ft UNITED STATES DEPARTMENT OF COMMERCE

PUBLICATION NBS TECHNICAL NOTE 766

''Am o<

A Gamma Ray Moments

Computer Code, GAMM0M-I

U.S. APARTMENT

OF

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NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and

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m& u^iU Of St«<U

A Gamma Ray Moments

§57^3 Computer Code, GAMM0M-I

Charles M. Eisenhauer, George L. Simmons,

and

Lewis V. Spencer

Center for Radiation Physics

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.

V °F ^

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued April 1973
National Bureau of Standards Technical Note 766

Nat. Bur. Stand. (U.S.), Tech. Note 766, 41 pages (Apr. 1973) CODEN: NBTNAE

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

Price $0.75 domestic postpaid or $0.50G.P.O. Bookstore.
Table of Contents

Page

I. Introduction 1

II. Theory 2

A. Moments Equations 2

B. Gaussian Technique Applied to Scattering

Integrals 6 III. Organization and Description of GAMM0M-I 9

A. General Structure 9

B. Discussion of GAMM0M-I Subroutines 11 IV. Input and Output Description 12

A. Card Input 12

B. Scattering Kernel Tape 16

C. Moments Solution Tape 17

D. Discussion of Output Data 18

V. Sample Problem 21 References 36

in

A Gamma Ray Moments Computer Code, GAMM0M-I*

f

Charles M. Eisenhauer, George L. Simmons,

and Lewis V. Spencer

In this paper we describe a computer code for generating

spatial-angular moments of gamma ray energy fluence in an infinite medium. The equation for moments of the energy

fluence is given and the techniques used for the solution are discussed. The structure of the code and of the main subroutines

is also given. Details of the input and output data are presented and the printout from a sample problem is included.

Key words: Computer code; gamma rays; gamma ray transport;

gauss quadrature; moments; shielding.

I, Introduction

In this note we describe a computer program to calculate spatial and

angular moments of the gamma ray fluence in an infinite medium. These

moments can then be used to reconstruct spatial and angular distributions

by techniques described elsewhere [1,2 J. Application of the method of moments to problems in gamma ray attenuation was first discussed by Spencer and Fano [3J. Later Goldstein and

Wilkins of Nuclear Development Associates collaborated with Spencer to produce a systematic tabulation of buildup factors for gamma radiation [4],

That reference also contains an excellent discussion of the solution of

the transport equation by the moments method. The data published in that

report have stood for almost twenty years as the main source of information on penetration of gamma rays from point isotropic sources. Although the data of reference [4] are sufficiently accurate for

many practical applications, they have certain limitations:

1. Tabulations were restricted to energies between 0.255 and

10 MeV.

2. Tabulations were limited to depths of less than 20 mean free

paths of the source radiation.

Supported by the Defense Civil Preparedness Agency, Washington, D.C.

_l20310 and Office of Naval Research, Arlington, Va. 22217

Present address: Science Applications, Inc., Huntsville, Ala. 35805.

Figures in brackets indicate the literature references at the end of

this paper.
3. Results did not include the effects of annihilation or fluorescent radiation.

4. Data are available for only about six elements and water.

5. Calculations were limited to sources whose angular distribution

is azimuthally symmetric with respect to a direction perpendicular to the plane of a slab.

6. The data were generated on an early computer, the SEAC, which

quickly became obsolete.

An effort is in progress at the National Bureau of Standards to produce gamma ray data which are not subject to these restrictions.

The first major production of moments data at NBS was accomplished

by a computer program written by Spencer and Lamkin [5] for the IBM 704

machine. Several reports [6,7,8] give penetration data generated by this

program in water and concrete media. The main application of these results, however, appeared in an NBS Monograph [9] on the calculation of

shielding from fallout gamma radiation. The data in this Monograph form

the technical basis for the Standard Method [10,11,12,13] used by the

Defense Civil Preparedness Agency (formerly Office of Civil Defense) to evaluate the protective capability of structures. This computer program

extended the range of source energies down to about .020 MeV in energy but

had the disadvantage that it was written in machine language.

