EFERENCE

NBS TECHNICAL NOTE 725

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U.S.

'ARTMENT

OF

OMMERCE

National

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173

A Neutron Moments

Computer Code,

Moment I
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

technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measure- ment system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Center for Computer Sciences and

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THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and accessibility of scientific information generated within NBS and other agencies of

the Federal Government; promotes the development of the National Standard Reference Data System and a system of information analysis centers dealing with the broader

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1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

2 Part of the Center for Radiation Research.

3 Located at Boulder, Colorado 80302.
Ml BUREAU OF STANDS

3 1972

ft\oo' a Neutron Moments Computer Code, %3

5 Moment I

Charles M. Eisenhauer, George L. Simmons, and

Lewis V. Spencer

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

<? WW \

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

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

• f

-

Issued May 1972
National Bureau of Standards Technical Note 725

Nat. Bur. Stand. (U.S.), Tech. Note 725, 38 pages (May 1972) CODEN: NBTNAE

Issued May 1972

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. C13.46:725). Price 50 cents.
Table of Contents

Page

I. Introduction 1

II. History of Neutron Moments Code 2

III. Theory 3

A. Moments Equations 3

B. Gaussian Technique Applied to Scattering Integral . . 5

C. Neutron Scattering Laws 8

D. Integration of the Scattering Kernel 10

IV. Organization and Description of MOMENT1

11

A. General Structure 11

B. Discussion of Neutron Subroutines 11

V. Input Instructions and Description of Input and

Output Data 14

A. Card Input 14

B. Input Cross Section Tape (LIB) 17

C. Scattering Kernel Tape 19

D. Moments Solution Tape 20

E. Discussion of Output Data 21

VI. Sample Problem 24

References 32

III
A NEUTRON MOMENTS COMPUTER CODE, MOMENT I*

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

In this paper we discuss a computer code for generating

spatial -angular moments of neutron flux in an infinite medium.

The equation for the flux moments is given and the techniques

used for the solution are discussed. The structure of the

computer code and of the main subroutines is also discussed.

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

Key words: Carbon; computer code; gauss quadrature; moments;

neutron transport; shielding.

Supported by the Office of Civil Defense, Defense Nuclear Agency,

and Office of Naval Research.

IV
I. INTRODUCTION

In this note we describe a computer program to calculate spatial

and angular moments of the neutron flux distribution in an infinite

medium. The method of moments has proved extremely accurate in the past

for predicting gamma ray distributions in infinite media, but some

difficulties were encountered in applying the technique to neutron

problems. Principal among these difficulties were:

1. The energy range of interest in application to reactor or

-2

weapons shielding problems extends from 15 MeV to 10 eV, a range of

in9 • 10 in energy.

2. Neutron cross section data are not smooth functions of energy,

but exhibit a great deal of structure. This problem manifests itself

in the attempt to guarantee . that the moments converge to the correct

result.

3. For elastic scattering, the neutron energy E is degraded to a

minimum value of E( ——

—J . For elements with high atomic weight A,

therefore, the energy degradation is very small and the number of solutions required to describe the distribution of the moments as a function

of energy is very large.

MOMENT I is an attempt to overcome the difficulties discussed above,

but still maintain practical computing economy. The program is written

in FORTRAN IV and has been made operational on the UNIVAC 1107/1108,

CDC-3600, and IBM-360/91. Our experience has indicated that the code

can be converted from one machine to another with only minor modifications.

These modifications involve the size of the memory available and consequently the size of the particular problem that can be solved.
II. HISTORY OF NEUTRON MOMENTS CODE

The method of moments was first applied by Spencer and Fano to

the solution of the transport equation for gamma rays. Application of

this method to solution of problems in neutron transport was difficult

at that time because neutron scattering cross sections could not be

expressed in a simple analytic form analogous to the Klein-Nishina

formula for the scattering of gamma rays. Therefore, large tabulations

of neutron cross section data were required. The subsequent development

of automatic computers with large storage capacity made the solution by

moments feasible for neutrons. A code called NUPAK was reported by

(2) (3) Certaine in 1954. A subsequent version called RENUPAK was

reported in 1959. A history of this code along with references is given

(4) by Weisbin.

