A UNITED STATES DEPARTMENT OF COMMERCE

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OCT 2 8 «9

NBS TECHNICAL NOTE 499

Multiple Scattering Corrections

for the Associated-Particle

Neutron Time-of-Flight Technique

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NATIONAL BUREAU OF STANDARDS

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UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

NBS TECHNICAL NOTE 499

ISSUED OCTOBER 1969

Nat. Bur. Stand. (U.S.), Tech. Note 499, 68 pages (Oct. 1969)

CODEN: NBTNA

Multiple Scattering Corrections for the Associated-Particle

Neutron Time-of-Flight Technique

Allan C. B. Richardson*

Nuclear Radiation Division

Center for Radiation Research

National Bureau of Standards

Washington, D.C. 20234

*The author's present address is: Office of Grants Administration, Environmental Control Administration, Department of Health, Education, and

Welfare, Washington, D.C, 20203.

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.

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

(Order by SD Catalog No. CI 3.46:499), Price 65 cents.
Abstract

The computer code, MAGGIE, for the calculation of multiple scattering and sample attenuation in neutron differential cross-section measurements, has been revised and corrected. The particular

case of the scattering geometry required by the associted-particle time-of-f light is considered in detail.

Key words: associated-particle, cross-section,

Monte Carlo, multiple scattering,

neutron, tirae-of -flight
MULTIPLE SCATTERING CORRECTIONS FOR THE ASSOCIATED- PARTICLE

NEUTRON TIME-OF-FLIGHT TECHNIQUE

Allan C. B. Richardson

I. INTRODUCTION

Measurements of fast neutron elastic and inelastic differential cross sections have, for many years now, usually been done using one of two tiraeof-flight techniques. The first of these requires a pulsed source of

neutrons, and energy separation of the various neutron groups is then

achieved by time correlation of the scattered neutrons with the incident

neutron pulse. This technique has the advantage of flexibility of incident

neutron energy and intensity, and the disadvantages of a relatively high

time-correlated background and a low duty cycle. The other, v and less

commonly used, technique utilizes the detection of a charged particle from

the neutron source reaction to £ag the incident neutrons in time and direction. Energy separation of the various scattered neutron groups is then

achieved by time correlation with the incident neutrons. This "associatedparticle technique" has the advantages of very small time-correlated background, inherent absolute determination of the incident correlated neutron

flux, and high duty cycle; but it suffers from limitations on the available

neutron intensity and energy. However, at energies and intensities where

this technique is applicable, it is the method of choice, since it is ca- pable of yielding results of high accuracy without the ambiguities introduced by the time-correlated backgrounds and the massive shielding required

by pulsed source techniques. The source reactions eligible for this method

are those involving very light nuclei, and therefore capable of producing

a light (and thus energetic) stable recoil nucleus. The T(d,n) He reaction,

producing 14-15 MeV neutrons at 90° over a rather wide range of incident

particle energies, is most commonly used; other reactions that have been

employed are D(d,n) 3 He and T(p,n) 3 He, both of which produce lower energy

neutrons. We will confine the discussion here to the T(d,n) 4 He case,

although the method described is more generally applicable.

The associated-particle technique has scattering sample requirements

that are quite different from those for a pulsed source. Instead of a relatively uniform incident neutron flux across the sample, the correlated

neutron beam is highly directional. The angular distribution about the

neutron beam line is usually well approximated by

I = I e

9 o

o

with 9 typically only a few degrees [lj. This property can be very useful [2]. It provides a high degree of neutron collimation without the

need for massive collimators, which, in the case of 14 MeV neutrons, can

produce substantial degradation of the initially monoenergetic neutron
beam. However, it forces the use of scattering samples of uniform thickness so that the cylindrical or spherical samples commonly used to simplify

multiple scattering corrections are immediately ruled out. Otherwise a detailed knowledge of the neutron beam shape and extremely accurate alignment of this beam with respect to the scattering sample is needed. This

alignment problem is further complicated because the center of the neutron

beam slowly moves back as the neutron producing target ages with use. In

measuring angular distributions of scattered neutrons it is of course also

desirable to reduce the amount of scattering sample not directly in the

neutron beam, so as to minimize multiple scattering.

A scattering sample in the form of a truncated cone, axis lying along

the neutron beam, best satisfies all of these requirements. The origin

of this cone is taken sufficiently far behind the source to allow for

finite spot size on the neutron producing target, to provide some flexibility in alignment, and also to make allowance for changes due to target

aging during a run. In order to make best use of the available neutron

intensity, one must also use the thickest sample possible. The limit is

set by either the angular resolution required at 90° or the time resolution required. A typical geometry is shown in figure 1.

In either case the resulting samples are sufficiently thick to require

a careful multiple scattering correction. None of the analytical techniques [3], useful at energies up to a few MeV, are adequate at 14 MeV, the

energy most commonly used for measurements of this type, since at energies

above 6-7 MeV the diffraction peaks in the elastic angular distributions

become too numerous. The only method of sufficient generality is the

Monte Carlo technique. A survey of existing Monte Carlo codes revealed

none for this particular geometry, but it was immediately apparent that

the code "MAGGIE," developed by Parker, Towle, Saras, et al. at Alderraaston [4,5] contained all of the other elements important to such a calculation. For example, this code easily accommodates the neutron source

distribution specified above. In addition, MAGGIE calculates an energy

spectrum at each angle, so that false peaks due to double scattering can easily be identified. This sophistication is often useful, for example:

the measurement of inelastic scattering from the very weakly excited

7.66 MeV level in 12 C is easily confused by double scattering from the

more easily excited 4.43 MeV 2 + level in this nucleus.