Accordingly, a gamma ray moments code written in FORTRAN IV language

was developed with the intent of eliminating all the restrictions listed

earlier. This code, called GAMM0M-I, is the subject of this report.

II. Theory

A. Moments Equations

In order to make this report self-contained we sketch the development of the moments equations for gamma rays. The derivation of the

moments equation for gamma rays given here is very similar to that

described in the documentation of MOMENT- I, a neutron moments code [ 14] . The main difference between the two codes is that it is convenient in the

neutron moments code to solve for the collision density moments

Y (E) = a(E) (E) , (neutrons)
where £(E) is the total cross section and (E) are moments of the

n-6 neutron fluence; whereas in the gamma ray code the quantity solved for is

the energy fluence,

In^(E) = E^(E) (gammas)

where E is the gamma ray energy in units of the rest energy of the elec2 tron (mc units) and s (E) are moments of the gamma ray fluence.

The gamma ray transport equation in one dimensional plane geometry,

including fluorescent and annihilation radiation, has the form

00 cos9— (E,z,uj) + n(E)$(E,z,<iu) =

f dE' dO' K(e'-E, w'-w)$ (e' , z,uJ) E 4rr

where

+ p(E-E

i )Y

i

[i. (E

, )$(E',z,^))dE

/ + S(E,z,to).

E. / i

~* 2 2 $(E,z,cu) is the gamma ray fluence (photons/(cm -ster-mc

unit))

,

u(E) is the total cross section (Thomson units per

electron)

,

w is a unit vector in the photon direction, k is a unit vector along the positive z axis, and

(tj«k = coso , _ -3

S(E,z,co) is the source strength (photons/(cm -ster2 . xx mc unit)),

/ —< / —

>

K(E -»E,(ju -"U)) is the Klein Nishina scattering probability per

2 unit path length (TU/electron/mc unit)

E. is the energy of either a fluorescent photon or an annihilation photon

H

(E) is the cross section for either photoelectric

absorption or pair production (TU/electron)

(1)

l

Y. is the yield of either fluorescent photons or annihilation photons, and

E. is the threshold for photoelectric absorption or

1 2 pair production (mc units)
The source term for fluorescent and annihilation radiation may include any process in which photons above a threshold energy E. are absorbed according to a cross section y. with the subsequent emission

of photons at energy E.. The calculations reported in reference [4]

treated these processes as pure absorption (i.e. Y. = 0).

Following the usual practice we find it more convenient to study

the energy fluence rather than the number fluence. We therefore let

I(E,z,^) = E^(E,z,uj) (2)

The transport equation (1) then becomes

CO

cosG ~ (E,z,U)) + u(E)I(E,z,io) = J

dE' j dQ

, K(E'-E,(5 / ^)I(E ,

, z,^u)

4tt (3)

V i ^i.(E'i(E )!(£',IE',z,(D) z,t3) + 1 E.6(E-E.)Y.J — dE ' + ES(E,z,m)

Contrary to earlier procedure, however, we use as an independent variable

the energy rather than the wavelength X of the photon.

If one now expands the energy flux in spherical harmonics,

ia...5>-II ^* $«,.>$«> ,<p>

< 4)

and defines spatial moments of the angular expansion coefficients as

n+1 °°

T^(E) =^T J

2 "

1"(E ' z)dz - (5)

-co the transport equation becomes a function of energy alone. However,

there is an infinite set of equations for I „ (E) . Here, n is the total

m- o cross section at the source energy. The source term can be obtained by

evaluating

n+1 °° YT (9,cp)ES(E,z,'o) dQ , V ' S>> = -^T-J "«» 2£+l I

4tt

m^" m

where Y£ is the complex conjugate of Y» . The moments equation then has

4
the form

H(E) I^(E) = Jk(E'-E) Pt(l-| + |.) C(E

' )dE ' +

E

/('.+l)

2 -m

Z A 2 2

CO + »o*lol 5(E" E

i

) Y

i J , ^(E')I

no

(E') f' + S^(E)

(7)

E

1 where

P^(cosS) is the Legendre polynomial

and

K(E'-E) = 0.375E jj-'+§' -1 +cos

2e]/E' :

