UNITED STATES PARTMENT OF OMMERCE

JBLICATION NBS TECHNICAL NOTE 748

An Adjoint

Gamma-Ray Moments

Computer Code

,

ADJMOM-I
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B0REAB :

: '

An Adjoint Gamma-Ray Moments

. Computer Code, ADJMOM-I

/<?7<^

C Z..

George L. Simmons

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

^/W-c^^->-^v «*' 7^2

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.

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

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued February 1973
National Bureau of Standards Technical Note 748

Nat. Bur. Stand. (U.S.), Tech. Note 748, 23 pages (Feb. 1973)

CODEN: NBTNAE

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

Price $0 .30 domestic postpaid or $0.20G.P.O. Bookstore.
Table of Contents

Page

1. Introduction and Theory 1

2. Description of Input 8

3. Input 8

4. Sample Input and Output 12

5. References 17

An Adjoint Gamma- Ray Moments Computer Code, ADJMOM-I*

** George L. Simmons

In this paper we discuss a computer code for generating

spatial-angular moments of the adjoint gamma-ray flux in an

infinite medium. The equation for the flux moments is given

and techniques used for the solution are described. Details

of the input data and a sample problem are also supplied.

Key words: Adjoint; buildup factor; dosimetry; gamma-ray

transport; moments method; shielding.

1. Introduction and Theory

This note describes a computer code, ADJMOM-I, which can be used to

calculate the moments (spatial and angular) of the adjoint gamma-ray flux

distribution in an infinite homogeneous medium. These moments can then

be used to calculate dose distributions for monoenergetic point isotropic,

plane isotropic, plane oblique, and point conical sources. The adjoint

moments method is simply the application of the moments method Ll] to

the adjoint gamma-ray transport equation. A particular feature of the

adjoint formulation is that the contribution of annihilating electron

pairs and fluorescent gamma-rays to the gamma-ray transport is quite

simple to include. Additionally the adjoint source may have delta

function angular characteristics thereby allowing the calculation of the

dose angular distribution.

The adjoint moments method for gamma-rays has been applied by

several authors C 2, 3,4, 5] but only a limited amount of results have been

Work supported by Defense Civil Preparedness Agency, Washington, D.C.

20310 and Office of Naval Research, Washington, D.C. 20390.

**Present address: Babcock and Wilcox, Lynchburg, Virginia 24505

Figures in brackets indicate the literature references at the end of

this paper.

1.
obtained. In the formulation of the adjoint equation, we begin with the

one dimensional gamma-ray transport equation for a plane monoenergetic

source located at z = with angular characteristics S(uo),

E

o

^'n -» -* r r do* -» cos9 ^ (E,z,ao) + u(E)N(E,z,oj) =J dE ' J — K(E,E' , cos©)N(E' , z ,ou ) +

E 4^

E _ 6(E-E )6(z) , f]f, n)l r + S(u)) j- + °^X dE' dQ' u (E')N(E' .z.ou

1 ) +

4tt 2tt

E

PP

2 4tt

o 6(E-E^ r + (JU k 4V^J dE ' J^' tJ- h (E')N(E', z,(I)'). (1)

E

1 ,

where the last two terms in the equation represent the contribution of

annihilating electron pairs and fluorescent gamma-rays, respectively.

Note that electron motion is not considered. In Equation (1)

-» 2 N(E,z,uu) is the photon flux (gammas/sr-sec -cm ),

2 E is the photon energy (m c units),

0) is the photon direction with <Jo°k = cos9 , when k is a unit

vector in the positive z direction.

E is the source energy,

o

u(E) is the total attenuation coefficient in Thomson units per

electron (TU/el), u(cm2

/g) = 0.400594 (Z/A) u(TU/el),

cos© = 1 - I

+ |, ,

E' is the initial energy of the gamma ray,

E is the final energy after scattering,

u (E) is the pair production cross section (TU/electron)

,

PP

2.
H,(E) is the photoelectric absorption cross section (TU/el), ph

E, is the fluorescent yield for K X-rays of energy E, ,

i EL is the minimum energy for ionizing K electrons,

K(E,E') = 0.375 E ||r + j- + 005% - 1|/(E') 3

.