A corollary need of any Monte Carlo neutronic calculation is an easily

accessible, but at the same time sufficiently general, file program for

the nuclear data required. This was available in a companion code to

MAGGIE, entitled MOULD [6J. We have therefore modified the code MAGGIE

so that it now handles the geometry required for associated-particle timeof-flight measurements. During the course of this modification we also

corrected a few coding errors found in the original version, and made

several additional modifications, principally updating the code to current computer syntax and capabilities.
d-*-

a DETECTOR

1

OS

-

: J

SCATTERING

SAMPLE

^-TRITIUM-TITANIUM

TARGET

-

!

.

3

TO NEUTRON

DETECTOR

Fig. 1. The experimental geometry for associated-particle measurements

at 14 MeV. The associated neutron intensity profile is indicated at the bottom of the figure.

3
Although a complete description of the code would be out of place

here (see references 4 and 5 for details), a general overview of the program will facilitate later discussion of specific details. The point of

view adopted is first to sample the scattering sample geometry and the

available nuclear interactions using straightforward Monte Carlo techniques. Then, at each collision point, the weighted probability of scattering and escape to each of 33 detector positions (angles) in a half-plane

lying to one side of the neutron beam axis is computed using the experimental data for elastic events and each of those inelastic processes of

interest. The use of weighted probabilities of scattering to each of the

detector positions at the final collision in the sample in place of a completely Monte Carlo approach results in a greatly reduced computation

time. This is because the small solid angle subtended by each of the

detectors (typically 10"3

) makes a final Monte Carlo scattering particularly inefficient. The output angular distributions obtained from applying this procedure to, typically, 1000 interacting neutrons are then re- flected about the experimental input data and used as input for a second

iteration. Two or three iterations are usually sufficient to obtain a convergent result.

In section II we give a description of the changes made in the code.

First, the new coding for the scattering geometry used in the associated

particle technique is described. Next we discuss a number of coding errors in the original version. Finally some changes are described that

simplify the code and adopt it to FORTRAN V syntax. In section III we give some results obtained using data from the scattering of 14 MeV neu- trons on natural carbon. A listing of those subroutines entirely re- written or having extensive changes is given in Appendix A. In figure 2

the calling sequence for all of the components of the program is shown.

A brief description of all of these subroutines appears in Appendix B.

II. DESCRIPTION OF MODIFICATIONS TO THE MONTE CARLO CODE MAGGIE

A. Changes Due to the New Scattering Sample Shape

Four subroutines are affected by changing the shape of the scattering sample. These are: 1) subroutine INPUT - those sections where

scatterer parameters are read in and the flux attenuation factor is calculated are affected. The flux attenuation factor is defined as the ratio

of incident flux along the axis of the sample to average flux in the entire

sample. 2) CRNEU, the subroutine that creates incident neutrons at the

entrance face of the scatterer by random sampling of the incident neutron

spatial distribution. 3) TRACK, the tracking subroutine. 4) FPATH, the

subroutine for calculating the probability of neutron escape from the

sample in the direction of each of the assumed detector positions, for

each collision point arrived at in TRACK.

It is worth noting here, although not necessary to what follows, that

the data used for a) the neutron track lengths in the Monte Carlo sampling

of the sample shape from subroutine EGMV, and b) the Monte Carlo sampling

of reaction type and angular distribution at each collision by subroutine

4
QZ \ o1-3<

ino

_l

a.<

(-3

Q1-3O

CO

h-3O

1-

CO»-

CC*

10CO3O

1-

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1-

— _/ z

Ul - o

'*•

X

Ul z<<z * X<

a.u

X

Ul

Ul UJoo< o

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ccu

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Ulzo

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ccz

-*>

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cc

^—

W

v >z

(9Ul <oz<

cc

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Ulz

cco r

y>

i-3

a.z

<— { 1-

cco

u.

cc z CO <Q o _ z>- w en

CD<

** <

_l

a.iS zoQZ<

CC

w

l-l os

cu -aoamocoH

COU

CD>

4-1 C

CD

CO

a)

!-i a

a) x:

j->

S-J O

4-4 £CJ

cfl

CD2

•H

CJ

CD H

CM

En
CR, as well as c) the transmission probabilities in the direction of the

detectors used in subroutines FPATH and NAPAN, are all drawn from the

MOULD nuclear data file tape and not from input experimental data. The

only experimental data employed are angular distributions used for the

calculation of the scores for the relative probability of scattering to

the various assumed detector positions in subroutine NAPAN. The elastic

angular distribution (for the incident neutron energy only) used here is

also automatically updated after each iteration by the subroutine AUTO.

Thus it may be necessary to update the angular distributions on the MOULD

file tape before and during the course of data-processing.

We now describe the changes due to the new scattering geometry in some detail for each of the subroutines affected.

1. INPUT

a. Cards 0207-0226 are changed to eliminate parameters required for

samples consisting of concentric cylindrical shells, with common axes perpendicular to the beam axis, and to substitute those needed to characterize a truncated cone sample, axis lying along the neutron beam, and

origin behind the neutron source. The new variables are HITE, FRAD, and

ANGLE; the height, entrance face radius, and half -angle of the sample,

respectively. HITE is immediately redefined as HITE/2, a more convenient

quantity in subsequent calculations.

b. Card 0245, the calculation of the maximum angle subtended by the

scattering sample at the neutron source, is changed to conform to the new geometry

c. Cards 0301-0303 have been replaced. The correct expression for

the flux attenuation factor for the new geometry is

F K(o) C h(r1 2 +r 1 Ar+Ar3 /3)

o ..11

NX (1-cose ) d

s m

where h is the half-length of the scattering sample, r, the radius of its

entrance face, Ar the difference in radii of the entrance and exit faces,

and d the distance from the source to the entrance face of the sample.

