applied nuclear physics division

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

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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\\\\\\\\

% •

\ •\\\\

"=""=?"FÔ

%

i^_

V*- —«1 "•*«.

-»«„,^^-ê/eoî^^•m

- —

^^<ê~ '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


(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


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


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


fRof • MYn-^07^)

o

<>•

5o

<>

( ) -^ i

2

c5 j \f

H20i

\

o <

/

Yrb -

0 2 4 6 8 10 12 14


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


UNCLASSIFIED2-01-059-1Q9

Fig. 9. Gamma-Ray Dose Rate and Dose Buildup


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


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


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/,


•

O

< )

o

4 )

cV'r

c > •«

H20

y•

rD

0 2 4 6 8 10 12 14


Fig. 12. Gamma-Ray Energy Buildup Factor as a Function


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


60

Fig. 13. Gamma-Ray Dose Rate and Dose Buildup


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,


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


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)


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


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).

applied nuclear physics division

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