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UCRL-7827 Propuls ion Systems and

Energy Conver'sion, UC-33 TID-4500 (30th Ed. )

UNIVERSITY OF CALIFORNIA

Lawrence Radiation Laboratory

L ive rmore , California

Contract No. W-7405-eng-48 '

A NEW DERIVATION OF MULTIGROUP CROSS SECTIONS FOR

BeO- AND GRAPHITE-MODERATED SYSTEMS

R. j. Doyas

E . H. Canfield

April 28, 1964

CONTENTS :

Page No.

ABSTRACT . i INTRODUCTION .

THE CRITICAL EXPERIMENTS . BASIC CROSS SECTION DATA .

Sources . Anisotropic Elas t ic Scattering . Inelas t ic Scattering

Be(n;y) . . Be(n,2n) Reaction . . . .

U-235 F i s s ion . . , . U-235 Capture

The F i s s ion Spectrum . Neutrons p e r Fiss ion: v(E) . The T r a n s p o r t C r o s s Section . Crys t a l Scattering .

THE SUPERNOVA CROSS SECTIONS . C r o s s Section Calculations . Resul t s . Comments and Fu tu re Ef for t s .

APPENDIX A - (n,2n) Neutrons v ia the 2.43 MeV E3e9"~od.e

APPENDIX B - Monte Car lo Correc t ion to the Brooks Data

APPENDIX C - The SUPERNOVA C r o s s Sections.

A NEW DERIVATION O F MULTIGROUP CROSS SECTIONS FOR :k

BeO- AND GRAPHITE-MODERATED SYSTEMS

R. 3 . Doyas and E . H. Canfield

Lawrence Radiation Laboratory, University of California .

Live rmore , Calilo.rnia . .

Apri l 28, 1964 - . , , .

Multigroup diffusion equation c r o s s sect ions a r e reder ived using updated

numer ica l techniques and recent bas ic c ros s - sec t ion measurements . The

bas ic s e r i e s of Spade (BeO-moderated) and Snoopy (graphi te-moderated) .

c r i t i ca l a s sembl i e s a r e calculated using these group constants , and the r e su l t s

a r e compared .with those f r o m ear1ie.r compilations. . .

>:: Work done under the ausp ices of the US S. Atomic Ene rgy Commission.

INTRODUCTION

Along with the development of f iner analytical techniques has come a

considerable amount of improved experimental 'data on neutron c r o s s sections JI -F

of in te res t to the Tory IIC design. Since cer ta in shortcomings of the present ly

used cross-sec t ion l ib ra ry1 have been apparent for some t ime, a program

was launched to develop a m o r e accurate and self-consistent s e t of c r o s s

sections for use in the production codes. This development has .been oriented

towards the simple -geometry Spade and Snoopy cr i t ica l experiments , in o rde r

that the effects of the'various refinements can be m o r e easi ly interpreted and

the usefulness of the complete se t evaluated.

This repor t covers the development of new 18-group c r o s s sections and

their application to beryllium oxide - and graphite -moderated homogeneous

cr i t ica l experiments. These. c r o s s sections, for basic modera tors and fuel,

will next be used, in conjunction with other experiments a t N Division to

ascer ta in the effective c r o s s sections of other s t ruc tura l mater ia l s for use

in a re-evaluation of the Tory IIC design; self- shielded c r o s s sections for the

heterogeneous c r i t s have been included for this purpose.

The principal improvements over the l i b r a r y tape c r o s s sections include

the following, l is ted approximately in o rde r of importance:

1) Generally updated c r o s s - section data.

2) Generation of a separa te se t of c r o s s sections for each M/U ra t io

of interest .

3) Incorporation of experimental angular scat ter ing distributions.

4) Crystall ine effects in neutron thermalization.

5) The use of fine-grouping for the initial c r o s s section calculation.

6 ) Improved t rea tment of downscatter in the Be(n,2n) reaction.

7 ) New values of v(E).

8) Trea tment of inelastic scattering.

This new SUPERNOVA se t of 18 group c r o s s sections, t r ans fe r ma t r i ces ,

etc. is designed for use in the s tandard ZOOM' diffusion code. (A slight 3

modification r ende r s i t suitable for the ANGIE diffusion code. ) The techniques

used in generating the SUPERNOVA se t a r e an expansion of those previously

* The Tory IIC i s a full-scale reac tor which will demonstrate the feasibility

of the Pluto ramje t reac tor system.

4 used in generating the NOVA se t of c r o s s sections which a r e superseded by

this repor t . Specific detai ls a r e discussed in the remainder of this repor t

but the essence i s .this:

The basic experimental and/or theoretical c ross-sec t ion data i s converted

to 189 energy group data by various neutronics' codes developed a t LRL. This ' v 5

fine - group data i s fed' into WHIZ, a zero-dimensional multigroup diffusion

code which calculates flux spec t r a typical of the var ious r eac to r s in question.

WHIZ uses these spec t r a to weight these fine-group c r o s s sections ( a s

descr ibed in Ref. 5) and calculate 18 group numbers suitable for use in ZOOM.

THE CRITICAL EXPERIMENTS

6 Kloverstron e t al. measured the c r i t ica l height of a s e r i e s of BeO-

moderated oralloy fueled assembl ies with B ~ o / u ~ ~ ~ rat ios of 247, 493, 986,

1920, 3826, and 7660. Recently, the measurements were extended to include

a B ~ o / u ~ ~ ~ = 120 assembly (pr ivate communication f rom Jack Morton). These

spade c r i t s employed fuel foils of varying thicknesses and separat ions, f rom

which extrapolations to homogeneous geometry can be made. Table I l i s t s

the experimental c r i t s , extrapolated heights ( s e e following), and the mater ia l

number densi t ies used in the machine calculations.

The cr i t ica l heights a r e quoted to f 0.1 in. , corresponding to about ~ 5 7 0

in B ~ o / u ~ ~ ~ ~ Each of the 4 9 3 ~ 1 , 9 8 6 ~ 1 , 1920: 1 and 3826: 1 c r i t s were repeated

with different foil thicknesses so that extrapolation to .zero foil thickness i s

s t raightforward ( see Fig. 5 of Ref. 6). The 120: 1, 247: 1 and 7660: 1 c r i t s

were each run with only one foil thickness; to Itextrapolate" them to the

homogeneous l imi t r ecourse mus t be made to the systeinatic changes in the 2 235 extrapolation. F igure 1 of this paper plots AB /mil of foil - vs B ~ O / U ; i t

indicates no extrapolation i s needed for the 120: 1 and 247: 1 sys t ems , and

shows that the extrapolation for the 7 6 6 0 ~ 1 i s ill defined, s o that the extrap-

olated height in Table I i s ve ry approximate.

The data in Table I includes cor rec t ions for the diaphragm. Neutron

room re tu rn was not observable within experimental accuracy in the 493:l

Spade experiment; the re fore no room return. cor rec t ions have been made.

An effective buckling decrement due to neutron reflection f rom the

pedestal on which the lower half of the Spade assembly r e s t s has been mea-

su red for the 247:l Spade, and has been calculated for the other Spades. 7

2 2 These cor rec t ions , in the form AB /B were obtained f r o m calculated values

Table I. Homogeneous Spades. . . . . . .

olatea Height Base

atoms/cc u235 ( c m ) (cmx cm) u~~~ u2 38 9 Be a n d 0 1 2

.Homogeneous Snoopys

olate;! Height Base

loz4 atoms/cc &2 3.5

( c m l ( c r n X c d u235 u238 C

.Heterogeneous Cr i t s

Foil thick. /sep. H C

M / u ~ ~ ~ ( m i l l i n - ) ( c m ) Base

( c m X c m )

, r ' ' . Table I. (Can't)

Heterogeneous Cr i t s

Foil thick. /sep. H C Base

M / u ~ ~ ~ (mil/in. ) . . , ( c m ) ( c m X cm)

3 82 6 1/2 57.658 91.44 A 91.44

1 2/4 61.468'

7660 l / l b 91.948 I. a

Hal f -area t r iangular fuel foils. b

Quarter - a r e a triangular fuel foils.

(Note: Because of the lack of a straightforward approach to the self- shi'eldi'ng of the t r iangular foils, these c r i t s a r e omitted in the calculations).

of AK/K a s explained in Ref. 7. The cor rec t ions , converted back to AK/K

in s imi lar manner , a r e l is ted in Table 11. They have been added to the

calculated values of K to give al l final values of K quoted in this report . (P lo t s

of K vs M/U for the l i b r a r y and NOVA data which have appeared previously . .

'do not include this correction:. .)

Table 11. Pedes ta l Correct ions

a (0.102 for 30 in. X 24 in. base c r i t s )

A s imi l a r s e r i e s of c r i t s (Snoopys) were done for graphite moderated 8 .

sys tems. These resu l t s a r e a l so shown in Table I. The dimensions shown

include cor rec t ions for block porosity, room re turn , control voids, and , . '

nonhomogeneous, fuel a s pe r Ref.'. 8.

BASIC CROSS-SECTION DATA

Sources . .