The remaining symbols are defined as in the original.

d. The original version of MAGGIE utilized a sample which was not

symmetric about the axis of the incident neutron flux. Thus the experimental sampling of scattered neutrons in the detector plane was not truly

representative of the scattering into 4tt , which is employed in the program to calculate the flux attenuation factors and which are used in turn

to infer the integrated cross-sections. This necessitated a small correction which was calculated with the help of a classification of the outgoing

6
Monte Carlo tracks vs. energy and the angle with respect to the scattering sample axis. Since the present scattering sample is symmetric about

the incident neutron flux the experimental sampling of scattered neutrons

in the detector plane is representative, and no correction is required.

Accordingly, cards 0331-0338 in INPUT, cards 1544-1548 in DATIN, and cards

1735-1743 in MAGGIE are deleted. Card 1734 of MAGGIE is replaced by the

statement

20 CONTINUE

The output of this table, by cards 3695-3715 of subroutine 0UTB3, is also

deleted.

2. CRNEU

Cards 3117-3119, 3140-3141, 3143-3147 and 3151 are replaced as shown

in the listing. The new coding creates neutrons randomly scattered over

the face of the conical scattering sample, in accordance with the specified spatial distribution. The old reference to multiple materials is not

deleted, but the sample is now designated "material one." Similarly, the

register containing neutrons which miss the sample is retained, although

for this geometry misses occur with very low frequency.

3. TRACK

The entire subroutine has been replaced. The geometry is illustrated in figure 3. The subroutine is entered with the starting point

(x ,y ,z ) and the direction cosines (cos9 , cos9 » cos9 ) already defined.

A random sampling of the neutron mean free path establishes the track

length to the next possible collision. The track vector is then extended

until it intersects the plane determined by the endface of the sample in

the direction of travel of the neutron, and the path length from the starting point to this intersection is computed. A comparison of this intersection point with the radius of the endface establishes whether the track

is in the direction of the endface or the curved surface of the sample,

and the program branches accordingly. If the track passes through the

endface, the path length calculated above is compared with the Monte Carlo

track length to the next collision and the coordinates of collision or escape from the surface of the sample as well as the time elapsed along

the track are computed in a straight-forward way.

We consider now the procedure used for a neutron headed toward a curved surface of the scattering sample. The coordinates of the endpoint of the previously determined Monte Carlo track length are first

found. These are used to calculate the projected distance, perpendicular

to the symmetry axis, of this endpoint from the symmetry axis, as well as

the length of the radius of the sample lying along this projection. Compar*

ison of these quantities determines whether a collision occurs within the

sample or not. If so, the coordinates are already known and the time

elapsed along the track is all that remains to be calculated. If not,

it is necessary to calculate the coordinates of the neutron's exit from
Fig, 3. The geometry for subroutine TRACK . The trajectory shown is that

of a neutron which, after collision at point x , yQ

, zq

escapes

from the curved surface of the sample at point x, y, z
the surface of the cone. Designate I the path length to the surface and

x, y, z the coordinates of the intersection of the track with the surface,

We have, referring to figure 3,

x - x = £cos9

o x

y - yJ = icos9

o y

z - z = JlcosQ , o z

so that the projected distance, r, from the symmetry axis of the cone to

the point (x, y, z) is given by

r

2 = y

2 + z2

= f sin2

6 + 2je(y cose + z cos9 ) + y + z , (1) x o y o z o o

where we have used the identity

cos

2

e + cos

2

e + cos

2

e = 1 . x y z

Since we know the parameters of the cone we can also calculate r from the

x coordinate (again see figure 3.):

r = r,+(h+x) tan9 , (2)

1 o

where 6 is the half-angle of the cone. Eliminating r from eq"ns. (1)

and (2) and arranging the terms as a quadratic in j>, we obtain

i

2

| l-cos2

9 (l+tan2

e Q

) I

+ 2l\y cose +z cose -cos9 (r.tanG +h+x )tan2

e ) I [_ o y° z xl O O OJ

+ ly +z - '-(r +(h+x i?L - u)tan9 " ^) i

' = 0. (3)

\_ J o o 1 o o J

The positive root of this equation is the required path length to the

surface, and the subroutine FONEC is then called to provide the x, y, and

9
BACKWARD

CALCULATE THE

DISTANCE TO THE

PLANE OF THE

ENTRANCE FACE ALONG THE TRACK

VECTOR, + ALSO

Y,2, R * ( Y* + Z* J 17* AT THE INTERSECTION WITH THIS PLANE.

CALCULATE THE

DISTANCE TO THE PLANE OF THE SAMPLE EXIT FACE ALONG THE TRACK

VECTOR + ALSO

Y, 2, R = (Y*+ Z*)"2 AT THE INTERSECTION

WITH THIS PLANE.

YES

YES -&< FONEC > \PATHLENGTH/

ISCOLL=Z

(PLANE ESCAPE ) CALCULATE

X AT SURFACE AND TIME

ISCOLL = 3 (COLLISION ) CALCULATE

TIME

RETURN RETURN

(QUADRATIC \ ERROR W EXITS /

ISCOLL: | (CURVED ESCAPE) CALCULATE

X, Y, Z AT SURFACE

AND TIME T

RETURN

Figo 4. Subroutine TRACKo The subroutines EGMV and FONEC return the

neutron mean free path and the coordinates at the end of a track, respectively,, RANDA returns a random number.

10
z coordinates. Error prints are provided for the cases of two positive,

two negative, imaginary, or indeterminate roots as solutions to eq. (3).

However none of these situations has occurred in many tens of thousands

of tracks, although at least some of them are mathematically possible.

Once I has been calculated, determination of the coordinates of escape

and the track time are straightforward. The subroutine also returns an

index that indicates the fate of the neutron, i.e., curved escape, plane

escape, or collision. A block diagram is shown on figure 4.

