)

j=i

=/?o(0.0057) (0.523) =0.0030Z?o. ANSWER.

The value is to be compared with 0.0032Z>o,

which was obtained by the circular method. The

agreement is surprisingly good, considering that

the length/width ratio of the roof slab is 2. The

largest part of the disagreement appears to come

from the term corresponding to the second corner,

with its length/width ratio of 6. The ratio of

values given by the two approximations for this

term is about 0.81.

EXAMPLE ?, : Consider the rectangular roof slab

of the preceding examples as two adjacent squares,

and calculate the contribution to the detector

response from the square farthest from the

detector. The detector is still considered to be

10 ft below the roof, halfway between center and

corner. The values of the parameters 77,, and turn

out to have the same magnitudes as before; the

only changes are in the g,, which affect the value

of m:

t «i Vi Ti 5.

1 30/15 10/15 0. 58 1 2 30/(-5)| 10/(-5) . 278 -1

3 10/(-5)| 10/(-5) . 205 — 1 4 10/15 10/15 . 40 1

The sign changes lead to increased cancellation,

so that the resulting solid angle fraction is reduced

to 0.063.

The values of w' and co" can now be calculated:

co' = l- 10/15

15/ ^15/ ^

5

V22+2='+l

1'3

Inserting these values into eq (42.5), we obtain

m= . 063 (2 1/8) = .165.

We next obtain the values

L«(150,a;') = .78,L,(150,w") = .54,

from figure 28.10, and make the final combination

indicated by eq (43.3),

D=DoL(150) (0.165) (0.78-0.54) =0.00023Z?o.

ANSWER.

This result is much smaller than the value

obtained in the preceding examples partly because

the solid angle fraction is much smaller and partly

because the radiation tends to go downward

rather than sideways after emergence from the

roof slab.

44. Further Discussion of Geometry Factors

It is possible to calculate geometry factors while

avoiding the use of approximations like those described in the preceding Sections. To accomplish this,

the circular source data obtained from expressions

typified by eq (27.1) must be replaced bv calculations using more general integral forms of types

eqs (16.2) and (16.3), with gs{d,<p) and gs{0) cor- responding to rectangles rather than circles.^** The "centered rectangle" case is the only one

that need be considered in this connection be- cause off-center cases are always obtainable as

linear combinations. Because e and 77 uniquely

3' For an alternative and different approach to the calculation of geometry

factors see Section 49 and references 29, 30, and 41.

78
specify a centered rectangle configuration, it is in

principle possible to make extensive tabulations

of geometry factors G{X,e,r}), using these variables

in place of co. Presumably G{X,€,r}) can be presented, for each X, by means of contour diagrams

similar to figure 41.2. In the future this may

prove to be a desirable way of presenting geometry

factor data, particularly for radiation directional

distributions which depend on an azimuthal angle

(p as well as a polar angle 6, as in the case of radiation emerging from a vertical wall adjacent to a

fallout field.

It is instructive to discuss once more the approximations of the preceding Sections, using the

functions' find gX^)- In figure 42.1, the

points within the rectangle correspond to {d,tp)

combinations for which gsid,(p) = l; while the

points outside the rectangle correspond to gs{d,ip) =

0. The circles similarly divide the points in the

plane into two sets, one giving ^s(0,<^) = l, the

other giving j/,(0,<p)=O. But that of sketch (a)

leads to a function g

1 gsid,(p)d<p which

looks like the solid line of figure 42.1, (a), right;

while the circular approximation (dashed line)

resembles a step function. Note that the area under both curves must be about the same, since

it corresponds approximately to the solid angle

fraction. The circular sector and annular sector

approximations are similarly compared with the

functions they represent in the sketches (b) and

(c), right, of the same figure. Thus all three

approximations are seen to be combinations of

step functions.

Presumably greater accuracy is obtainable by

use of approximations which involve more terms;

but the greater effort is a serious impediment, and

other sources of error tend to be large enough to

make such refinements have questionable significance.

45. Acknowledgments

Many people have had a part in the preparation

of this manuscript. I am particularly indebted

to James Lamkin, who did most of the computer

programming and operation, and to Charles

Eisenhauer and Neal FitzSimons for almost continuous discussions of problems arising in the

analysis of different types of structures. In

addition, hand computations have been carried

out by a number of students, particularly students

of Ottawa University, in Ottawa, Kansas.

