applied nuclear physics division
TRANSCRIPT
UNCLASSIFIED
Contract No. W-7405-eng-26
Applied Nuclear Physics Division
ORNL-2191^Physics
TID-J+500 (12th ed.)
A MONTE CARLO STUDY OF THE GAMMA-RAY ENERGY FLUX,DOSE RATE, AND BUILDUP FACTORS IN A LEAD-WATER
SLAB SHIELD OF FINITE THICKNESS
S. Auslender
Date Issued
JAN 101957
OAK RIDGE NATIONAL LABORATORY
Operated byUNION CARBIDE NUCLEAR COMPANY
A Division of Union Carbide and Carbon CorporationPost Office Box X
Oak Ridge, Tennessee
UNCLASSIFIED
M/»TlNMflf„E
1113'*St 03S0330 „
UNCLASSIFIED -11-
INTERNAL DISTRIBUTION
ORNL-219^
r
Physics
49. c P. Keim
50. c. S. Harrill
51. c. E. Winters
52. D. s. Billington
53. D. W. Cardwell
51*. E. M. King
55. A. J. Miller
56. E. P. Blizard
57. w. M. Breazeale (consultant)58. D. D. Cowen
59- M. J. Skinner
60. R. B. Murray
61. R. R. Dickison
62. J. A. Harvey
63. A. Simon
6k. F. C. Maienschein
65. J. A. Lane
66. R. N^jLyon
67. J. Smolen
68. H. Moran
69. C A. Goetz
70. A. T. Futterer
71. R. W. Peelle
72. c. D. Zerby
73. R- Davis
Ik. C. E. Clifford
75- F. L. Keller
76. D. R. Otis
77. ORNL - Y-12 Technical Library,Document Reference Section
1.
2.
3.
5.6.
7-26.27.28.
29.30.
31.32.
33.34.35.36.37.38.
39.ko.ki.k2.h3.kk.k5.k6.hi.kQ.
C. E. Center
Biology LibraryHealth Physics LibraryCentral Research LibraryReactor ExperimentalEngineering LibraryLaboratory Records DepartmentLaboratory Records, ORNL R.C.A. M. WeinbergL. B. Emlet (K-25)J. P. Murray (Y-12)J. A. Swartout
E. H. TaylorE. D. ShipleyA. H. Snell
M. L. Nelson
W. H. Jordan
S. J. Cromer
G. E. BoydR. A. CharpieS. C. Lind
F. L. Culler
A. Hollaender
J. H. Frye, Jr.M. T. KelleyR. S. LivingstonK. Z. MorganT. A. Lincoln
A. S. Householder
EXTERNAL DISTRIBUTION
78. R. F. Bacher, California Institute of Technology79« H. Goldstein, Nuclear Development Corporation of America80. Division; of Research and Development, AEC, 0R0
8I-677. Given distribution as shown in TID-U500 (12th ed.) under Physics category(200 copies - OTS)
UNCLASSIFIED
A MONTE CARLO STUDY OF THE GAMMA-RAY ENERGY FLUX,DOSE RATE, AND BUILDUP FACTORS IN A LEAD-WATER
SLAB SHIELD OF FINITE THICKNESS
S. Auslender
The gamma-ray energy flux, dose rate, and buildup factors in a lead-
water shield of finite thickness have been calculated by a Monte Carlo
method. The calculation included 1-, 3-, and 6-Mev photons incident on the
slab both along a normal and at an angle of 60 deg. (Calculations for an
angle of 75-1/2 deg were also performed, but they are not included here
since the attenuation was quite large in spite of the large buildup factor.
For the same reason the results of 1-Mev photons incident at an angle of 60
deg were also omitted.) The buildup factors for energy and dose obtained
in this calculation were compared with those obtained by use of the moments
method for monoenergetic, plane monodirectional sources normally incident
upon a semi-infinite, homogeneous medium.
Table 1. Normal Thicknesses of a Lead-Water
Slab Shield
Region
Pb
HgO
Total
Thickness
mfp
cm 1 Mev 3 Mev 6 Mev
11.58 9.O89 5.65U 5.878
55.81 2.542 1.382 0.996
k-j.ko 11.630 7-035 6.874
H. Goldstein and J. E. Wilkins, Jr., Calculations of the Penetration ofGamma Rays, NYO-3075 (June 30, 195*0.
-2-
Table 2. Dose Equivalent for Incident Flux
E(Mev) D^ !-**£*7/cm2sec
1 0.001923
3 0.004368
6 0.007206
The thicknesses of the lead-water shield in centimeters and in mean free
paths for the various incident gamma-ray energies are given in Table 1.
