D. K. Trubey DO NOT TRANSFER TO ANOTHER PERSON

lUunmuBw ^S&m/ma

3 445^ 03biSII

CHTRJIl RESEARCH LfBHART ^COLLEGDOU

ORNL-2998UC-34 - Physics and Mathematics

THE SINGLE-SCATTERING APPROXIMATION

TO THE GAMMA-RAY AIR-SCATTERING PROBLEM

D. K. Trubey

CENTRAL RESEARCH LIBRARYDOCUMENT COLLECTION

LIBRARY LOAN COPYDO NOT TRANSFER TO ANOTHER PERSON

If you wish someone else to see thisdocument, send in name with documentand the library will arrange a loan.

OAK RIDGE NATIONAL LABORATORY

operated by

UNION CARBIDE CORPORATION

for the

U.S. ATOMIC ENERGY COMMISSION

Printed in USA. Price

Office of Technical Services

Department of Commerce

Washington 25, D.C.

$0.50

LEGAL NOTICE "

Available from the

This report was prepared as an account of Government sponsored work. Neither the United States,

nor the Commission, nor any person acting on behalf of the Commission:

A, Makes any warranty or representation, expressed or implied, with respect to the accuracy,

completeness, or usefulness of the information contained in this report, or that the use of

any information, apparatus, method, or process disclosed in this report may not infringe

privately owned rights; or

B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of

any information, apparatus, method, or process disclosed in this report.

As used in the above, "person acting on behalf of the Commission" includes any employee or

contractor of the Commission, or employee of such contractor, to the extent that such employee

or contractor of the Commission, or employee of such contractor prepares, disseminates, or

provides access to, any information pursuant to his employment or contract with the Commission,

or his employment with such contractor.

\

ORNL-2998

Contract No. W-7^05-eng-26

Neutron Physics Division

THE SINGLE-SCATTERING APPROXIMATION TO THE GAMMA-RAY

AIR-SCATTERING PROBLEM

D. K. Trubey

DATE ISSUED

JAN 20 1961

OAK RIDGE NATIONAL LABORATORY

Oak Ridge, Tennesseeoperated by

UNION CARBIDE CORPORATION

for the

U. S. ATOMIC ENERGY COMMISSION

l"|4|mi"llirA|liETI{i,E!E"GY Sy5TE««S LIBRARIES

inn 11ill IIII III

3 "4M5b D3blSME 5

-iii-

ABSTRACT

The result of a calculation using the single-scattering

approximation, with no exponential attenuation or buildup, has

been compared to the results of a Monte Carlo calculation of

the dose rate and number flux in air from a monoenergetic gamma-

ray source. The comparison shows that the simple approximation

is generally adequate for unshielded detectors.

The Single-Scattering Approximation to the Gamma-Ray

Air-Scattering Problem

D. K. Trubey

It has been hypothesized that a good approximation to the scattered

flux (or dose rate) in air is the singly-scattered flux (or dose rate)

with the exponential attenuation and buildup neglected. The neglect of

both these factors, of course, tends to compensate. Such a simple method,

if accurate, would be very useful in computing skyshine from a reactor or

some other source.

This approximation was used by Keller et al. for neutrons scattered

in air isotropically in the center-of-mass system, and their results were

in good agreement with those of a Monte Carlo calculation. Since Monte

Carlo calculations of the air-scattered flux and dose rate for gamma rays are

2also available, the purpose of this paper is to compare these results with

the proposed simple approximation, thereby further investigating its validity.

Consider a line beam of monoenergetic gamma rays emitted from a point

source in a uniform infinite medium. The line-beam concept is selected so

that the effect of direction can be easily assessed. The photons are emitted

at the source S (see Fig. l) at an angle y, travel a distance R,, are scattered

through the angle 0, and arrive at the detector located at point P. If exponential

attenuation and buildup are neglected, the photon flux from a monoenergetic

1. F. L. Keller, C. D. Zerby, and J. Hilgeman, Monte Carlo Calculations ofFluxes and Dose Rates Resulting from Neutrons Multiply Scattered in Air,ORNL-2375 (Dec. 15, 1958). ———