People who helped prepare the charts and

graphs included John Hubbell and Hollis Hill.

Martin Berger and U. Fano have been helpful in

making decisions regarding the form and organization of the manuscript.

Finally, I would like to express appreciation to

the staffs of the Office of Civil and Defense Mobilization and the Defense Atomic Support Agency

for their continuous support of this work.

Appendix A. Computations

46. Preparation of Data: Use of Digital

Computers

To obtain most of the data presented in this

manuscript, it was necessary to rely on digital

computer programs. A family of these programs

was written for the IBM 704 computer at the

National Bureau of Standards, using the SAP

system of orders. Because so much of the desired

data involved angular distributions of a fairly

general type, many of these computer programs

involved extensive development and experimentation. Figure Al is a block diagram showing the

system in its final form, and indicating by means

of arrows the order followed.

The first to be constructed was the general

MOMENTS routine, whose output is a large

tabulation of values for coefficients /^", (Ef) , as

well as dose integrals Dl,°i, for a given monoenergetic gamma ray source in a given material. Here,

Ei is the i'th scattered energy and the indices n,

I, lo refer to the n'th spatial moment, the Vth

Legendre coefficient of the directional distribution,

resulting from a plane source emitting gamma rays

having initial directional distribution Pi^{cos Oo),

where is the initial obliquity angle relative to

the normal to the source plane [12]. The calculation is carried out using photon wavelength (in

Compton units) as a variable. For most calculations the basic integration interval was AX=0.025.

This interval was doubled at X~Xo+2, and re- doubled at X«;Xo+4, to keep the total down to

manageable proportions. For l<lo<l2, values

of n up to 12 were calculated; while for ^o=0, the

calculation was extended to n = 18. Output was

MOMENTS

ABSTRACTOR

SORT

SEEK

ISOTROPIC PARAMETERS SLANT PARAMETERS

ISOTROPIC DISTRIBUTIONS SLANT DISTRIBUTIONS

Figure Al. Flow diagram for the computer programs.

79
to magnetic tape, and totaled the order of 10^

numbers. After a number of time-saving modifications of the code, calculations for a single source energy and material took about one half

hour of machine time.

The purpose of the ABSTRACTOR code was

to make a selection of data from the tape generated

by the MOMENTS code. Two types of selection were considered, one essentially linear in the

photon wavelength and the other essentially logarithmic and including all the source wavelengths

to be used. Both selections included the integral

data, treating it as if it corresponded to spectral

energies. (Four types of integral data were included: Data for the total dose, the total scattered

dose, the once-scattered dose, and the multiply

scattered dose.) By selecting about 1/10 of the

scattered energies, it was possible to put on a

single magnetic tape results for the following source wavelengths:

Xo =0.05, 0.075, 0.1, 0.125, 0.15, 0.2, 0.25,

0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 1.2, 1.6, 2.0,

2.4, 3.0, 4.0, 5.0, 7.0, 9.0, 12.0, 15.0.

On another tape was put data for the source wavelengths 0.384, 0.434, and 0.77, corresponding to Co^° and Cs^". Initial calculations were for penetration in H2O. These were followed eventually

by calculations for concrete, and still later by

calculations for Fe.

Following the abstraction, the SORT code re- ordered the data from

{E,(Un.{l{Em)

to

(£0(^0(^(^^(71))))),

because it was thought that the tape search time

would be cut down by having all the spatial

moments in a group together. (This was probably

an ineffective modification, all things considered.)

From the basic moment-coefficient data it was

possible to construct spatial and directional distributions corresponding to many different source

configurations. Of these configurations, two types

had special importance: (a) those corresponding

to isotropic sources (Zo=0), and (b) those corre- sponding to plane sources emitting at a fixed

obliquity angle do relative to the normal to the

source plane. Accordingly a tape search subroutine called SEEK was written, which, for

specified Eg, Ef, I would yield a set of moments

describing one or the other type of distribution.