For a plane source located at z = 0, i.e. S(E,z,cu) - S(E,cos9) 6(z)/4rr,

equation (7) reduces to

,(E)Ini (E) = J K(E^E)P^(l-| + |/ )I

iTL (E')dE' + 8^ £ 6 (E-E^S*

E k

(8)

+

( (2^)"° L^ +1)In-l,t +l (E) +«n-l f *-l (E)]+ ^noVE )

where

.1 St (E) =

J d(cose)P

t (cose)S(E,cos6) ( 9 )

and

no

S = Y,

J u' (E')I (e'^E'/E' n kJ_/ k no

E k

The computer code GAMM0M-I is designed to treat monoenergetic sources, that is. sources for which
S(E) = 6(E-E

q

) . (10)

To eliminate the difficulties of the 6-function we express the moments

g as a sum of a 6-function and a continuous function I (E)

:

n

I . (E) = I

S

» (E) + S E 6 (E-E )C . m- n</ o o o n-0

where C are a set of coefficients that can be generated by recursion:

n-t

Cnl -jhn^l+l K-l,l+l + Cn-l,^-l ] < 12 >

with C0l = VE > and the C i are a somewhat more complicated set given by

tl = cfe, [ ^+1 > Cn-l,^l + lCtl,l-J + 5 no

Sn (13)

The equation that is actually solved in the computer code is therefore:

OD ,(E)I^(E) = J

K(B'-.E)P

t (l-| + |,) I^(E')dE' * E^E^E^l^ + ±)C^

E

(14)

k

J< - where S is given by

n

sj" = Y -jS u/(E )C . + f I (E')u' (E')^'l

n k o k o n-t' l/no k E J k

The contributions from the 6-f unctions, which are associated with the

primary and secondary sources, and are given in equation (11), is also

calculated by the code.

B. Gaussian Technique Applied to Scattering Integral.

The method used by earlier workers [2,3] to obtain solutions

to Equation (14) was to assume a grid of energies (usually with constant
intervals of the wavelength in Compton units and proceed with the numerical

solution from high to low energies. With this procedure the number of

points in the numerical integration of the scattering kernel is re- stricted to the number of solutions previously obtained. However, the

forward peak in the Klein-Nishina scattering distribution makes it desirable to obtain better definition in the region of integration for which

E' * E. To obtain better definition we evaluate the integral in this

region by Gaussian Quadrature as in the neutron code.

The application of Gaussian Quadrature to Equation (14) is very

similar to the procedure described in Reference [12] for solution of the

moments equations for neutrons. Again, to make this report self-contained we reproduce that discussion here.

This technique has the advantage that its accuracy depends only on

the smoothness of the energy fluence function 1 (E ) in the integrand,

rather than on the smoothness of the entire integrand, which is generally

more rapidly varying due to the scattering kernel. We rewrite Eq. (14)

in the following manner:

E. o l(E)I

n-t (E) " I

KCE'-E^d-! +^)I^(E')dE' = J

K(E'-E)P^

(1-J + |jl^ flOdrf

E Ej

(1-6 )[i no 7 o + (2^i) [l+l K-iMi W+llli,t-i^

(15)

\

^ + ho 5

° k + E

oK(E^E)P^(l-i

+ - ) Cnt + 5^ I 5(E-E

k

)Ek

Snk

In this equation E. is the solution energy just above E. Letting R equal

the right hand side of Eq. (15) and noting that

E.

ti(E)I

S (E) =

[ n(E')I

S (E') 6(E'-E)dE /

,
we may rewrite Eq. (15), E

i

J

C^(E') 6(E'-E) - K(E'-E) \(l-\ +|/)]l^(E')dE' = R . ( 16 ) E

This integral is evaluated using the Gaussian technique by approximating

the term in the square brackets by the sum of two delta functions.

