PB 151370

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PENETRATION OF GAMMA

RAYS FROM ISOTROPIC

SOURCES THROUGH

ALUMINUM AND CONCRETE

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U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical <v2©tc

ft

MAY 11, 1959

PENETRATION OF GAMMA RAYS FROM ISOTROPIC SOURCES

THROUGH ALUMINUM AND CONCRETE

Martin J. Berger and Lewis V. Spencer

The work described in this Technical Note

was sponsored by the Bureau of Yards and

Docks, Department of the Navy.

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature. They are for sale by the Office of Technical Services, U. S. Department of Commerce, Washington 25, D. C

DISTRIBUTED BY

UNITED STATES DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25. D. C.

Price 50 cents

Penetration of Gamma Rays from Isotropic Sources through Aluminum

and Concrete

Martin J. Berger and Lewis V. Spencer

ABSTRACT

Semi-analytical expressions, with numerically specified parameters,

are given which represent the gamma ray dose distribution in infinite

aluminum or concrete media, for sources that are monoenergetic (with

energies between 10.22 Mev and 0,034-1 Mev), isotropic, and have the form

of an infinite plane, point, disk or spherical surface.

1. Introduction

This publication presents recent results of a program' of gamma ray

penetration calculations now in progress at the National Bureau of

1/ 2/ Standards. This program is based on the use of the moment method.

\J Previous unpublished reports on this program include: J. H. Hubbell, Dose

due to distributed gamma ray sources, November 1956. L. V„ Spencer and

J. C. Lamkin, Slant penetration of gamma rays in water, July 1958.

L. V. Spencer and J; C. Lamkin, Slant penetration of gamma rays; mixed

radiation sources, February 1959-

2/ L. V. Spencer and U. Fano, J. Res. NBS 2yS, 446 (1951 ) - See also U. Fa no,

L. V. Spencer and M. J. Berger, Encyclopedia of Physics, 38 , II, Springer,

Berlin, July 1959.
It provides basic information about the attenuation of radiation in extended

homogeneous media, which is needed for Civil Defense shielding studies.

The present report deals with the penetration of gamma rays from isotropic sources through aluminum or concrete* The physical quantity computed

is the gamma ray dose as a function of the distance from the source. The results apply to all types of isotropic sources (plane or point sources, disk

sources, spherical sources, etc.).

The dose as function of the distance from a point-isotropic source has

previously been calculated. 3/ iBJU The new calculations differ from this work

J/ H. Goldstein and J. E. Wilkins, Jr., AEC Report NYO 3075 (1954).

ij Unpublished report, J. H„ Hubbell, Dose due to distributed gamma ray

sources, November 1956.

in the following respects: (a) A wider range of monoenergetic sources is

treated, extending from 10.22 Mev down to 0.034-1 Mev. (b) The results are

presented in sernianalytical form, with numerically specified parameters,

rather than in completely numerical form. This has the advantage that the

basic calculation, for a plane isotropic source, can readily be applied to

other source types by simple analytical manipulations.
2. Plane Isotropic Source

Notation:

E = source energy

o

z = distance from source plane

m(e) = gamma ray attenuation coefficients n = ll(E )

ll (e) = energy absorption coefficient for air

•enK

p = source strength: number of gamma rays emitted per second

from a unit area of the source plane

.00, .-.,

z 6/

DD(z) = absorbed air dose*^ at a distance z from the source plane.

• (z) = J

(e /s)ds = exponential integral"

'PL

-^5/For a tabulation of the exponential integral, see, e.g., Tables of Sine,

Cosine and Exponential Integrals? WPA, 1940° -6/J For a definition of absorbed dose, see Nat„ Bur„ of Standards Handbook 62,

Report of the International Commission of Radiological Units and Measurements (1956).

The absorbed air dose can be represented by the following formula:

D

PL<

Z

> = \ hlJ^oK ElM + 1

K

?L>

J-cn<V E

o {h^ + O^ M
If E is expressed in units of 100 ergs, u. in cm /g, K

p

in cm sec ,

and z and \± in reciprocal but otherwise arbitrary units, then D

pT (z)

7/ -1

has units of rads-^ sec „ The first and second term in (l) represent

1/

1 rad corresponds to an energy absorption of 100 ergs per gram of the

medium (in the present case air).

the contribution to the dose by unscattered and scattered gamma rays,

respectively.

The dose depends on the atomic number of the medium primarily through

the attenuation coefficient \i , and much less sensitively through the parameters A , A , EL and B . The atomic number of aluminum Z = 13) is close

to that of concrete (z rr , . ~ 13.4-) so that the same set of parameters effective

can be used for both materials.. Table 1 lists these parameters (obtained

through a moment calculation for aluminum) for various source energies.

Also shown are the energy absorption coefficient for air, and the attenuation coefficients for aluminum and concrete,. The latter two quantities,

when expressed in cm /g, are very close to each other..
3. Other Source Geome t rie s,

There are simple relations between the dose distributions for different

8/ source geometries and which hold under the following conditions:

8/ Cf. Reactor Handbook, Vol. I, Chapter 2.5, AECD 3645 (1955).

(a) The detector and source are isotropic. (b) The medium is homogeneous,

(c) The boundaries are far enough removed to be unimportant. We shall apply

a few of the more important of these relations.