Two codes were then constructed to determine

actual distribution parameters. In the first, called

ISOTROPIC PARAMETERS, the isotropic distribution moments weie used to calculate the

coefficients of a power series representation of the

point, isotropic build-up factor. In other words,

the (plane source) moments were turned into point

source moments, and were used to determine

coefficients A, in the representation ^ /r"(S0=4^ g ^,W)^e-W), (Al) f

of the rth Legend re coefficient of the point iso- M

tiopic source angular distribution. The Aj co- ^

efficients, for a specified set of l,Ei, and Eq values, were punched on an IBM card, together with the

descriptive l,Ei, and Eq values. It may be of

interest to note that this calculation involved a matrix inversion, which was accomplished in a very simple way by hand, using the bi-orthogonal

polynomial system of Ref. 12. Equation

(Al) refers also to the integral data, which, in

fact, was utilized almost exclusively.

In the second program of this type, called

SLANT PARAMETERS, calculations were made

of the constants A% A% B^j, and B°j in a representation

S A^je ±2 A'^je )

(A2)

of the I'ih harmonic coefficient of the angular

distribution corresponding to a plane oblique (or

slant) source at specified initial obliquity angle do. In this calculation the values of Bq, Bq were

fixed at the value (mo/cos do) and the values of

Bl, B\ could be assigned arbitrarily. The other

constants were all determined so that the moments

of the approximate distribution agreed with the

known values and with the known value of the

source strength. The coefficients for "even" and

"odd" component distributions, as indicated by

the superscripts, were calculated separately and

punched on separate IBM cards, together with

identifying data for Eo, E,, cos 60, and a parity

indicator which was zero or one. The calculation of slant distribution parameters

was a fairly tricky business, because the B/s were

determined as the roots of polynomial equations.

These roots could always turn out to be imaginary,

signifying an oscillatory and therefore nonphysical

distribution. The arbitrary Bi coefficient was

sometimes given a variety of values in the search

for a reasonable, real set of roots.

Many sets of parameter cards were punched,

mostly corresponding to integral (dose) data, with

^=0. Sets were prepared for values of I up to

1=7 for isotropic sources, dose data, and various

Ei spectral energies. Vames for I greater than

zero were not calculated for oblique sources, nor were calculations made for more than a very few

Ei values. This was partly because most relevant

shielding data did not require these details, and

partly because the slant parameter calculations were expensive as well as tricky, and required careful examination of results.

Tne final machine programs calculated spatial

distributions, and in the case of the isotropic source cases, directional distributions. Specifically, the SLANT DISTRIBUTIONS program

80
could calculate penetration data for an arbiti'ary

combination of source obliquities, source energies,

and scattered energies, so long as these obliquities

and energies were represented in the deck of

binary parameter cards. In general, dose parameter cards for the source energies already mentioned were available, and the source directions

included cos do — ±1,±.9, . . . ±.1, 0. The ISOTKOPIC DISTRIBUTIONS program

was much more complicated. Initially, this program determined the total intensity or an angular

distribution resulting from a Legendre sum, for a point isotropic source or a plane isotiopic source. Superposition of different source energies was possible. All the angular distributions for the fission source were obtained with this early form of the

code. The resulting angular distributions were

mostly not directly usable because of large oscillations; it was a case of too few coefficients to

describe a peaked distribution. These angular

distributions were smoothed by adding angular

functions of the forms

a(cosh /3-cos d)-'"'-J2 (^ +1/2) a,P, (cos d), 1=0 Point Isotropic Source, (A3)

and

a (cos 6)-'e-»"''°'^-^ {l +l/2)biPiicos 6),

Plane Isotropic Source. (A4)

where a, /3 are arbitrary constants, and the a/s

and .hi's are the expansion coefficients of the

accompanying function. Itwill be seen that additions of this type do not affect the values of the

first 8, machine-calculated Legendre coefficients.