!i(E')6(E'-E) - K(E'-^E) P^(l-| + |,)= ^(e'-E.) + a^S (e'-E|;) . (17)

Note that the function used in equation (17) could be modified by some

Q function of E' in order to make the remaining function I (E') in the

n integrand of equation (16) more smoothly varying. We have not made any

attempt to explore such modifications. Inserting the right hand side of

equation (17) into (16) the integral is readily evaluated to give

alA( V +a2^ (EP

= R (18)

*

The three parameters a . , ct and E are determined by conditions on the

lav 2As 'V first three moments of the scattering kernel:

E.-E

x u(E)5.

o-I. =

j (E'-E) j [h(E')8(E'-E) - K(E / -E)P_

L (cos®)]d(E'-E)

(19)

o

= ^(E

iE ) j + *

2l

(ElE)i

•

for j = 0,1,2

Solving this system of equations gives

-^(E.-E) + I*; El

- E + ~Z i , I

± + (n(E)-I

o )(E -E).

-l

x + (I

Q-u(E))(E

i -E)

V = -*— • and

(20) \ " E

i
In this application of the Gaussian technique to the integral in Eq. (16)

the argument of one delta-function is not fixed beforehand. This quadrag ture is exact for any energy fluence function I (E') which is of degree

two or lower in the range between E and E.. *

x The energy fluence moments at E can be obtained by solving Eq. (12).

•k Generally, E is somewhat lower than E and therefore an interpolation

must be performed to obtain the value of the moments at E. Also, in

evaluating the integral on the righthand side of Eq. (15), an interpolag tion is performed to obtain I (E'). GAMM0M-I uses linear interpolation

g for the solution at E and quadratic interpolation to obtain I (E').

Since the energy fluence is slowly varying with energy, these interpolations should introduce little error into the result.

III. Organization and Description of GAMM0M-I

A. General Structure

GAMM0M-I is structured as a five segment overlay. The structure of this overlay is shown in Fig. 1. The function of each segment may be

described briefly as follows:

Segment 1. Reads all input data except dose response

coefficients

.

Segment 2. Calculates discontinuity energies and solution

mesh parameters.

Segment 3. Calculates scattering kernel and writes the

results on Tape 8. Segment 4. This is the main solution part of the program.

The Gaussian procedure is used. Also the effect

of fluorescence and annihilation is calculated

in this segment. Drum storage is required.

Segment 5. This segment prints the results and calculates

dose moments for the response functions that are defined by means of input data. Also the moments

are written on tape 9 for later processing.
GAMMA

CNSULT

INTPI

INTWT2

1 LOCATE

Segment 12Segment Segment

3 \ /

Segment Segment

4 5

I INPUT EN DISC STOREX

)

CMPUTE OUTPUT

CMBINE KERNEL SOURCE

EIDISC GAUSS f Tape 9

J /~\ LOCATD

( Tape 8 SECOND

CROSSX

ADRUM 35)

FIGURE 1. Overlay structure for GAMM^M-I computer code

10
This overlay procedure is employed on the UNIVAC 1108 and should be

easy to implement on other computers. Note that the interpolation and

integration routines are available to each segment.

B. Discussion of GAMM0M-I Subroutines

GAMMA

This routine calls in the main subroutines in the following

sequence

INPUT - reads in input data related to energy grids, cross sections,

and parameters related to the source angular distribution and the

linkage between moments.

ENDISC - identifies the energies at which primary source energies

exist and anticipates the existence of discontinuities in the

scattered energy flux moments due to these sources. ST0REX - calculates Legendre coefficients of the scattering kernel

and stores them on tape.

SEC0ND - calculates the source terms for fluorescent and annihilation

radiation both for the continuous spectrum and the discrete lines.

COMPUTE - solves the moments equation numerically for a plane source and a fixed solution energy.

OUTPUT - writes moments on tape for each energy and calculates dose

moments for plane and point sources. INPUT

This subroutine reads in all data related to cross sections, energy

grids, source characteristics, and moment linkages. A detailed description of the input data is given in section IV. ENDISC

This subroutine sets up the energy grid for which solutions are to be calculated. Allowance is made for double-valued solutions at energies

for which discontinuities in the scattered energy fluence occur # The

fineness of the grid is determined by input parameters except that 70

intervals are automatically assumed below the lowest single-scatter cutoff. The subroutine calls CMBINE which combines energy grids determined

for each fi-function source into a final energy grid.