2. R. E. Lynch, J. W. Benoit, W. P. Johnson, and C. D. Zerby, A Monte CarloCalculation of Air-Scattered Gamma Rays, ORNL-2292, Vol I (Sept. 10, 1958),

-2-

source of energy E at P from all scatterings of the line beam is

T *« dR]}(E,¥,X) = H|(E,fl) -^ , CD0 R2

where

N = electron density (cm •*),

t? = Klein-Nishina scattering cross section (cm ),

From the law of sines,

Rl R2 Xsin0 sin? sin;* - (¥ + 0)_

Differentiating,

dRl _ X sin(¥ +0) cos0 -X cos(^ +0) sin0^~ " sin2(I +0)

The numerator of Eq. 3 equals

Xsin [(f +0) -0J = Xsin? ;

(2)

(3)

therefore pRp

dR = d0 ,* (k)Xsinf

andif-I

*£ (K.frt JL, • (5)few . 4 u ^C^)^0

But 0 = ¥ + 0; therefore

***'x) - dtg J n(B'9) aa • (6)

*The derivation of this equation was taken from Richard Stephenson, Introductionto Nuclear, Engineering, p. 198, McGraw-Hill Book Company, Inc., New York, 195*+•

IJi')MI1IIW!'l'.WIIMWWPl

-3-

Now define

>

£2 (E,0) d0 =K(E) F(E,¥) (7)

1

where K(E) is a function of energy only and f(eJi) is a fraction. This equation

was evaluated numerically for six angles and six energies by using the ORACLE com

puting machine. The value for S! = it is easily determined from

TIlim

m°»°I

sintt *-** sin(rt -?)

The functions F(E,J) and K(E) are shown in Fig. 2.

The dose rate is computed in a similar manner with a conversion factor for

flux to dose rate included in the integral. Since the energy and angle are

related by the well-known Compton formula, the conversion factor, T, is a

function of the angle 0. So that a comparison could be made with the Monte

Carlo data, the conversion factors given in Ref. 2 were used, the factors being

averages over certain energy intervals. The pertinent functions C(E) and D(E,S?)

defined by

dft<*>}V—• it

00

dfl(E,0) T(E,0) d0 = c(e) d(e,M)

M

d0

(8)

(9)

are shown in Fig. 3. The functions F(E,2) and D(E,J|Q can be fitted reasonably

well by a hyperbola, i.e., by

F(E,I) y(i - u)(1 - uy)

where y = 1 - (l/n) and u is a function of energy. The fit is somewhat bettero

if the denominator also contains a y term. (Several fitted curves are compared

with computed values of K(E,¥) and D(E,¥) in Table 1. The parameters of the

-it-

fitted curves are given in Table 2.) The dose rate per initial photon per second

may then be computed from

Dose(E,X,2) - \ °(E) D(E^} . (10)sing

The resulting fluxes and dose rates in air for six energies and various

angles, as a function of distance, are shown in Figs, k - 15. The points

shown are the data of Lynch et al. (i.e., from the Monte Carlo calculation),

and the curves are given by Eqs. 6 and 10. The results show the simple dose-

rate approximation to be very good for all angles and energies and the flux

approximation to be good to about 60 deg for large distances but low as much

as a factor of two for the back-scattered radiation. For an isotropic source,

however, these angles are unmportant. If the detector is shielded in any way,

however, these results probably are not valid since energy and angular distri

bution at the shield are important.

-5-

Table 1. Comparison of Fitting Functions to

Values of F(E,¥) and D(E,y)

y(i -ux) y(i -u2) y(l -u3 +U]+)

E <Mev) I (deg) F (E,Y) 1 - u^yD(E,J)

1 - u2y 1 -u y + u^y2

0.6 15 0.7617 0.743 0.7039 O.656 0.705' 30 0.5683 O.568 0.4637 0.464 0.4726o 0.3448 0.345 0.2107 0.257 0.211

