A UNITED STATES DEPARTMENT OF COMMERCE

PUBLICATION yy>

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NBS TECHNICAL NOTE 745

Neutron Dose

and Fluence Distributions

in an Infinite Air Medium

U.S.

EPARTMENT

OF

COMMERCE

National

Bureau

of

Standards

Neutron Dose and Fluence Distributions

in an Infinite Air Medium

George L. Simmons and Charles M. Eisenhauer

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

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.

.< °>«

**fAU Of *

U.S. DEPARTMENT OF COMMERCE, Peter G. Peterson, Secretary

NATIONAL BUREAU OF STANDARDS, Lawrence M. Kushner, Acting Director,

Issued January 1973
National Bureau of Standards Technical Note 745

Nat. Bur. Stand. (U.S.), Tech. Note 745, 39 pages (Jan. 1973) CODEN: NBTNAE

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

Price $ .65 domestic postpaid or $ .45 G.P.O. Bookstore.
Contents

Page

I. Introduction 1

II. Results 5

A. Benchmark Problem (Air density 1.29 g/liter) 8

B. Comparisons with Sn Calculations (Air density

1.11 g/liter) 8

C. Simple Representations for Fluence and Dose

Distribution 14

D. Angular Distributions 19

III. References 34

in

Neutron Dose and Fluence Distributions in an Infinite Air Medium

**

George L. Simmons and Charles Eisenhauer

The moments method is applied to the problem of calculating the neutron dose and fluence distributions in an infinite

medium of air. These calculations are compared with Monte

Carlo and Discrete Ordinates (S ) results. Simple parametric

representations for the distributions are given which facilitate the calculation of dose and flux distributions in air

with a different density.

Key words: Benchmark problems; dose distributions; moments

method; neutron penetration; shielding; weapons radiation.

I. INTRODUCTION

The moments method for solving neutron and gamma-ray transport

problems is ideally suited for obtaining simple, but accurate representations of neutron dose and fluence distributions. In this note, we present

data on neutron transport in air for three point isotropic sources of

interest in shielding applications. These sources are: 1) fission source,

) 14-MeV source, and 3) a typical thermonuclear source. These results

are compared with similar data obtained using the Monte Carlo and Discrete Ordinates (S ) techniques. We also give simple parametric representations for the dose and fluence which may be used to calculate these

quantities at various distances in air of arbitrary density.

The cross section data used in these calculations are taken from

Work supported by the Defense Civil Preparedness Agency,

Washington, D.C. 20310

**Present address: Babcock & Wilcox, Lynchburg, Va. 24505

1.
ENDF/B-I compilation for oxygen [l] and nitrogen [2] and are the sai

as those used by Straker in Shielding Benchmark Problem 3.0 [3], Air

was assumed to be 797 nitrogen and 217o oxygen by volume, yielding nuclear

densities of 4.25 x 10 and 1.13 x 10 nuclei/barm-cm, respectively,

for an air density of 1.29 grams/liter. Our calculations require that

some prescription be given for interpolating on the basic cross section

data. We used the specification given on the ENDF/B data file, namely,

linear log-log interpolation on the cross sections and linear interpolation on the Legendre expansion coefficients of the elastic scattering

angular distribution.

Calculation of spatial distributions by the moments method involves

two steps. First, spatial angular moments of the fluence are obtained.

These moments are then used to reconstruct distributions at various

depths. We used the computer code MOMENT- I [4] to calculate spatial-

-4

angular moments at 510 energies between 15 MeV and 10 eV. The solution

grid used in these calculations is given in Table I. All of the source

Table I. Solution Energy Grid Specification

Upper Inte rvals

Energy (MeV) Lethargy Number

15.0 0.0314 35

5.0 0.0196 200

0.1 0.0316 60

0.015

1.0x10"

1.0x10

0.0769

0.1023

125

90

spectra used in these calculations were normalized to one neutron emitted.

The fission source energy distribution used has the form

Figures in brackets indicate the literature references at the end of

this paper.

2.
S(E) = 0.453 Sinh(/2.29E) exp (-E/0.965). (1)

The 14 MeV source was assumed to be represented by a histogram between

15 and 12.2 MeV. The thermonuclear weapon spectrum is given by Straker

[5] and was also represented in histogram form.