Basic c ross -sec t ion data were drawn p r imar i ly f r o m three sources: 9 the b a r n book and i t s supplement! O and a compilat ionby Rt J. Howerton. 11

The breakdown is given in Table 111. Reactions with such high thresholds a s

t o be insignificant in r eac to r calculations have been ignored. . .

Table 111. Sources of c r o s s - section data.

E lement Reactions . . . Source (Ref. No.)

Oxygen ( E l 9 ",P . .

( E l n,a 9

( E l . n,n 10

. .. (El@) n1n . 1 1 .

Beryl l ium ( E l 9 n,a

( E l n ,n 1'0

' (E l n, 2n 12 . .

0- ( E l n, Y

10

u , (E,o). 12 n ,n

u235 0- ( E l n1n

13

16 20 ,keV - 1.0 MeV

1 3 0.01-1.0 eV

17 1.0-10 eV

. . . . . . , . . . . 16 20 keV-10 . , MeV 2 38 .. ( E ) . . . n1n

; i o ( E l @ ) . - .

2 35 . n n . . ;: . . . . . . .

Same a s . U . . , u ( E l O) 9 . n,n

Table 111. Sources of c ross-sec t ion data. (Con't . )

Element Reactiona Source (Ref. No;)

Carbon

9

below 1 keV

above 1 keV

l /v determined experimentally to be 1 3 ppm boron equiv.

a The c r o s s section notation used he re conforms to that suggested by the

AEC Nuclear Cross Section Advisory ~ r o u ~ .

Anisotropic Elas t ic Scattering

The angular distributions of neutrons sca t te red in elast ic collisions

have been measured for many of the common reac tor mater ia l s . Much of 13

this data i s compiled in Ref. 19. Joanou e t al. have analyzed the existing

experimental data for a number of e lements using a Legendre expansion

through 1 = 6 . Data i s a l so presented which has been Itsmoothed. This

la t te r s e t for Be, 0 , and C was used in SOPHIST-5,20 an IBM-'7090/7094

code which converts data in this f o r m to t r ans fe r ma t r ix elements appropriate

to the ZOOM and ANGIE formulation, and a lso computes anisotropic sca t te r 2 1

ma t r i ces appropriate f o r use in DSN.

The inc1usio.n of these distributions has a p,rofound effect on the moder -

ation of fast neutrons. It resu l t s in much s t ronger downscatter t e r m s

(par t icu lar ly for oxygen) than existed in the l i b r a r y c r o s s section set. This

inc reases K by 3-5% for a l l the Spade c r i t s and thus r ep resen t s a significant

improveAent over the l i b r a r y set . In fact, the s t rong backward peaking in

the oxygen a t cer ta in l l resonances l t causes such perturbation on the flux "

spec t rum a s to cause considerable deviation f rom "fission t a i l t t and " 1 / ~ "

behavior in the 0.3-2.5-MeV range, even to causing r eve r sa l s in the slope

a of the E + - v s - E - curve. (See Fig. 7:) Therefore , the analytic flux representat ion used throughout the fine-group data compilation had to be regarded a s not

suitable over this energy range - part icular ly a s r ega rds downscatter, which

i s ve ry dependent on f l u x shape within a group. The f l u x generated in WHIZ,

therefore , was used a s input to a second round of SOPHIST-5 calculations

for beryl l ium and oxygen, to generate a new se t of t ransfer coefficients. This

i terat ion proved sufficient, and was. the only instanc.e where such a s tep was

found to be necessary.

Inelastic Scattering

22 According to the Weiskopf level density formula, the probability

density for the inelastic sca t t e r of a neutron a t energy E' to energy E is;

where o = compound nucleus tempera ture j

where A = atomic m a s s of the s c a t t e r e r , and

C = constant such that

. .

Trans fe r ma t r i ces . based on the above distribution have been determined

fo r the inelastic sca t t e r react ion of U 235 and U 238 with the FORTRAN code

UTRA'NS. 23 ' ~ n e l a s t i c sca t t e r for the modera tors i s insignificant to these

applications because' of the high th re shold.ene rgie s involved, and was not

calculated; even the uranium u with i t s lower threshold is r a the r . . n ,ns

insignificant h e r e because of ' the high M/U rat ios of the c r i t s . 4 , ' . ' . . . . . . .

This reaction i s l is ted a s a l / v c r o s s section, but seve ra l sources

(e. g o , ' ~ e f s . 11 and 24) indicate a constant c r o s s section a t 1 m b when the

l / v representat ion reaches this value - this i s m e r e l y to imply that there

should be some nonvanishing probability fo r this react ion even a t v e r y high

energ ies , but does not' represent known experimental values. of cr (E) a t n, Y

these energies . If the 1-mb value had been used, ' the 247:l B e 0 c r i t would

suffer a reduction in K of 'about -0.5870, and the 7660: 1 case -0.7470, but we . .

s e e no justification in such a repr=ese'ntation.'

-11-

Be(n, 2n) Reaction

The experimental Be(n,2n) data and the fit to the data used by General- 13Atomics is reproduced in Fig. 2. The SUPERNOVA calculations use the

same fit except that ( as indicated by the dashed curve) the values near theV

experimental threshold have been increased so that the three data points

below the GA curve cutoff at 2.7 MeV are not completely ignored (thetheoretical threshold is 1.85 MeV). The rapid fall-off of the fission spectrum

above 3-4 MeV causes the reactivity calculations to be quite sensitive to the

values of the cross section assumed below 3 or 4 MeV, and no high degree

of accuracy can be claimed for the SUPERNOVA choice of (n, 2n) cross section

(which contributes about 10% to the reactivity of Spade crit calculations).

Other "reasonable" fits to the (n, 2n) cross section data were tried. It was

found easy to raise or lower the K-vs-M/U curve for the Spade crits by a

few percent, with almost no resulting change in slope.The spectrum of the two neutrons emerging from the (n, 2n) reaction

is even less well known than is the cross section and recourse must be made

to theoretical arguments. The SUPERNOVA numbers include the followingassunnptions:

9*(1) For reactions proceeding through the 2.43 MeV Be level

(2.7 MeV in lab), the two neutrons emerge with energies

El = 0.83 E' - 2.187E2 = 0.0189 E' + 0.473.

(2) When energetically possible, 50% of the (n, 2nl reactions go through9*this Be   level. ,·· 4. -··· ..,

(3) Fot all other modes, all neutrons come off with equal probabili:iesin allteriergy int6rvals 0<E<E' - 1.7.

(4) All >einerging neutrons have average lab cos ines, B = 0.11.The arguments leading to ( 1) above are outlined in A*pendix A.

" 25Assumption(2) approximates the branching ratios measured by Mal:iop: et, al.,and seems to be assumed, though not explicitly stated, in the Britifih "Barn

..24 26Book," but disagrees somewhat from APEX-467, which allows up to 80%

9*of reactions through the 2.43 Be level. Assumption (3) agrees with APEX-467 and with the British. Assumption (4) is also used by the British, and is

consistant with the crude arguments in Appendix A. To recapitulate, forrk *

reactions through the 2.43 Be' level

Fig. 2. Expeiimental (n,2n) cross-section data for ~ ' e ~ . We Lse the dashed line near threshold and .the . . solid line (General Atomics' f i t ) elsewhere. . . .

and for all other reactions

= 0 E 3 E' - 1.7.

For incident energies E 1 > 2.7,

For El < 2.7

P(E -- E)dE = Pother (El + E)dE.

Calculations of the (n,2n) transfer matrix on the basis of the above distributions

were done numerically with the FORTRAN code INTUNE. 23

I rum be in'^ seems to indicate a somewhat lower effective threshold

than we have used here, and also offers an alternative scheme for calculating

the neutron energy distributions.

The problem of handling the production of neutrons by the (n,2n) process

in codes like ZOOM and WHIZ i s best met through the use of a firftitious

negative group absorption cross section equal in magnitude to the (n,2n)

cross section and a positive scatter cross section equal in magnitude to

twice this value. The appropriate transfer matrix i s then associated with

the (n,2n) events by considering this reaction as a separate material in the

fine group WHIZ problems; after the collapsing to the Safonov grouping has

occur~dd, the (n,2n) llmaterialtl i s combined with the Be, the transfer matrices

being weighted with their respective scatter cross sections.

u~~~ Fission

A plot of the data used for the capture and fission of u~~~ (from ~ i k n e r 16)

v above 20 keV i s shown in Fig. 3,

The resonance region represents a more complex situation; several

sets of recent experimental fission data l4*l 5s28)exist which agree fairly

well in gross, but not as much in specific, details, Moreover, these sets

are all analyzed by their respective authors in terms of resonance fission

0.1 I

E (MeV)

2 35 Fig. 3. Fission and capture cross sections above 0.0-1 MeV for U ,

-

paramete r s , a s determined by a r e a analysis of smooth curves fitted to the

pointwise data over each fission resonance. Choosing an adequate smooth

curve fi t becomes difficult when the data has appreciable sca t te r and/or

poor resolution. In addition, subjective e r r o r s a r e introduced when an ill-