The resulting, hand-calculated angular distributions are those of figures 26.1 and 26.3. Because of the work involved in obtaining these distributions from the machine output, it appeared

highly desirable to include methods for obtaining

such distributions in the machine code itself. Accordingly, a series of modifications was incorporated in the code: (1) For the plane, isotropic

calculations, the unscattered component was calculated exactly, and the scattered component was

represented by a simple Legendre sum. (2) For

the point isotropic calculations, the Legendre coefficients were first calculated. Then the ^=3 and

^=5 coefficients were used to obtain coefficients

a and /3 for a function a(cosh /3 —cos 0)""^'^ so as to make the corresponding coefficients in the Legendre sum of that function equal to the known coefficients. Finally, the angular distribution was

calculated using the expression

/?(cos e)=a (cosh iS— cos e)~^''^

+Z; (^+ 1/2) [D,-a,]/^, (cose). (A5)

;=()

(3) Provisions were made to extend all calculations

to include ring source, circular soiu'ce, and circular

clearing source types, and another modification

was made to provide for the determination of both

unscattered and scattered components of the point

isotropic source (total dose).

Since this code provided for superposition of different source energies, and for modification of the

output angular distribution (Legendre sum only)

to correspond to different reference axes, it made

possible calculations for different spectra as well as the calculations of angular distributions incident

on vertical walls (See eq 27.7). The angular distributions in Appendix B were calculated using

this code.

Because of the extensive development work

that accompanied the writing of these codes, they

included many features which turned out not to

be useful or desirable. They have never been put

in such a form that general circulation appeared

desirable. An effort is presently underway to

rewrite the best features of this series, using the

Fortran system, for general use.

47. Preparation of Data : Hand Computations

For the most part, the hand computations were

perfectly straightforward integrations over the

machine-produced data. One exception is to be

noted: Integration of expressions eqs (27.11),

(27.12), and (27.16) was accomplished in the most

obvious way only once, namely for the fission

data. Thereafter it was noted that one could

obtain the same result in a far easier way by using

integral data such as La, Sa, and Pa. The advantage is twofold: The azimuthal integration is carried out analytically and the obliquity integration does not involve extremely peaked functions. The trick involves using an identity such

as

dcosOlid, cosd) =S{d)dSa,cosd<0, (A6)

to write the expression for in the form

Wb{d,l—cos a)=- [ dSa cos-'f-^SL^n , T' J Said. l-Blna) LV2'<' — " J

(A7)

where dSa—d[Said,o:)].

In future calculations it would be highly desirable to arrange it so that all the integrations

aie carried out by machine. This will probably

involve angular distribution output either in the

form of punched cards or on magnetic tape.

48. Comments on Accuracy

In discussing the material of this manuscript in

regards to accuracy, it is convenient to distinguish

three types of error: (1) errors due to inaccurate

input int'oniiation; (2) errors due to the procedure

used to calculate distributions and produce the

graphs; and (3) errors due to procedures for using

tlie data, which are traceable to tlie inadequacies

of tlie sclieniatization.

81
It is clear that the use of a particular approximate fission spectrum, namely that for 1.12 hour

old fission products, to represent all practical situations, is by far the largest error of the first type.

The magnitude of such errors can be estimated

roughly from data such as that in figure 26.6.

For planning purposes the shielding factors may

be defined relative to a particular spectrum. This

doesn't alter the fact that the data may not represent an experimental situation exactly, but it does encourage a shift of attention to other parts

of the problem, since then the validity of the

spectrum and the accuracy of shielding factor calculations are considered separately.

The other major type of input information is the cross sectional data. We expect small inaccuracies in the differential scattering cross section

due, e.g., to atomic binding corrections in the

Klein-Nishina expression, to have negligible effect

on the accuracy of the calculations. On the other

hand, errors in the total cross sections which, generally speaking, are expected to be less than 1 percent in magnitude, can result in errors in the

barrier factors at the deeper penetrations which

are up to 10 times larger, i.e., 5 to 10 percent.

A change in air density has the effect of a change

in total cross section; and this can be brought

about by temperature or altitude variations. Finally, not all data was prepared using precisely

the conversion factor of eq 26.1 ; there were differences of a few tenths of a percent. For this

reason, and due also to the use of overlays in

preparing the graphs, the data are not expected

to be internally consistent to better than about

5 percent.

Generally speaking, the attenuation data represented in the barrier factors can be calculated

with much greater accuracy than the angular distributions which are used to obtain geometry

factors. Calculations of barrier factors for point

and plane isotropic sources, i.e., the L{X) and

P{_X) curves, are probably accurate to within 5

percent, assuming that the input data is exactly

correct. Considering the different sources of error together except for the spectrum assumption, these curves should be correct to within, say 8 percent.