In obtaining the fluence from the moments, we could perform reconstructions at all 510 energies. However, this would produce more data

than can be easily managed. Also, since many of the calculational tools

used in shielding analysis produce multigroup results, it is desirable

to give our results in a comparable form. For these reasons, we have

found it convenient to define multigroup moments as

E.

, J 0„ (E)dE , (2) n CE.-E ) J *n,o'

J J £j-1 n = 0,2,4, . .

where E. and E. are the group boundaries for the j— energy group

(see Table II). Consistent with this procedure, we can weight these

multigroup moments by a dose response functions, C., and sum over all

groups:

22

D = L AE. j C. (3) n

i=1 J n J

thereby obtaining dose moments (See Table II for a set of dose response

functions for several detectors of interest). The dose moments can then

be used to reconstruct dose distributions.

We have applied two different procedures in reconstructing distributions from moments: 1) function fitting and 2) plural series approxi3.
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4.
mations [6], The first procedure has the advantage that the representations obtained for the distributions are relatively simple and are

therefore easily used in point-kernel type calculations. The plural

series technique, on the other hand, has the advantage that it enables

us to estimate the truncation error associated with using only a finite

number of moments in reconstructing the distribution. The function

fitting representations make use of the form

-E R £ , -E R/p.

4ttR D(R) = D

Q

e ° + ^ c^G^R/^) e

^

( 4) i=l

while the plural series representation uses the product of exponentials

and U-polynomials [6] and has the form

-£ R

3 10

4ttR

2

D(R) = D e ° + (S R)

k T A./X

k+1 Y a..U

k

(S R/X .)e"

S

oR/X

j (5)

o o Z^jjZ^ijnoj

j=l i=l

Here E is the mean-free-path for neutrons at the highest energy considered, i.e. 15 MeV, and k is a parameter which is specified for the fit

and is usually set equal to unity. In both cases, D is the value of

o

2

4rrR times the unscattered flux or dose at R = 0. D is easily calcu-

° lated from Equations (2) and (3) , provided that the source energy

distribution, S(E) is substituted for the moments, (E)

no '

II. RESULTS

We first compare our results with existing calculations. Theoretically, in the reconstruction of distributions from moments, an infinite
number of moments are required to obtain the correct distribution. Since

only a finite number of moments (six to ten) are available to characterize the distribution, we estimate the truncation error due to negle(

of remaining moments.

The apparent convergence of the distribution, using successively

more moments in the fit, has been used in the past to assess the truncation error. That is to say, distributions are constructed using N

moments and N+l moments, and the difference between the two distributions

is examined. Generally, as more moments are used, these differences

become smaller. However, there is no theoretical justification for

assuming that the distributions converge to the correct value.

Spencer [6] has developed independent procedures for estimating the

truncation error when plural series approximations are used to represent

the distribution. These procedures are based on a more rigorous theory

which gives criteria guaranteeing convergence, and which gives rigorous

bounds to the truncation error which involve the norm of the unknown

function. We had no difficulty satisfying the convergence criteria; but

as in Ref. 6, we rely on estimations rather than rigorous values for the

norm and therefore also for the truncation error bounds. The truncation

error bounds represent maximum errors, rather than probable errors. They

are therefore a conservative estimate of the truncation error.

For the data presented in this report, we have relied heavily upon

the convergence of the distribution as the method for selecting the best

fit to the moments. We have then applied Spencer's approach to determine

rigorous error bounds on the selected fit. In Table III, we give a

typical distribution which illustrate both the convergence and error

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bound data for the distribution. Under the heading CONVERG, the percentage differences between an 8 and 10 moment fit is given. Note that the

truncation error bound (BOUND) calculated as indicated in References [6]

and [7], is larger than the convergence indicator (CONVERG). This gives

us confidence that when convergence is good, the convergence indicator is

a realistic estimate of the error that can be attributed to distribution

construction. However, if this indicator is large, then we must rely

solely upon error bound calculations to indicate the truncation error.