< a defined non- r e sonance background i s subtracted out, and when a "wing"

correct ion i s estimated. Nevertheless , f ission c r o s s sections computed f rom

resonance pa ramete r s thus obtained adequately reproduce the resonance peaks

in the original c r o s s section data. But the computed values of the valleys

come out much too low. A se t of group c r o s s sections computed f r o m

resonance pa ramete r s by the SOPHIST - z Z 9 code resu l t s in par t ia l resonance

integrals about one-half a s l a rge a s those determined f rom integration under

the original data points. (The SOPHIST-2 equations a r e formulated on the

bas is that the single -level Bre i t - Wigner assumption holds; pa ramet r i c data

a r e a l so generally determined on the basis. ) Some of the discrepancy could

be removed by the formal introduction of one o r m o r e negative energy (i. e. , bound level) resonances, but SOPHIST-2 is not equipped to handle these. The

remainder cannot be explained by some missing l /v contribution to the fission

c r o s s section, and i s usually ascr ibed to interference effects between

neighboring resonance s o

Since the valleys do contribute an appreciable par t to the f iss ion in the

resonance region, the SUPERNOVA c r o s s sections a r e not calculated f rom - resonance pa ramete r s . Instead, the pointwise data (not published in the

original r epor t s ) were obtained f r o m the LRL l 4 exper imenters (0.03-60 eV)

and f rom the Saclay (F rance )15 group (4.4 eV - 20 keV). Each of these se ts

of data were numerical ly integrated in SOPHIST-2 (which was modified so a s

to handle data f rom tape in blocks of up to 1500 data points) over a number of

smal l energy intervals which span the region of overlap (4.4-60 eV) of the

two sets . The fine-group average Saclay data c r o s s sections for each

interval ,

a r e plotted in Fig. 4. Also plotted i s the ratio, [F (Saclay) - T ~ ( L R L ) ] / F

B (Saclay). It i s seen that below 10 eV, the Saclay data yield higher c r o s s F

sections than the LRL data does. We find

E(eV) GLL -646 -1618

235 Fig. 4. His togram of average values of the Saclay U f iss ion c r o s s s e c - - . t ions ,

=F (Saclay) ' , f o r 24 f ine-energy.groups. between 4.5 and 60 eV. Also - shown i s the r a t i o [r (Saclay) - rF (LRL) ] b,F (Saclay). F

whereas, above 10 eV, we find

i"," dE vF( Saclay)

In view of the degree of uncertainty evidenced by the plot of fine-group . , . .

rat ios , this la t te r resu l t s e e m s to indicate sat isfactory agreement between 2 ,

the two se t s of data above 10 eV a s they stand. 14 This i s surpris ing in view of the conclusions reached by the LRL group

in their analysis of the pa ramete r s f r o m the LRL, Saclay, and Columbia

e'xperiments. They found that when al l three se t s of data were normalized to

the parametr ic data of Shore and ,Sailor3' a t the 4.9-, 6.4-, and 7.1 eV , . .ar.. resonances, the Saclay pa ramete r s in the 5-40-eV range averaged 870 higher

than the LRL pa ramete r s ( the standard deviation of individual resonances fo r

each se t was about * l4% f rom the average of the three se t s , though the l a r g e r

r.esonances agreed bet ter than this). Next, LRL renormalized the i r experiment 235

by car ry ing down in energy to fie on to.:theBNL-325 U fission c r o s s section

curve in the 0.03-0.15-eV range. This caused them to reduce their original

c r o s s sections and values of I' by 1870, and they suggest that the Saclay and F Columbia data be s imi lar ly reduced.

Indeed, below 10 eV we have found the integrated Saclay data to be some

1870 higher than the integrated LRL data. However, above 10 eV we have

seen that the Saclay resu l t s a s they stand a r e in essent ial agreement with LRL

's' resu l t s , instead of being 1870 l a r g e r ( o r 2670 l a r g e r i f one allows for the average

8% higher Saclay pa ramete r s p r io r to renormalization of the LRL data).

a About al l we can rea l ly conclude f rom al l this is that pa ramet r i c

representat ion of u~~~ fiss ion data is of r a the r l imited value. F o r our

purposes, the pointwise Saclay data was used a s i t stands above 10 eV, where

i t appeared to possess l e s s sca t te r than the LRL data; below 10 eV, and down

to 0.25 eV, the pointwise LRL data appeared to be the s t ronger , and was

therefore adopted.

A significant depar ture f rom the ZOOM l i b r a r y c r o s s section se t occurs 2 35

in the use of m o r e recent data for U capture, par t icular ly in the resonance

regions. The origins of the c r o s s sections used in the l i b r a r y se t a r e los t in 4

antiquity. The NOVA c r o s s sections, based on the curves in BNL-325,

Second Edition, Supplement 1 (below 1.8 eV), on the experiment of Shore and 30

Sailor ( 1.8- 3.2 eV), and est imated f rom BNL-325 resonance pa ramete r s

(above 3.2 eV) were much m o r e rea l i s t ic , and much higher than those of the

l i b r a r y set . This i s apparently the p r imary cause for the 77'0 drop in the

calculated valup o f K for the 247:l Spade c r i t in going f rom the l i b i a r y se t to

the NOVA c r o s s sections.

uncertainty in the radiative capture of u~~~ i s 'even g rea te r than that

fo r fission. The capture i s difficult to measure directly. Values obtained by

subtracting fission'and scat ter ing f rom total c r o s s sections a r e uncertain,

because e r r o r s in the original data lead to l a r g e r relative e r r o r s in the

difference. And a s we have seen above, data f rom different sources m a y

differ in normalization, accuracy, etc.

Most effort of late i s focused on determination of a = uC/uF, the rat io

of radiative capture to fission. In resonances, one expects a = r /rF, but V

values of I' a r e ve ry approximate, and the,technique i s of l i t t le use in the Y '

valleys.

SUPERNOVA values of uc above 20 keV a r e based on the data in Ref.

16, which fi ts the experimental values of a of Divens and ~ o ~ k i n s . ~ ~ F r o m

.0.10'-20 k d . ~ , the a curves of ~ e f . ' 18 a r e used. These , a s sume a =..0.5 f r o m

50 eV to 1 keV wi th .a smooth extrapolation f rom 1 keV to the 20-kev 'data .

~ r o k 10-50. eV, the curves a s sume that the pa ramet r i c rat io I' /I' gives . . v F9

a in the resonances (often average values of r a r e used, since measured Y

values a r e sca rce ) and that a = 0.174 (the thermal value) in the valleys. . .

Below 1 e'V, a was obtaink'd f r o m BNL-325,' 'In the important range

f r o m 1'-10 eV, we use a measurement by ~ r e o k s ' ~ of q = v/( l + a ) , the

average number of prompt neutr'ons emkrging pe r total absorption event.

- 19-

Brook's data cannot be used a s i t stands; he presents only q/+, where +' is an unmeasured scat ter ing correct ion to the data. Therefore, a calculation

of this correct ion factor was done with the IBM 7090 Y O G I - N ~ ~ Mqnte Carlo

code ( s e e Appendix B for the detai ls of the calc'ulation). The calculated values

of the scattering cor rec t ion (which. a r e appreciable) and the resultant values

of q a r e shown in Figs . 5-a, b. These values of q a r e the bas is for the 235

SUPERNOVA U capture c r o s s sections f r o m 1-10 eV. It i s seen that even

af ter ra is ing Brook's data by correct ion fac tors approaching two in some

regions, the resultant q curve i s disturbingly low, r a r e l y approaching the

thermal value, q = 2.07.

'That portion of Brook's data f rom 10-50 eV was - not used in generating

the SUPERNOVA numbers. I ts use would have fur ther depressed the a l ready

low calculated values of K for the f a s t e r (lowest M/U ratio) sys tems by severa l

percent. This was undoubtedly the motivation behind the a r b i t r a r y decision

to not use this data; the rationalization was that the sudden drop . in q above

30 eV made the data suspect.

Unfortunately, repor ts of even higher values of a-have 'now appeared,

too recent ly to include in detail in this report . De Saussure, Weston, Givin,

Russell and Hockenburg ( to be published in Nuclear Science and ~ n ~ i n e e r i n ~ )

have made d i rec t measurement of a fo r u~~~ f r o m 4 eV to 2 keV. P re l imina ry

analysis of their da ta leads to capture c r o s s section's considerably l a r g e r

than those derived.from Brook's data over the range of overlap below 30 eV.

It s e e m s that in the future, SUPERNOVA c r o s s sections above will have to be

substantially increased, wreaking fur ther havoc with our calculations of the

fas te r c r i t s .

The Fiss ion Spectrum

The spec t rum of neutrons produced f rom ei ther u~~~ o r u~~~ fissions 235

i s assumed to be that fit to experimental U data given by Cranberg e t al. 33

N(E)dE = ke - E / ~ ' ~ ~ ~ s i n h d m dE V

where k i s a normalizing constant. . .

The SUPERNOVA fi, which sum to unity, a r e just

F i g s . 5 a and 5b. V a l u e s of T-, = 2.414/(1 t a ) d e r i v e d f r o m B r o o k ' s U 2 3 5

d a t a f o r 1-50 eV. A l s o shown a r e the Monte C a r l o v a l u e s of. P, ' t he s c a t t e r - ing c o r r e c t i o n u s e d t o d e r i v e T-,.