90 0.2372 0.209 0.1272 0.148 O.O96135 0.1192 0.081 0.0449 0.055 0.026

1 15 0.7193 0.697 O.6352 0.555 O.63630 0.5093 0.511 O.3623 O.362 0.37660 0.2953 0.295 0.1399 O.185 0.140

90 0.1973 0.173 0.0745 0.102 O.O58135 0.0947 0.065 O.O3O9 O.O36 0.015

2 15 0.6^74 0.6235 0.5282 0.413 0.53030 0.4247 0.429 0.2426 0.242 0.26160 0.2346 0.231 0.0794 0.113 0.08090 0.1504 0.131 O.O385 0.060 0.030

135 O.0682 0.048 0.0142 0.021 0.007

4 15 O.5589 0.541 0.4145 0.294 0.41930 0.3412 O.348 0.1589 0.159 0.17260 O.1807 O.176 0.0455 0.070 0.04690 0.1115 O.097 O.0207 O.O36 0.017

135 0.0486 O.034 0.0070 0.012 0.004

7 15 0.4797 0.460 0.2957 0.199 0.299

30 0.2790 O.279 0.1017 0.102 0.10660 0.1429 0.134 0.0266 0.043 0.026

90 O.0861 0.072 0.0112 0.022 0.0093

135 O.O368 0.025 O.OO38 0.0074 0.0021

12 15 0.4028 0.391 0.1980 0.121 0.19730 0.2255 0.226 0.0592 0.059 0.064

60 0.1123 0.104 0.0153 0.024 0.01590 0.0666 0.055 O.OO65 0.012 0.0053

135 0.0281 0.019 0.0024 0.0041 0.0012

-6-

Table 2. Parameters Used in Fitting Functions

E (Mev) Ul U2 U3 u4

0.6 0.7366 0.8270 I.692O 0.7581

1.0 0.7908 O.8863 1.8109 0.8464

2.0 0.8494 0.9360 1.8902 0.9074

4.0 O.8930 0.9622 1.9315 0.9405

7-0 0.9226 0.9774 1.9473 O.9522

12.0 O.9417 0.9875 1.9508 0.9536

UNCLASSIFIED

2-01-059-607

Fig. 1. Geometry for Single-Scattering Calculations

1.0

0.9

0.8

0.7

0.6

Uj 0.5

0.4

0.3

0.2

0.1

0

-7-

UNCLASSIFIED

2-01 -059-605

E (Mev) K(E)(cm )

0.6

x 10~268.400

1

2

4

7.015

5.372

4.010

7

12

3.119

2.425

\ N. ^S?

X >nP«\x/.

xS^

^==^0 30 60 90

^(deg]120 150 180

Fig. 2. The Function E{E^) for Various Energies.

UNCLASSIFIED

2-01-059 -606

180

Fig. 3. The Function D{£,¥) for Various Energies.

(0

10"

£ 10"

(0

I icT

10

(0

-9-

UNCLASSIFIED

2-01-059- 593

-H— ^—f-f-+-f-f-4-4-+-H 1

AIR DENSITY = 0.00(25 g/cm3f0, SOURCE ENERGY = 0.6 Mev

1 1 1 1 1 1 1 1 1 1

.. O. • nATA OF 1 YMC.H fit nl

E QUATION 6

c ?*••

nH^ ni- « »»

1 deg

^^"•<>^

-»< 1

•"J* _< J""""«*|

C^

^*m t15 deg

v"° "^"^"*—«^< )

^< • "^X)^30 deg^<^jh

s^ »

c ) ""•*•<>i | 60 deg

*^. 90 deg

"»-~^» 135 deg"""180 deg

, L

6 7 8 9 10 15 20 30 40 50 60 70 100

X, SOURCE -DETECTOR SEPARATION DISTANCE (ft)

150 200

Fig. 4. Air-Scattered Gamma-Ray Fluxes as a Function of Distancefrom a 0.6-Mev Line-Beam Source

-10-

10

10

10

T_ <o

10

10

10'

-4

UNCLASSIFIED

2-01-059-594

i i i i i i i i i i i

AIR DENSITY = 0.00125 g/cm3f0, SOURCE ENERGY = 1 Mev

1 1 1 1 II 1 1 1 1

5

2

-5

I 1 1 II II II 1. O. • DATA OF LYNCH, et al. .