On the other hand, the S{d), and W{X,d)

are appreciably less accurate; and it is difficult to

assess quantitatively the accuracy of these curves even though some of the major sources of error are known. Perhaps for guidance purposes it may

be useful to give "educated guesses" that excluding

the assumption of spectrum, errors in S((i) may

be in the range of 10 to 15 percent; errors in

W{X,d) for small X are perhaps 10 percent and

for large X may be two or three times larger; and

errors in S' {X) can be expected to increase roughly

linearly with X, from 0 at jS"=1 to perhaps as much as 100 percent at S' = 10-*.3»

Errors in S(d) stom from iiiodenitely rough caloulalions of tlie angular

distribution; errors in Il'(-V,d) come largely from the .spectrum and anguler

distribution assumed as source for tlie wall penetration; and errors in .S"(.V) are dua to its calculation as a small difference between large numbers.

Turning to angular distributions and geometry

factors, the inherent inaccuracies of the calculations can be seen visually in the Co®° and Cs^^^

angular distributions in Appendix B, which have

not been smoothed. The distributions of figures.

26.1 and 26.3 originally looked very much the

same but were arbitrarily smoothed with french

curves. Errors in the range of 20 to 25 percent,

perhaps even greater, can be expected at places

in these data. However, the integrations which

yield geometry factors improve the situation somewhat. Generally speaking, we expect the geometry factors to be quite accurate when they are

large and to approach the accuracy of the angular

distributions from which they were obtained when

they are very small. Thus, from right to left the

geometry factor curves vary from no error to

errors of perhaps as much as 25 percent, with

the Lb and Lc curves being more reliable than

this because they were not obtained from angular

distributions. Combinations of geometry and barrier factors, such as in figures 28.18, 28.19, and

28.20 have different accuracy in different regions

as determined by the separate errors inherent in

the two ingredient factors.

Finally, in regards to the utilization of the

data, there are several comments which can be

made. Use of water data to represent concrete

gives results normally within the accuracy of the

calculations if the 0.9 factor differences are cor- rectly applied. But it should be remembered that near the source, where the total contribution of

scattered photons is a relatively small part of

the detector response, the difference between water

and concrete is negligible, certainly not as large

as 10 percent; and at great penetrations the differ- ence can be larger than 10 percent. Errors due

to estimation of boundary effects probably approach the magnitude of tlie total boundary effect.

And errors due to the use of circular source data

to represent rectangular source geometry factors

will vary greatly with the situation. They can perhaps be estimated by the use of several calculations using different superpositions of data, as was indicated in the problem worked out in

vSection 43.

49. Alternative Data Types and Procedures

The use of circular source data for computation of geometry factors is only one of several

possible procedures involving different data types.

For example, in Section 44 it was indicated that

geometry factors for rectangular source types

could be calculated directly.

An interesting alternative procedure has been

explored by Berger, Hubbell, and Lamkin [29,

30, 41]. This consists of using the Legendre coefficients of the directional distributions rather than

the directional distributions themselves. If, in

eqs (16.2) and (16.3) the dose angular distribution (D) is written in the form of a Legendre

series, the detector response Ds takes the form

of a sum of products in which one factor is a

82
Legendre coefficient of the series and the other

factor is the corresponding coefficient of the

Legendre seiies representation of gs- Hubbell,

Lamkin, and Bach [41] have prepared extensive

tables of the gs Legendre coefficients to make

such calculations feasible; but tables of Legendre

coefficients for the relevant dose angular distributions have never been prepared.

The Legendre coefficient approach needs more

attention. It is easy to apply, and has as its

main limitation a tendency for the series of products to converge slowly when the angular distribution has a sharp peak, as it often does (figs.

26.1 and 26.3), and when the solid angle fraction

of gs is at the same time small, as it often is. This limitation may prove moie apparent than real.

Appendix B. Data for GO*' and CS^^'

50. Introductory Comments

Procedures for the calculation of the data represented in the figures Bl through B50 have

already been described. The main differences

from the fission source data stem from the use

of desk computers in that case for some calculations performed by digital computer in the case

of Co*° and Cs'^'', the procedures involved not

being entirely identical.