3. lo Point Isotropic Source

Notation:

r = distance from point source

K = source strength: number of gamma rays emitted per second

D

p

(r) = absorbed air dose at a distance r from the source

The general relation between point- and plane-source distributions is

d

pT w = - is [-fc Vz

> ] .. (2; z=r
By applying this relation to Eq. (l) and inserting the appropriate

source normalization constant we find that

n ( ) - K

PT^en (E o )E

o "V . d

pt (p) - r^

e +

4-77T

KDTu (e)E^ f -B.pir -B_u r

j. PT^en o o { . _ l^o . «• n 2^o + ~2 Ho r \A lBl e + A^ e

47ir

If r is expressed in cm, K in sec , and the units for the remaining

quantities are the same as in the case of the plane source problem, then

D_ (r) again represents a dose in rads sec

From Eq. (3) one can derive an expression for the dose build-up

factor B(r) (ratio of the total dose to the dose contributed by unscattered

radiation). We find that

-(B,-l) Hor -(B,-l)^r1

V l-Vl- +A2

B

2

B(r) =1 + nr \A B e + A Be£ *

J.

This expression may be compared with results previously obtained by Goldstein

3/ and Wilkins. According to Figure 1 which contains plots of B(r) vs. E ,

for different values of u. r, the two sets of calculations are in good agreement, insofar as they cover the same range of source energies. This is

interesting in view of the fact that the methods of calculation differ. Both

make use of the numerical flux moments, calculated according to identical
equations, but the construction of the flux from the moments was done

differently. Goldstein and Wilkins used the method of polynomial expansion developed in Reference 2. The present calculations are based

9/

on a technique called "function-fitting"—^ which we believe to be

9/"^Described in the Encyclopedia article mentioned in footnote 2.

somewhat more accurate, and which leads to a representation of the type

of Equation (l) which is convenient for analytical manipulations.

3.2. Isotropic Disk Source

Notation:

a = radius of disk source

z = distance from source along axis of disk

K_ Tpit, = source strength: number of gamma rays emitted per second from

a unit area of the disk

DnTOI/ (z, a ) = absorbed air dose at a point on the axis of the disk, at a

distance z.
Using the general relation

lOT

D

DISK

(z ' a) = 2lT h

D

PT ^) rdl> (5)

and the appropriate source normalization, we' obtain the result

D

DISK

(z ' a) = 2

K

DISK^en (E

o )Eo {

E> z) " ^^^l +

)lz

2

+a 2 + -iKnTWu (EJE. (Al [e

X °-e lo " ,f ] + (6)

B

^z "B i^Vz "*'

2 "DISK^WV'o VI

-Buz -BuW . + A

2

[e

2^°-e 2 ° ]}.

With a as well as z expressed in units reciprocal to those of \± , and with

K „ in cm sec , D (z,a) is in rads sec
3.3. Isotropic Spherical Surface Source.

Notation:

r = radius of spherical surface containing the source

o r = distance from center of sphere

K = source strength: number of gamma rays emitted per second

SPH

from unit area of source

D (r r ) = absorbed air dose at a distance r from the center of the

SPH ' o

J

sphere.

Using the relation

D (r,r ) = -^

JDpT (|r-r I ) - D

pT

(r+r ) X

SPH ° r [ PL ' o

• PL o J
and the appropriate normalization, we have the following result:

(i) r > r

W^.V =^ Wen(E o )E

o ¥Vr"r

o

)] +

+ ^ Wen(E o )E

o {

A

l e "^"^iVo1 + V ^VW

(ii) r < r

o .

(8)

W1 '^ = t

KSP#en (E

o )Eo Elk%*>] +

+ ^ KSPH^n (E o )Eo{A

l e lV°3inh(BlV) + A

2

e ^^sinhfB^r)}

"I With K_p„ in sec , and the other quantities in the same units as in the

plane source problem, D uis in rads sec

10
4-* Comments

The results of this paper are intended chiefly for applications

to situations where an analytical representation of the dose distribution is useful. Tables with complete numerical results for the dose

distributions from point-isotropic and plane isotropic sources will be

published in later reports. Further work is also in progress to obtain

not only the spatial distribution of the dose, but also the directional

distribution of the radiation giving rise to the dose.

We are indebted to Mr. J. Lamkin and Mrs. I. Reingold for help

with the computations.

11
Al Concrete Air

E ^0 ^0 ^en

(E

o)

A

l A

2

B

l B

2 (Mev

)

(cm /g) (cm /g) (cm /g")

10.22 .0228 .0228 .0144 0.00085 0.27904 1.55034 0.99218

6.81 .0253 .0256 .0157 0. 00689 0.39173 1.46162 0.99425

3.41 .0333 . 0342 .0196 -0.03842 0.63979 1.39280 0.98916

2.04 .0^27 .0441 .0232 -0.14505 0.89909 1.30878 0.98127

1.28 .0543 .0562 .0266 -0.35610 1 . 25634 1.21554 0.97062

0.852 .0663 .0686 .0287 -0.69202 1.73577 I.14244 0.95916

0.511 .0832 .0862 .0297 -1.35130 2.62614 1 . 08074 0.94685

0.319 .101 .104 .0290 -1.71335 3.23857 1.06 037 0.93355

0.213 .117 .121 .0272 -1.61858 3.39072 1.06061 0.92969

0.128 .143 .149 .0240 -0.91086 2.87029 1.07050 0.94280

0.0730 .201 .214 .0246 0.33281 1.33242 1.66 004 0.98954

o. 0426 .443 .474 .0546 0. 30688 0.37299 1 . 80194 1.05416

0. 0341 .745 .793 .100 0.18547 0.18908 1.91352 1.06999

Table 1. Parameters of dose distribution

12

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U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

A. V. Astiii, Director THE NATIONAL BUREAU OF STANDARDS

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