Table 2 gives the tissue dose equivalent for the incident flux (no shield
present).
The results of the calculations for 1-Mev photons are presented in
Figs. 1 through 4, those for 3-Mev photons in Figs. 5 through 10, and those
for 6-Mev photons in Figs. 11 through 16. An index to the figures is shown
in Table 3.
The normalized energy flux for the 1-, 3-> and 6-Mev energy groups is
plotted in Figs. 1, 5, and 11, respectively,as a function of the normal
thickness of the shield in centimeter units. The uncollided energy flux,
normalized to unity at the initial boundary, is also plotted. The third
curve in each figure is the energy buildup factor, BE, which is the ratio
of the two energy flux curves:
_ 0e/eo0I 0E
Be -tA *<*T»-tAe-V* Eo0ie-
where
0g(Mev/cm2 sec) « energy flux at the point of interest in the shield,
10'
10
10
10
10
10
2
10
5
2
UNCLASSIFIED
2-01-059-101
C — IMITIAI DunTAM CMTD^V- IIMIMHL rnuiwiN LINCnOI
= INCIDENT NUMBER FLUX
- FMCDr.V CI MYch9£ —i--- • • .-w~
Xpb = 1.37 cmXH 0 = 14.09 cm
BE = ENERGY BUILDUP FACTOR\t\\\\\\\\
% •
\ •\\\\
"=""=?"F^O
%
i^_
V*- —«1 "•*«.
-»«„,^^-^e/eo^i^^•m
- —
^^<^e~ 'A
^BF
10
5
2
10
5
2
10"2
5
2
10"3
5
,-1
,-410
5
2
10"
5
2
,-610
0 10 20 30 40
/, NORMAL THICKNESS (cm)
50 60
Fig. 1. Gamma-Ray Energy Flux and Energy Buildup Factoras a Function of the Normal Thickness (Centimeters) of a
Finite Lead-Water Slab Shield: 1-Mev Normally Incident Photons.
ot-o<
CL3Q
m
>-
en
uj
Uj<*5
-K-
5
UNCLASSIFIED
2-01-059-102
— MONTE CARLO, FINITE SLABO H ^^ MnMFMTC MFTunn
2
-10
• f
"2"
Db
SEMI-INFINITE SLAB,NORMAL INCIDENCE
(Rpf • NYO-3075)
o
o
0
Pb H50 —""• r2
5
t tr-jjL0r = 11.63 mfp
•
c ) / <>
oi \^*-—•*"""""
•
2
k
V
0 2 4 6 8 10 12
fi0r, OBLIQUE THICKNESS (mfp)
Fig. 2. Gamma-Ray Energy Buildup Factor as a Function
of the Oblique Thickness (Mean Free Paths) of a
Finite Lead-Water Slab Shield. 1- Mev Normally
Incident Photons.
14 16
10'
2
6
5
2
10
10*
10
5
2
103
5
2
102
5
2
10
5
UNCLASSIFIED
2-01-059-103
n — nncr datcu — UuoL nAlL
D0 = INITIAL DOSE RATE
<t>f= INCIDENT NUMBER FLUX/ i0/ ^T W.WUIv/t-vJ vim/ i" // V 11 \*\w OC W
XPb —1.37 cm
-V— XH 0 = 14.09 cmBr= DOSE BUILDUP FACTOR—w—
\\\\11
\v^\\Y*—
,- pi, \\ 11 n' " l\ i ipU
i *—•• _V mmmm* ^^^^
^l_-~»«»>-0/'A9/
^«
a~}'A—-D ' /d>-" LU 'TJ
~B
-210
5
2
10
5
-3
-410
5
2
10
5
2
10
5
-5
-6
-710
5
2
10-8
-9
0 10 20 30 40
A NORMAL THICKNESS (cm)
5010
60
Fig. 3. Gamma-Ray Dose Rate and Dose Buildup Factor as aFunction of the Normal Thickness (Centimeters) of a Finite
Lead-Water Slab Shield: 1-Mev Normally Incident Photons.
-6-
UNCLASSIFIED
2-01-059-104
o
MONlL L-AKLU, MINI I t bLAbH20l MOMENTS METHOD,
SEMI-INFINITE SLAB
NORMAL INCIDENCE
b J (Ref.: NYO-3075)
n
2
10
•
o
o
•Pb— 1
-LO —^m' ^•^ rc.