(Th i s spec t rum was used for fine group flux weighting calculations f o r energ ies

> i MeV, and was used a l so for the fine group f iss ion spectrum. )

Neutrons pe r Fiss ion: v(E)

.The number of neutrons given off pe r f i s s ion has long been known to be

a function of the energy of the neutron causing the fission. F o r u ~ ~ ~ , Meadows -

34 and Whalen give f o r p r o m p t n e u t r o n s . .

v(E) = 2.414 + (0.097 * 0.008)E (E in MeV).

We add 0.016 delayed neutron p e r f iss ion and use

v(E) = 2.43 + 0.1 E ( E in MeV).

WHIZ, ZOOM, and ANGIE have provisions f o r a different v f o r each

group

vJ = ( C + DEJ) ZOOM and WHIZ

= (C , + c2cJ) ANGIE.

Our v(E) gives C := C1 = 2.43

D = C = O . P 2

E~ = cJ = average energy for f iss ions in group t t j . l t

This energy dependence inc reases K by only a few tenths of 170 in the fas tes t

Spade o r Snoopy, and by even l e s s in the o thers , but i s included for comple te - ness .

Notice that the new v = 2.43 i s some 27'0 lower than, the v = 2.4858

cur ren t ly in use in conjunction with the exist ing l i b r a r y tape numbers . A -270'

change in v means -2% in K. 3 5 Recent measu remen t s fo r u~~~ a t 1.5 MeV give v238( 1.5) = 2.65 + 0.09,

whereas our v (1.5) = 2.58. ZOOM, ANGIE, and WHIZ have provisions 2 35 fo r only one .set of v values. In o r d e r to get the c o r r e c t f iss ion contribution - f r o m u ~ ~ ~ , the I1rawtt u~~~ c r o s s sect ions f r o m WHIZ have been adjusted a s

* I

follows to give the SUPERNOVA set.

2.65 u ( r a w ) u (238) = - F 2.58 F

uc( 2 38) = uc( raw) + ' ( 1 - - :I::) aFi raw).

These correct ions keep the total absorption c r o s s section invariant, but

i nc rease the neutrons p e r actual u~~~ fission by the des i red amount.

It is interesting to note that there i s no experimental evidence ( s o f a r

a s we can determine) to refute the possibility that +(El might de.part markedly

f r o m the postulated average v(E) = 2 .43 t O.1E dependence in thk resonance

region.

The Transpor t C r o s s Section

In monoenergetic asymptotic diffusion theory with l inear anisotropic

scat ter ing, the diffusion constant i s given by

where K is determined f rom

. a

and = average value of lab scat ter ing cosine, ' 2 = 2 + 2 = total c r o s s a s . section. (The above expression i s derived, except for a typographical e r r o r ,

on page 396 of Glasstone and Edlund. 3 6

The validity of applying these formulae to individual groups in multi-

group calculations i s open to question; nevertheless i t is commonly done and .

i s done herein.

In the' ZOOM and ANGIE codes, the quantity called the "group macroscopic

t ranspor t c r o s s se.ctiontt i s 'defined in t e r m s of the'diffusion constant for that

group such that

(dropping energyindices for convenience): This .leads to

If the charac ter i s t ic equation for K . i's expanded in powers of 2 / 2 the above a definition gives

L

Alternatively, a s Ca + C (i. e. , pure absorber )

[Some confusion a r i s e s in the l i te ra ture because various authors define

- - ZTR

C( l - p) o r Cs( l - p) o r C - pCs.

Then in the general case they cannot (and do not) use the simple relationship

D = 1/3CTR. Only in the case of pure scattering does D C / K 2 = 1/3CTR . a

f o r these definitions of C T R9

for only then do al l the variously defined C TR become identical. ]

It i s obvious that the diffusion constant appropriate for a homogeneous ,

mixture i s a function of the composite c r o s s sections for the mixture. And

only in the limiting cases of pure scat ter ing o r pure absorption i l lustrated

two paragraphs.above i s our C a l inear function of these c r o s s sections. TR

Unfortunately, ZOOM and ANGIE a r e coded to calculate a CTR for a m i x t u r e ,

as a l inear sum of (masroscopic) t ranspor t c r o s s sections which a r e input

f o r the individual mater ia l s . If the individual ma te r i a l C i s based on the TR

proper t ies of that mater ia l alone ( a s i s the case with the cu r ren t l i b r a r y tape

numbers) the mixture C thus obtained will in general be in e r r o r . The TR

SUPERNOVA t ranspor t c r o s s sections for each mater ia l have been modified

( a s explained below) s o that the l inear summation in ZOOM o r ANGIE will

.give, the "correct CTR ( S O f a r a s the theory discussed in this section applies)

fo r the par t icular mixture for which the c r o s s sections a r e quoted.

WHIZ'^^^ been modified so a s to calculate values of C by solving TR

the charac ter i s t ic equation fo r K and now requi res values of ( instead of

uTR) in the fine group input for each mater ia l . (Values of F for e last ic

scat ter ing come f r o m SOPHIST-5; for (n,2n), (n ,n l ) , and crystal l ine scat ter ing

they come f r o m theoret ical es t imates) . WHIZ calcu1,ates fine group CTR1s,

fo r the mixture, solves the diffusion equation, and calculates collapsed group

values of F and the various c r o s s sections fo r each ma te r i a l and f o r the . i

mixture. F r o m these values , for the mixture WHIZ calculates C f o r the - TR

mixture. F r o m the c r o s s sect ions, p,!s and number densi t ies i t calculates

i f0.r. each mater ia l , sums these to get values of X for the mixture a s i s

TR .done in ZOOM and.ANGIE, and finds the rat io of the to the "summa-

i tion" ZTR fo r the mixture. The individual mater ia l f i icroscopic t ranspor t

c r o s s sections a r e then multiplied by this rat io ( a correct ion of severa l percent

a t t imes ) and a r e presented a s the SUPERNOVA t ranspor t c r o s s sections.

These cor rec ted c r o s s sections, when used in ZOOM i n d ANGIE insure that

the group diffusion constants for the mixture a r e what the theory outlined at

the beginning of this section predicts.

Crys ta l Scattering

One well- recognized weakness in e a r l i e r t rea tments of thermal sca t t e r -

ing has been the assumption of a gas model for the modera tor , even though

the modera tors of in te res t to us a r e crystal l ine, with bonds that profoundly

affect the scat ter ing of neutrons with incident energies comparable to o r l e s s

than these crystall ine bonds.

'CRYSTAL SOPHIST i s an unpublished IBM 7090/7094 code developed

a t LRL to calculate scat ter ing ma t r i ces for simple cubic Bravais la t t ices

using Debye Model theory (See Ref. 37 for theory). Neither ~ e 0 ' n o r graphite

a r e Debye crys ta l s ( n o r a r e they simple cubics), s o the best choice of Debye

6 fo r each i s difficult. ,

F o r .graphite ;. a. Debye 0 = 1020% was used. ,The model was checked

by a WHIZ calculation of the flux spec t rum for one of the poisoned graphite

a s s e m b l i e s studied by P a r k s e t a1. 38 The resu l t s agreed well with the

experimental data and the calculations of Pa rks .

The B e 0 Debye model i s fur ther complicated by the t ransi t ion f rom

the average crystal l ine atomic m a s s of 12.5 to the higher-energy gas model

m a s s of 9 (Be) o r 16 (0). CRYSTAL SOPHIST postulates a smooth t ransi t ion

of the fo rm

where Mc

= crystal l ine m a s s = 12.5 for B e 0

M~ = g'as m a s s = 9 o r 16

E ' t E ' =, sum of initial and final neutron l ab energ ies

k 6 ~ = Debye tempera ture in energy units

Z = unknown parameter . m

Some experimental values of inelastic ( in the lab) scattering c r o s s

sections for B e 0 a t various tempera tures have been reported by Begum e t al . 39

Calculated values for three tempera tures were obtained f rom CRYSTAL

SOPHIST for var ious choices of 8 and Zm. The requirement of a fi t to the D

experimental data pins down 8 to within 2 o r 3'7'0 and Zm to about 20%. The D .

excellent fit obtained with 8 = 1000°K and Z = 1.3 shown in Fig. 6, was the D m

bas is of our choice of these values for BeO.

To calculate t ranspor t c rossasec t ions for the crystall ine energy region

one needs the appropriate average values of the l ab scattering cosines. The

theory embodied in CRYSTAL SOPHIST (private communication f rom R.. N.

Stuart) 'suggests that be' calculated from;

whe r e 6E C(E) = -

kT = moderator tempera ture (energy units)

k O ~ = Debye tempera ture (energy units)

A = effective atomic m a s s

= function tabulated in Ref. 40. ' e~

This relation was used up to that energy where reached the f r e e gas value,

2/ 3 ~ .

The use of CRYSTAL SOPHIST represents another ma jo r depar ture of

the SUPERNOVA c r o s s section se t f rom that of the ZOOM. library., The

c rys t a l model makes l i t t le change in the calculated K for fas t sys t ems ( e . g. ,

247:l) but lowers the m o r e thermal sys tems (e . g . , 3826~1) by about 1'7'0. F o r -

a , v e r y thermal s y d e m there would of course be no effect: the the rma l flux

will be Maxwellian for any moderation rrlechanism sophisticated enough to

impose det'ailed balance.