In developing the equation adjoint to Equation (1), i.e. the forward

equation, we note that the forward equation may be simply written as

H N = S, where His the forward transport operator, N is the flux and S

is the source term. The adjoint equation can be represented as

t f f H N = S , where we use the dagger to indicate the adjoint analogues

to the forward equation. If these two equations are to be adjoint, then

in operation notation we require

(N, H N ) = (N , HN). (2a)

Clearly, a consequence of this relation is the requirement that

t t (N,S ) = (N ,S) . (2b)

It is this property of the adjoint flux which makes it useful in

calculating dose distributions for many types of sources. That is to

say, if we use the product of the energy deposition coefficient,

u j(E)L8], and photon energy, E, as the adjoint source, the energy

deposition distribution for the forward problem may be obtained by performing the integrals indicated in Equation (2b). Particularly if

S(E^, Z ) = 6(E- EQ> 6(Z > 4n

and
l Eu (E)6(z)

S (E,a) f z) = -^

then

E

o D(z) =

f dE' E'u^E') dQ'N(E' ,z,<o')

J ed «J , o 4tt

OS ' = J

dE' 6(E'-E

)J dQ'N (E» ,z,uj' ) (2c)

4tt

It is the property of the adjoint flux which allows the computation of

dose distributions for many monoenergetic types.

The derivation of the adjoint transport equation is well known and

will not be given here. The reader is referred to Reference [6] for a

detailed discussion of the formulation of this equation as well as for an

extensive bibliography on the subject. The equation for the adjoint flux

for a detector with angular response R(m) is

E

-cos9 ^ N (E,z,w) + ti(E)N

t (E,z,^) = j" dE'

j |^- K(E' , E, -cos9 ) X o

X N (E',z,U)') + En

ed (E)6(z)R(^) + ^ [ dQ- N

f

(l, Z ,w') +

4tt ^f^»V,J). (3)

4tt

where E is the minimum final energy attained by a gamma ray of energy

E undergoing a single Compton scattering.

Note that Equation (2c) may be obtained by applying Equation (2a)

to Equations (1) and (3), with R(m ) = S(m) = 1.
If we define adjoint moments as

N

t (E) = M^ (AJ1_Y P dz z

' n.j&.nT ; 4trEn! \2l+lJ J dQ' Y^(e,cp) V? ft, z ,-&)', (4)

£

4tt

then Equation (3) can be transformed into the following adjoint moments

equation

^k,, ra

< E> J

drT k< e '- e> v^-M^r N

i,*, m

(E,)

+

+ (£

Z -m

2 f J . ft) n-1 , £-i ,m

26

H (E) r PP

J&O U(E)

u(D. n,o,o

+ (i) k ho

^ph(E> Mill N (E. ) . Lu(Ek

)J n,o,o

v k'

(5)

where R is the spherical harmonic coefficient of the angular response

function R(uj) .

The adjoint moments code ADJMOM-I calculates solutions to equation

(5). Whereas in the forward moments solution, the solution procedure

starts at E = E and proceeds downward to some minimum solution energy

max

E . , in the adjoint moments solution, the solution procedure starts min' J ' r with E (typically 10 keV) and goes upward in energy to some maximum

energy E (typically 15 MeV) . The technique used in ADJMOM-I to solve

Equation (5) is similar to the one that is used in the neutron moments

code, MOMENT- l[7]. It involves the use of Gaussian Quadrature. We want

to obtain solutions at N energies logarithemically spaced between

5
E and E . t max min*

E . = E, < E, < E_ < ... < E_

T . ( B. = E

mm 1 2 3 N-l N max'

where we require that E. )

Ei+i

i 2E + 1 To illustrate the use of Gaussian

Quadrature, we consider Equation (5), for the m = case and an isotropic

detector

H<E)H <B) - dE'|*(E',E) P/l+|-|,) u(E)

u(E').

n»<,.*«•>

J l-1

E

i-1

= 6 6. u ,(E) +

no io ^ed dE-f-K(E-,E) P/l+H) [mh] \.l l '"> +

+ 26Z0 %£L\*I£L

£o E Lu(l).