E QL A Tl 0 VI €

5

2

-6

( )

5 "N >-

I ' BEAM ANGLE =

2

-7

o-^^1 deg

i »•

5

i !"*

2

-8

S5Ck "( t _

«L^^-—. ( i

15 deg

1 ^ ^«w -. «

*5i^i 55

„ fIs" C)( '^- 60 deg

2

-9

t P ( )

]. 90 deg

180 deq

5

2

-10

7 8 9 10 15 20 30 40 50 60 70 100

x, SOURCE -DETECTOR SEPARATION DISTANCE (ft)

150 200

Fig. 5» Air-Scattered Gamma-Ray Fluxes as a Function of Distancefrom a 1-Mev Line-Beam Source

10

5 6 7 8 9 (0 15 20

-11-

I I I I I I FFF

UNCLASSIFIEO

2-01-059-595

AIR DENSITY = 0.00125 g/cm3f0, SOURCE ENERGY = 2 Mev

O, • DATA OF LYNCH, et ol.

EQUATION 6

30 40 50 60 70 100 150 200

x, SOURCE-DETECTOR SEPARATION DISTANCE (ft)

Fig. 6. Air-Scattered Gamma-Ray Fluxes as a Function of Distancefrom a 2-Mev Line-Beam Source

-12-

10"4 r =F-4--+-4- MM

UNCLASSIFIED

2-01-059-596

-m 1

- AIR DENSITY = 0.00125 g/cm

5 fQ, SOURCE ENERGY = 4 Meu

I I I I I II !4-l \-

O, • DATA OF LYNCH, et ol. -2

m-5

- EQUATION 6

5

2

|io-6OJZa.

s 5o

k, ''"""BEAM ANGLE =2

03in 1'1

^"""M~^w

1 dec

u

c

o 5<»^„^

SZ

X "%,<

*^< »^

=> 2_iu_

o: 10"*•«•.< 1

-1 >^

^•».(, 15 deg

i

<i

S 53

•-::<^ i""*•«» 30 deg

• ( »2

10" 9

.^^^ i >

>

<•

*">>.«, 60 deg

(5

5.^ 1

•«.^*.*>, 135 deg

180 deg

2

1—1—

7 8 9 (0 15 20 30 40 50 60 70


100 150 200


10

10

10"

\

10

10

2

10

10"

•13-

UNCLASSIFIED

2-01-059-597

=F 1 II M Mill 4

AIR DENSITY = 0.00125 g/cmfg, SOURCE ENERGY = 7 Mev

)

1 111 11C , • DATA

EQUA

OF LYNCH, et al.TION 6.

i—

"*"•"•«.,. BEAM AN GLE = _

^1 »*«^"" 1 deg

^••11

^»^

*""•> •^"-~.

"""I»^,

^fc!-.

II v^^-^

•"»!

:;>!! "^-^•^^^ <

..^^ < ^>->«-

< ^*i'

"""" 30 deg

1 r^; I3LI !*" «;---<>

^ 1

•^•^»

s>^»«. 7l***^ ~90deg

*^-135 deg180 deg

7 8 9 10 15 20 30 40 50 60 70 100 150

X, SOURCE-DETECTOR SEPARATION DISTANCE (ft)

Fig.-8. Air-Scattered Gamma-Ray Fluxes as a Function of Distancefrom a 7-Mev Line-Beam Source

200

-14-

10"4 r L 1 1 1 1 i i i

UNCLASSIFIED

2-01-059-598

1 1 l~-

5 £0, SOURCE ENERGY = 12 Mev

I I I I I I I I I 1 1

O. • DATA OF LYNCH, et a\.2

10" 5

• E:q UATK)N 6

5

z

o <0O

o> 5

3

o

2<uto

J

^v^BEAM ANGLE =° 1 deg

—1 r^.

c/>

£ 5 """""••k^Q.

X

3 2

< -ao: 10 8i

^^^« >

^< •

5

S 5

2

10"9

•"»! l

-:J!-X^ ""•^•^c )^-( 1

^ 30 deg

> ^**

5>««^^

i |( ) •>J»

( 5 60 deg

2

180 deg/' I!'"90 deg

^"•^135 deg

7 8 9 10 15 20 30 40 50 60 70

x, SOURCE-DETECTOR SEPARATION DISTANCE (ft)

100 150


200

•«SJ." ^«T*i!I*WB,')!K?Jj*lW^^»»#Sa^W«K*a.