( >
5
•
C) / «>
2
OA \4**^
•
{•
0 4 6 8 10 12 14
fi0r, OBLIQUE THICKNESS (mfp)16
Fig.4. Gamma-Ray Dose Buildup Factor as a Function of
Oblique Thickness (Mean Free Paths) of a Lead-Water Slab Shield. 1-Mev Normally Incident Photons,
10'
10'
icr
10'
icr
10'
10
UNCLASSIFIED
2-01-059-105
r — INITIAL PHOTON ENE
INCIDENT NUMBER F
ENERGY FLUX
2.05 cm
25.91 cm
ENERGY BUILDUP FA
'or\/
L0 no!
$t = LUX
{ *f -
v= APb5 \
Vv AMo0
vt BF = :tor
V#
t
\ V\\y v
\"**•"•—. ^_ V^o */ :
*- • r-
e-t/\ ^^****«
-m. Ph i 11 r\"• \ U ' 1 InU
HcE
-*—^.-
S^
10
-210
10"
o 10 20 30 40
/, NORMAL THICKNESS (cm)50 60
Fig. 5. Gamma-Ray Energy Flux and Energy Buildup
Factor as a Function of the Normal Thickness (Centimeters)
of a Finite Lead-Water Slab Shield: 3-Mev Normally
Incident Photons.
orOh-O<n- 10Q_Z)O_J
Z>QQ
>-
° 5LU
UJ
1
UNCLASSIFIED
2-01-059-106
— 1
O [
V10NTE CARLO, FINITE SLABJ Ol MHMPMTC MFTUnn
1
• Db
SEMI-INFINITE SLAB,NORMAL INCIDENCE
(Rof • Mvn_^n7c;)
o
c ) •
o
< »
c > < »
(
< \^r
•*••>
Ph . H20
/
)'
r Q ^
0 2 4 6 8 10 12 14
H0r, OBLIQUE THICKNESS (mfp)
Fig . 6. Gamma-Ray Energy Buildup Factor as a Function
of the Oblique Thickness (Mean Free Paths) of a
Finite Lead-Water Slab Shield: 3-Mev Normally
Incident Photons.
16
10'
10
10'
10
10c
2
2
5
2
10
10
UNCLASSIFIED
2-01-059-107
\V\
— W Ph 11 r\\\ 112U
n -^»»
^=•—
1 "•*•»••
-°,/a/cPl
^^**»
\ ^^^^^m
»-'''%Do'
D = DOSE RA"fE
D0 = INITIAL DOSE RATE
--Do/4>i -INCIDENT NUMBER FLUX
0.004368 (mr/hr)/(^/cm2-s2.05 cm
ec!
^Pb —XH20 = 25.91cm
Br = DOSE BUI LDUP FAC"for
-B,
-210
-310
5
2
10
5
-4
2
10~5
5
2
10-6
5
2
10
5
2
10"
-7
-910
0 10 20 30 40
/, NORMAL THICKNESS (cm)
50 60
Fig. 7. Gamma-Ray Dose Rate and Dose Buildup Factor asa Function of the Normal Thickness (Centimeters) of a Finite
Lead-Water Slab Shield: 3 Mev Normally Incident Photons.
or
Oh-
<
Q_Z>
Q_J
GO
UJ
CO
O
^
CQ
-10-
UNCLASSIFIED
2-01-059-108
yj
MONTE CARLO, FINITE SLABo h ^ 1 iwiniwiCMTC MFTunn
2
10
• 1
'2W
Db
SEMI-INFINITE SLAB,NORMAL INCIDENCE
fRof • MYn-^07^)
o
<>•
5o
<>
( ) -^ i
2
c5 j \f
H20i
\
o <
/
Yrb -
0 2 4 6 8 10 12 14
fi0r, OBLIQUE THICKNESS (mfp)
Fig. 8. Gamma-Ray Dose Buildup Factor as a Function of
Oblique Thickness (Mean Free Paths) of a Finite
Lead-Water Slab Shield. 3-Mev Normally
Incident Photons.
16
-11-
20 30 40
/, NORMAL THICKNESS (cm)
UNCLASSIFIED2-01-059-1Q9
Fig. 9. Gamma-Ray Dose Rate and Dose Buildup
Factor as a Function of the Normal Thickness (Centimeters)
of a Finite Lead-Water Slab Shield: 3-Mev Photons
Incident at a 60-deg Angle.