N

f

(1) +o), 6. ^rn,o k lo E

^v(E ) r-,.fv\ in

tiCEl

.nCE^J

Nn.o<V +

+ -S?r ^E)[ (i+1)Nl-i,«i (E) +

»I-i.i'.i (,>]- (6)

t

Since the right hand side of equation (6) does not contain N -(E), we

set it equal to some known quantity R. We can then rewrite Equation (6)

in a more compact form,

R

u(E) L

6(E E) E u(E) V^E E'

; V(E')

n-n N' ,(E')dE' (7)

'i-1

The integral is evaluated by writing the term in square brackets as a

sum of two delta functions, namely

fifE'-E^ - ^- K< E '^ E

> P Cl4- —

)

6(E E) E n(E) V" '^E £<-

Mi*L

li(E'). = \/CE'-E..1

)

+

Pnx6(E, - E;,i ) (8)
Equation (7) then becomes

This procedure is somewhat different from that of Reference [7] in that

a dependence on n has been included. The three parameters a „ , P » * and E . may be determined by requiring that the first three moments of

the scattering integral be given correctly,

E-E. - i~l TB M . _. §_ _n

J ^i.jg

CE -E') J £l K< E 'i E

tJi >.p (i+L-L)piiILl d ( E.E') (10) *

; E u(E) Vi+

E E

,; l4i(E') J

a ^ * ; UU;

o Calculation of the first three moments of the functions on each side

of equation (8) yields a system of equations:

" * .J- • - I

J

„ = a .(E-E. ,)

J + B „(E-E „)J

a = n \ 2 jo n,£ njt, l-l ^n,£ K n,i/ ' J u » i

» ^ (11)

After obtaining a , B . and E Equation (9) can be solved for n , X- n , J6 iijA + it £• N , (E .). Generally E is not precisely equal to E and an interpon-,- Mr "* n , Jo n , iy lation must be performed to obtain the solution at E.

An interesting feature of Equation (5) has been pointed out by

Morris in Reference [5l for the n = i = case. If an energy deposition

coefficient is defined as

E

^(E) , n(E) - | „pp(E) - f

^(E) -

J dE> -^(E'.E), (12)

E

c

+ then N

Q Q

(E) = 1.0, for all E. This definition of M-ed

(E) is equivalent

to the Mtt-CE) defined by Hubbell [8]. By using this form for ^ed

(E) we

7
have an energy absorption coefficient which is consistent with the energy

loses that are contained in the solution of the moments equations.

2. Description of Input

ADJMOM-I is written in FORTRAN-IV and is operational on the UNIVAC1108, CDC-3600, and IBM-360/75 and 360/91. It requires 40K decimal words

and two tape drives, one of which may be a scratch tape or disk. Several

cases may be processed at one time with the output from each case being

saved on the remaining tape.

The following are the input instructions for ADJMOM-I for the

UNIVAC-1108 version. The only difference between the three versions of

the code is in the assignment of the input and output units--which may

be different at each installation.

3. Input

Card 1 (15)*

NOUTI Number of energies at which adjoint moments will be

punched out.

Card 2 (16F5.0)

(EOUTI(I), 1=1, NOUTI) Energy list for which adjoint moments

will be punched out on cards. (MeV)

Card 3 (I1.15A4)

IPROB Problem type, IPROB = 1, Complete case; IPROB = 2

? Use previous solution grid; IPROB = 3, Read new solution grid.

(TITLE(I), 1=1, 15) Title card for case.

* This" is the format for the card or cards.

8
Card 4 (815)

NMUS Number of total cross sections.

NSORS Number of energies for energy deposition coefficient.

LS Maximum angular expansion coefficient used, LS-1 = i

max

MINNO Number of integration points for Equation (11).

MAXNO Number of integration points for scattering integral.

NPXS Number of partial cross sections. Pair production and

each fluorescent X-ray are considered to be partial cross

section reactions.

KEYIND Normally = 0, KEYIND = 1 causes the calculation of adjoint

moments for the energy deposition coefficient in Equation

(12) to be bypassed.

NEDI Number of energies for specifying the solution mesh.

Card 5 (3E10.0)

ETOP Maximum energy in the calculation. (MeV)

EBOTM Minimum energy in the calculation. (MeV)

XO Coefficient for converting input cross sections to units

of TU/el. (Thomson units per electron)

Card 6 (8E10.0)

(EMUS(I), TABMU(I), I = 1,NMUS) EMUS(I) is the energy in MeV,

TABMU(I) is the corresponding total cross section.

2 TABMU usually has units of cm /g. The EMUS list must

be in descending order.

Card 7 (8E10.0)

(ESORS(I), 1=1, NSORS) Energy list for tabulation of energy

deposition coefficients (MeV).

9
Cards 8, 9, 10, 11, and 12 are required for NPXS ) 0.

Card 8 (1215)

(NPART(I), 1=1, NPXS) Number of energies for tabulating the

I-th partial cross section.