A moments code for neutron transport was also developed at the

National Bureau of Standards. Results of calculations made during the

development of this code have appeared in several publication. Results

calculated for water were contributed to the comparative study of different transport methods conducted by the American Nuclear Society in

1963. Results calculated for sodium were used by Miele. Calculations of spectra from fission neutrons and 14 MeV neutrons in air were

quoted in a status report by Straker.

A discussion of the NBS codes for generating moments and COnStrUCting spatial distributions was given in a 1968 unpublished report.

In that report calculations for a point isotropic photofission source

in carbon were presented. Since the publication of that report several

improvements have been made in the code.
1. The processing of input cross section information has been

simplified and speeded up.

2. The calculation of the scattering kernel has been changed in

order to improve convergence of the moments.

3. General numerical procedures for interpolation and integration

have been streamlined.

III. THEORY

A. Moments Equations

The neutron transport equation in one dimensional plane geometry

has the form

cos9|i- (E,z,uj) + S(E)$(E,z,3) = j dE' ; dQ' K(E'->E,u)'-*o)$ (e' , z,uu'

)

4tt (1) + S(E,z,u>) .

where

$(E,z,cu) is the neutron flux (neutrons/cm -sec-MeV-ster)

,

S(E) is the macroscopic total cross section (cm ),

u) is the neutron direction relative to the positive

z axis, and 'u-k = cos9,

-> -3

S(E,z,uo) is the source strength (neutrons /cm -sec-MeV-ster ),

and K(E ->E,uo -^u)) is the scattering probability per unit path length

(probability/cm-MeV-ster)

.

If one now expands the flux in spherical harmonics,

CO -t *».*.5)-Z I /^f*>, z) ^(P,cp)

, (2)
and defines spatial moments of the angular expansion coefficients as

^.n+1 oo

no n.

z $„ (E,z) dz , (3)

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

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

no o cross section at the highest energy considered. The source term can be

obtained by evaluating

-,n+l °°

S>E

> = TT

n

dz

J Mk tf* <9,tp) s(E - z ><°> <3 (4 >

4tt

where yA is the complex conjugate of yA . The moments equation then has I

the form

V

c m

2(E) S^(E) K(E%E) Pt

(coS9) $TM (E')dE' +

+ s /(^i)2 -

2 ^ , ,.. (E) +E /<

2 2 £ -m x m

(20+1) n" 1 ^+lV" "° 2^+1 n - 1 ^_1

-,m + S (E) . n,-0 (5)

where is the scattering angle in the laboratory system. For a plane

->. isotropic source located at z = 0, i.e. S(E,z,uo) = S (E)6 (z)/4tt :

E(E) $ .(E) = K(E'->E) P., «. s ,-,/*,-./ . no J. -0 (cos® ) $ » (E )dE +

E

ryt (6)
(1 ^°)S° [(t+l) i

n. 1(t+1

(E) Hi

n. 1]W (E)] +

+ S 8 S(E) . o no

Since the evaluation of the integral in Eq . (6) must be done

numerically, we would like to make the integral as smooth as possible.

In order to reduce the effect of the variations with energy associated

with the integration of the scattering kernel, we take advantage of the

properties of , namely

*oo<

E> " "A

and define collision density moments

Y^(E) = S(E) $^(E) . (7)

'The collision density moments equation corresponding to Eq. (6) is

V(E

>

I' 'WT1

*l (COS9) rnl

(E

' )dE' (8)

^maw) C(t+1) Vi,w<E' +tVi,t-i'E»

+ S 6 S(E) . o no

B. Gaussian Technique Applied to Scattering Integral

The usual method of solving Eq . (8) or Eq. (6) assumes a grid

of energies between some maximum energy, E and the lowest energy of
interest and proceeds with the numerical solution of the problem, working

from high to low energies. As indicated in the introduction, some compromise must be struck between covering the range of interest with an

adequate number of points and being able to reach the lowest energy with

a minimum number of points. This problem is aggravated at high energies

by forward peaking of the elastic scattering cross section and the

implication, therefore, of high probability for small energy loss

collisions.

To avoid this problem, we evaluate part of the scattering integral

by the technique of Gaussian Quadrature. This technique has the advantage

that its accuracy depends only on the smoothness of the collision density

function Y . (E ) in the integrand rather than the smoothness of the

entire integrand, which is generally rapidly varying due to the scattering kernel. We rewrite Eq . (8 ) in the following manner,

E. E

^(E > - 1 "so?!