THE SUPERNOVA CROSS .SECTIONS

C r o s s Section Calculations

Essent ial ly one mus t know the final flux distribution in o r d e r to prop-

e r ly flux weight the raw c r o s s section data and come up with multigroup

GLL-646 -1621 TEMPERATURE (OK)

1

Fig. 6. Experimental and calculated values of the B e 0 inelastic ( lab system) scattering c r o s s section for . incident neutron energy E = 0.00265 eV, and temperatures ranging f rom 100-900°K.

7- 0 - EXPERIMENTAL - Ref. 39

X - CALCULATED - OD= 1000°K 6 - z =1.3

m ( E = 0.00265 eV)

5 -

Q9 4 -

0

3- 0

0

2 - R

- 0

0

I I I I I I I 100 2 00 30 0 400 500 600 700 800 9 00

constants with which to determine the flux distribution (and reactivity) in the

standard multigroup formulations of a reac tor . Thus one i s rea l ly faced with

an i terat ive problem; i t s severi ty may, however, be reduced through the use

of f iner groupings. A ve ry fine grouping, such a s was used in the generation

;+ of the NOVA c r o s s section set ,4 will a lmost completely eliminate the need for

any prescience in setting the f i r s t flux guess. This procedure, however,

becomes costly of t ime and effort, so we have t r ied he re to match some

reasonable guess of the initial spec t rum with a moderately fine group scheme

(189 groups total). In only one instance was a second iteration on the flux

necessary , a s descr ibed below.

One ve ry useful tool which e a s e s machine t ime requirements on many- 5

group problems i s the WHIZ code, a zero-dimensional diffusion program with

multigroup buckling which flux weights c r o s s sections, then collapses them

into any des i red c o a r s e r grouping. SUPERNOVA numbers were obtained for

each Safanov group by collapsing f rom 10 o r 11 subgroups of roughly equal

le thargy width.

Since WHIZ handles only 130 groups i t was necessa ry to run tandem

problems in o rde r to cover the ent i re energy range. F o r convenience the

energy range was divided into three parts : fas t ( 10.5 MeV- 3.1622 keV),

intermediate (3.1622 keV- 10 eV), and thermal ( 10-0.01 eV). Separate WHIZ

tapes were prepared for each of these ranges. Except a s otherwise noted

herein, fine group c r o s s sections for WHIZ were generated by SOPHIST-2 29

by weighting experimental c r o s s section data with "reasonable" flux guesses .

(Only for U- 238 were c r o s s sections calculated f rom resonance pa ramete r s

by SOPHIST - 2. ) The requisite l a rge energy t ransfer a r r a y s for e last ic

scat ter ing were entered direct ly f rom tapes prepared by SOPHIST-520 (high- 37

energy) and CRYSTAL SOPHIST (low-energy).

The high-energy-group WHIZ was run f i r s t , with fine grouping extending

down one Safonov group into the intermediate energy region so a s to provide

sources for that region, and SUPERNOVA numbers were obtained for groups

. 12- 18. These were used to make the WHIZ tape for the intermediate region

(again with fine grouping extending into the lower energy range), and SUPER-

NOVA numbers were obtained for groups 7-11. Finally, WHIZ tapes were

run for the thermal region, using SUPERNOVA numbers in the higher energy

groups.

. I t was found that the c r o s s sections obtained .in the top two energy .regions

were independent of the c r o s s sections' used. for lower energies , so that no

recycling of this kind was needed.

A look a t the flux distributions obtained f rom the various WHIZ problems

(F igs . 7 - 10) suggests the possibility of improving the SUPERNOVA se t by using

these fluxes a s bet ter weighting functions for the various fine group input

quantities to WHIZ. This was found necessary only for the high-energy region

in the B e 0 sys tems where l a rge anisotropic scat ter ing effects gave marked ,

unexpected changes in the flux distribution, even causing r eve r sa l of the flux

slope. A single i teration proved sufficient and was the only instance where

such a s tep was necessary . In the thermal region the resultant flux spectrum

i s quite different for each M/U ratio. To check the importance of the initial

flux guess , WHIZ problems were run with tapes based on different thermal

spec t ra . The resu l t s agr-eed, showing that the WHIZ groupings were fine

enough so that a single se t of fine group c r o s s seclions and t ransfer a r r a y s

sufficed for getting SUPERNOVA numbers for the var ious M/U ratios. There -

fore we decided to make and use only one WHIZ tape for each energy region,

a g rea t saving in t ime and effort.

This decision does involve some inaccuracy in the resonance region. 2 35

T h e f i n e - g r o u p U c a p t u r e a n d f i s s i o n c r o s s s e c t i o n s u s e d i n m a k i n g t h e

WHIZ tape were calculated in SOPHIST -2 using i t s resonance flux depression

option for homogeneous mixtures of modera tor anq absorber . The flux used a s

a weighting function in the fine group calculation i s

where @ ( E ) = input flux to SOPHIST-2,

u(E) = c r o s s section being weighted,

u = modera tor scat ter ing c r o s s section pe r atom of absorber . S

The SOPHIST-2 was run with numbers hopefully appropriate to the B ~ O / U 2 35

= 247 Spade, the fas tes t Spade c r i t assembled a t the t ime. The use of these

2 4 7 ~ 1 fine group c r o s s sections (which incorporate flux depression fac tors of

seve ra l percent in the m o r e prominent resonances) in WHIZ calculations for

m o r e dilute sys tems was excused on the bas is of the decreasing importance

of the resonance region for these m o r e thermal sys tems. Subsequently, these

Figs . 7a and 7b. Scalar flux spectra for the various Spade (BeO) crits plotted from the fine group WHIZ output a s EQ(E) - v s E.

Figs. 8a and 8b. Scalar flux spectra for the various Snoopy (graph rits plotted from the fine group WHIZ output a s E6(E) v s E. - > .

Fig. 9. Low-energy scalar flux spectra for the various Spade (BeO) crits plotted from the fine groups WHIZ output as @(E) - vs E.

Fig. 10. Low-energy scalar flux spectra for the various Snoopy (graph- ite) crits plotted from the fine group WHIZ output as @(El vs E.

7

numbers were a l so used to calculate c r o s s sections for the m o r e recent

120:l Spade where the resonance region i s quite important. Recalculations

with a m o r e appropriate flux depression factor might yield somewhat smal le r u235

capture and fission c r o s s sections for the 1 2 0 ~ 1 Spade and the 301:l

Snoopy. >

2 35 Table 1Va gives the fine group U fission and capture c r o s s sections

(and a] used in WHIZ fo r a l l SUPERNOVA calculations. Table IVb shows \

2 35 the resonance inte.grals calculated for a ve.r.y diliite (33,000:l C/U

sys tem, and experimental values for the f iss ion resonance integral f r o m

Ref. 41. (Unfortunately, no such data i s available 'for capture. )

A single se t of fine-group c r o s s sections might suffice a s input to

WHIZ, but the standard 18 ZOOM groups a r e too coa r se to expect a single

set..of 18 group numbers to be adequate for a l l M/U rat ios . Therefore

SUPERNOVA c r o s s sections were calculated for each of the var ious Spade

and Snoopy M/U ratios. This i s a major and important depar ture f rom the

present l i b ra ry set which uses one se t of c r o s s sections for all systems. ,

Using the 247: 1 numbers to calculate the 3826:l c r i t gives a value of K nearly

5% below that calculated f rom the 3826:l set . It i s strongly recommended

that c r o s s sections applicable to the M/U rat io of in te res t be used wherever

possible.

In addition to the c r o s s sections calculated for the Itextrapolated height"

homogeneous c r i t s , SUPERNOVA numbers were generated for each of the

c r i t s using the fluxdisadvantage factor feature of WHIZ (which now incorporates . 42

the SLLAB routine). F o r each of the c r i t s , WHIZ calculated fine-group

flux disadvantage factors appropriate to the foil thickness and spacing,

(excluding the three t r iangular foil c a s e s which we do not attempt to calculate),

multiplied the Oy c r o s s sections by these fac tors , and proceeded a s before.

The final SUPERNOVA numbers obtained a r e to be used a s effective homogeneous

c r o s s sections; the self-shielding factors a r e incorporated in them and a r e

not available separately. '

All of the SUPERNOVA 18-group c r o s s sections and t r ans fe r coefficients

a r e now available on the SUPERNOVA l i b r a r y tape. ~ a d r o s c o ~ i c numbers

a r e l is ted fo r mixtures corresponding to each of the homogeneous and

homogenized-foil Spade and Snoopy c r i t s of Table V ( the mixtures a r e calcu-

lated f rom the number densi t ies of Table I). Microscopic numbers a r e given

for the individual e lements for a l l the homogeneous c r i t s , .and for the 120: 1,

247: 1, and 493: 1 foil spades.

- 34-

T a b l e I v ~ . F i n e - g r o u p s t r u c t u r e a n d u~~~ g r o u p d a t a .