A. Benchmark Problem (Air density 1.29 g/liter)

We have calculated the multigroup f luence and Henderson dose at

several distances for the fission and 14 MeV sources, and compared them

with Monte Carlo calculations by Straker [3], In Table IV, we present

a comparison of the multigroup fluences for the two sources for various

distances from the source point. Generally the two methods compare very

favorably, except in regions where the statistical error of the Monte

Carlo results is large. Table V shows a similar comparison for the

Henderson dose.

B. Comparisons with S Calculations (Air density 1.11 g/liter)

A large tabulation of fluences and dose has been made by Straker [5]

for the three source types of interest, as well as for other source

distributions. These calculations were made using the ANISN code (S P,)

for a 5000 meter sphere. Figures 1, 2, and 3 give comparisons of the

multigroup spectra for the three sources of interest, as calculated by

the S and moments methods. The major discrepancies are in the groups

8.
~ 10

z

o

a:

h-Z>

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ctI-UJ

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0.01

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i—rrn 1

1—r~n

FISSION SOURCE

L

R = 1200 m

T

"L

R = 1800 m

MOMENTS

Sn

J

L_

I I 1

I

,

l__

i r~

L__l

J L_L_L J L_Ll

0. I.0 10 ENERGY (MeV)

Comparison of moments and S^ (Ref. 5) calculations

from a fission source at two depths in air.

9.
1 1 — l±J > <-> — w tr.

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0) ai-i 00 00

m 1-1 1-1 l/~!

H z NO <* <!- <n CM CM 1-1 M r-» £ i 1 1 1 »J eg vO Q r-» NO ON <t o m 1-1 ^ u in 2 <f o NO cn CM 00 rv. 60 u • • • • • 3 CM i-i ON m CM ON U~| i-i O

i-< CO a >v

r-l > u-| <* cn CM CM i—l \*t 01 05

¥ § 1 i i i 1 1 2 U cn •* i-H i-i i-H u-i i—l

"««. 01 o CM ON NO CM 00

m

Oi to 4J M< c s

-I cn CM —I rv. CM i—i

h u-i -* cn CM CM i—l o 2 u v-r 2 1 i i I 1 1 + M 3 00 o 1-1 in o <* CM o <r

01 CM s CM o i—i NO u~i ON en

W Z

pi

s-r- i—i -* en i—l r-~ CM 1-1

O 0) CO oc 01 CJ2

enI CM1

1—11 i-l1 o+ z 01 (J o -I NO en 00 •vt CM o 3 0) m i-H On NO 1-1 ON en M i-H •u wM

rH 1

CM

CM NO <r CM 1-1 en i-i

X 2

U"l1 cni eni CM 1-4 i-H1 ?

CM v* o <* 00 en NO CM NO <r

Pi i—l 2 >0

m -( NO CM ON cn • 1= > >* CM r-» U1 CM i-l en i-4 M

01

i-l U"l o- en CM i—l i—i rH

£> + 1 i 1 1 i 1 nj TO y o ON 00 U-| ON ON cn H U

01 2 NO o cn CM en r~. ON

4J

m 00 NO en i-H

m r^

0) 2 2 m cn cn CM 1-1 i-i i-H

i-H IA cn Tt CM NO ON r-. t~» O 00 m cn CM -* 00 r«.

II cn ON NO en i-i cn ON &

TO •a CM 00 NO NO NO •—I -I c • i-i cn O cn i—

r-i 3 CM o ^^ i—l oo NO <* CM i—i O

OQ =5 i i i t ^S60 >«—

o CM oo NO NO NO i-i U • • r-l en o cn i-H a> m CM • • • c i-i i-( 00 NO o- CM r-l w couu3

01 z

01ou3

CO

r-l > y~s 01 2 o

CO>

0) 2

Pi

cnu

a) 2 u

tu 2o 2

<NJ

O

<t 2 <—t

enu

0) § u

CI) 2o 2 o O o 2

r-l

COu

a) % u

eu 2

r>. - vn C5

U"N 2

coU

01 o2 U

01 2

vt 2

CO C^

CM 2

COu

0)