1 Veo*> *nt

(E '> dE ' = f T^pfK <«»-»*,*<"'»' +

E E. (9)

+ gSn^i) [(^+1) Vi,WE) + %-m-i (e)] +Eo

6 no

s(E >

•

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

the right hand side of Eq. (9) and noting that

E.

l

Y(E) =

J YCe') 6(E / -E)dE/

, E

we may rewrite Eq. (9),
E. K(E'->E) [6(E'-E) - SaSg^<«,*>] V(E

< )dE , = R . (1Q)

This integral is evaluated using the Gaussian technique by approximating

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

6(E'-E) - K^f P^CcosS)) = au

6(E'-E

±

) + a^fi (E'-E*) . (11)

Clearly, Eq . (10) becomes

The three parameters a. » , a . , and E„ are determined by conditions on

the first three moments of the scattering kernel :

, ?i"

E

• K(e'->E) P (cos3)

6. -Ij =

J (E'-E) J [6(E /-E) ^^ ] d(E'-E) (13)

= au

(E.-E)J +a^(E;-E)J

for j = 0,1,2 .

Solving this system of equations gives

-I^E -E) + I1

ED = E +

l\ + (1-I^)(E.-E)
-l\ + (/-D(E.-E)

1 o 1

-t 1

*K = 1

- Xo V

and (14)

This application of the Gaussian technique to the integral in Eq. (10)

differs from the usual Gaussian Quadrature approach in that the argument of one delta- function is not fixed beforehand. This quadrature is

exact for any collision density which is of degree two or lower in the

range between E and E.

.

The collision density moments at E. can be obtained by solving

*

Eq. (12). Generally, E, is somewhat lower than E and therefore an interim polation must be performed to obtain the value of the moments at E.

Also, in evaluating the integral on the righthand side of Eq. (9), an

interpolation is performed to obtain ¥ „(E'). MOMENT I has the

capability of using either first or second order polynomials for these

interpolations. Since the collision density (as opposed to the flux)

is slowly varying with energy r these interpolations should introduce

little error into the result.

C. Neutron Scattering Laws

The scattering kernel, K(E'h>E), is the sum over all possible

scattering reactions that can contribute to the integral. MOMENT I has

the flexibility of considering elastic, discrete level inelastic and

continuum inelastic scattering events, as well as (n,n'p), (n,2n'),

(njn'a), and(n,n'3a) reactions. For both elastic and discrete level

8
inelastic scattering, the contribution to the scattering kernel may be

represented as

- - (1-fM. ) 2E. .(E) r

K(E'-»E) =) )

1 iJ

) (l4)f. ,,(E') P ,(coS9 ) , (15)

l j l lj £=o

where

M. is the atomic weight of the i isotope (AMU),

E..(E') is the scattering cross section for the j interij

. .th . . -L action in the i isotope (cm ),

f„ ..(E) is the Legendre polynomial expansion coefficient

for the ij interaction, normalized such that

f ,.(E') = 1. o,ij

The scattering angle, ®.., in the LAB system is related to the scattering

angle, 9. ., in the center of mass system by the following relationships,

M

i +1 IT ^..(E')-l [=T

"*u • -T- /t - * ^fe /f (16 > 1

COs9

lj 2M.v!.(E') [(1 ^i )2 I " 4

V

lj

(E,) + 1] ' (1?)

1 1J

V

with the ratio of velocities of the center of mass after and before a

collision being given by

V,,) v1 -"H- (18)
Q. . being the Q value for the ij interaction, (MeV).

For elastic scattering, (Q = 0), V. . is unity, but for inelastic

scattering V. . is less than unity.

The remaining interactions are treated by assuming that the reactions

are isotropic in the LAB system and that the secondary neutron distribution is a Maxwellian. The kernel can then be written

K. .(E'-»E) = E. . (E') -f- exp (-E/T.) , (19)

i

with the nuclear temperature, T. , being given by the prescription

/M.

T.(E') = A., /-f- (20)

with A. . being characteristic of the particular nuclei and particle

considered. This procedure has been adopted primarily because the tabulation of continuum type reaction data is simplified, without loss of

accuracy, by the use of an analytical expression.