4. FPATH

This subroutine has also been entirely replaced. The same simplifying

assumption is made as in the original coding--that is, the dimensions of

the scattering sample are considered to be negligible compared to the

flight path to the detector. This has the effect that the angle and flight

path to any particular detector position may be considered to be the same from any scattering point in the sample. The quantities to be calculated

are the neutron path length, p, in the sample from any point (x, y, z) to

the surface of the sample and the scattering angle, 8,, in the direction

of a particular detector, at the angle / . The geometry is shown in figures

5 and 6. All paths are assumed to be parallel to the x-y plane, and

all detector positions satisfy the condition y > 0.

We note that the locus of the intersection of a cone with a plane

parallel to its axis is an hyperbola, and write, using the notation shown

in figure 5,

a

2 b*

or Y

2 = X

2 tan2

- z , (4)

since b = z and a = z/tan9.

Now, referring to figure 6, we may also write

Y = p sinij; + y

and X = p cosijf + x , (5)

r

i where x /= R + h + x

= + h + x. tanG

Again we reduce the problem to a quadratic in path length, p, after

35

11
ru

U & a)

<d o c

4J ctj cti 4-> ^ r-J

CtJ 4-)

a o

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a) c «\ C -H i—l cd cd .—1 CO S-i a -rl CD X! c

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QJ 0) C 4J o H 43 -a

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43 4J d co d

co a) 4J c ^ cd o 4J

<4-i -a c

a) CD >> 4-> X) u cd •H U U U

CD o n o e -h •H c o 13 a; co (1) 00 -h x: XI

4J CO

a> at si c <4-1 CO H -H o cd m

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r—

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p tfl a)

c3 d) 4-J w a x: 4-1 .a j-j cd M 4-J u c >. CO •H o xj u 4J a)

0) X)

X

4-Ju CD 4J

4->

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CO

m •H M-l

CJ a) '4-1 o

to '^ 0) X) 0)

0J CO

u

X! CD cd cd -u H r—1 '4-1 U o '-4 -G •H ,4=1 3

00 4J

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r3 c OJ o cd a, cu

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C CTf 4-1 4-> 0) M-J X c to o 4-) o cd •H

•rH o j-) 3

4-1 cd o n ,£ /—

tU Q) J-) ^ to a, 1T» a to X < 3 •h ^-X

•H

13
combining eq's (4) and (5) to eliminate the variables X and Y to obtain:

P p p p p (sin y-cos ij/tan 9)

+ 2p(ysinij/-x 'cosijftan 9)

+ (y

2 -x

/2 tan2 9+z2

) = . (6)

For each value of z there will be a pair of angles, a and B (see

figure 6 , which define the limits of the hyperbolic curve. These angles

are given by

i/ 2

tan a = (r2 -z

2

) -y

h + x

1/2

2 2 (r -z ) -y

tan B = r • (7) r h - x

The subroutine is diagrammed in figure 7. After testing for the

special case of exit perpendicular to the beam axis, the program tests,

using eq's (7), for exit through the endfaces vs. the curved sides for the

exit angle, y, and branches accordingly. The path length in the former case is a straightforward calculation, and for curved escape is the positive root of eq. (6). It should be noted that in the limiting case of

cylinder (tan9=0) this subroutine, unlike all of the others, does not

work. A modified subroutine for a cylindrical sample is given in the

Appendix following the listing of FPATH for a truncated cone.

B. Correction of Coding Errors [7]

1. INPUT

a. Statements resulting in LGR=0 in the table look-up for the mean

free path (following card 0308) have all been changed to give LGR=1.

LGR=0 references not a mean free path, but instead the last tabular value

of collision probability in the previous MOULD table. The revised look-up

supplies the mean free path at the lowest tabulated energy for any neutron

at or below that energy.

b. On the card following statement 5001, LRG has been replaced by the

correct variable, LGR.

c. Card 0441 is replaced by

IMINM = IMAXM+1 .

This change yields sequential storage of supplementary ranges of experimental angular distributions, and thus avoids loss of needed storage space

in the array SUPVAL (I, J).

14
CALCULATE

TAN ft TAN 8a

RETURN

NO

FORWARD

YES - CALCULATE

0d,P

\

RETURN

BACK

CALCULATE

COEFFICIENTS OF THE

QUADRATIC

IN p(EQ.6)

YES

CALCULATE

P (LINEAR SOL'N)

7

CALCULATE

P

(QUAD. SOL'N)

1

RETURN RETURN

Figo 7„ Subroutine FPATH, for a truncated cone,

cylinder is similar

15

YES

CALCULATE

P

The subrouti

\

RET

ne fo

URN

r a
2. EGMV

LGROUP=0 has been eliminated for the reason given in La. above.

3. NAPAN

a. The angle used for the interpolation of the experimental distributions, as well as for calculation of the neutron energy after collision

is the laboratory angle, calculated in the subroutine FPATH. For this reason the experimental angular distributions must be read in at laboratory angles. Also the center-of -mass energy calculation on cards 3494 and

3497-3501 (in the case of center-of -mass MOULD angular distribution tables)

is in error. This has been changed to a calculation in laboratory coor- dinates by replacing card 3494 by

GO TO (36,11,11,11,11,11,11,11,36,11,11,11,11,11,11,11) NFORM

and deleting cards 3497-3501.

b. The sequence of cards 3502-3512, which picks the appropriate

experimental angular distribution values, does not refer to the correct

angular distributions, nor to the correct neutron energy-- that before

collision. It has been replaced by the sequence:

11 IFQMAX(J).LE.O) GO TO 15

IMINM=IMIN(J)

IMAXM= IMINM+IMAX ( J ) -

DO 50 JJ=IMINM,IMAXM

IF(EIN-ENVAL(J,JJ))50,51,52

50 CONTINUE

51 KK=JJ

GO TO 19

52 IF(JJ.LE.IMINM) GO TO 15

KK=JJ-1

c. On the card following statement 5001, LRG has been replaced by

the correct variable, LGR.

d. LGR=0 has been eliminated for the reason given in la. above.

e. The option that provides for suppression of printout of various

scores, through setting the indicators IANALA, IANALB, IANALC , and IANALD

negative, should not suppress storage of these scores by NAPAN. The time

saved is negligible and the elastic scores, DEB(I,L), are required by the

automatic iteration scheme, AUTO. Cards 3528-43 have been revised to

eliminate the dependence on these printout indicators.