CJ2

•u

(11 2

CSJ 2

V+O2+

01u

u 2

0>u

0) 2

r-l 2

r~. < > 2

II Pi

CO T3c3

^>

a >

<ii ^261 N-^ U

0)cw

en en en

NO

OnO i-i

O

NO

00

-* -I cn cn cn

-Ct NO i-l i-H i—I On 00 ON

cn cn cn cn cm

r-l r-l CM U1 i-l CM On en en en en cm cm cm

oncn

r-l CM NO

CM CM CM CM 1 CM o 1 en r^

00 NO

O

O

CM CM CM CM i-l I o

r^ cn cm u"> i-i CM CM CM CM CM

NO

CM CM CM

CM

m r-l

r-l CM CM ON

ON

i-l CM CM u-i 00

NO

U~|

mCM00

u-i en00

1-1

1-1 oo1-1 NO

ui

r-l 1—1 r-l CM u-i i-H en

CM CM CM CM CM «H r-i NO

00 o1—1

r-l ON

ON ONON

CM

1—

NO

00 CM CM u-i ON i-H ""I 1 rH 1 o i i ON 1 o

ON in o NO ino

ON

r—l CM CM CM CM1 CM

r-l

CM CM

U-I in

O

NO

00 o1-1

Om

CM cn CM en l-» ON ON

r-l CM CM CM CM CM —I NO ON cn

ON NO

ON CMO fv oo

CM CM r-i cn NO r~ 1—1

00 NOcn NOc

NOcn

i-i

r-i

i-l r-4 00 NO >* CM i—i O

CM

CM

00

i—i CO

NO

cnNO

NOo

NO

cn

CM

r-l

r-l

r-l

u

0) 4J

00u3ou§

COco

•H

CO

CO •H

Ft, >IH

rQ

CO H

eg -j

pi 3

CO •3 CJ O Pi u O

cu rH 2 1-1 C3 n o uc 0) a u e c 2 2 * +

12.
2 2

4ttR x Henderson Dose (Rad-cm /Source Neutron)

R Fission Source 14 Me'V Source

(meters) Momen t s Monte Carlo Moments Monte Carlo

71 4.15-09 3.98-09 5.74-09 5.97-09

142 4.59-09 4.60-09 5.60-09 6.03-09

284 3.83-09 3.70-09 5.33-09 4.75-09

380 2.93-09 2.65-09 3.72-09 3.86-09

472 2.14-09 1.98-09 2.92-09 2.99-09

567 1.47-09 1.32-09 2.19-09 2.20-09

662 9.85-10 8.40-10 1.61-09 1.60-09

755 6.49-10 5.60-10 1.16-09 1.14-09

850 4.17-10 3.30-10 8.18-10 7.80-10

1000 2.03-10 1.75-10 4.58-10 4.40-10

1135 1.04-10 7.51-11 2.66-10 2.42-10

1275 5.17-11 4.25-11 1.48-10 1.38-10

1420 2.84-11 2.18-11 7.98-10 7.55-11

Table V. Comparison between Monte Carlo and moments

methods calculations of Henderson dose for

a point isotropic fission and 14 MeV source

in air (density 1.29 g/liter).

13.
6 and 9. At these energies there exist minima in the air cross section

which may be difficult to represent in the multigroup procedure [ 5]

Consequently, the streaming through these minima will not be calculated

correctly. This effect will become more pronounced as the sourcedetector separation is increased. In Table VI we show a comparison of

the Henderson and Snyder-Neufeld dose for the three sources of interest.

Generally the two calculations compare favorably, with the moments

results tending to be higher than the S results. We feel that this is

due to the increased streaming that is predicted by the moment calculations, as well as to the leakage effect that is associated with the

finite sphere used in the S calculations.