D. Integration of the Scattering Kernel

In Fig. 1, we indicate the l> = o component of the scattering

kernel for a particular solution energy from energies for which solutions

have been previously calculated. The contributions from various interactions are clearly indicated; the discontinuities are recognized by the

code and for each cross section contribution an integration mesh,

independent of the solution mesh, is determined. Typically, the integration over each individual contribution is done by using a prespecified

10
number of points. Provisions are made for allowing overlay of interactions with the resulting discontinuities. This procedure results in

an integration scheme which treats each interval and discontinuity

correctly, without resorting to smoothing a discontinuous function.

IV. ORGANIZATION AND DESCRIPTION OF MOMENT

A. General Structure

MOMENT- I is written in such a way that it may be easily overlayed in order to maximize the amount of core available for data storage,

In Fig. 2, the structure of the code is shown with each level suitable

for overlaying and the subroutines associated with each level.

On CDC 3600 and UNIVAC 1107/1108 computers this overlay

structure can be used to expand the effective size of the available core

The following is a brief description of each subroutine.

B. Discussion of Neutron Subroutines

MOMENT I

This routine calls in the main subroutines. Tapes are identified by numbers appropriate to the computer installation. The subroutine REMTIM, which prints out the CPU and input-output times elapsed,

is called at several points. The following subroutines are called in a

linear sequence:

INPUTA - reads in input data related to cross sections and

energy grids.

INPUTB - reads in input data related to angular and linkage

parameters.

SIGMAX - generates partial cross sections for material of

11
interest from input cross section data.

CMPUTN - computes moments for a given energy.

OUTNUT - processes moments for output on tape, cards and/or

printer.

INPUT A

This subroutine reads in data related to cross section and

energy grids. It reads the cross section tape (LIB) and selects

appropriate materials. A description of the input data is given in

Section V-A.

INPUT B

This subroutine reads in data describing the angular characteristics of the source. A description of these input data is given in

Section V-A.

SIGMAX

This subroutine organizes the writing of scattering kernel

information on the tape LIBCX. It calls NEIMSH AND NKERNL.

NEIMSH

This subroutine determines the integration spacing for the

scattering kernel. Energies at which each partial interaction turns

on and off are determined. This list is merged with a list of energies

related to the solution mesh to produce a list of energies at which

potential discontinuities in the integrand may occur. Between each of

these energies an integration mesh is generated, determined by input

parameters such as MAXN0 and MINN0.

NKERNL

This subroutine calculates the contribution from each partial

12
interaction to the scattering kernel. Contributions for elastic scattering and inelastic scattering by discrete levels are calculated by

calling NCR0SS. Inelastic scattering via the continuum is calculated

by calling C0NTIN. The scattering kernel is written on tape LIBCX.

GAUSSN is called to calculate the appropriate Gaussian integration

parameters.

NCR0SS

This subroutine calculates the scattering cross section for fixed

initial and final energies and for a fixed excitation energy Q.

C0NTIN

This subroutine calculates the inelastic scattering cross section

for fixed initial and final energies according to the formula

K(E'-E) =S(E')^-e- E/T

, 1

L

where

T = A ./M/E '

and A is a normalization constant.

CMPUTN

This subroutine solves the moments equation, namely,

E

v(E) Ki^n VcoS3

> *nl<E '> dE ' +S

l

(E)

E

S

+ s(iTau5T t«-*> Vi,« +l Vi,m]

13
where

Y^CE) = 2(E) ^(E)

are moments of the collision density.

V. INPUT INSTRUCTIONS AND DESCRIPTION OF INPUT

AND OUTPUT DATA

A. Card Input

SUBROUTINE INPUTA

Card Type 1 (20A4)

TITLE Alphanumeric title for problem.

Card Type 2 (1615)

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

KEYIND =0 Do entire calculation.

KEYIND = 1 Generate partial cross sections and store on

LIBCX tape for later use .

KEYIND = 2 Compute moments using partial cross sections

stored previously on tape LIBCX.

NW Number of elements in mixture.

LS Maximum number of Legendre harmonic coefficients calculated by code (I = 0, 1, ... LS-1).

MAXN0 The minimum number of intervals in the scattering kernel

for any partial interaction

MINN0 The minimum number of intervals in the scattering kernel

between the energy ranges of partial interactions.

NIS0 Number of isotopes on the cross section library tape.

NSPEC Number of energies at which source spectrum is tabulated.

If NSPEC = 0, a U-235 fission spectrum is used.