Card 9 (8E10.0)

(YIELD(I), 1=1, NPXS) Photon yield for each partial cross

section. Fluorescent yield for u , and 2 for u ph pp

Card 10 (8E10.0)

(ECUT(I), 1=1, NPXS) Lowest energy for each partial cross

section, MeV. 1.022 MeV for pair production and edge

energy for fluorescent gamma-rays.

Card 11 (8E10.0)

(ESTART(I), 1=1, NPXS) Energy for the gamma-ray produced by

I-th partial cross section interaction, MeV.

Card 12 (8E10.0) NPXS of these card sets are required. (1=1, NPXS)

(EPAKT(J,I), SIGMA (J, I), J = 1, NPART(I)) EPART is the energy

in MeV and SIGMA is the corresponding partial cross

section. SIGMA has the same units as TABMU in Card 6.

Card 13 (415)

NLO Number of harmonic coefficients of the source angular

distribution. 1 for isotropic detector.

LZRO Index of source harmonic when NLO = 1. Otherwise

arbitrary.

MZRO Index of azimuthal harmonic. Usually MZRO = 0.

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

to be obtained.

10
Card 14 (413,4(16,313))

(N(I), L(I),LINKH(I),LINKL(I), I = 1 , NNL) where

N(I) n-index for the I-th moment.

L(I) ^-index for the I-th moment.

LINKH(I) I-index for the higher I linkage moment.

LINKL(I) I-index for the lower I linkage moment.

Card 15 (8E10.0)

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

distribution. For an isotropic source, CNL(l) = 1.0.

Card 16 (4(E10. 0,110))

(ED(I),NED(I), I = 1.NED1)

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

NED(I) Number of equally spaced intervals (logarithmically)

between ED(I) and ED(I+1).

Cards 1 thru 15 are required for the calculation of adjoint moments

for the energy deposition coefficient given by Equation (12). The

following cards are required in order to do calculations for additional

energy deposition coefficients.

Card 17 (15A4)

(TITLE(I), I = 1,15) Title card for additional calculation.

Card 18 (I5.E10.0)

NSXXX Number of cross sections to be read. If NSXXX < 0, then

card 19 must be supplied. Otherwise the cross sections

for energy deposition will be assumed to be tabulated at

the energy grid of the previous case.

DEN Changes cross sections to units of TU/el.

11
Card 19 (8E10.0) Not required for NSXXX > 0.

(ESORS(I), 1=1, ABS (NSXXX)) Energly list for tabulation of

energy deposition cross section, MeV.

Card 20 (8E10.0)

(STSORS(I), 1=1, ABS(NSXXX)) Energy deposition cross section

corresponding to the energy list, ESORS.

If NSXXX = 0, then the code assumes that the next card is a new

problem card, i.e. Card 3. When the calculation is completed with the

energy deposition coefficients given on Card 20, then the code expects

a title card (Card 15) and continues on with Card 16, etc. The code will

terminate when it encounters an end of file or the end of a data set,

whichever is appropriate for the machine being used.

4. Sample Input and Output

Input data for the calculation of adjoint moments for concrete

using Kj. for air taken from Reference [8],

CARD COLUMNS

Card iiniiilii2222222222333333333344444444445555555555666666666677777777778

Type 123456789012345678901234567890123456789012345678901234567890123456789OI23456789O

1 12

2 1.0 0.80 .662 0.60 0.40 .279 0.20 0.10 0.08 0.06 0.04 0.02

3 1 ADJOINT TEST PROBLEM—ALUMINUM- -SKIP CONSISTENT PORTION

1+ 17 17 4 40 40 3

5 1.0 0.01 5-181057

6 1.0 0.06l4 0.8 O.0683 0.6

6 0.4 0.0922 0.3 0.1030 0.2

6 0.1 0.162 0.08 0.189 0.06

6 0.04 0.514 0.03 1.03 0.02

6 0.01 25.8

7 1.0 0.8 0.6 0.5 0.4

7 0.1 0.08 0.06 0.05 0.04

7 0.01

13 1 10

14 0000 1101 2020

14 4050 3304 4275

15 1.00

16 1.0 45 0.10 70 0.01

17 ALUMINUM WITH MU-K (AIR) DETECTOR AS ADJOINT SOURCE

18 17 5.0012

20 0.02797 0.02890 0.02958 0.02971 0.02952 O.O2876 0.02677 0.02502

20 0.02338 0.02427 0.03053 0.04062 0.06689 0.1480 0.51200 1.27100

20 4.631

12.