n

C. Simple Representations for Fluence and Dose Distribution .

For point kernel calculations, it is useful to have a simple set of

parameters which can be used to represent the dose and fluence for the

three sources. Since we are interested in simple representations of

these distributions, the results of the function fitting calculations

are used. We note in passing that function fitting and plural series

techniques give the same distribution over the range of interest. In

making these tabulations, we restrict ourselves to representations of

the distributions by 11 or fewer parameters. In Tables VII, VIII, and

IX, we give coefficients for the dose quantities in Table II. The form

used in calculating the distributions from the coefficients is

? -E R -E R/p. 7

4ttR D(R) = D

Q

e + ) a (£

QR/p )e °

X (rads-cm ) (6)

i=l

14.
R

4nR x Henderson Dose

Fission Source

Moments Sn 4.50-09 4.49-09

(Rads-cm

14 MeV

/Source Nei

Source

ltron)

Thermoi

Weapon

Moments

2.41-09

uiclear

Source

(Meters)

150

Moments

5.69-09

Sn 6.07-09

Sn 2.49-09

300 4.05-09 3.93-09 4.88-09 5.06-09 1.95-09 2.03-09

600 1.81-09 1.76-09 2.59-09 2.60-09 8.55-10 8.85-10

900 5.92-10 5.82-10 1.09-09 1.08-09 3.08-10 3.13-10

1200 1.72-10 1.68-10 4.04-10 3.94-10 1.02-10 1.01-10

1800 1.29-11 1.16-11 4.48-11 4.24-11 9.91-12 9.12-12

2400 9.30-13 7.19-13 4.32-12 3.85-12

3000 6.80-14 4.27-14 3.87-13 3.17-13

3600 5.08-15 2.52-15 3.34-14 2.46-14

4200 3.84-16 1.49-16 2.81-15 1.84-15

4800 2.92-17 8.43-18 2.32-16 1.26-16

2 2 4ttR x Snyder-Neufeld Dose (Rads-cm /Source Neutron)

Thermonuclear

R Fission

Moments

Source

Sn

14 MeV Source Weapon Source

(Meters) Moments Sn Moments Sn

150 6.80-09 7.40-09 8.40-09 8.25-09 5.47-09 5.49-09

300 7.08-09 7.19-09 7.01-09 7.22-09 4.42-09 4.55-09

600 3.82-09 3.62-09 4.05-09 4.05-09 1.76-09 1.85-09

900 1.30-09 1.28-09 1.84-09 1.79-09 6.15-10 6.39-10

1200 3.86-10 3.82-10 7.10-10 6.88-10 2.01-10 2.02-10

1800 2.89-11 2.70-11 8.24-11 7.87-11 1.93-11 1.84-11

2400 2.02-12 1.66-12 8.11-12 7.39-12

3000 1.41-13 9.78-14 7.35-13 6.20-13

3600 1.03-14 5.68-15 6.36-14 4.86-14

4200 8.10-16 3.31-16 5.35-14 3.65-14

4800 6.84-17 1.83-17 4.40-16 2.48-16

Table VI. Comparison between S_ calculations and moments method calcul.

tion of Henderson and Snyder-Neufeld dose for a point isotropic

fission, 14 MeV, and thermonuclear weapon source in air (density 1.11 g/liter). S data taken from Reference 5.

15.
FISSION SOURCE IN AIR

HENDERSON (RAD) SNYDER-NEUFEL.D (RAD)

D :2. 711910-09 K= 1 3.843843-09 K= 1 i ai pi ai Pi

1 7.U24213-10 .21b36 3.032895-08 .48881

2 -4013338-U9 .89b73 2.651148-08 1.46643

3 d.l90b79-09 1.38896 -3.700481-08 .63225

4 3.651962-09 1.3091<+ 6.949531-10 1.85179

b 2.613979-U9 1.67617 0.000000 1.00000

D : MID-PHANTOM (RAD) CONCRETE KERMA (ERG/G)

8.776196-10 K= 1 3.453603-08 K= 1 i ai Pi ai Pi

1 4.018b30-09 . 49140 9.839580-09 .21622

2 4.2Ub045-U9 1.47119 -5.994808-08 .89933

3 -b.U63U73-09 .66646 3.863627-08 1.39454

4 l.lU9b93-10 1.6b908 1.162011-07 1.35447

b 0.U0U000 1.00000 3.363860-08 1.67387

TISSUE KERMA (ERG/G)

2.752741-07 K= 1 ai pi

9.387321-08 .21780

-5.611913-07 .90590

8.124505-06 1.40472

-6.808031-06 1.41258

3.164664-07 1.66805

AIR KERMA (ERG/G)