11
ST0REX

This subroutine interpolates for the total cross section at each

energy and writes detailed calculations of the scattering kernel on to tape. It calls EIDISC and KERNEL.

EIDISC - This subroutine sets up the energy grid for each integration over the scattering kernel. The fineness of the grid is determined by the input parameters MAXN0, which sets the minimum

number of intervals over any single scatter region and MINN0 which

is only used when the problem involves more than one monoenergetic

source

.

KERNEL - This subroutine calculates the required number of Legendre

coefficients of the scattering kernel.

SEC0ND

This subroutine calculates the contribution of the annihilation and

fluorescent secondary source radiation to the scattered energy fluence

moments. This contribution is analogous to the contribution from singlescatter radiation generated by the primary source. This subroutine also

calculates the contribution from photons at the discrete secondary source energies and adds them to the direct radiation from the primary source. CMPUTE

This subroutine solves equation 14 for moments of the energy fluence

Solutions proceed from the highest energy downward with numerical integration of the scattering integral.

OUTPUT

This subroutine contains options to print or write on to tape the

energy fluence moments for scattered and direct radiation. Dose moments

are produced by integration over the energy fluence moments, for up to ten dose response functions. Dose moments may be printed and/or punched

on cards.

IV. Input and Output Description

A. Card Input

The following cards are read in the subroutine INPUT:

12
Card Type 1 (II, 12A6)

IPR0B Type of input data

IPR0B = 1 All input data required.

IPR0B = 2 No source data required; source is assumed to be the same

as in previous problem.

IPR0B =3 No cross section or source data required; these data are assumed to be the same as in previous problem.

IPR0B = 4 Only data for source angular distribution is required;

all other input data are assumed tn be the same as in previous

problem.

IPR0B = 5 End of calculation (See card type 22).

TITLE Alpha numeric title for problem.

Card Type 2 (1115)

NMUS Number of energies for which total cross sections are to be read in.

NS0RS Number of primary source energies.

LS Maximum number of Legendre coefficients calculated by code

(1= 0,1,... LS-1).

MINN0 The minimum number of intervals in the scattering kernel between

the energy ranges of single scattering interactions.

MAXN0 The minimum number of intervals in the scattering kernel for any

single scatter interaction.

NPXS Number of partial cross sections such as pair production or photoelectric absorption. NPXS > only if annihilation or fluorescent

radiation is to be included in calculation.

I0UT Parameter determining type of output information.

I0UT = Fluence moments printed and written on unit 9. Plane and

point dose moments printed. Plane dose moments punched.

I0UT = 1 Fluence moments written on unit 9 and printed only for

primary and secondary source energies. Plane and point dose moments

printed. Plane dose moments punched.

I0UT = 2 Fluence moments printed. Plane and point dose moments

printed.

I0UT = 3 Fluence moments printed only for primary and secondary

source energies. Plane and point dose moments printed. Plane dose

moments punched.

13
KEYIND Index which allows part of calculation to be skipped.

KEYIND =0,1 Do entire calculation.

KEYIND = 2 Compute moments using scattering kernel data stored

previously on tape mounted on logical unit 8. NRN Run number (arbitrary).

IC0NT Obsolete parameter (arbitrary) . NPRINT Number of solution energies for which detailed printout of

solution is desired. (For diagnostic use.)

Card Type 3 (3E10.0)

ET0P Highest energy for which solutions are to be calculated. Ordinarily

ET0P is equal to the highest primary source energy.

EB0TM Lowest energy for which solutions are to be calculated.

2 SO Factor necessary to convert energy from input units to mc units.

(Normally input energies are in MeV units and S0 = 1.95692).

X0 Factor necessary to convert cross section from input units to

il] 1

2 Thomson units. (Normally input cross sections are cm /g units and X0

is then (.400594 Z/A)

Card Type 4 (2015) (Only if NPRINT > 0)

(IXEN(I), 1=1, NPRINT) Indices of solutions for which detailed printout

is desired.

Card Type 5 (8E10.0) (Only if IPR0B < 2)

(EMUS(I), TABMU(I), 1=1, NMUS) List of energies and tabulated cross sections

.