14
NPRINT Number of solution energies for which detailed printout of

solutions is desired. (For diagnostic use).

IPX Print index. IPX = 0, bypass input cross section printout;

IPX > 0, printout complete cross section input.

NED Number of energies at which solution grid changes (NED > 2).

NSIG Number of energies at which grid of total cross sections

tabulated in the program changes (NSIG ^ 2).

Card Type 3 (1615) (Only if NPRINT > 0)

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

printout is desired.

Card Type 4 (I10.2E10.4) (One card for each element in the problem).

IDENTP(I) Identification number for each element in mixture.

CONC(I) Concentration of each element in atoms /barn- cm.

Wl(I) Ratio of mass of each element to the neutron mass.

Card Type 5 (6E12.5) (Not required for NSPEC = 0)

(ESPEC(I), 1=1, NSPEC) Energies at which source spectrum is

tabulated.

(SSPEC(I), 1=1, NSPEC) Number spectrum of source.

Card Type 6 (4 (El 0.0, IIP))

ED(I) Energy at which grid of solution energies changes

(ED(1) = ET0P).

NQD(I) Number of equally spaced lethargy intervals between ED(I)

and ED(I +1).

Card Type 7 (4(E10.0, IIP))

ERATE(I) Energy at which grid of total cross section tabulation

changes

.

15
NREP(I) Number of intervals between ERATE(I) and ERATE(I-fl).

SUBROUTINE INPUTB

Card Type 8 (515)

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 4 only when neither

source nor detector response is azimuthally symmetric.

NNL Number of (n,£) combinations for which solutions are desired,

KP Input- Output index.

KP PRINT PUNCH TAPE

£=0 only -t=0 only none

1 t=0 only t=0 only all I

2 all I £=0 only all I

3 all I all I all I

Card Type 9 (413, 4(16,313))

N(I) n- index of i moment.

L(I) -t- index of i moment.

LINKH(I) i-index of higher-t linkage moment.

LINKL(I) i-index of lower-'t linkage moment.

Card Type 10 (5EI5.6)

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

d'istribution. The subroutine INPUTB then calculates the linkage coefficients and the C , values.

Vet

16
B. Input Cross Section Tape (LIB)

For the CDC-3600 and UNIVAC 1107/1108 systems, LIB is tape

unit 7 and for the IBM 360/75-91, LIB is tape unit 1. LIB must be

available for KEYIND < 2. The following is a description of the BCD

tape format for a particular isotope.

Record 1 (3X, 515, 13A4)

IDENT Isotope identification number (it corresponds to

IDENTP in the INPUTA Data).

NX Number of partial cross sections.

NTO Number of energies for which the total cross section

is tabulated.

NA Number of energies for which the absorption cross

section is tabulated.

NFL Number of energies at which angular harmonics for which

elastic scattering are tabulated.

(A1(I) , 1=1 ,13) Alphanumeric data which describes the cross section.

Record 2 (3X.16I5)

(NT(I) , 1=1 ,NX) Each entry indicates the number of energies for

which each partial cross section is tabulated.

Record 3 (3X,5E15.7)

(Q1(I) ,1=1, NX) Q value for the I partial cross section. For

continuum type secondary distributions, the

following prescription is used:

Continuum Q = 100 + A,

(n,2n) Q = 200 + A,

17
(n,n'p) Q = 300 + A,

(n,N / a),(n,n'3a) Q = 400 + A.

Here A is the parameter in Eq. (20).

Record 4 (3X.5E15.7)

(ELO(I) , 1=1 ,NX) Lowest energy for cross section tabulation of the

-rth . , I partial cross section.

(EHI(I) ,1=1 ,NX) Highest energy for cross section tabulation of the

T th . , I partial cross section.

(EIS0(I) , 1=1, NX) Energy below which the I interaction is isotropic

In the following records, the energy lists must be tabulated in descending order.

Record 5 (3X,5E15.7)

(ET(I) ,I=1,NT0) Energy list for total cross section

Record 6 (3X,5E15.7)

(ST(I) , 1=1 ,NT0) Total cross section corresponding to each energy

ET(I).

Record 7 (3X,5E15.7)

(EZ(I) , 1=1 ,NA) Energy list for the absorption cross section

Record 8 (3X,5E15.7)

(SA(I) , 1=1 ,NA) Absorption cross section corresponding to each

energy, EA(I).