E u u F a = uc/uF . G r o u p max c

.1 89 '10.5 X l o 6 0 .0 , 1.876 0 .0

-188 0.0 1.847

187 , 0.0 1.696

186 0.0 .lo 337

185 0.0 1.192

184 . . ' 0.0 1.177

1 8 3 '. 0.0 1.183 .

18 2 0,. 0 1.208 Y'

181 0.00705 1.241 0.00568

180 0.02095 1.283 0.01633

179 0.03246 .. 1.317 0.02465

178 0.040.07 1.336 0.02999

177 0.046 6 3 1.352 0.0 3449

176 0.05324 1.36 0.03915

175 0.06 1.36 0.04412

174 0.06,725 . 1.357 , O . 04956

1 7 3 0.07744 1 .333 0.05809

'1.7 2 0.08803 . 1.306 0.06740

171 0.09848 1.280 ,O. 07694

: 1.70 0.1089 1 .253 0.,0869 1

169 0.1168 1 .233 0,. 0947 3

168 0.1223 1.220 . . 0.1002

,167 . 9.0 x l o 5 0.1275 1.206 0.1057

166 . . . 8 . 0 X 1 05' 0.1384 . 1.208 0.1146

, 1 6 5 5 0.155 1.225 0.1265 , ' . '7..0 x 1 0 , . , . 164 6.0. .

1 6 3 5.5

162 5.0 ...

,161 . 4.5

160 4.0 . .

159 3.6

158 3.1622

0.1677 1.238 0.1355 . .

0.1758 . . . 1.246 0..141 1

. . 0.1918 1.265 ' 0.1516

.0.216.3 , ' : 1. 1 .295 , 0.1670

, 0.2381 , 1:323 0: 1800

0.2.58 1 1.348 0.1915

0.2746 . . - 1.368 0.2007

157 2.9 v 0.2857 . ' 1.382 0.2067 . .

2 35 Table IVa . Fine -group s t ruc ture and U group data . , (Continued)

I ' E u u a ,= u ./u

i Group max c F c F I .

-36- .

Table I V ~ ; F ' i ie-group s t r i c t u r e and u~~~ group data. ( ~ b n t i n u e d )

E u . Group max . . c

1 2 3 7.0 1 1.750

104 9 . 0 X l o 2

i 0 3

102

i o i

100

9 9

9 8

9 7

96

9 5

Table IVa. Fine -group s t ruc ture and u~~~ group data. (Continued) . .

I I E 0-

0 - ~ a = u / 0 - Group max c c F

90 2 1 I 9.768 19.52 0.5004

Table IVa. Fine-group s tructure and u~~~ group data. (Continued)

E 0- Group . max c "F a = . ~ c / ~ F

235 ,Table IVa. Fine-group s t ruc ture and U , .group data. (Continued)

E . .

0- ,.% - a = i 7 / u F Group m ax C .

a . . . .

Table IVb. Integral data. : , .

. ,

.' E: (eV) - ' ~ e s o n a h c e ~ e i c t i o n . c integral Source

Calculated uF 0.44 276.8 SUPERNOVA

Calculated u : 0.4 154.47, SUPERNOVA C

Experimental uF , 0.44 27 1 Ref. 42

~ x ~ e r imenta l u F 0.44 274zt11 . Ref. 42

T o i l l u s t r a t e the changes resu l t ing f r o m dif ferent M/U r a t i o s , Appendix

C lists the SUPERNOVA mic roscop i c c r o s s sec t ions and t r a n s f e r co'efficients

. f o r t h e individual ma t e r i ab s (I3e9$. o1 6 9 ~ ~ ~ 5 , and u ~ ~ ~ ) in the homogeneous .

247:l and 3826:l Spades.

Tab l e V. SUPERNOVA mult ip l ica t ion f a c t o r s , K , f o r Spade and Snoopy c r i t s .

F o i l F o i l K Spade s i z e spac ing B a s e , (W/AV = 0.2 in

B ~ O / U - 2 3 5 (mils) ( in. ) ( c m X c m ) K groups 6-10)

120 Homogeneous 60.96 X 60.96 0.9620 1.0075

J 16 1 - 60.96 X 60.96 0.9659 1.0079

247 Homogeneous 60.96 X 60.96 0.97107 1.00415 8 1 60.96 X 60.96 0.97323 1.00395

49 3 Homogeneous 60.96 X 60.96 0.98768 1.00817

-1 - 4 1 60.96 X 60.96 0.98870 1.00825 8 2 60.96 X 60.96 0.98467 1.00326

19.20 Homogeneous

1 1 1 2 2

1920 - A 2 2 1920 3 3

38 26 Homogeneous 91.44 X.91.44 1.00296 1.00665 1 2 91.44.X 91.44 1.01356 1.01723

.. . . 2 4 91.44 X, 91.44,. 1.01054 1.01417

.76 60 Homogeneous 91.44 X 91.44 1.01571 1.01776

. 301 Homogeneous 107.95 X 107.95 0.95871 1.00228

4 4 0.5 107.95 X 107.95 0.959 24 1.00171

6 0 3 Homogeneous 123.19 X 123.19 0.96218 0.99280

C 2 0.5 123.19 X 123.19 0.96536 0.99531

. .. 1206 Homogen.eouS 123.19 X 123.19 0.9849 1 1.00363

,+, .- 2 1 123 .19 x 123.19 0.98377 idoo200 .

. . . 2355 . . . Homageneous 123.19 ~ ' 1 2 3 . 1 9 .0.99518 1.00603

1 1 123.19 X 123.19 0.99036 ' ldOQ103 - . .

-41 -

Results

ZOOM calculations of K for each of the Spade and Snoopy c r i t s of Table

I a r e given in Table V and a r e plotted'vs - ~ / ~ ~ ~ ~ ' i n ~ i ~ . 11'. The K for the. .

extrapolated height t'homogeneous " c r i t s a r e replotted in Fig. P 2, along with

the corresponding values fro'm the l i b r a r y c r o s s sections and the values for

Spade c r i t s f rom the NOVA, set. (The K for a l l Spades include the pedestal

cor rec t ions of Table 11.) Note that in going f rom the l ib ra ry numbers to the

SUPERNOVA se t the slopes of the curves have changed sign. This effect

(evidenced also by the s imi l a r behavior of the NOVA curve) i s caused in near ly

equal pa r t s f rom weighting the low energy u~~~ c r o s s sections with fluxes

appropriate to the M/U in question instead of using a single se t for all M/U

rat ios , and f rom the increased resonance region u~~~ capture in the SUPERNOVA

set.

. Unfortunately, the new curve for K i s not a constant 1,O ( the swing in

K between fa s t and thermal sys tems remains a t m o r e than 4%). The

s imi lar i ty in slope of the Spade and Snoopy curves indicate that the l ikely

chief villain i s that mater ia l common to the two c r i t s , Oy, and pr imar i ly ' '

u~~~~ The U 235 thermal data i s well documented, and the SUPERNOVA K's

for the m o r e thermal c r i t s a r e gratifyingly close to unity, so i t i s likely that

the thermal group c r o s s sections a r e pret ty good.

The medium fission energy for the.247: 1 Spade i s in group 6; for the thermal

Spades ( 3826: 1 and 7660: 1) i t i s in group 2. The corresponding shift in the

Snoopys i s f r o m 8 . to 2. In o r d e r to r a i se K by a few percent for the fas t

sys tems but without much change to the thermal K, i t s eems necessa ry to

increase the effectiveness of U235 in the res0nanc.e energy region ( f rom a

few eV t o . a few hundred 'ev) . It i s just this energy range ( a s previously

mentioned) where 'experiments a r e difficult and existing se t s of data a r e

often inconsistant.

The calculated K fo r the fas t c r i t s could be ra i sed relative to the thermal

sys tems by lowering the values of a ( the ra t io of radiative capture to fission) 2 in the resonance region. Unfortunately, what experimental data there a r e

indicate that the values, of a used in generating the SUPERNOVA se t ( s e e

, a l ist ing in Table IV) a r e a l ready to low in the resonance region.

In l ieu of pushing the u~~~ cap tu re ' c ross section da tabe low the believable

l imi ts , we offer an al ternate means of achieving the same result . If v ( the

. .

0.008/1 r . . . X - SPADES

0.97 H x 0 - SNOOPYS

H - HOMOGENEOUS 0.016/1 X 0 0.002/0.5

Foi l th ickness (in.)

0.96 0.0004/0.P o*oox/y = Foil spacing (in.)

8 H / u ~ ~ ~ = 120 247 493 986 1920 3826 7660

GLL -646 -1628 - M/U 235

Fig . 11. Values of K for the var ious Spade and Snoopy c r i t s calculated f r o m the SUPERNOVA c r o s s sec- t ions , plotted v s M / u ~ ~ ~ (da ta f r o m Table V). -

Fig. 12. v a l u e s of K for the "homogenized" Spade and ~ n o o ~ ~ c r i t s calculated f r o m the l i b r a r y , NOVA, and SUPERNOVA c r o s s sec t ions , ' plotted - vs M / u ~ ~ ~ .

number-of nascent neutrons per f iss ion) i s increased severa l percent in the

resonance energy groups , the calculated K-vs-M/U - curves for both Spades

and Snoopys become very good. Table V and Fig. 13 give the calculated K

for each of the c r i t s considered (Spades and Snoopys, homogeneous and foil)

when A v = 0.2 (2: 870 of 2.43) i s added to groups 6- 10. (This i s n e a s i l y done in

ZOOM by setting EJ = 2 in the C + D E ~ = 2.43 t 0. 1Ej t e r m for j = 6- 10.)