0.0777 0.5 0.0841 0.0120 0.15 0.1340

0.255 0.05 0.334

3.24 0.015 7.66

0.3 0.2 0.15 0.03 0.02 0.015

2 3 1 4 5 , 8 6 6 9
IRROB= 1

JS NSORS LS

17 17 It 1.00000

ADJOINT TEST PROBLEM—ALUMINUM—5KID CONSISTFNT °ORTTOM

>I0 MAXNO N=XS KETTNO NE01

TOTAL CROSS SECTIONS

.100+01 .611-01

.300+00 .103+00

.600-01 .255+00

.150-01 .766+01

ENERGY-- SIGUA .800+00 .200+00

.500-01

.100-01 TOTAL CROSS SECTIONS ENER3Y--SI

.196+01 .318+01 .157+01

.517+00 .531+00 .391+00

.117+00 .132+01 .978-01

.291-01 .397+02 .196-01 ENERGIES FOR ENTERING DO E DATA .100+01 .800+00 . 00+00 .600-01 .500-01 .100-01

.683-01 .600+00 .777-01 500+00 .oil-Ol .111+01 .922-11 .120+00 .150+00 .131+01 100+11 .162+01 .811-11 .189+00 .331+00 .100-01 .511+10 300-01 .113+11 .210-01 .321+01 .258+02

GWA .351+00 .117+01 ,103+00 978+10 .136+01 .783+01 .178+00

.622+00 .291+00 .691+00 196+10 .039+01 .157+01 .979+10

.173+01 .783-01 .266+01 587-11 .631+01 .391-11 .168+02

.131+03

.500+00 .100+00 .300+10 200+10 .151+11 , 1 10+00 .800-01

.300-0! .200-01 .150-11 100-01

00000 .oooin .11101+01

00000 .33333+00 .33333+00

10000+01 .00000 .33333+00

00000 .10000+00 .13333+00 66667+00 .33333+00 .20101+00

10000+01 .10001 .20100+00

00000 .12857+00 .57113-01 60000+00 .10000+00 .11129+00 66667+00 .33333+09 .11286+00

10000+01 .11010 .11286+00

.1000+01 15 .1000+00 70 .1000-01

I ENERGY(I) TOTAL! ) «U-ED(I> I ENERGY(T) TOTAL! I) wu-enm I ENERGY! I) TnTALdl MU-En(I)

(MC»«2) (TU/EL (TU/EL 1 (WC**2) (TU/EL) (TU/EL ) (MC*«2> (TU/EL! (TU/EL ! 1 1.956926 .3181 7 .110335 2 1.859311 .325981 .19f93 3 1 .766568 .331111 .142456

1 1.678117 .312298 .113523 5 1.594729 .350761 .194612 6 1.515178 .350091 146372

7 1.139597 .367122 .115910 3 1.367788 .375946 .196507 Q 1.200561 389650 .147178

10 1.231737 .393592 .117653 11 1.173146 .102718 .118'17 12 1.119628 .111761 ,148663

13 1.059028 .121011 .113890 11 1.006203 .130171 .119235 16 .956012 .130015 ,14937b

16 .908321 .119289 .119261 17 .363116 .158863 .199166 IB .910967 .453642 ,1 49191

19 .779066 .178567 .118936 20 .740215 .138089 . 118^01 21 .703283 .497891 .147642

22 .663202 .5077 5 .117009 23 .631371 .517807 .116415 24 .613203 .529111 . 145072

25 .573111 .5385 1 .115281 25 .511526 .518993 .111635 27 .817369 .550670 144066

28 .191557 .5705 1 .113581 29 .167038 .581680 .113'05 39 .143741 .503013 , 1 42Q41

31 .121607 .601516 .112811 32 .100576 .616313 .112333 33 .331595 .628431 .143138

31 .361610 .610836 .113729 35 .313573 .653589 .111530 36 .3'6436 .666641 116660

37 .310152 .679755 .116832 38 .291681 .693228 .118371 39 .270982 .700797 16302U

«0 .266016 .726999 .158312 91 .252717 .791618 .161125 42 .240130 .762665 170395

13 .228161 .781119 .177179 11 .216780 .800080 .104197 45 .295966 .819471 192370

»S .195693 .839331 .200816 17 .189360 .858622 .212081 49 .133233 .873357 .225415

19 .177303 .398515 .238162 50 .171566 .919197 .252036 51 .166014 .940324 .265151 52 .160612 .961936 .231813 53 .155111 .986501 .298512 54 .161414 1.020371 .326082