4.324899-08 K= 1 ai Pi

9.665720-08 .49046

1.955809-07 1.47138

-1.497707-07 .77937

6.125755-09 1.84231

0.000000 1.U0000

NON-ION. SILICON KERMA

D :3.71U73l-09 K= 1 i ai Pi

1 9.UU3S85-10 .69297

2 -1.33b7b4-08 .98297

3 1.402235-08 1.15605

4 1.U73952-08 1.44839

b 2.47bb67-09 1.69788

ION. SILICON KERMA

4.329498-09 K= 1 ai Pi

7.347080-10 .21288

•4.782786-09 .88542

1.069465-08 1.37297

2.156281-09 1.00379

3.044102-09 1.66981

SNYDER-NEUFELD (REM)

3.210515-08 K= 1 ai Pi

-3.173191-07 1.13018

4.050881-07 1.25199

5.584109-08 1.64183

0.000000 1.00000

0.000000 1.00000

2 f

k

4ttR" X Dose(R) = D

oexp(-£ R) + ^ a

1 (E

Q R/pi

) exp(-E

() R/p1

) i=l

Table VII. Coefficients for calculating dose distributions in air for

a point isotropic fission source. The dose distributions

have units of (dose units-cm^ per source neutron). lb is

133.059 meters for an air density of 1.11 g/liter.

16.
14 MeV SOURCE IN AIR

HENDERSON (RAD)

D : b.HbUUl9-09 K= 1

i ai Pi

1 -6.64b2b2-10 .25261

2 9.663165-09 1.05069

3 6.827164-09 1.62923

4 -1.427677-08 1.14732

b 9.676135-09 1.72521

SNYDER-NEUFELD (RAD)

7.000024-09 K= 1 ai Pi

-8.955006-09 .15094

7.164796-09 .69763

4.898157-08 1.34196

2.860304-08 1.70389

-6.504560-08 1.31416

TISSUE KFRMA

6.360022-07

a

i -7.657200-06

1.172000-06

8.068461-07

-1.694500-06

1.003330-06

(EkG/G)

K= 1Pi

.25311

1.05276

l.o3244

l.lb221

1.72744

MID-PHANTOM (RAD)

D : 4.9U0017-09 K = 1 I H Pi

1 -4. /*33d9-10 .23b46

2 l./21b64-09 .97933

3 3.U92364-09 l.bl858

4 -2.612731-09 1.26748

b 4.960620-09 1.71004

CONCRETE KERMA (ERG/G)

1.580005-07

ai -1.517908-08

6.707914-08

8.164871-08 1 -9.334668-08 1

K = 1 Pi

.23996

.99806

.54762

.16231

1.620361-07 1.71573

AIR KERMA (ERG/G)

2.660009-07 K= 1 ai Pi

-2.829879-08 .23338

7.271133-08 .97071

1.485257-07 1.50521

-1.071440-07 1.31727

2.252018-07 1.71239

NON-ION. SILICON KERMA

D : 7.500026-09 K= 1

i ° H Pi

1 2.734338-08 1.63360

2 -1.368610-08 1.40440

3 1.971405-09 1.8645b

4 -b.Bb4663-12 2.35862

b O.OOOUOO 1.00000

ION. SILICON KERMA

8.600030-08 K= 1 ai Pi

6.418379-09 .21030

7.669295-09 .87471

4.430919-08 1.35636

3.612633-08 1.59763

7.979537-09 1.74196

SNYDER-NEUFELD (REM)

5.640019-06 K = 1 at pi

-9.282823-09 .26209

4.560220-07 1.09012

2.022771-07 1.69037

-5.393797-07 1.13266

1.088691-08 l.bl020

4ttR

2

x Dose(R) = P exp(-E R) +^ ai (2 R/p

i

) expC-^R/f^)

i=l

Table VIII. Coefficients for calculating dose distributions in air for

a point isotropic 14 MeV source. The dose distributions

have units of (dose units-cm2 per source neutron).

133.059 meters for an air density of 1.11 g/liter.