16
4. ABSYND

LGR0UP=0 has been eliminated for the reason given in la. above.

5. CREN

For the same reason set out in section 3a. above, the center-of -mass energy calculation (cards 2960-66) has been replaced by a calculation in

laboratory coordinates by deleting cards 2961-66 and replacing card 2960

with

880 GO TO 800

6. OUTPUT, 0UTB3, 0UTB4

The sum over energy, at each detector angle, stored in the vector

DES(J,33) is required by subroutine AUTO. So as to allow calculation of

this quantity in subroutine 0UTB4, even when no B4 printout is requested,

a print indicator, IFPR, has been added to the call for subroutine 0UTB4.

This is accomplished by the changes shown in the listing for cards 3721,

3808A, 3873, and 3975-4002.

7. AUTO

a. The quantity "chi-squared, " on card 2084, should be

/ \ 2

_ 33 la(9) -j(9) , 2 _ v \ exP calc/

x " r o-(e)

1 exp

and not as formerly written, with the denominator squared.

b. It should be noted that the normalization of the Monte Carlo output on cards 2065-2068 and 2088 is not, in general, correct; but only holds

for detectors positioned at equal increments of cos9 between -1 and +1.

For this reason, if subroutine AUTO is used, the input detector positions

must satisfy this criterion.

8. FPATH

The original version of this subroutine contained two errors. These

do not apply to the new version of FPATH reported here, but are listed for

completeness. Card 3356 of the original should read

SINOM = SQRT(1-C0S0M**2) ,

card 3358 be deleted, and a new card inserted following card 3359

FP1 = W0RK16**2+W0RK17**2-FP2**2

17
C. Other Modifications

1. General

a. Common storage assignments have been handled by combining all

common statements into one package, processed by the UNIVAC 1108 "Procedure Definition Processor" using the assembler directive FCOPY (Fortran

copy). This package, designated CINC1, is then included at the time of

assembly in all subroutines making use of common variables by means of the

statement

INCLUDE CINC1

placed immediately following the subroutine name definition. This process,

which effects a considerable economy in the size of the source deck and

listing, is characteristic of the UNIVAC 1108 Assembler and FORTRAN V;

however, equivalent procedures are often available to other systems. In

some subroutines dimension statements still appear for those few variables

not in common storage. A block of common storage has been specifically

assigned to the subroutine ABSYND variables DATA and IDATA. This carries

the dummy label BLANK. The details of these changes in memory storage

allotment will be obvious upon examination of the listings.

b. The size of many modern computers obviates the need for linkage.

We have deleted references to CHAIN(I,J) and incorporated the balance of

subroutine PRELUDE into the main program MAGGIE. Similarly, references to

CHAIN(I,J) in subroutines DATIN and AUTO have been replaced by the appropriate CALL and/or RETURN statements.

c. The final F in the names of all library functions (such as SIN,

COS, EXP, MIN) has been deleted, as it is not compatible with FORTRAN V.

In addition, such functions as FLOAT and INT have been eliminated by using

the mixed expressions allowed by FORTRAN V.

d. Disc storage read and write statements have been modified in the

main program MAGGIE and subroutine WRTBT to conform to 1108 FORTRAN V

usage.

e. A number of formats have been altered. Some of the changes are merely different spacing options, but others are required by the new scattering geometry. A list of card numbers of the affected statements

follows

0226

0249

0410

1719

2111

3639

On card 2735 in subroutine CR the variable N has been replaced by the

correct variable NSECS. 10

0226A 0226B

0258 0345

0450 0451

1963 1972

2115 3638

3828 3833
f. A new random number generator, subroutine RANDOM, entry RDM, has

been incorporated, and the required changes in the functions RANDA and

NRANDA are shown in the listings.

2. MAGGIE

a. All of the subroutine PRELUDE, except for calls to subroutines

CLOCK and CHAIN, has been inserted following card 1654., A call to subroutine RANDOM immediately following the read-in of OCT, the starting

value of the random number generator, initializes this generator. OCT is

printed out following the call to subroutine INPUT.

b. All references to subroutine CLOCK have been deleted.

c. Card 0530 is eliminated by changing cards 1776, 1778, 1787, 1789,

and 1792 to read

GO TO 1

3. ABSYND

a. Card 0525, which assigns the logical tape unit carrying the MOULD

nuclear data tape, now reads

NUCDAT=9

On cards 0525, 0530, and 1394 the variable name TAPE has been changed to

ATAPE to avoid confusing the compiler. These changes may not be required

by other installations.

b. Cards 0553 and 0554 set all storage for the variables DATA and

IDATA to zero before each ABSYND run.

c. Card 1398 is modified so as to print only those action numbers

processed.

4. CR

The variable names NACT and Q have been changed to NACTV and QV, so

as to avoid conflict with the array names NACT(I) and Q(I). Similarly,

the variable name COS is changed to C0Z, so as to make the library function

COS available.

5. CREN

a. The variable names NACT and Q have been changed to NACTV and QV,

for the reason given above.

b. Cards 2832-2837, 2878-2881, and 2911-2915 have been deleted, since

they are not needed. In order to accomodate these deletions card 2910

19
has been amended to read

IFCNVCOS.LT. 1) GO TO 208

and card 2916 to read

P = (AS+SQRT(AS**2+A*QV*(1+A)/EIN+A**2-1))/(1+A) .