55 .115517 1.056139 .351926 56 .110837 1.093295 .385190 67 .136231 1.131335 .416616

58 .131870 1.170751 .119550 59 .127603 1.211590 .433Q 45 68 .123471 1.253751 519345

61 .119178 1.297132 .557305 62 .115612 1.351792 .605c 32 63 .111871 1.419241 ,686041

61 .108251 1.190055 .731815 65 .101718 1.561102 .810'99 66 .191350 1.642450 872694

67 .098079 1.721111 .918830 69 .099905 1.035581 1.054479 69 .001834 1.956012 1 169436

70 .083862 2.081315 1.292397 71 .085097 2.221097 1.423165 72 .083204 2.366322 1 664416

73 .080512 2.522108 1.711607 71 .077917 2.693769 I.901274 76 .076336 2.9165Q6 2 fl90?24 76 .072916 3.157862 2.335696 77 .170506 3.119986 2.592'36 70 .060302 3.791919 2 07H469

79 .166091 1.008117 3.172193 80 .063953 9.339717 3.409-<29 01 .081833 4.699895 3 064011

82 .059881 5.137379 1.238172 83 .057943 5.537917 4.681°91 84 .056060 6. 077405 5 220427

85 .051251 6.669623 5.303616 n6 .052408 7.319955 6.151603 37 .050801 8.032691 7 16390^

88 .019156 8.815229 7.91,2955 09 .147565 9.679111 3.798', 07 91 .146026 10.616675 9 737305

91 .111537 11.651073 10.768296 92 .143095 12.786255 11. 900'05 93 .141701 14.132141 .13 142680

91 .010351 15.399203 11.506623 95 .139046 16.906170 lfi.010616 96 .037782 18.654141 17 753420

97 .036560 20.532615 19.681100 93 .135377 22.710578 21.806137 00 .031232 25.158549 21 .161394

100 .033121 27.619263 26.739176 101 .032052 30.507823 29.595'75 192 .0^1015 33.661923 32 746Q11

103 .030012 37.112109 36.221615 101 .129140 90. "83836 90.063035 195 .920101 15.227101 44 3047OT

106 .027191 19.909615 IB. 985021 107 .126311 55.176927 51.150111 193 .026160 60.779237 59 851277

109 .021636 67. 071921 66.110837 110 .123839 79.116123 73.092060 111 .123060 01.679272 81 .744316

112 .022321 90.135813 89.198835 113 .021509 99.967899 98.528''31 ll" .021900 109.76615O 100 .026839 115 .020221 121.130632 120.385306 116 .019569 133.671711 133.6717H

13
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14
ADJOINT SOLUTIONS FOR ALL ENERGIES — L=0 viOMEMTS ONLY ALUMINUM WITH MU-K (AIR) DETECTOR AS ADJOINT SOURCE

N= L=

1 9.3015932-01 2

7 9.0621759-01 8 13 8.7326291-01 11

19 8.2825965-01 20 25 7.6920976-01 26

31 6.8800371-01 32 37 5.8631907-01 38

13 1.6661925-01 11

19 3.6836711-01 50 55 2.979930B-01 56

61 2.1307302-01 62

67 2.0118153-01 68

73 1.8066150-01 71

79 1.6779026-01 80 85 1.6210001-01 86

91 1.6013581-01 92

97 1.6139138-01 98

103 1.6362187-01 101

109 1.6801283-01 110 115 1.7322998-01 116

N= 2 L=

1 7.1511280-01 2

7 7.8226851-01 8 13 8.0619899-01 11

19 S. 1001150-01 20 25 7.7815758-01 26

31 7.0031613-01 32

37 5.7556128-01 38 13 1.2133286-01 11 19 2.9205906-01 50 55 2.0716683-01 56

61 1.1661176-01 62 67 1.0531106-01 68

73 8.1107879-02 71

79 6.7853761-02 80 85 6.0706110-02 86

91 5.7192032-02 92

97 5.5989278-02 98 103 5.5797118-02 101

109 5.6713989-02 110 115 5.8160866-02 116

N= 1 L=

1 7.8117711-01 2 7 8.6319231-01 8 13 9.2925972-01 11

19 9.6705010-01 20 25 9.1361890-01 26

31 8.1533522-01 32

37 6.7019210-01 38

"3 1.6082703-01 11 19 2.9011053-01 50 55 1.8825136-01 56

61 1.2215691-01 62

67 7.9860761-02 68

73 5.7132292-02 71

79 1.5117333-02 80 85 3.8713195-02 86

91 3.5608851-02 92

97 3.1326097-02 98 103 3.3897887-02 101 109 3.1291025-02 110 115 3.5015581-02 116