•15-

15 20 30 40 50 60 70

X, SOURCE- DETECTOR SEPARATION DISTANCE (ft)

UNCLASSIFIED

2-01-059-599

150 200

Fig. 10. Air-Scattered Gamma-Ray Dose Rates as a Function of Distancefrom a 0.6-Mev Line-Beam Source

10

10

10"

£ 10

a:

i 10"

<

10

10

-16-

UNCLASSIFIED

2-01-059-600

—M4

- AIR

I I I I I I I I I h

DENSITY = 0.00125 g/cm3f0, SOURCE ENERGY = 1 MevI | , | | | i i i | 1 1

O. • DATA OF LYNCH, et a\.

( >^^*'-

-1 ^W " RFAM ANGLE

= 1 deg

""'^^

c r*-4i^^

4 f—•

"•« »^

1! "-•^ 15 deg

°—

"*"()l*^^-(l***^.

•^ """"^^ "*"< >».

•^X^ •^*"< l~

>«^ <r-60 de 3

* u --!> ^>>* 90 de

^x I">^ 135 deg

"*>**»« 180 deg

5 6 7 8 9 10 15 20 30 40 50 60 70

,r, SOURCE-DETECTOR SEPARATION DISTANCE (ft)

100 150 200

Fig. 11. Air-Scattered Gamma-Ray Dose Rates as a Function of Distancefrom a 1-Mev Line-Beam Source

ww mmmmmmmmmmm

10

10

10

J= 10

a:

Oo

>-<

10

10

10

-17-

UNCLASSIFIED

2-01-059-601

lilt 11 1 1 1 1 1

AIR DENSITY = 0.00125 g/cm3£0, SOURCE ENERGY = 2 Meu

1

1 1 1 1 1 1 1 1 1 1 1

Q.« DATA OF LYNCH, et al.

EQUATION 10

(^">,>» BEAM ANGLE =

' 1

1 1**

—

<) i

_|•-*.

1 »»> <>"*"^15 deg

<5--

*•>( ~l »^v^ ^*^ _, _( >*" 30 deg

"*>-0,>-^ ( >«,

"' '"-^. 60 deg,. 1

_1 t?^.^^ 90 deg

1»^"*•».{> 135 deg

180 deg

I6 7 8 9 10 15 20 30 40 50 60 70 100


150 200


-18-

10-10

i ! 1 1

UNCLASSIFIED

2-01-059-602

1 1 1 1 1

b S0. SOURCE ENERGY = 4 Mev

1 1 1 1 1 1 1 1 1 1 1i I i 1 1 1 1 1 1 1 1

O. • DATA OF LYNCH, et al. .2

-,.

n.;u

--i!^^—

2

-12

( 3"* BEAM ANGLE =

5_^|

2

v'o-^^^

i >—

Mk

5

2

-14

'— 15 deg

4 >

^

t i~,

5*"^^. -t r* .i

~~« >-^

*^ " "*•().

2

.-15

i ^

""""'k^. ^*( )^_^>-< >„

""-J ' "•il"*•••• i i

5•"( ).

—•< ».c )"•» -•°.^90 deg

2

,-16

•*''J*»135deg"^180 deg

10

10

\

I-

oo

<q:

<52

10

10

10

10

6 7 8 9 10 15 20 30 40 50 60 70 100


150 200

Fig. 13- Air-Scattered Gamma-Ray Dose Rates as a Function of Distancefrom a 4-Mev Line-Beam Source

m- •mwm&mwmmwm&wmmm*'*

-19-

10-10

=T-4-4-4-4

UNCLASSIFIED

2-01-059-603

MINI 1

b f0, SOURCE ENERGY = 7 Mev

1 1 1 1 1 1 1 1 II 1 1-1 1 1 1 II 1 1 1 1 1

0,« DATA OF LYNCH, et al. ,_2

-11

<)IQUATIOPA 10

—^—.

b

2

-12

BEAM ANGLE=

••«.,, 1 deg

5 *1•^

"°^-^_2

-13 1 1—«^

b . 1)

i >»

*"» [v^.15 deg

-14 !*"" _

b

<r*"30 deg

-15

)"*•

^'<>^1 »^.