The next 5 types of cards are required only if fluorescent or annihilation radiation is to be calculated.

Card Type 6 (2015) (Only if IPR0B S 2 and NPXS > 0)

(NPART(I) , 1=1, NPXS) Number of tabulated cross sections for each

reaction.

Card Type 7 (8E10.0) (Only if IPR0B < 2 and NPXS > 0)

(YIELD(I), 1=1, NPXS) Yield for each reaction.

Card Type 8 (8E10.0) (Only if IPR0B < 2 and NPXS > 0)

(ECUT(I) , 1=1, NPXS) Threshold energy for each reaction.

Card Type 9 (8E10.0) (Only if IPR0B < 2 and NPXS > 0)

(ESTART(I), 1=1, NPXS) 6-function energy of photon produced by each

reaction.

14
Card Type 10 (8E10.0) (Only if IPROB < 2 and NPXS > 0)

(EPART(I,J), SIGMA(I,J), J = 1, NPART(I)) List of energies and tabulated

partial cross sections. (One set for each reaction.)

Card Type 11 (8E10.0) (Only if IPR0B = 1)

(ES0RS(I), STS0RS(I), 1=1, NS0RS) List of energies and strength of each

primary source. Card Type 12 (415)

NL0 Number of harmonic coefficients of the source angular distribution.

LZR0 Index of source harmonic when NL0 = 1; otherwise arbitrary.

MZR0 Index of azimuthal harmonic. (MZR0 = unless both source distribution and detector response include azimuthal dependence)

.

Card Type 13 (4l3, 4(16, 3l3))

N(I) n- index of the i moment

L(I) t- index of the i moment

LINKH(I) i-index of higher--f, linkage moment

LINKL(I) i-index of lower--t linkage moment

Card Type 14 (8E10.0)

(CNL(I) , 1=1, NL0) Harmonic coefficients of source angular distribution.

The following cards are read in the Main Program GAMMA:

Card Type 15 (215) (Only if KEYIND > 1)

NENSP Number of energies at which solutions are to be obtained.

NED Number of discontinuities in solution mesh.

Card Type 16 (3(13, IP2E11.6)) (Only if KEYIND > 1)

NEIN(K) number of integration points for K solution.

ENS(K) energy corresponding to K solution.

EMUN(K) total cross section corresponding to K energy (TU/el)

Card Type 17 (3(13, 1P2E11.6) (Only if KEYIND > 1)

NSW(K) number of solutions, uniformly spaced in lethargy (E./E.

f" Vi i i — x constant) in K solution interval.

ED(K) highest energy in K solution interval.

DED(K) ratio of energies of successive solutions in K interval.

Subroutine OUTPUT

Card Type 18 (215)

NDOSE number of energies at which dose response coefficients are tabulated

15
NOFD number of types of detector responses to be calculated

Card Type 19 (8E10.0)

2 ECONS factor to convert energies in input units to energy in mc units

(see Card Type 3)

(CONS(I), 1=1, NOFD) factor to convert energy absorption coefficients

from input units to units of TU/electron. (see Card Type 3)

Card Type 20 (8E10.0)

(EDOSE(I), 1=1, NDOSE) energy list for dose response coefficients.

Card Type 21 (8E10.0)

(SIGMAD(I, J) , 1=1, NDOSE) energy absorption coefficient for I energy

i ~th _ , and J type of dose response

Card Type 22 (blank)

End of calculation

B. Scattering Kernel Tape

For the UNIVAC 1107/1108 systems, the results of the calculations of

the scattering kernel are written tape unit 8. This tape must be available for all values of KEYIND. The tape is written in the binary mode.

Binary records are written in the following form for each solution:

Record 1 WRITE (8) ENS(N), NIN, LSP, ND

f~Tt where ENS(N) is energy for the N solution

NIN is the number of integration points in the integrand

LSP is the number of Legendre coefficients LS, plus 4

ND is the number of intervals over which the integration mesh

is uniformly spaced.