N= 6 L=

1 B. 5078599-01 2 7 9.7265792-01 8 13 1.0837852+00 11

19 1.1620786+00 20 25 1.1183911+00 26

31 1.0217316+00 32

37 7.8837130-01 38

13 5.1611796-01 11

19 3.0118916-01 50 55 1.8155112-01 56

61 1.1019971-01 62

67 6.7253970-02 68

73 1.5985371-02 71

79 3.1601128-02 80 85 2.8875010-02 86

91 2.6090209-02 92

97 2.1885036-02 98

103 2.1121697-02 101

109 2.1615391-02 110 115 2.5107189-02 116

9.2662128-01

9.0155029-01

8.6652711-01

8.1966195-01 7.5685067-01

6.7273276-01

5.6782015-01

1.1691851-01

3.5599018-01

2.8716973-01

2.3538126-01

1.9905290-01 1.7803038-01

1.6666125-01

1.6137798-01 1.6026071-01

1.6162583-01

1.6120061-01

1.6889862-01 1.7326508-01

7.5201155-01 7.8730936-01

8.0888111-01

8.0711383-01

7.6836896-01

6.8276173-01

5.5037665-01 3.9193683-01 2.7670228-01

1.9563831-01 1.3812606-01

1.0022738-01 7.8100751-02

6-5153381-02

5.9352615-02

5.6906215-02

5.5870483-02 5.5880116-02

5.6966121-02

5.7755027-02

7.9769676-01

8.7530300-01

9.3853977-01 9.6738237-01

9.3255955-01

8.2122833-01

6.3352362-01 1.2195739-01

2.7116978-01

1.7570071-01

1.1292857-01 7.1978909-02

5.1181301-02

1.386921*5-02 3.8006553-02 3.5323156-02

3.1188853-02 3.3911326-02

3.1103613-02

3.1653015-02

8.7016282-01

9.9152789-01 1.1010752+00

1.1656690+00

1.1355888+00

9.9266165-01

7.3771321-01

1.6975608-01

2.7857871-01

1.6801761-01 1.0053879-01 6.2522288-02 1.3100290-02

3.3178761-02

2.8227151-02

2.5827206-02 2.1753611-02

2.1113031-02

2.1685191-02

2.1752153-02

39 15

212733

39

15

5157

63

69

75

8187

9399 105

111

15

2127

33

39

15

5157

6369

75

3137

93

99 105

111

3

9 15

2127

33

39

15

5157

63

69

75

3187

9399

105

111

3

9 152127

33

39

15

5157

6369

75

31"793

99 105

111

9.2293172-01

8.9661017-01

8.5950755-01 8.1061310-01

7.1103692-01 6.5619871-01

5.4743611-01

1.2736286-01 3.1101125-01 2.7753541-01

2.2802336-01

1.9447211-01

1.7522279-01 1.6574872-01

1.6033610-01 1.6048258-01

1.6191998-01

1.6190311-01 1.6977903-01

7.5348556-01 7.9203523-01

3.1060601-01 3.0386033-01

7.5733601-01 6.6377132-01

5.2193253-01

3.6811559-01 2.6168322-01

1.3178352-01

1.3051939-01 9.5699990-02

7.5679086-02

5.5191379-0? 5.9123158-02

5.6682172-02

5.5790959-02

5.6015196-02

5.7202513-02

3.1111562-01 3.3695801-01

9.1665216-01

9.6611717-01 9.1940667-01 7.9505791-01

5.9950205-01 3.3823507-01 2.5277147-01

1.6393786-01

1.0535482-01 7.0739622-0?

5.?065?97-02

4.2737628-0?

3.7369923-0?

3.5086164-02

3.4081511-0?

3.3959136-0?

3.1526252-02

3.9079051-01 1.0105681+00

1.1166165+00 1.1671029+00 1.1197609+00

9.5775081-01

6.9395422-01

4.2199510-01 2.5615146-01

1.5546172-01

q. 2947872-0? 5.3439340-0?

4. 0894215-0?

3.2464064-0? 2.7666001-02

2.5601671-02 2.1616881-0?

2.1130715-0?

2.1763622-0?

4 1016

22

2831

101652

58

61

7076

3288

91 100 106

112

110162?

28

31

1016

5258

61

7076

8288

91 100 106

112

11016

2328

31

1016

5258

61

7076

82

88

94 100 106

112

10

162?