-12-
UNCLASSIFIED
2-01-059-110
10'
-MONT
H20~
Pb
E CARLO, FINITE SLAEo MOMENTS METHOD,
SEMI-INFINITE SL;
NORMAL INCIDENCE
(Ref.: NYO-3075)
^B
•
o
9 •
Ph 1 -^ ru
o
!IU- -
H90-»
c )yT •
*i
( Y •
o/ <
/•
i
DCOh-O
o_
Q
00
UJ
OQ
10
tf
0 4 6 8 10 12
fiQr, OBLIQUE THICKNESS (mfp)14 16
Fig. 10. Gamma-Ray Dose Buildup Factor as a Function of
Oblique Thickness (Mean Free Paths) of a Finite Lead-Water
Slab Shield. 3-Mev Photons Incident at a 60-deg Angle.
-13-
20 30 40
/, NORMAL THICKNESS (cm)
UNCLASSIFIED
2-01-059-111
Fig. 11. Gamma-Ray Energy Flux and Energy Buildup Factoras a Function of the Normal Thickness (Centimeters) of a
Finite Lead-Water Slab Shield: 6-Mev Normally Incident Photons.
Oh-
<u.
Q_IDQ
CD
>-
orLJ
10
Uj03
1
-Ik-
UNCLASSIFIED
2-04-059-112
MONTE CARLO, FINITE SLAB
o H20
• ph
IVIUIVIQIM 1 O IVILinUL/,
SEMI-INFINITE SLAB,NORMAL INCIDENCE
•
O
< )
o
4 )
cV'r
c > •«
H20
y•
rD
0 2 4 6 8 10 12 14
fi0r, OBLIQUE THICKNESS (mfp)
Fig. 12. Gamma-Ray Energy Buildup Factor as a Function
of the Oblique Thickness (Mean Free Paths) of a
Finite Lead-Water Slab Shield. 6-Mev Normally
Incident Photons.
16
10
10'
10v
10
10'
10'
10
-15-
UNCLASSIFIED
2-01-059-113-2
•^=\'•
\»\\\\\\\#
_Jv\\•—
\D/$Ii • m ._• h ——».«_^
°6>-"y</>7^^^*«
TJ
•i « ,^
£=D0SE RATE
DQ = INITIAL DOSE RATE
4>f = INCIDENT NUMBER FLUX
DQ/4>I =0.007206 (mr/hr)/(y/cm2se r\LI
Xpb= l.9^cm
XH 0 = 35.96 cm
^-=DOSE BUILDUP FACTOR
-m Ph • 11 n^ \ U • 1 1oU
B,ur
10
-310
-410
-510
,-610
-710'
10•8
-910
0 10 20 30 40 50
/, NORMAL THICKNESS (cm)
60
Fig. 13. Gamma-Ray Dose Rate and Dose Buildup
Factor as a Function of the Normal Thickness (Centimeters)
of a Lead-Water Slab Shield: 6-Mev Normally Incident
Photons.
orO
^ 10u.
0_
3Q
00 5UJ 3
o
tf
-16-
UNCLASSIFIED
2-01-059-114
MONTE CARLO, FINITE SLAB
o H20
• Pb
MUMtlMlb Mt 1 MUU,
SEMI-INFINITE SLAB,NORMAL INCIDENCE
V nei.: in 1U~ J\JI sJ )
•
o
l1
o
c > I
( > y\- Dk -— H20
V •
rD
4 6 8 10 12 14
ix0r, OBLIQUE THICKNESS (mfp)16
Fig. 14. Gamma-Ray Dose Buildup Factor as a Function
of Oblique Thickness (Mean Free Paths) of a
Finite Lead-Water Slab Shield. 6-Mev Normally
Incident Photons.
10
5
2
106
5
10
10H
5
2
103
5
2
102
5
2
10
5
2
1
-17-
UNCLASSIFIED
2-01-059-115-2
10
5
-310
5
2
10
5
2
10
5
-4
-5
1
t=\ D = DOSE RATE\\
•
DQ = INITIAL DOSE RATE
(jij = INCIDENT NUMBER FLUX
°o/$I = 0007206 (mr/hr)/(v/cm2.seA
- / n)-
Xpb —1.97 cm
4- XH 0 = 35.96 cmBr = DOSE BUILDUP FACTOR
—ft—\ii %
\\— Ph \ \ 11 /~»
1 •
\\11'?w
MS\ V—
—\E_- =>^D/h' -•—
s^=B>r»
r ^-*
-2'A/^—^—DQe
2
10~6
5
2
10"7
5
2
10~8
5
-9
0 10 20 30 40
/, NORMAL THICKNESS (cm)
10
50 60
Fig. 15. Gamma-Ray Dose Rate and Dose Buildup Factor asa Function of the Normal Thickness (Centimeters) of a Finite
Lead-Water Slab Shield: 6 Mev Photons Incident at a 60-degAngle.