The figures given here are in the same sequence

as the corresponding fission source figures in the

main text ; and data for the two source types have

been placed adjacent to one another.

No attempt has been made to analyze the obvi- ous similarities between the figures for the different source types. It is clear that many of the

geometry factors are quite insensitive to source spectrum.

83
84
COS 6

Figure B2. Cs-137, concrete.

85
COS 6

Figure B3. Co-60, concrete.

86
COS 6

Figure B4. Cs-137, concrete.

87
X.psf

Figure B5. Co-60, concrete.

88
X.psf

Figure B6. Cs-137, concrete.

89
0 20 40 60 80 100 120 140

X,psf

Figure B7. Co-60, concrete.

90

92
COS 9o

Figure BlO. Cs-137, hackscatter from concrete.

93

Figure B12. Cs-lS7, H2O.

95
96
X

X,psf

FiGUKE B14. Cs-137, concrete.

97
98
99
X,psf

Figure BIT. Co-60, concrete.

100
0 20 40 60 80 100 120 140

X,psf

Figure B18. Cs-137, concrete.

101
d,ft.

FiGUBE B19. Co-60, H2O.

102
103
0 100 200 300 400 500

X,psf

FiGUKE B21. Co-60, concrete.

104
X,psf

Figure B22. Cs-137, concrete.

105
106
X,psf

Figure B24. Cs-137, concrete.

107
X.psf

FiGUEE B25. Co-60, H2O data for l{d,cosd), concrete data for s{X,cos9o)-

108
0 100 200 300 400 500 600

X.psf

Figure B26. Cs-137, H2O data for l{d,cos0), concrete data for s{X,coseo).

109
FiGtTBE B27. Co-60, concrete.

110
Figure B28. Cs-137, concrete.

Ill
112
Figure B30. Cs-137, concrete.

113
CD

Figure B31. Co-60, concrete.

114
FiGUHE B32. Cs-137, concrete.

115
Figure B33. Co-60, H2O.

116
Figure B34. Cs-lS7, H2O.

117
FiGTJEE B35. Co-60, H2O.

118
Figure B36. Cs-137, H2O.

119
120
Figure B38. Cs-137, H2O.

121
Figure B39. Co-60, H2O.

122
Figure B40. Cs-187, H2O.

123
ill

Figure B41. Co-60, H2O.

124
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132
133
References

[I] T. Rockwell, III (Editor), Reactor shielding design

manual (McGraw-Hill, New York, N.Y., 1957).

[2] Reactor Handbook, 1, AECD-3645 (USAEC 1955).

[3] H. Goldstein, Fundamental aspects of reactor shielding,

(Addison-Wesley Publ. Co., Reading, Mass., 1958).

[4] B. T. Price, C. C. Horton, and K. T. Spinney, Radiation Shielding (Pergamon Press, Inc., New York,

N.Y., 1957).

[5] Reactor Physics Constants, ANL-5800 (Office of

Technical Services, U.S. Dept. of Commerce,

Washington 25, D.C., July 1958).

[6] S. Glasstone (Editor), Effects of Nuclear Weapons,

(Pergamon Press, Inc., New York, N.Y., 1957).

[7] A method for Evaluating the Protection Afforded Buildings Against Fallout Radiation (ODM, Washington,

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[8] W. Malich and L. A. Beach, Radiation protection

afforded by barracks and underground shelters,

NRL Rept. 5017 (U.S. Naval Radiological Defense

Laboratory, San Francisco, Calif., Sept. 18, 1952).

[9] R. R. Putz and A. Broido, A computation method for

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shielding and source configurations (Proj. Civil,

Univ. of California, Dec. 1957).

[10] F. C. Brooks, et al.. Radiological Defense Planning

Guide, TOI-58-26 (Tech. Operations, Inc., Burlington, Mass., 1958).

[II] Assessment of the protection afforded by buildings

against gamma radiation fallout (Home Office,

London, England, 1957).