Record 2 WRITE (8) (NKIND(K), K= 1, NIN)

where NKIND(K) is an index which indicates when the integration

mesh changes

NKIND > 0: number of energies with constant integration mesh

NKIND = 0: no change in integration mesh

Record 3 WRITE (8) (EIN(K) , K = 1, NIN)

where EIN(K) is the list of energies in the integrand

16
Record 4

WRITE (8) (EDI(K), K = 1, ND)

where EDI(K) is the list of energies at which the integration mesh

changes

Record 5 WRITE (8) ((XKIN(K,L), L = 1, LS) , K = 1, NIN)

where XKIN(K,L) is the (L-l) harmonic coefficient of the cross

section for the K energy in the integrand

C„ Moments Solution Tape

For the UNIVAC 1107/1108 systems, moments are written on tape unit

9. This tape must be available for IOUT (input card 2) values less than

2. The following describes the BCD records written for a single problem.

Record 1 (12A6)

TITLE Problem title

Record 2 (515)

NENS number of solution energies

NSORS number of primary source energies

NOFD number of different detector responses

NNL number of (n,£) combinations

NDOSE number of energies at which dose response coefficients

are tabulated

Record 3 (1615)

(N(I), L(I) , 1=1, NNL) n and t indices of each solution

Record 4 (1P10E13.7)

(ENS (I), 1=1, NENS) energy list

Record 5 (1P10E13.7)

(EMUN(I), 1=1, NENS) list of total cross sections

Record 6 (1P10E13.7)

(CNL(I), 1=1, NNL) list of C . values for each (n,£) combination

n-L Record 7 thru 2*NNL + 6 (1P10E13.7)

(SOLN (NENS + K, I), K = 1, (NSORS + NPXS)) Direct radiation

moments for primary and secondary sources for the I (n,-t) combination

17
(SOLN(K,I), K = 1, NENS) Scattered radiation moments for the I

fc

(n,£) combination.

D. Discussion of Output Data

The data preceded by (NPRINT > 0) are printed only for those solutions for which detailed printout is requested. These solutions are chosen by means of the input parameters NPRINT and IXEN. These data are

for diagnostic use. The printout consists of the following blocks of data :

Input data from INPUT subroutine.

Tabulation of energies and total cross sections in input units.

Tabulation of energies and total cross sections in units of electron

mass and Thomson units, respectively.

(NPXS > 0) List of secondary source energy, threshold energy yield

number of tabulated partial cross sections, list of energies and cross sections for each partial interaction.

List of source energies (primary and secondary) and source strengths

(strength of secondary sources is initially set at zero).

List of normalized source energy and strength.

List of indices for each moment and values of linkage coefficients.

List of solution index, number of integration points, energy, and

total cross section for each solution.

List of index numbers of solution points, energy and ratio of

successive solution energies associated with each discontinuity.

Discontinuity refers to an energy below which the solution energy

grid changes.

(NPXS > 0) List of moment (n,£) indices and contribution from

integration over scattered moments.

Energy and partial cross section for each primary scource.

List of moment (n,£) indices and contribution from scattered

moments and primary sources. List of moment (n,£) indices and contribution from scattered moments

and primary sources, weighted by radiation yield and appropriate

cross section ratios.

List of energies for which secondary sources may contribute to

18
scattered moments. (Note that the SOURCE subroutine calculates a non-zero contribution only when

(2E

2 + 1)* E < E

2

'

where E is the energy of the secondary source.

(I = IXEN(J)) The following information is printed only for those

energies for which detailed printout has been requested.

(n,^) index, source contribution, and source contribution

at solution discontinuity.

Parameters associated with this solution energy.

Index of solution, number of integration points, integrand

energy

.

Legendre coefficients of the scattering cross section, for each

interval of integration.

£ index, three Gauss parameters and three obsolete parameters, for each ^-value.

Energy and contribution from numerical integration for each

(n, -^) combination

Energy and moment solution for each (n,^-) combination.

Printout of input parameters.

List of energies and dose response coefficients.

2 List of energies in mc units and dose response coefficients in

Thomson units per electron.

(IOUT = or 2) Energy, total cross section, solution index, LZRO,

and MZRO, followed by a list of (n,£) indices and fluence values

for each (n,^) combination.