6 . AUTO

a. The multiple elastic sums are not required, and so cards 2069-2072

are deleted.

b. The quantity RATIO is not used, and has been eliminated from

cards 2077 and 2078.

c. The do-loop 2083-2085 is redundant, since ELM2(J) is also set at

card 2097. Cards 2083-2085 are therefore deleted, and card 2080 is modified to reflect this change.

d. The three do- loops in the sequence 2087-2098 have been combined

into a single loop.

e. The variable name EXP (J) has been changed to EZP(J) to avoid conflict with the library function EXP.

f. Often it is desirable to separate an iteration procedure into two or more consecutive runs. Cards 2073-2082 have been modified to allow a run to be made using the partially corrected output of a previous run as input, instead of the experimental data. When this option is to be used

the second field of the input card specifying the number of iterations,

NTERM, should be non-zero. This is then followed by the cards specifying the experimental values for the angular distribution, FCVAL, which

are removed from their usual position and replaced by the partially cor- rected output of the previous run.

g. In the case of rapidly varying angular distributions such as at

14.1 MeV, the usual iteration procedure is not as rapidly convergent as

the "physical" method. For this method the calculated multiple scattering is first subtracted from the experimental input, and then the balance

of the iteration (sample attenuation) is performed as usual by reflecting

the output for single scattered neutrons about the input. The new coding

on cards 2091-2097 reflects this change. For the first two iterations the

iteration improves only the multiple scattering, the correction being

applied both times to the experimental input. After two iterations multiple

scattering is well enough known to allow the iterative procedure full play,

so the correction is applied each time to the previous input, instead of

the experimental values. This procedure yields much more rapid convergence,

three iteration Droviding better convergence than six of the previous

method.
III. AN APPLICATION TO 14.1 MeV NEUTRON SCATTERING

The code has been applied to a typical associated-particle scattering

geometry for 14.1 MeV neutrons on carbon. The scattering sample was a truncated cone of half -angle seven degrees, length 3.193 cm., and entrance

face radius 2.059 cm., with its center located 20 cm from the neutron

source. The experimental input used was simulated using published results

for carbon at this energy. The results of a typical run are shown in

Table 1, and on figures 8 and 10. The cross-section data used for the

Monte Carlo scattering were those of Slaggie and Reynolds [8]. In Table 1 the quantity o~(9) represents the true angular distribution, and the total

and multiple outputs are the results calculated using it and the specified

sample shape. The code varies a(9) until the total elastic output matches

the experimental input to the accuracy required. The calculated angular

distribution for neutrons undergoing multiple scattering is shown in

figure 8. It can be seen that the result converges quite rapidly. Further

iterations resulted only in statistical variations about the values obtained after three iterations; in fact, the largest change after two iterations, that at cos = 0.625, represents only a 1.5% change in the cross

section.

In figure 9 we show the part of the correction, exclusive of multiple

scattering, that depends upon sample shape. Also displayed for comparison

is a calculation of the effect expected due to attenuation in the sample

for an isotropic angular distribution. As can be seen, this correction

is not independent of the input angular distribution. The quantity

plotted, Aa/c, is the difference between the singly scattered output and

the input, divided by the input, for each angle. This geometrical correction is contributed to about equally by varying path length in the sample

and the change in the total neutron cross section with energy as a function

of the neutron scattering angle. The correction varies from a few percent

to about ten percent, and is directly dependent upon the accuracy with which

the total neutron cross section is known over the range of energy exhibited

by the neutron recoil. On the other hand, the correction for multiple

scattering is larger, ranging from about 50% at the backward minimum to

4% at zero degrees, but depends mainly upon the experimental input data.

Over most of the practical angular range of measurement it is a quite

appreciable 15-20%. Figure 10 shows the input distribution used and the

corrected output distribution obtained after three iterations of approximately 2000 interacting neutrons each. The total correction is, of course,

largest in regions where the cross section exhibits minima. It varies

from a few percent to a maximum of only twenty percent at measurable angles,

since the geometrical and multiple scattering corrections tend to be in

opposite directions, except at the extreme backward angles. Several

examples of spectra showing elastic-inelastic and multiple inelastic effects

in the energy spectra are given in the original references [4, 5]; the

present version of MAGGIE also yields similar results.

21
<

>

4J

cdco

u

o

o

M-lc

CJ

<Dc

o

cd

CJ •Ha

•u

S-i O

<w

0)

CDoo

CO

x:H

H

z

w

H-l w

P4X

W

ol-l H

c/i W

<

H

O

H

Ui-i H

CO<

W

O

H

cr>b

coo

CQs

<fm H H^cOHH(NN\o<t'-ip-i\oir)fOiriu*|tON^OIN CvJ^O <TC^r^cMC0<tr^^>-Hcn<finin^CNOC0incncMCNCN<fvD'-| 00O OOHHrHHHHNcNNCMCMOJC\ltN^H>4 iH r^ p^ ^ H N <N <f o in m a\ r-* cm O N fl H 0D vo ^DOl •* Ol fl O

cm oo in

^csj<f\ocoo^^<finvof^r^coooooQor~r^r^aA'--<<t- ONinc,o<fr«.-.^-in<-i>--<vo

cm csicN(NcM^onc^roc^c^rnooc^c^(r)(r)rno^(^oo<f<f<tinv£)r-- ooo cm moo—i OOOOOOOOOOOOOOOOOOOOOOOOOOOO^^r-i—i—icM