[12] U. Fano, L. V. Spencer, and M. J. Berger, Penetration

and diffusion of X-rays, Handbuch der Physik,

38 II, 660 (Springer-Verlag, Berlin, Germany, 1959).

[13] G. M. Corcos, On the small-scale, nonhomogeneity

of fallout deposition (Univ. of California, Oct. 30,

1958).

[14] K. H. Larson, Adherence of fallout to trees and

shrubs, USNRDL Letter Report (Unpublished)

(U.S. Naval Radiological Defense Laboratory, San

Francisco, Calif., Jan. 1959).

[15] H. Nishita and K. H. Larson, Summary of certain

trends in soil-plant relationship studies of the

biologically availability of fallout debris, Rept.

No. UCLA-401 (Univ. of Cahfornia School of

Medicine, Los Angeles, Calif., July 28, 1957).

[16] E. P. Cronkhite, et al.. Response of human beings

accidentally exposed to fallout radiation, J. Am.

Med. Assoc., 159, 430-4 (Oct. 1, 1955).

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Francisco, Calif., Oct. 1956).

[18] A. T. Nelms and J. W. Cooper, U^" fission product

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Physics 1, 427-441 (Sept. 1958).

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the slow neutron fission of U^ss, USNRDL-TR-187

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from fallout, USNRDL-TR-318 (U.S. Naval

Radiological Defense Laboratory, San Francisco,

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Handb. 62 (1956).

[25] R. C. Bolles and N. E. Ballou, Calculated actidties

and abundances of U^^s fission products USNRDL456 (U.S. Naval Radiological Defense Laboratory,

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[26] C. F. Miller, Gamma decay of fission products from

the slow-neutron fission of U^'^, USNRDL-TR-187

(U.S. Naval Radiological Defense Laboratory, San

Francisco, Calif., July 11, 1957).

[27] J. Moteif, Miscellaneous data for shielding calculations, Rept. No. APEX-134, (USAEC 1953).

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Dosimetry (Academic Press, Inc., New York, N.Y.,

1956).

[29] M. J. Berger, The dose received from partially shielded

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[30] M. J. Berger and J. C. Lamkin, Sample calculations

of gamma ray penetration into shelters: Contributions of skyshine and roof contamination, J. Research NBS, 60, 109 (1957) RP 2827.

[31] F. H. Attix and V. H. Ritz, A determination of the

gamma ray emission of radium, J. Research NBS,

59, 293 (1957) RP 2801.

[32] R. T. McGinnies, X-ray attenuation coefficients from

10 kev to 100 Mev, NBS Circ. 583 Suppl. (Oct. 30,

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the penetration of gamma rays (USAEC Rept.

NYO-3075) (1954).

[34] M. J. Berger and D. J. Raso, Monte Carlo calculations

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12, 20 (1960).

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in existing urban buildings in the United States,

Project Civil Report (Inst, of Eng. Research, Univ.

of Calif., Richmond, Calif., Jan. 1958).

[36] F. Titus, Penetration in concrete of gamma radiation

from fallout, AEC Rept. ITR-1477 (Oct. 22, 1957).

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Radiations from a rough infinite plane, USNRDLTR-108 (U.S. Naval Radiological Defense Laboratory, San Francisco, Calif., Jan. 18, 1956).

[38] C. E. ClifiFord, J. A. Carruthers, and J. R. Cunningham, Scattered gamma radiation from a simulated

fallout field using Cs-137, DRCL Rept. No. 296

(Defense Research Chemical Labs., Ottawa, Canada,

Jan. 1959).

[39] E. T. Clarke, J. F. Batter, and A. L. Kaplan, Measurement of attenuation in existing structures of radiation from simulated fallout, Rept. TO-B 59-4

(Tech. Operations, Inc., Burlington, Mass., Apr.

27, 1959).

[40] W. B. Boast, Illumination Engineering, Chapt. 7, Determination of illumination from surface and

line sources, 82-100 (McGraw-Hill Book Co.,

New York, N.Y., 1953).

[41] J. H. Hubbell, R. L. Bach, and J. C. Lamkin, Radiation field from a rectangular source, J. Research

NBS 64C (Eng. and Instr.), No. 2, 121-138 (Apr.-

June 1960).

134

U.S. GOVERNMENT PRINTING OFFICE; 1962 O—622827


U.S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

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