Record 9 (3X,5E15.7)

(EEL(I) ,1=1 ,NT(1)) Energy list for elastic scattering.

Record 10 (3X.5E15.7)

(SEL(I) , 1=1 ,NT(1) ) Elastic scattering cross section corresponding

to each energy EEL(I).

18
Record 11 (3X, 5E15.7)

(EFL(I) , 1=1 ,NFL) Energy list for elastic scattering harmonic

coefficient.

Record 12 (3X,8F9.6)

(F1(I) ,F2(I) , . .. ,F8(I),I=1,NFL) Angular expansion coefficients,

f , for each energy EFL(I).

Each partial cross section is now tabulated and requires 2*NX-2 records,

one record for the energy list and another for the cross section list.

The format is (3X,5E15.7).

(E(I) , I=1,NT(J)) Energy list for the J partial cross section.

(S(I) , 1=1 ,NT(J)) Cross section list for the J partial cross

section.

Following these records an end of isotope record must be written; it has

the form

(3H99,75X) End of isotope record.

C. Scattering Kernel Tape

For CDC-3600 and UNIVAC 1107/1108 systems, LIBCX is tape unit 8

and for the IBM 360/75-91 it is unit 2. LIBCX must be available for all

values of KEYIND. The tape is written in BINARY mode and with the

exception of the first and second solution energies, has the following

form:

For the I solution energy, there are NEIS(I) records corresponding to each integration region calculated by EIMESH. The record is

written in the form

WRITE (LIBCX) E(I), S(I), NINTS , (EI(J) , WATES(J), (XKIN(J,M),

M = 1,LS), J = 1, NINTS)

19
where E(I) is the energy of the I solution,

S(I) is the total cross section at energy E(I),

NINTS is the number of integration points,

EI (J) is the integration energy,

WATES(J) is the Simpsons rule integration weights, and

XKIN(J,M) is the scattering kernel value at energy EI(J) and

M-l harmonic coefficient.

After all NEIS(I) integration records have been written on the tape, the

Gaussian weights are written in the form:

WRITE (LIBCX) (A1(J), A2(J), E2(J) , J = 1, LS)

The first energy solution does not require either of these records. The

second solution energy requires only the Gaussian weights and, therefore,

the first record on the tape is the Gaussian Weights for the second

solution energy.

D. Moments Solution Tape

In all versions of MOMENT- I, LSAVE is tape unit 10. This tape

must be available for KP(input card 8) values greater than zero (0). The

following describes the form of this BCD tape that would be written for

a single problem.

Record 1 (20A4)

(TITLE (I), 1=1, 20) Problem title

Record 2 (215)

NENS Number of solution energies

NNL Number of (n,£)combinations

20 
Record 3 (1613)

((N(I),L('I),I = l.NNL)

Record 4 (IP5E15.7)

(S(I),I=1,NENS)

Record 5 (IP5E15.7)

(E(I) ,I*1,NENS)

Record 6 (IP5E15.7)

(n,-L) array for the NNL combinations

Total cross sections

Solution energy list

Source strength at each energy

C for each (n,-c) combination

no

(S0RS(I),I=1,NENS)

Record 7 (IP5E15.7)

(CNL(I),I=1,NNL)

Record 8 thru NNL+7 (IP5E15.7)

(SOLN(I, J) ,I=1,NENS) The solutions for the N(J),L(J) moment

E. Discussion of Output Data

The data preceded by (IPX > 0) are printed only when the input

parameter IPX is equal to a positive integer. They are useful only when

one wants to examine the details of the input cross section 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 INDXPR. These data

are useful for trying to find out why the code is not working properly.

Printout consists of the following blocks of data:

Input data from INPUTA subroutine.

Label for each element for which data is read from the LIB

cross section tape.

(IPX >0) List of energies and total microscopic cross sections

read from LIB tape.

21
Information on partial cross sections.

Solution energies and interpolated macroscopic total

cross sections.

Tabulated energies and total macroscopic cross sections.

(IPX >0) List of energies and macroscopic partial cross

sections for each interaction.

(IPX >0) List of energies and Legendre coefficients

(t=l, 8) for elastic scatter.

Input data from INPUTB subroutine.

List of linkage coefficients.

(NPRINT >0) Parameters generated in EIMESH subroutine.