^o"

1 is

17.
THERMONUCLEAR SOURCE IN AIR

HENDERSON (RAD)

DQ

: 1.598693-09 K= 1

i <H Pi

1 2.23bU7g-10 .53182

2 2.229882-09 1.35445

3 2.699256-09 1.69711

4 2.9bl245"10 1.75437

5 U.OUOUUO 1.UOU0U

SNYDER-NEUFELD (RAD)

2.462538-09 K= 1 ai Pi

5.087558-09 1.37551

-8.094390-09 .47979

9.294106-09 .63088

9.040126-11 1.60195

5.399257-09 1.70926

TISSUE KERMA (Ek&/G)

1.77218^-07 K= 1 ai Pi

-3.41836Q-08 1.41642

5.06356«-08 .60201

2.976032-07 1.37311

3.183777-07 1.70691

0.000000 1.00000

MID-PHANTOM (RAD)

Dn : B.90b8b7-10 K= 1

i ° ^ Pi

1 -7.337294-10 .21165

2 1.531144-09 .88030

3 1.1609b6-09 1.36502

4 -8.230029-10 .99819

b 1.135242-09 1.70155

CONCRETE KERMA (ERG/G)

2.863221-08 K= 1 ai Pi

4.211789-09 .61293

3.432219-08 1.35017

3.735728-08 1.69671

2.190751-09 1.77446

AIR KERMA (ERG/G)

0.000000 1.00000

4.131295-08

a-i 4.733615-08

•7.026025-08

7.407690-08

4.939778-08

0.000000

K= 1 Pi

.39377

.57382

.62382

.70684

.00000

NON-ION. SILICON KERMA ION. SILICON KERMA

1.139916-08 K= 1 ai Pi

-1.102444-10

-3.306599-12

3.814736-09

7.002446-09 1.45429

4.738565-09 1.68985

I k

4ttR

2

y Dose(R) = D exp(-£ R) +) a^^/^) exp(-S R/p1

) 1=1

D : 2.1U4669-09

i ai

1 -2.849184-09

2 l.b/8367-09 1.44882

3 2.b92974-09 .436b0

4 1.771836-09

b 3.Bbbb46-09

K= 1 Pi

.48294

1.28652

1.70903

1.35895

.40243

1.17663

SNYDER-NEUFELD (REM)

1.842464-08 K = 1

°-i Pi

-1.560135-08 .47091

2.265075-08 1.41273

1.859109-08 .o5473

1.240604-08 1.28815

3.849104-08 1.71016

Table IX. Coefficients for calculating dose distributions in air for a PTI thermonuclear weapon source. The dose distributions have

units of (dose units-cm^ per source neutron). X^ is 133.059

meters for an air density of 1.11 g/liter.

18.
Here 1/S = 133.905 meters which corresponds to an energy of 15 MeV and

o an air density of 1.11 grams/liter. These same coefficients may be

applied to calculations at different air densities by using simple scaling

procedures based on equivalent mean free paths. In Tables X, XI, and

XII, the distributions obtained by using the coefficients in Tables VII,

VIII, and IX are given.

In Tables XIII, XIV, and XV, the coefficients which represent the

multigroup spectra for each source are given. The form for these dis-

.th tributions for the j energy group is

5 (7)

2

• • -2 R A -£ R/p.

4ttR (R) = e +) a . (2 R/p.)e 1 (neutrons/MeV/source neutron)

o /__j i o i

i=l

Tables XVI, XVII, and XVIII give the distributions obtained using these

coefficients. Note that for the thermonuclear weapon source, at very

low energies and close to the source, the distributions are negative.

This indicates that the coefficients for these particular energies should

be used with caution at distances less than one mean-free-path. Indeed

in this particular region, it is extremely difficult to fit the moments

and obtain suitable distributions over the entire range.