We can find no experimental measurement of v in the resonance region

to t e s t the hypothesis that the average v i s higher there (or even different).

Of in te res t , however, a r e recent experimental measurements by Cowan 43

e t a l . of relative abundances of a few fission fragments f rom various U 2 35

resonance f iss ions. His l imited r e su l t s show that the fission fragment

distribution differs f r o m resonance to resonance and for the few isotopes

measured w a s noticeably different f r o m thermal f iss ion resu l t s ; but the

measurements were nowhere near complete enough f o r a determination of V .

Those reluctant to consider such an a r b i t r a r y change in v a s proposed

may be happier to consider that identical r e su l t s may be obtained in the

following manner: the effect on K of changing v for a group by a given per -

centage i s exactly the same a s would have resulted f r o m changing the group

f i ss ion c r o s s section by the same percentage and adjusting the group capture

c r o s s section in the opposite direction so a s to keep the total c r o s s section

invariant. Note, however, that this yields values of a for these groups

which a r e lower than experimental data s e e m to warrant .

Comments and Future Efforts -

One obvious shortcoming in the old l ib ra ry se t of c r o s s sections when

used to calculate the Spade and Snoopy c r i t s i s that only one set of U 2 3 5

c r o s s sections is available (a set whose thermal-group values s e e m designed

for ve ry fas t c r i t s ) . This defect i s cor rec ted in the NOVA and SUPERNOVA

se t s .

The explicit handling. of anisotropic scat ter ing in the SUPERNOVA se t

has increased K fo r Spades by 3-5.70. La rge r values of a ( r a t io of capture

to f iss ion in u ~ ~ ~ ) have decreased K fo r the f a s t s p a d e s and Snoopys by about

6-770. Lower Be (n, 2n) c r o s s sections have reduced the Spades nearly 170.

A lower value for the thermal v reduces a l l K by 270. Crystall ine kffects

lower K for the more the rma l c r i t s by about 170. More consistent evaluation

of t ranspor t c r o s s sections have changed individual group leakages by a few

percent , but the overal l change in K is 'probably l e s s than 170. In general

X - S P A D E S - S N O O P Y S

H = HOMOGENEOUS F O I L THICKNESS(in.1

O . O O y ~ = F O I L SPACING ( i n . )

Fig . 13. Values of K fo r the va r ious Spade and Snoopy c r i t s ca lcula ted f r o m the SUPERNOVA c r o s s s e c - t ions , but with A .= 0.2 added to groups 6-10 (da t a f r o m Table V) .

the basic c r o s s section data , the theory, and the techniques used in generating

the SUPERNOVA c r o s s sections a r e substantially better than those used in

preparing the l i b r a r y set.

It might reasonably be expected that all the modifications made in

generating the SUPERNOVA c r o s s section se t would a l so resu l t in improve-

ments in the calculations of other pa ramete r s - prel iminary calculations of

a number of these effects have been done. Each of these a r e a s will be reported

on in the future in detail , but the following comments can be made here:

1) Neutron lifetime a s compared with experimental prompt decay

constant measurements with pulsed neutron sources on the Spades -

the ZOOM l i b r a r y discrepancy has been halved.

2) Reactivity of sheet and bar experiments in Spade assembl ies - again,

the e r r o r resulting f rom use of the l i b r a r y tape c r o s s sections i s

cut in half o r l e s s .

3) Hot BOX^^ calculations of the tempera ture effect on reactivity -

significant improvement in the calculated value of this effect; the

mos t sensit ive ( 1 120: 1) B ~ O / U - 2 3 5 Hot Box i s now in essent ial

agreement with the experiment.

F u r t h e r effort will be directed a t obtaining c r o s s sections of var ious

ma te r i a l s which have been checked against the sheet and ba r experiments;

c r o s s sections a t elevated t empera tu res for use in dealing with Hot Boxes;

and, finally, a complete set will be developed for a study of the Tory IIC

configuration.

Effort i s underway to improve t ranspor t c r o s s sections by bet ter theory,

and by numerical comparisons of leakages and fluxes f rom ZOOM and DSN.

The detai ls of fitting B e 0 and graphite to a Debye model c rys t a l a r e

s t i l l uncertain, and al l crystal l ine effects a r e sti l l somewhat tentative.

ACKNOWLEDGMENTS

The authors would l ike to thank Mrs. N. Groteguth for h e r considerable

help with the calculations, plotting, and data handling necessa ry f o r this

compilation. .

APPENDIX A

9* (n,2n) Neutrons via the 2.43-MeV Be Mode

. . 25

Marion e t al. have measured angular distributions for the inelastic 9 9- Be (n ,n ) Be reaction a t 3.5, 4.1 and 5.0 MeV. Thei r resu l t s can be

1 reasonably fi t by an angular scat ter ing probability

- - d p P C = 6 = Average center-of-mass cosine of neutron. C C

Upon converting to the lab f r a m e one gets ( to 1st o rde r in p)

P (p)dp = ( 1/2 + 0.359p)dp = , l a b distribution for neutron N1

and

9 :: Pq,(p)dp = (1/2 - 0.137p)dp = l ab distribution for Be .

Elementary calculations invoking conservation of energy and momentum

give

= 0.82 E l - 2.187 + 2p [ 0 . 8 1 ( ~ ' ) ~ - 2.187 E ' ] C

= l ab energy of neutron

f o r an initial lab energy E t with the neutr.on emerging a t center-of- . : - . . :

.ma&s cos.ine.-p . We approximate these 'by evaluating. the brackets a t E ' 3 MeV C

(excitation of 2.43-MeV level i s strongly peaked. around E ' - , 3 for fission -

neutrons) and by assuming p - - pc = 1/6. This gives our l ab energy and angle C

distributions as:

-48 -

9 :: We mus t now consider the decay modes of the Be 2.43-MeV level.

. .

10- l6 9 :: sec Be f ~ e ~ + N - a + a + N

Mode (A): ~ e ~ * + = ~e~ t N ( Q = 0.768 MeV)

One easi ly establ ishes that the neutron lab energy i s

\ r 9>k

where E i s the l ab energy of the Be , p i s cosine -of angle of neutron in ,p + 6

Be f rame. We use Eq* =

0.17 E ' - 0.243 a s before and average over p . C

Then

We use this average a s the l ab energy distribution for the neutron emitted

through mode (A). . .

Mode (B): ~ e ~ * + a + ~e~ + a t a t N

2.49 fW 2.432 'A ' , X 'lo-'z1 s i c

9 :: . . Be

1..570

Here things a r e cbfnplicated ,. . . ,by-the fact that the inte,rmediate s ta te is

energet ical ly forbidden classically. ' However, Heisenberg 's uncertainty

principle shows that the energy of such a short lived s tate can not be precisely

pinned down.

AEAt 2 R = 6.58 MeV-sec. -2 1

F o r At = 2 X 10 sec , AE - 0.33 MeV i s the uncertainty in the energy of the 5

a t He . We assume the a has 5/9 (0.33) - 0.2 MeV energy in the ~ e e * f r ame . 5 9 96

If the He i s emitted isotropically in the Be f r a m e , and the N i s emitted - 3

isotropically i n the He f r a m e , one ncan show that the average lab energy of

the neutron

enle ley and ~ u n z ~ ~ find that approximately 7% of the decays proceed via

mode (A), 9370 via (B) and claim (C) very unlikely. Therefore, we get for the 9:x

average lab energy of the neutron emerging f r o m the 2.43-MeV Be level

Angular Distribution's

F o r the angular distribution previously discussed we find the average

lab cosine fo r the neutrons which excite the 2.43-MeV leve l ' ( l / 4 of the total

when energy permi ts ) . .

. - If we a s s u m e t h e other neutrons come off with the average p = 2 / 3 ~ associated

with isotropic elast ic scattering, the average value of p . i s

This i s perhaps the bas i s for the distribution used in the Br i t i sh "Barn Book. ' I

APPENDIX B

Monte Carlo Correct ion to the Brooks Data

The experimental arrangement i s sketched in ,Fig. 14-a. In the exper-

iment two thick uranium disk ta rge ts a r e used, and th ree time-of-flight runs

a r e made to measure :

a. BROOKS EXPERIMENTS

DETEC TOR

I t - ! ! a. BEAM TUBE

URANIUM DISK

\ , : SOURCE PLANE, :

b. YOGIREGIONS

4

. .

Fig. 14. The Brooks exper iment and the Monte Ca r lo representa t ion. . . . .

. . . . , . . .. . . . . ,. .. .