Energy, total cross sections, solution index, LZRO and MZRO,

followed by (n,£) indices and fluence values for source energies

only . List of energy, interpolated dose response coefficient (normalized)

fluence (0,0) moment, cumulative dose integral for each type of

response.

Normalizing constant for which total energy absorbed (i.e. the zero-zero moment for total dose) is equal to unity. The deviation

19
of the moment from unity is a measure of inconsistency in cross sections introduced by interpolation procedures in the code.

Title of calculation and index of dose response coefficient.

Dose moments for a plane isotropic (PLI) source. Values are given

for total, direct (including secondary sources) and scattered

dose.

Dose moments for a point isotropic (PTI) source. Values are given

for total, direct (including secondary sources), and scattered

dose.

20
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REFERENCES

[l] U. Fano, L. V. Spencer, and M. J. Berger, "Penetration and Diffusion of X-rays," Handbuch der Physik , XXXVIII (Springer- Verlag,

Berlin, 1959).

[2] L. V. Spencer, "Plural Series Representation of Functions, J_. Res . Nat . Bur . Standards (to be published).

[3] L. V. Spencer and U. Fano, "Penetration and Diffusion of X-Rays.

Calculation of the Spatial Distributions by Polynomial Expansion,"

J. Res. Nat . Bur . Standards 46, 446 (1951).

[4] H. Goldstein and J. Ernest Wilkins, Jr., "Calculations of the

Penetration of Gamma Rays," AEC Report NYO-3075, June 30, 1954.

[5] L. V. Spencer and J. C. Lamkin (unpublished).

[6] L. V. Spencer and J. C. Lamkin, "Slant Penetration of y

_ Rays in

HO," NBS' Report 5944, unpublished.

[7] L. V. Spencer and J. C. Lamkin, "Slant Penetration of y-Rays:

Mixed Radiation Sources," NBS Report 6322, unpublished.

[8] L. V. Spencer and J. C. Lamkin, "Slant Penetration of y-Rays in

Concrete," NBS Report 6591, unpublished.

[9j L. V. Spencer, "Structure Shielding Against Fallout Radiation from

Nuclear Weapons," NBS Monograph 42, U.S. Gov't Printing Office,

June, 1962.

[lOJ C. Eisenhauer, "An Engineering Method for Calculating Protection

Afforded by Structures Against Fallout Radiation, " NBS Monograph

76, U.S. Gov't Printing Office, July, 1964.

[ll] "Design and Review of Structure for Protection from Fallout Gamma

Radiation," PM- 100-1, Office of Civil Defense, Feb., 1965.

[12] E. T. Bramlitt and C. M. Eisenhauer, "A Discussion and Tabulation

of the OCD Standard Method for Fallout Gamma Radiation Shielding

Analysis," TR-70, Office of Civil Defense, December, 1971.

[13] "Shelter Design and Analysis, Volume 1, Office of Civil Defense,

June, 1968 (Rev: 1969, 1970, 1971).

Ll4] C. Eisenhauer, G. L. Simmons and L. V. Spencer, "A Neutron Moments

Computer Code, Moment I.," NBS Technical Note 725, June, 1972.

36 >' S. GOVERNMENT PRINTING OFFICE : 1973—511-324/216
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-7 66

2. Gov t Accession No.3. Recipient's Accession No.

4. TITLE AND SUBTITLE A Gamma Ray Moments Computer Code. GAMM0M-I

5. Publication Date April 1973

6. Performing Organization Code

7. AUTHOR(S)

Charles M. Eisenhauer, George L. Simmons, and Lewis V. Spencer

8. Performing Organization

10. Project/Task/Work Unit No. 2400434 - 2400439

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

Office of Naval Research, Arlington, Va. 22217

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.)

In this paper we describe a computer code for generating spatial-angular moments of gamma ray energy fluence in an infinite medium. The equation

for moments of the energy fluence is given and the techniques used for the solution are discussed. The structure of the code and of the main sub- routines is also given. Details of the input and output data are presented

and the printout from a sample problem is included.

17. KEY WORDS (Alphabetical order, separated by semicolons) Computer code; gamma rays; gamma ray transport; gauss quatrature; moments; shielding.

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