10
or
oi-u<Ll_
Q_
Q_J
00
LU
COo
10
-18-
UNCLASSIFIED
2-01-059-116
c£ 5
I II
- IVIUINI 1 L tAKLU, l-IIMI 1 L bLAB
u H20-
• Pb .
IVIUIVILIM lb IVItL i nuu,
SEMI-INFINITE SLA
NORMAL INCIDENCE
Ru>
(Ref.: NYO-3075)
r > L, » u n *1 D MpU *
•
o
I
( ) j/ •
c
° <
) <>/
0 2 4 6 8 10 12 14 16
fjL0r, OBLIQUE THICKNESS (mfp)
Fig. 16. Gamma-Ray Dose Buildup Factor as a Function of
Oblique Thickness (Mean Free Paths) of a Finite
Lead-Water Slab Shields. 6-Mev Photons
Incident at a 60-deg Angle.
Table3.IndextoPlottedData
ParametersPlottedDataa>b
Energy-AngleofIncidenceComparisonShield(Mev)(deg)ThisCalculationData
Fig.No.Fb-H20Pb13•606003^001e-t/xBED/0EDoe-tB9ce/\/0xBrBEBr
1XXXNNN
2XXX00
3XXXNNNkXXX00
5XXXNNN
6XXXN0
7XXXNNH8XXX00
9XXXNNN10XXX0011XXXNNN
32XXX00
13XXXNNN
IfcXXX00
15XXXNNN16XXX00
ITXXXN0C018XXX00
a.Nindicatesaplotasafunctionofthenormalthicknessincentimeters,while0indicatesaplotasafunctionoftheobliquethicknessinmeanfreepaths.Fora0-degangleofincidencetheobliqueand
normalthicknessesareidentical.
Thedefinitionsofthesymbolsareasfollows;0E=energyfluxatthepointofinterest,Eo=initialenergyD0=initialdoserate,Br=dosebuildupfactor,
A,=relaxationlength.
ActuallyaplotofD0e-i*/X70i
0j=numberfluxattheinitialboundary,Be=energybuildupfactor,D=doserateatthepointofinterest,t=distancebetweentheinitialboundaryand
thepointofinterest.
voI
-20-
E0(Mev/V) = initial photon energy,
0 (7's/cm2 sec) = number flux incident on shield,
t(cm) = distance between the initial boundary and the
point of interest in the shield,
\(cm) - relaxation length,
e-t/X _ uncoiiided energy flux at the point of interest.
In a similar manner the normalized dose rate and dose buildup factor,
Br, for the three energies are plotted in Figs. 3, 1, 9, 13> and 15. Here
D/0T DBr Do6-t sece/X/0^. Dq e-t seceA
where
D(mr/hr) « dose rate (tissue) at the point of interest,
D0(mr/hr) = dose rate (tissue) of the uncoiiided incident photons at
the initial boundary.
The various energy buildup factors as a function of oblique thickness,
in mean free paths, are presented in Figs. 2, 6, and 12. Corresponding dose
buildup factors are given in Figs, k, 8, 10, 1^, and 16. Data from the re
sults of the moments method solution1 are also plotted for purposes of
comparison, although it must be remembered that those calculations were for
infinite homogeneous media and are not directly comparable to the present
calculations for a finite two-region slab. The calculations for lead do
agree reasonably well for the first few relaxation lengths where the effects
stemming from the dissimilarity of the slabs should be least.
The dose rate and dose buildup factors resulting from an earlier Monte
Carlo calculation for 3-Mev photons normally incident upon a one-region
-21-
8-mfp-thick lead shield2 are also presented (Figs. 17 and 18) as an example
of a case intermediate between the Monte Carlo calculation for a two-region
finite shield and the moments method solution for the one-region semi-infinite
shield. A comparison of the dose rate in the one-region finite shield with
the dose rate in the one-region semi-infinite shield (calculated from build
up factors reported in Ref. l) shows good agreement. The characteristic dip
near the final boundary of the finite lead shield is apparent.