-^ H »,

"""-^^^ »^^b

, 1^i Y~

*~4^90 deg2

-16

_ V/135 deg

*», ,1^180 deg"""0

10

10

J; 10

>-<a:

10

10

10

7 8 9 10 15 20 30 40 50 60 70


100 150 200


, '-20-

10

10

r'°

UNCLASSIFIED

2-01-059-604

T~l 1 1 1 M 1 II 1

AIR DENSITY = 0.00125 g/cm3£0, SOURCE ENERGY = 12 Mev

1

5

)—

2 . n.« nATA OF I YNCIH ot n\ .

t)-"

EOUATK5N 10

5

2

r,z

( 5^„ BEAM ANGLE =

V*J^ 1deg

5

*•< >«*

2

r13""''r*"*"-^

Z5**< t——

5c )

2

"*< u

-14 <)* „15 deg

—1 r*

5

,>~***^N^

?

<J** "(5**^ 30 deg

-.3 <)

—« i—

^'"**»^1

5""^60 deg

1)

2135 deg/

-16

180 deg'

1

^<

"**Gi15 »_

"""t^ 90 degi

- 10

oQ

>-4a:

10

10

10

5 6 7 8 9 10 15 20 30 40 50 60 70 100 150 200

X, SOURCE- DETECTOR SEPARATION DISTANCE (ft)


*W*W»*\ WIlWIWIMWMWIilWWWWiWH m

\ !

INTERNAL DISTRIBUTION

1. C. E. Center 92.

2. Biolo,gy Library 93-

3- Health physics Library 94.4-6. Central Research Library 95-

7. Reactor Experimental 96.Engineering Library 97-

8-64. Laboratory Records Department 98.

65. Laboratory Records, ORNL R.C. 99-66. A. M. Weinberg 100.

67. L. B. Emlet (K-25) 101.

68. J. P. Murray (Y-12) 102.

69. J. A. Swartout 103.

70. G. E. Boyd 104.

71. R. A. Charpie 105.

72. W. H. Jordan 106.

73. A. H. Snell 107.

74. C. E. Winters 108.

75. D. S. Billington 109.

76. F. L. Culler 110.

77. J. L. Fowler 111.

78. J. H. Frye, Jr. 112.

79- C. S. Harrill 113-118.80. A. Hollaender 119.

81. A. S. Householder 120.

82. C. P. Keim 121.

83. M. T. Kelley 122.

84. S. C. Lind 123.

85. R. S. Livingston 124.

86. K. z. Morgan 125.

87. M. L. Nelson 126.

88. H. E. Seagren 127-

89. E. D. Shipley 128.

90. E. H. Taylor 129.

91. P. M. Reyling 130.

131.132.

EXTERNAL DISTRIBUTION

ORNL-2998Physics and Mathematics

TID-4500 (15th ed.)December 19, i960

D. PhillipsM. J. Skinner

T. V. Blosser

V. R. Cain

A. D. Callihan

G. T. Chapman

G. deSaussure

G. M. Estabrook

L. B. Holland 1L. Jung s

J. Lewin

T. A. Love

A. D. MacKellar

R. E. Maerker

F. C. Maienschein

J. M. Miller

F. J. Muckenthaler

R. W. Peelle

S. K. PennyA. Simon

E.G. Silver

D. K. TrubeyD. R. Ward

CD. ZerbyW. Zobel

C. A. Willis

A. J. Miller

C. A. Preskitt

F. B. K. Kam

H. E. Stern

P. F. Gast (consultant)Herbert Goldstein (consultant)L. W. Nordheim (consultant)R. F. Taschek (consultant)T. J. Thompson (consultant)E. P. Blizard

133. Division of Research and Development, AEC, ORO154-743. Given distribution as shown in TID-4500 (15th ed.) under Physics and

Mathematics category (75 copies - OTS)

-21-

Documents

D. K. Trubey DO NOT TRANSFER TO ANOTHER PERSON