2

1 4

* C - = 1 + = ( B ) - 5

3

(A)

1) YA, the count ra te with the f i r s t target a t positioriA;

2) YAB, the count rate with the f i r s t target a t A and the second. a t B;

3) Y, the count ra te with the second ta rge t a t A:

The t ransmiss ion ratio, YAB/yA8 i s a measure of. the reactions in the second

target. If N = number of incident neutrons which would hit A ( i f nothing-at B)

and NF, NC, NS = number of these neutrons removed f rom beam by.f iss ions,

captures , and sca t t e r s a t B,

'AB - N - .NF - NC - NS then - -

N y~ NF + NC

where p -PJ + Nc + N~), / (N= + NC) may be considered a scattering correction. F

Y is simply related to the number of prompt fission neutrons c rea ted in the

second ta rge t by the detector efficiency K,

Solving from-above for N in t e r m s of Y F

A B / ~ A , One gets

This i s the expression given by Brooks, but he makes l i t t le attempt to calculate

p. Instead he presents values of T/p, with es t imates of the energy regions in

which one might expect 1 < P < 1.05.

Since the ta rge t i s not smal l in t e r m s of mean f r ee path (the t ransmiss ion

approaches ze ro at some of the resonances) , a correct ion factor cannot be

derived on the bas is that a scat tered neutron simply escapes. In o rde r to

render the data useful, a Monte Car lo representation of the t ransmiss ion

experiment was se t up a s sketched in Fig. 14-b and solved in the YOGI 32

program. Region 5 approximates a void; regions 2, 3, and 4 were assigned

essent ial ly infinite capture c r o s s sections. The c r o s s sections in region 1

were fi-om the sources used in this repor t for fission and sca t te r data; the

capture was calculated f rom the resonance pa ramete r s of Ref. 10 below 10 eV, 235

and f rom Ref. 18 above 10 eV. Since the samples contained 93% U , 6%

u ~ ~ ~ , and 170 u ~ ~ ~ , capture fo r u~~~ was included. (The T in this experiment 2 34

includes the effect 0 f . 1 ~ ~ ~ ~ and U capture; this was l a t e r subtracted out to . . 235

get the U capture . c r o s s . sections determined with this a p p r o a c h ; ) Finite

energy zones were used, the width being a compromise between problem

running t ime and the fineness of the correct ion desired. The energy breakdown

i s detailed in Table VI, together with the number of par t ic le 'h is tor ies pe r run

and the l a rges t observed fractional variance for the energy meshes of region

1.

Table VI. YOGI problems.

Energy No. of Par t ic le La rges t Var i - Span (eV) E( eV) Histor ies ance ( reg . 1)

1-6 0.1 . .. . . 30,000 . .

0.046

6-10 0.1 20,000 0.046

- -- -- - --

Figure 5 i s 'a plot of the YOGI values for the scat ter ing cor rec t ion P and

the resul tant values of q.

where Ci = capture fraction in region i f rom YOGI. The -.5% var iances i n .

the 'C.. cou1.d be reduced.by longer running t imes in the Monte C,arlo c.alculation, 1

but the e x t r a effort s e e m s unwarranted, since the above .equation indicates

that p. i s relatively iqsensit ive .to sma l l e r r o r s in the C.. The Brooks data 1'

e r r o r flags -a re general.ly.much m o r e than 570, s o l i t t le improvement i n the

resultant T curve could be obtained f r o m bet ter Monte Carlo s ta t i s t ics , o r

f rom the use of a f iner energy m e s h i n YOGI.

F.rom the values of T , we ,calculate ., . .

( v , = 2.41 4 ' since the experiment counts only prompt fission neutrons). It i s

obvious that unc'ertaintie s in lead to g rea te r uncertaint ies in a, part icular ly

fo r smal l a. Fortunately, Brook's data appears s t rongest in the regions of

la rge values of the c r o s s sections and/or a , so that the energy regions where

the derived data i s mos t uncertain tend to be regions of l e s s e r neutronic

importance.

APPENDIX C

The SUPERNOVA Constants

I. Enertrv Groups and F i s s ion Spectrum

Upper i energy ( e v ) fi TT1

18 >' lo7 - 0.1867 4.7

17 m x lo6 0.5033 1 .,5

a Option which 'can be used to increase v by Av = 0.2 -- 8 7 ~ of 2.43, in groups 6 - 10.

11. Representative SUPERNOVA C r o s s Sections . .

On the following pages a r e the microscopic c r o s s sections and t r ans fe r

235: U238; Be9, aid 016 for the z45:1 and j8 t6 i l - , coefficients for U , . . , . . . . . . . . , . . c r i t i c a l assemblies . The complete s e t of SUPERNOVA cros ' s 3&ctions, too .

. , . .

lengthy to include herein, i s available on ' th i SUPERNOVA l i b r a r y tap&". . . . . . . . . . ' . . . . . -.

. . . .

ZOOM SUPERNOVA ZOOM 247 - 1 u 2 35

C SIGMA F S I G P A T R S I G M A S S I G M A CELL F A C T O R GROUP

ZOOM SUPERNOVA Z C O M 247 - 1 U235

TRANSFER COEFFIC IENTS . . .

ZOOM S U P E R N O V A ZOOM 247 - 1 2 38

N 46 ,29785999

C SIGMA T R SIGMA S S I G M A CELL F A C T O R

ZOOM SUPERNOVA ZOOM 247 - l u~~~ ,

....... ... . . . . . . . . . . . -. .... . . . . .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - - - - - . . . . . . . . . . . . . . - -- T R A N S F E R C O E F F I C I E N T S

Loon SUPERNOVA ZOOM 9 -.

247 - 1 Be

C SIGMA F SIGMA T R SIGMA S SIGMA CELL FACTOR

ZOOM S U P E R N O V A ZOOM 247 - 1 ~e~

4 .

ZOOM - S U P E R N O V A ZOOM 247 - 1 0

N 49 , 68860000

. C SIGMA F S I G M A TR S I G M A S S I G M A CELL FACTOR

Z O O M . SUPERNOVA ZOOM 247 . - 1 016 . .

Z O O M S U P E R N O V A ZOOM 3826 - 1 . uZ3' N 46 , 18000000

C SIGMA F S I G M A TR S I G M A S SIGMA CELL F A C T O R

ZOOM S U P E R N O V A ZOOM 3 8 2 6 - 1 u~~~

idAiusi'E R... CtjEF.F.iC ENT.s ...... - . . . -. . - .... - - . ... - ....-.. - .. - - ........ -. . . . . . - .... - - ..... -- .- . -- ... - . .- -. - - . -- -- -. . - - ............ - - - - . - . . . -. .-. -- -- -- -

ZOOM S U P E R N O V A ZOOM 38'26 - 1 u238

N 45,13419999.

C SIGMA F SIGMA T R S I G M A S S I G M A . C E L L F A C T O R

ZOOM SUPERNOVA ZOOM 3826 - 1 u238

ZOOM SUPERNOVA ZOOM 3826 - 1 Be 9

C SIGMA F SIGMA TR SIGMA S SIGMA CELL FACTOR,

ZOOM SUPERNOVA ZOOM . 3 8 2 6 - 1 Be 9

..... - .... - . - .................. .- ... -- ...... - ............. T R A N S F E R C O E F F I C I E N T S

18 17

ZOOM S U P E R N O V A ZOOM 3 8 2 6 - 1

C SIGMA F S I G W A . T R SIGMA S SIGMA CELL FACTOR

ZOOM SUPERNOVA ZOOM 3 8 2 6 - 1 016

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T h i s r e p o r t was p r e p a r e d a s a n a c c o u n t o f Government s p o n s o r e d work. N e i t h e r t h e U n i t e d S t a t e s , n o r t h e Com- m i s s i o n , n o r a n y p e r s o n a c t i n g on b e h a l f o f t he ,Commiss ion :

A. Makes a n y w a r r a n t y o r r e p r e s e n t a t i o n , e x p r e s s e d o r i m p l i e d , w i t h r e s p e c t t o t h e a c c u r a c y , c o m p l e t e n e s s , o r u s e f u l n e s s o f t h e i n f o r m a t i o n c o n t a i n e d i n t h i s r e p o r t , o r t h a t t h e u s e o f any i n f o r m a t i o n , appa - r a t u s , method, o r p r o c e s s d i s c l o s e d i n t h i s r e p o r t may n o t i n f r i n g e p r i v a t e l y owned r i g h t s ; o r

B. Assumes any l i a b i l i t i e s w i t h r e s p e c t t o t h e u s e . o f , o r f o r damages r e s u l t i n g from t h e u s e o f any i n f o r - m a t i o n , a p p a r a t u s , - m e t h o d , o r p r o c e s s d i s c l o s e d i n t h i s r e p o r t .

A s u s e d i n t h e a b o v e , " p e r s o n a c t i n g on b e h a l f o f t h e Commission" i n c l u d e s any employee o r c o n t r a c t o r o f t h e Com- m i s s i o n , , o r employee o f s u c h c o n t r a c t o r , t o t h e e x t e n t t h a t s u c h employee o r c o n t r a c t o r o f t h e Commission, o r employee o f s u c h c o n t r a c t o r p r e p a r e s , d i s s e m i n a t e s , o r p r o v i d e s a c c e s s t o , any i n f o r m a t i o n p u r s u a n t t o h i s employment o r c o n t r a c t w i t h t h e Commission, o r h i s employment w i t h s u c h c o n t r a c t o r .