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TRANSCRIPT
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
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\
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.
(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
x, SOURCE -DETECTOR SEPARATION DISTANCE (ft)
100 150 200
Fig. 7. Air-Scattered Gamma-Ray Fluxes as a Function of Distancefrom a 4-Mev Line-Beam Source
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
Fig. 9. Air-Scattered Gamma-Ray Fluxes as a Function of Distancefrom a 12-Mev Line-Beam Source
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
x, SOURCE -DETECTOR SEPARATION DISTANCE (ft)
150 200
Fig. 12. Air-Scattered Gamma-Ray Dose Rates as a Function of Distancefrom a 2-Mev Line-Beam Source
-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
x, SOURCE -DETECTOR SEPARATION DISTANCE (ft)
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
x, SOURCE -DETECTOR SEPARATION DISTANCE (ft)
100 150 200
Fig. 14. Air-Scattered Gamma-Ray Dose Rates as a Function of Distancefrom a 7-Mev Line-Beam Source
, '-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)
Fig. 15. Air-Scattered Gamma-Ray Dose Rates as a Function of Distancefrom a 12-Mev Line-Beam Source
*W*W»*\ WIlWIWIMWMWIilWWWWiWH m
\ !
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-21-