(NPRINT >0) List of energies, weights and scattering kernels

for the numerical integration term.

(NPRINT >0) Parameters generated in GAUSSN subroutine.

List of solution indices, number of partial interactions

contributing to the scattering integral, indices of the

previous solution, energies, and total cross sections.

This list is printed as LIBCX tape is being written.

List similar to the previous list, written after LIBCX tape

has been generated.

(NPRINT >0) Source terms.

(NPRINT >0) Numerical integration data.

(NPRINT >0) Gauss parameters.

(NPRINT >0) List of energies and solutions at the GAUSS energy

before final interpolation on solutions.

22
List of indices and energies printed as each solution is

generated.

Message indicating that an input energy parameter has been

reached and that the energy mesh for solutions may change

Title.

List of total flux moments for each energy and each (n,-t)

combination. Plane source.

Title.

List of scattered flux moments. Plane source.

Last data card.

23
VI. SAMPLE PROBLEM

This section contains an example of a typical printout for a sample

problem. In this problem the parameters IPX and NPRINT were set equal

to zero. The printout does not therefore include data obtained by setting these indices to some positive value.

The sample problem gives results for the penetration of neutrons

from a photo-fission source in carbon. Results for this type of source

have been discussed in References 4 and 8. Ordinarily, at least six

spatial moments would be calculated for t = harmonic coefficients in

order to construct spatial distributions of the flux. However, for

purposes of illustration, only three spatial moments are included in the

sample problem. A comparison of moments calculated for this problem

(4) with earlier results calculated by Weisbin is given in Reference 9.

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REFERENCES

1. L. V. Spencer and U. Fano, "Penetration and Diffusion of X-Rays.

Calculation of the Spatial Distributions by Polynomial Expansion,"

J. Res., Nat. Bur. Stand. 46_, 446 (1951).

2. J. Certaine, "A Solution of the Neutron Transport Equation - Introduction and Part I," NYO-3081, NDS-15C43 (1954); J. Certaine, "A

Solution of the Neutron Transport Equation - Part II," NYO-6268

(1955).

3. J. Certaine, E. D. Dufour, and G. Rabinowitz, "RENUPAK, an IBM-704

Program for Neutron Moment Calculations," NDA-2120-3 (1959).

4. C. R. Weisbin, "A New Moments Solution of the Neutron Transport

Equation," Pegram Nuclear Physics Lab., Columbia Univ., (1969).

5. R. L. Ashley and W. E. Kreger, ed. "Neutron Attenuation in Optically

Thick Shields ," ANS-SD-1, (American Nuclear Society, April 1964).

6. A. W. Miele, "Neutron and Gamma Ray Attenuation in Sodium Shields,"

NAA-SR-MEMO-12467, Atomics Int., June 30, 1967.

7. E. A. Straker, "Status of Neutron Transport in the Atmosphere,"

ORNL-TM-3065, Oak Ridge National Laboratory, July 1970.

8. C. Eisenhauer and L. V. Spencer, "Neutron Flux from a Point Isotropic

Source in Carbon Calculated by the Moments Method," NBS Report 9868,

National Bureau of Standards (unpublished).

9. G. L. Simmons, "Neutron Penetration in Infinite Media," Ph.D. Thesis,

Catholic Univ. of America (1972).

32
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34 U. S. GOVERNMENT PRINTING OFFICE: 1 972 —^81 -333 U^l)
FORM NBS-114A (1-71) 2. Gov't Accession

No.

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-725

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

A Neutron Moments Computer Code, Moment I

5. Publication Date May 1972

6. Performing Organization Code

7. AUTHOR(S)

C. Eisenhauer, G. L. Simmons, and L. V. Spencer

8. Performing Organization

10. Project/Task/ Work Unit No. 2400434, 2400432,

240Q439

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

1 1. Contrac104O3

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12. Sponsoring Organization Name and Address Office of Civil Defense, Defense Nuclear Agency, and

Office of Naval Research

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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 discuss a computer code for generating spatial-angular moments

of neutron flux in an infinite medium. The equation for the flux moments is

given and the techniques used for the solution are discussed. The structure of

the computer code and of the main subroutines is also discussed. Details of the

input and output data are given and the printout from a sample problem is included,

17. KEY WORDS (Alphabetical order, separated by semicolons)

Carbon; computer code; gauss quadrature; moments; neutron transport; shielding.

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