2331

4016

5258

61

7076

823891

100 106

112

9. 1906197-01 8.9110619-01 8.522H27-01

8.0115516-01 7.3076150-01

6.3918332-01

5.2699809-01

1.0800022-01

3.3211116-01 2.6816237-01

2.2128377-01

1.9039030-01

1.7281220-01

1.6503609-01

1.6015597-01

1.6078°30-01

1 .6227124-01

1 .6561152-01 1.7068269-01

7.6185267-01

7.9641917-01

8.1148442-01 7.9920131-01

7.4504729-01

6.4350322-01 1.9929637-01

3.1082912-01

2.1693332-01

1.7454041-01

1.2358292-01 9.1626836-02

7.3307776-02

6.4033352-02

5.8503679-02

5.6507207-02

5.5716936-02

5.6170540-02 5.7451311-02

3.2474045-01 3.9843682-01

9.5303228-01 9.6326225-01

9.0116208-01 7.6697696-01

5.6512703-01

3.5031155-01 2.3417758-01

1.5285634-01 9.8419238-0?

6.6956084-02

4.9961131-02

1.1668026-02

3.6821351-02

3.1886397-02

3.10011 q6-02 3.1022215-02 3.1658130-02

9.1181719-01

1.0297921+00

1.1293379+00 1.1662617+00

1. 1008500+00 9.2018635-01

6.5011932-01

3.7233151-01

2.3363311-01

1.1363395-01 8.6036202-02

5.4920359-02 3.8982126-0?

3.14930?9-0?

?. 7179939-02 2.5413372-02 2.4562353-02

?. 4460312-02 2.4849194-02

^ 9.1500361-01 6

11 3.3591110-01 1? 17 3. 4458630-01 13 2^ 7.9117033-01 24 29 7.l7no783-01 30 35 6.2212105-01 36

41 5.0669771-01 12 47 3.9436406-01 1» 53 3.2094283-01 54 59 2.5931736-01 60 65 2.1509488-01 66

71 1.8675317-01 7? y l.70«3796-01 78

8"^ 1.6117282-ni 84 89 1.6022175-01 90 95 1.6111118-01 96

101 1.6267162-01 10?

107 1.6641227-01 103

113 1 .7160795-01 114

5 7.7110826-01 6

11 3.0045241-01 1?

17 8. 1170019-01 18 23 7.9342929-01 24

2

n 7.3147155-01 30 35 6.2206493-01 36

11 4.7346782-01 42

47 3.2408021-01 48

53 2.3252011-01 54 59 1. 64 n 3?71-01 60 65 1 .1713192-01 66

71 3.7930139-02 72 77 7.1238601-0? 7"

8"^ 6.2853771-02 34 89 6. 79O2136-02 9n 95 5.6361551-02 96

101 5.5735133-0? 102 107 5.6311751-02 108

113 5.7711333-02 111

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15
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16
The author would like to thank C. Eisenhauer for many helpful discussions concerning this work.

5. References

[l] J. Res. NBS 46, 6 (1951) RP2213.

L2J E. B. Breshenkova and V. V. Orlov, Atomnaya Energiya, 10 , 173,

(Feb. 1961).

[3] F. W. Krueger, Kernenergie, 13, 47 (1970).

[4] G. L. Simmons, Unpublished Project Report, Catholic University of

America, Nuclear Science and Engineering Dept., (April 1970).

[5] E. E. Morris and A. B. Chilton, Trans. Am. NucL Soc. I, 405 (1970).

[6J J. Lewins, "Importance, the Adjoint Function ," (Pergamon Press,

Oxford, 1965).

[7] Nat. Bur. Stand. (U.S.), Tech. Note 725, 34 pages (May 1972).

[8] Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. (U.S.), 29, 80 pages

(August .1969).

17

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1. PUBLICATION OR REPORT NO. NBS Tech. Note 748

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

An Adjoint Gamma- Ray Moments Computer Code, ADJMOM-I

5. Publication Date February 1973

6. Performing Organization Code

7. AUTHOR(S)

George L. Simmons

8. Performing Organization

10. Project/Task/ Work Unit No. 2400434

7.400439

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NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

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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 the adjoint gamma-ray flux in an infinite medium. The equation for the flux

moments is given and techniques used for the solution are described. Details

of the input data and a sample problem are also supplied.

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

Adjoint; buildup factor; dosimetry; gamma-ray transport; moment methods; shielding.

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