<t" CM <T CJ\ o oo—4oooocMr-)von—40inLncnininLncnr^cNjoioin-+incy>LnoooLO

cm en <r r^ cx» <-icn-^-inLn<f<NOQ0incncMcM(N<tr^—i oo o o cm en .-< co^-t ^HHH^(\l(NOJNCN|C>J(NNr^Hr-lH^Hr-lWlNCM<l, ^OCTi<f(N(nO

cm m in

ooincooor^-cMv£)CMinocoincn\^cMcoLnin<tr^o<j"<tc,o—< oj co o> vo <f h rs cMr^a>o--fiNtnr^ocM<j-<f<foo—icyvi^-d-cMr-jOr-HCMinooo^cMvooor^inai OOOrHHHHHNNlNNtMCVINi-IHH^HHHHHCNIN^^CftsfNstH

-I ts fo in

omomoioomomomoinomomomomomOLOomomOLno or^incMor^incMor^incMor~-incMOcMinr^ocMinr>-ocMinr^-ocMinr^o ocor^—iincx3cM\oocor^r-iinc)ocMvoovocMooin>--<r^ooo^ocMcx)ini-ii^cno oo>cooor^vovoinin<fcoo^c\i^^ooOr^—4CMcooo<t-ininvovor^-oooocyiO

—iNn-cfi/ivor^oocnOr-i cm co <f in ^o r-~ cocriO'-icMcovtinvor^oocjNO^^cMcn —Ir-tCMCMCMCMCMCMCMCNCMCMPOcncnn

22
a:

< 2

or

GO

rr<

0.5

1

1 1 o 2 nd ITERATION

1 - 3rd ITERATION

MULTIPLY SCATTERED NEUTRONS

o ~ °- z o _

o

o

o

o

o

o _

> o o o o n —

a

o

o

o

"

11

1.0 0.5

cos 0, ab

-0.5 -1.0

Fig. 8. The angular distribution of multiply scattered neutrons on carbon at 14.1 MeV for two successive iterations. The data for

the third iteration are also given in Table 1, after suitable

normalization.

23
0.30

0.20

0.10

A <r(0) = f(0)

° <r(0) = CONSTANJ

o A

n A

0.00

-0.10

-Q20

A oA ©

A

000 ° o o A

Aa^^^^A

o A A ^A

oA

o o AA A A9 r

o ** A A A A ^

0.5

COS 0,

-0.5

lab

-1.0

Fig. 9. The correction due to sample shape. The quantity Aa/a is the

change in the normalized angular distribution after subtracting multiple scattering.

24
1.0

en

>-<

GO

<

b .01

.001

1.0

- THE EXPERIMENTAL INPUT

• CORRECTED OUTPUT

CARBON, 14.1 MeV

0.5 o

COS 0i

-0.5 -1.0

lab

Fig. 10. The experimental input and corrected output for carbon at 14 MeV,

25
REFERENCES

1. Monier, L. F. C, Tripard, G. E., and White, B. L., Nucl. Instr. and

Meth. 45, 282 (1966).

2. Marshak, H., Richardson, A. C. B. , and Taraura, T., Phys. Rev. 150 , 996 (1966).

3. A useful summary is given by M. Walt in Fast Neutron Physics , Vol. II,

Marion, J. B., and Fowler, J. L., Ed. (1960).

4. Parker, J. B., Towle, J. H., Saras, D., and Jones, P. G., Nucl. Instr.

and Meth. .14, 1 (1961).

5. Parker, J. B., et al, Nucl. Instr. and Meth. 30, 77 (1964).

6. Parker, K. , AWRE report 0-70/63; Kerr, W. M. M. , AWRE report 0-81/64

(1964); Miller, S. M. , and Parker, K. , AWRE report 0-55/65.

7. Some of these coding errors have also been previously noted in private

communications. We are indebted to J. B. Parker, Aldermaston, for

noting the substitution of REDEN for EIN in subroutine NAPAN, as well

as the errors in subroutine FPATH. H. Horstmann and H. Schmid, Geel,

have noted, in addition, the difficulty involving the center of mass calculation of neutron recoil energy in subroutines NAPAN and CREN.

8. Slaggie, E. L., and Reynolds, J. T. , KAPL-3099, (1966).

26
APPENDIX A

Listings of new and extensively revised coding,

Name Page

CINC1 28

MAGGIE 3A 30

INPUT 36

CRNEU 44

TRACK 46

NAPAN 49

FPATH (truncated cone) 53

FPATH (cylinder) 55

OUTPUT 56

AUTO 58

RANDOM 61

RANDA 61

NRANDA 61

A complete listing is available by request from the Center

for Radiation Research, Neutron Physics Section, National Bureau

of Standards, Washington, D. C. 20234.

27
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h- o »-^ <M cr y: 0. Ld

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id en o < CC % cr cr T-t cf to * 2 z » < .—. 3 o ^ vD o

en * «> «-» CL cf _J 3 cr ^» z

i- i- o ^ CM Z to o »-• 3

< D ^, o <-. ^^ •» r- 3 X Ld 2 O o .-t * K »-^ ^L to LU z

Z 3 o —CM O o cr in z <

*. «k *-i Ld «-' < LO c .H <-> •

LU Z w cd O z *-" 3 Kr to UJ O »-• z z » «. o to cr UJ _l

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H- o 1—1 —«. ^-l _l ^~ 3 t-t to to < o » o s: o o to 1^ i- LU

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28
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APPENDIX B

A Brief Description of the Components of MAGGIE

1. MAGGIE (Main Program). This program calls the various subroutines

required for the analysis, retrieves neutrons from disc storage when

required, outputs track parameters if desired, and records the

various fates of tracked neutrons.

2. RANDOM (and RDM). This subroutine contains the random number generator, RDM, as a separate entry. RANDOM is called at the beginning

of program MAGGIE to enter the starting value for RDM.

3. DATIN serves to set all of the output arrays to zero, and calls

ABSYND.

4. ABSYND reads the required nuclear data from the MOULD tape and puts

it in encoded sequential storage in the array DATA-IDATA for use during the Monte Carlo tracking.

5. INPUT reads and processes samples and experimental angular distribution data from card input and calculates most of the flux attenuation factor.