D. Angular Distributions .

Angular distributions for dose or f luence can also be calculated by

the moments method. Generally the procedure that is used for point

sources is to calculate spatial distributions for Legendre harmonic

coefficients 0»(E,z) for values of t up to about 8. In theory, -the

angular distributions at any distance can then be calculated by perform19.
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31.
ing a Legendre sum of the harmonic coefficients for the appropriate

values of E and z. In practice, the number of coefficients available is

insufficient - particularly for high energy neutrons - and spurious

oscillations or even negative values occur. These oscillations can be

reduced considerably by using an extrapolation technique devised by

Spencer [6] for estimating harmonic coefficients for higher £ values.

Although this technique does not eliminate oscillations, the remaining

peculiarities are very often limited to negative values of cosQ, where

the angular distribution has relatively low values and is therefore of

little practical interest. The angular distribution near the source,

particularly for point sources, tend to exhibit trends associated with

the single scattered angular distribution. Thus near the source, one

expects that the angular distribution would exhibit oscillations similar

to those of the scattering kernel.

In Figure 4 we show a comparison between moments and S calculations of the Snyder- Neuf eld dose angular distribution for a fission

source. These comparisons indicate that the moment results tend to be

more peaked forward than the S results. Consequently more neutrons

would transport further from the source. This phenomena also contributes

to the higher dose values at large distances that is predicted by the

moments calculations.

The authors wish to express their thanks to Dr. Lewis V. Spencer,

whose encouragement and advice helped make this paper possible.

32,
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I 1 I SNYDER NEUFELD RAD DOSE ANGULAR DISTRIBUTION

POINT ISOTROPIC FISSION SOURCE IN AIR

MOMENTS

Sn

1800 m

J. .0 -0.8 -0.6 -0.4 •0.2 0.0 0.2

cos e

0.4 0.6 0.8 1.0

Figure 4. Comparison of moments and Sn (Ref. 5) calculations of the

angular distribution of Snyder-Neufeld dose in rads at four

depths in air.

33.
REFERENCES

[l] J. T. Reynolds and E. L. Slaggie, KAPL-M-6452 (1965).

[2] C. W. Craven, Oak Ridge National Laboratory (June 1966).

[3] E. A.Straker, Shielding Benchmark Problem 3.0 Neutron Spectrum

from Point Fission and 14 MeV Sources in Infinite Air, 0RNL-RSIC-.

(ANS-SD-9) Oak Ridge National Laboratory, Oak Ridge, Tennessee

(August 1970).

[4] CM. Eisenhauer, G. L. Simmons, and L. V. Spencer, A Neutron

Moments Computer Code, Moment I, NBS Technical Note 725, National

Bureau of Standards (May 1972).

L5] E. A. Straker and M. L. Gritzner, Neutron and Secondary Gamma Ray

Transport in Infinite Homogeneous Air, ORNL-4464, Oak Ridge

National Laboratory (1969).

L 6 J L. V. Spencer, Plural Series Representation of Functions, to be

published in J. Res. Nat. Bur. Stand. (U.S.) (Phys. and Chem.).

L 7J L. V. Spencer, Bounds to Truncation Errors in Biorthogonal Polynomial Approximations, with Illustrative Applications to Gamma Ray

Transport Distributions, J. Res. Nat. Bur. Stand. (U.S.) 76B,

(Math. Sci.) No. 1 (1971).

34.
[form NBS-114A (i-7i) 2. Gov't Accession No.

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS TN-745

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Neutron Dose and Fluence Distributions in an Infinite Air Medium

5. Publication Date January 1973

6. Performing Organization Code

7. AUTHOR(S)

G. L. Simmons and C. M. Eisenhauer

8. Performing Organization

10. Project/Task/Work Unit No. 2400434

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No.

12. Sponsoring Organization Name and Address Defense Civil Preparedness Agency

Washington, D. C„ 20310

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) The moments method is applied to the problem of calculating the neutron dose and

fluence distributions in an infinite medium of air. These calculations are compared

with Monte Carlo and Discrete Ordinates (S ) results. Simple parametric representations for the distributions are given which1 facilitate the calculation of dose and

flux distributions in air with a different density.

17. key words (Alphabetical order, separated by semicolons) Benchmark problems; dose distributions;

moments method; neutron penetration; shielding; weapons radiation.

18. AVAILABILITY STATEMENT

fx~| UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

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(THIS REPORT)

X

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21. NO. OF PAGES

39

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