The dose buildup factors for the 3-Mev incident photons in the two-region
and one-region finite shields (Figs. 8 and 18) are identical to within 2 mfp
of the lead-water interface. It is apparent that at this energy and angle of
incidence the back scattering is Important in lead for a distance of about
h cm (Xpjj =2.05 cm at 3 Mev). The scattering in the water tends to compensate
for the finite thickness of the lead-water slab. It can also be concluded that
the back-scattered flux at that interface is about 12$ of the total flux and
about 20# of the scattered flux. The differences for the finite and the semi-
infinite one-region shields (Fig. 18) are partially due to differences in the
cross section data used for the calculations. An extensive discussion of the
errors in the calculations from the moments method is included in Ref. 2.
Consistent errors may exist that will bias the answers obtained with that
method. The Monte Carlo data is much less subject to bias than to statis
tical fluctuation. (The large fluctuation of the data in Fig. 9 may
indicate a relatively large error, but this is still probably less than 10$.)
Since the Monte Carlo calculations for the two-region finite shield and
the moments method solution for the one-region semi-infinite shield were in
2. S. Auslender, unpublished work.
-210
-310
.-410
-510
-610
-22-
UNCL2-01-
ASSIFIED-059-117
1 1 1 1 1
D = DOSE RATE
DQ= INITIAL DOSE RATE
<£7 = INCIDENT. NUMBER FLUX
fy =0.004368 (mr/hr)/(y/cm2-\.DK= 2.05 cm
n /
NNDo/( actJ
s\ \ L
\\ \
V \\ \ >
\
v\Dtf-^fij-^\ ')
'D/+I
nk1 D
k < 1
\ Ii
• MONTE CARLO, FINITE SLAB
MOMENTS METHOD, SEMI-INFINITA E
SLAB (REF: NY0-3075) \ t
0 2 3 4 5 6
fi0r, OBLIQUE THICKNESS (mfp)
Fig. 17. Gamma-Ray Dose Rate as a Function of Oblique
Thickness (Mean Free Paths) of a Finite Lead Slab Shield:
3-Mev Normally Incident Photons.
-23-
10
UNCLASSIFIED
2-01-059-118
• MONTE CARLO, FINITE SLABMOMENTS METHOD,O
SEMI-INFINITE SLAB (Ref.: NYO-3075)
c )
O
J «•" V
4/*
- Ph
/t
r D
orOh-O<
3 10CO
UJ
OQ
tf
0 4 6 8 10 12
^0r, OBLIQUE THICKNESS (mfp)14 16
Fig. 18. Gamma-Ray Dose Buildup Factor as a Function ofOblique Thickness (Mean Free Paths) of a FiniteLead Slab Shield. 3-Mev Normally Incident Photons.
-2k-
general agreement, the striking differences in the buildup factors near the
lead-water interface and near the final boundary should be enlightening.
In Figs. 2 and k, for example, the energy and dose buildup factors for 1-Mev
photons in a finite two-region shield increase sharply in water to a value
about halfway between those for semi-infinite one-region shields of water
and lead. At this energy the finite water region is 2-1/2 mfp-thick and
at first the buildup increases almost at the same rate as it does initially
in the semi-infinite water shield. This confirms that at this energy, which
is below the minimum in the total cross section, the uncoiiided radiation
dominates in the penetration of the lead, as it should.
In Figs. 6 and 8 the increase of the buildup factors in the water is
very small since for the primary 3-Mev photons for both lead and water the
major portion of the total cross section is attributable to scattering.
Hence, the difference in the behavior of the flux in water and in lead is
due to that small portion of the flux which is absorbed in the lead but is
scattered in the water. For 6-Mev photons (Figs. 12 and Ik) the absorption
cross section of lead is no longer so small that the absorption in it is
almost negligible. In this case the buildup factor increases rather abruptly
in the water almost to the buildup factor for 6-1/2 mfp at water calculated
using the moments method.
It is evident from the results of the calculations for radiation incident
at 60 deg (Figs. 9, 10, 15, and 16) that the practice of using only normal
incidence data for shield designs can lead to a poor approximation. This
problem becomes most acute when the number of mean free paths across the shield
is small or when the angular distribution is such that a large portion of
-25-
the radiation is not normal or nearly normal to the slab. Figure 16, which
shows the flux depression near the initial boundary and the large buildup
factor into the slab, is an excellent exaomple of streaming (or short-
circuiting).