6. CRNEU creates random incident neutrons at the entrance face of the

sample, in accordance with the input source distribution.

7. TRACK tracks neutrons in the sample, specifying coordinates of

collision or escape.

8. EGMV. This subroutine computes the mean free path, velocity, and

lethargy group number.

9. FONEC calculates coordinates at the end of a track from initial

position, direction cosines, and track length.

10. CR. This subroutine, using random sampling of the information stored

by ABSYND from the MOULD data tape, determines all of the parameters

of a collision.

11. CALC determines some constants used in subroutine NAP AN.

12. NAPAN. This subroutine calculates and scores the probability of

detection at each detector for each collision in the sample.

13. FPATH calculates the path length in the direction of each detector

for each collision.

14. CREN is an abbreviated version of CR used by NAPAN that determines

only the neutron energy.

62
15. TWIST chooses new direction cosines after a collision.

16. WRTBT stores any secondary neutrons produced, for recall at the end

of the current tracking.

17. OUTPUT, 0UTB3, 0UTB4 and APL0T3. These subroutines print the results

of the calculation.

18. AUTO performs the calculations required for iterative correction of

the elastic angular distribution, prints the current output, and

calls DATIN to begin the next iteration.

19. END, and its entries EXIT and EEX1T designate normal vs. error exit

conditions

20. TAPLAB returns the tape logical unit label.

21. SRFORT is a subroutine for skipping tape records.

63
APPENDIX C

Input requirements for MAGGIE-NBS

FORMAT VARIABLE

1. 012 OCT

2. 12A6 HEAD(I)

COMMENTS

Octal starting value for RDM.

Arbitrary heading. Column one should

be a 1.

3. 110

4. 110, E10.4

5. 110, E10.4

NMATS The number of materials in the sample.

This is always = 1.

IDATACMAT5) Number of different nuclides in the

material

DATA(MAT4) Density of the material.

IDATA(K2) Nuclide reference number (i.e. position

on the MOULD tape : see output of MOULD

for this).

DATACK4) Proportion of this nuclide in the material

This card is repeated for each nuclide

in the material.

6. 2110 1FB4

NSENSE

7. 3F10.4 HITE

FRAD

ANGLE

8. 2110 J0BFIN

ITMAG

9. F10.4 SAMPLE

10. F10.4 STRTE

Positive for B4 output.

Positive for track print.

Length of the samples, in cm. Entrance radius of same, in cm. Half -angle of same, in radians.

Number of independant samples (= 1 if

ITMAG > 0).

Positive for automatic iteration.

Number of neutrons in each sample.

Starting energy in MeV. If the starting

energy is set = a fission spectrum is

assumed

11. F10.4

12. 110

DIST

NSTDP

Distance from the center of the sample

to the source (negative), in cm.

No. of start distribution points to be

read in (^25).

13. 6F10.4 ANGS(I) Three pairs to a card, the angle and start

distribution for that angle. The start

distribution need not be normalized, and

this card is repeated until NSTDP pairs

are read in.

64
14. 110 NLWREV The number of Law Reference numbers for

inelastic scatter, i.e. the number of

Law Reference numbers to different

angular distributions on the MOULD output

(P.C.N.'s* 4-15 only) (NLWREV < 50).

15. 2110 LAWREV(I)

LWNO(I)

One pair per card, the above Law Reference

numbers and the monotonical ly increasing

"Monte Carlo" law numbers allocated.

Any inelastic law not given here will be

printed in the B3 section results under

Law zero.

NOTE: The following cards are not required if IFB4 < 0.

16. 4110 IANALA

IANALB

IANALC

IANALD

Four markers, for the tables: A) Complete

multiple scatter analysis. B) Elastic

events only. C) Inelastic events only.

D) Multiple elastic events only. A

negative marker suppresses the table.

A positive marker = N produces, in addition

to the table, a graph for the Nth counter.

17. 110 NANAL Number of actions (index I below) (i.e.

different neutron processes, such as elastic, inelastic from the 1st excited

state, etc.) to be processed by the subroutine NAPAN.

18. 6F10.4 NUCL(I)

NACl(I)

The nuclide reference number (as in card

5), and the P.C.N. for each action, are NANAL such pairs.

There

19. 6F10.4 FCCOS(J) Thirty-three values of cosine, including

-1 and +1, monotonical ly increasing.

20. 6F10.4 FCVAL(I,J) Thirty-three values of the angular distribution (index J) corresponding to the

above values of cosines. Repeat for each

action (index I) in the same order as in 18,

NOTE: The following three items are repeated, as a group, for each action,

21. 110 IMAX ( I

)

Number (index L) of supplementary ranges

for this action (even if zero).

The "P.C.N.'s (Particular Classification Numbers) are listed starting

on p. 8 of AWRE report no. 70/63, "The Aldermaston Nuclear Data Library

as at May, 1963", K. Parker.

65
NOTE: The following two items are repeated for each supplementary range

If, for any action, there is no supplementary range these items

are omitted.

22. F10.4

23. 6F10.4

24. 110

25. 6F10.4

26. 110

27. 6F10.6

28. 2110

29. 6F10.4

ENVALQ, L)

SUPVALCL, J)

NOCOU

COUNT(I)

NENSP

ENSP(I)

NTERM

IFCONT

SAMP(I)

Upper limit of the range in MeV. These

must be listed in order of decreasing

value

Thirty-three values (index J) of the

distribution (range L) at the cosine

values FCCOS(J).

Number of counters (^33).

Cosines of counter locations.

Number of output energy points (^33).

The output will be classified into the

bins formed by these points.

Values of output energy points in MeV.

Number of iterations (<10)

If this is a continuation of a previous

run, for further iterations this should

be non-zero (see Section II.c.6.f).

Number of neutrons for each iteration,

except the first.

66
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