SPECTRUM AVEP AGED CROSS SECT1 ONS DEDUCED FROM …

SPECTRUM AVEP AGED CROSS SECT1 ONS DEDUCED FROM

BURNUP DATA AND THEIR APPLICATION TO METHODS VERIFICATION(^)

D. E. Christensen

B. H. Duane

R. C. L i i k a l a -

R. P. Matsen

BATTELLE-NORTHWEST PAC1 FIC NORTHWEST LABORATORY

Richland, Washington

N O T I C E

This report was prepared as an account of work sponoo~d by the United States Government. Neither the United States nor the United States Atomic E n q Commission, nor my of thek employees, nor any of the& contractors, subcontractors, or their employe^,

makes any warranty, e x p m or implled, or assumes any legal liability or respondbllity for the accuracy, com- pleteness or usefulness of m y Information, appantus, product or process diilosed, or represents that its use would not infringe privately owned r m a .

(a) This paper i s based on work performed under United States Atomic Energy Cornmi ssion Contract AT(45-1)-1830.

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DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

The determination of r a t i a s of spectrum averaged cross sec t ions from

i so top ic concentrations is described along wi t h t h e i r appl icat ion t o

ver i f i ca t ion of ana ly t ica l , reactor design too l s . The determination

. i s made using spec ia l ly developed l e a s t squares computer programs which

f i t the mathematical transmutation re1 a t ionships t o i so top ic concen-

t ra- t ions measured i n . i r r ad i a t i on experiments. These data a r e u t i 1 i zed

as a bas is f o r evaluating the accuracy of calcula t ional methods used t o

p red ic t bu rnup behavior of nuclear fuel s . Data .obtained from i r r ad i a t i on

experiments using p1 utoni um-a1 uminum a1 loy fuel a r e given t o i 11 u s t r a t e

the techniques and demonstrate how these data a r e used t o ver i fy calcu-

l a t iona l methods.

INTRODUCTION

The accurate prediction of fuel burnup i s important f o r producing

...- r e l i a b l e and low cos t power w i t h nuclear reactor power systems. The

accurate calcula t ion of fuel burnup i s complicated by the number of

var iables w h i ch a r e present i n the mathematical representation of

the processes which a f f e c t the transmutations occurring i'n the fuel . . .

The success of the burnup calcula t ion depends on 'the va l i d i t y o i t h e

theore t ica l and mathematical models along w i t h the nuclear data used

a s input i n the calcula t ion. The accuracy of the burnup prediction

can be ver i f i ed through cor re la t ions of experimen'tal burnup data ,and

resul t s from burnup calcula t ions .

The usual approach t o solving the burnup problem i s t o acquire

data on the var ia t ion of fuel i so top ic composition as a function of

' i r r ad i a t i on time and use these data to eval'uate the meri t of burnup

calcula t ions . In terms of the theoret i cal and mathematical ' transmutation

re la t ionsh ips which describe these data , the time integral of the f lux

and the time dependent spectrum average nuclear cross sect ions a r e

- problem var iables . An approach was developed a t Battel le-Northwest

t o build upon t h i s data base of time dependent i so top ic compositions

by el iminating one of these var iables from the problem. The approach

was t o e l iminate the f l ux-time variable from d i r e c t consideration. . . ~.

,. The transmutation equations a r e c a s t as r a t i o s t o el iminate. the

fl.ux-time var iable and these equations ' f i h e d t o t h e measured i so top ic .

concentrations t o ' y i e l d , r a t i o s of spectrum averaged neutron cross

sect ions (hereaf te r re fe r red to simply as c r m s sect ion r a t i o s ) . The

. -9 ; : :' f i t t i ng - - i s accompl i shed using specially developed 1 eas t squares f i t t i n g .. / T

.; 35: . . . . .. , .. . . .

techniques:. I

The in ten t was to generate data which' focused on evaluation of the

theoretical methods and/or the neutron cross sections used i n b u r n u p

ca-l cul ations. More speci f i cal ly , the principal ob jec t i ve was to obtain

data which can be used to verify the accuracy.of the s ta r t ing point

calculation of neutroni c reactor analysis, name'ly , the mu1 tigroup neutron

spectrum in a uni t ce l l of the reactor. Thus, the isotopic data from

which the cross section rat ios a re deduced have t o be obtained from

fuels resulting i n those regions of the reactor which are typical of the

u n i t cell of i n t e re s t . Thus, a basic assumption i s t ha t the data used

i n the analysis come from samples i r radiated i n the same nuclear environment. $ -

The technique i s not applicable fo r samples from two locations n f the

reactor where neutron spectra are d i f fe rent for a given i r rad ia t ion such

-- as near control rods o r leakage boundaries and i n the center of fuel

bundl es.

' The burnup data to t e s t th i s technique have been obtained by chemical

assays and mass spectrometric measurements of a1 uminy-pl utoni urn (A1 - P u ) ,

urani m oxide (UOZ) , and mixed urani m-pl utoni m oxide (U02-Pu02) fuel

assembl ies i r radiated i n D20 and H20 moderated power reactors. These ma .

measurement techniques are described in Section I I . . .

The lack of a s t a t i s t i c a l analysis method with suf f id ien t . ve r sa t i l i t y . -

t o analyze the data made it necessary to fc.i..rst solve the s t a t i s t i c a l

problem as described i 'n S,ection 111. The data analysis method i s ,

described in Section 1.V and an application fo r A1-Pu alloy fuels which . .

were i r radiated i n the Plutor-ium ~ e c ~ c l e Test Reactor (.PRTR) i s a lso given.

-'.,,! The u t i l i r a t i o n o f t hese data i n veki f i c a t i o n o f burn,upi d c u l a t i o n s

i s descr ibed i n Sect ion V where the r e s u l t s o f , the c o r r e l a t i o n f o r

t he A1-Pu a l l o y f u e l a re g iven i n Table IV. -

P - TRANSMUTATION EQUATIONS

The equations used i n deducing cross sect ion r a t i o s .from ' burnup , .

data are the d t f f e r e n t i a1 equa t lons ,

These equations equate the concentrat ion change o f an isotope w i t h t ime i

t o the loss by decay, the loss by absorption, and the ga in by capture.

Other sources of an iso tope such as ekternal add i t ions or from f i s s i o n

are y c l u d e d from the equations so t h a t only unreplenished f ue l mate r ia l s

are being considered. Also, any c o n t r i b u t i o n from the decay o f a parent

iso tope i s n e g l i g i b l e f o r the isotopes considered and has been ignored. The

i : isotope concentrat ions N which are invo lved are N , 2 8 ' 25 N ~ ~ , and N , f o r

the uranium isdtopes o f -ass 235, 236,' and 230, and tj4', N40, N ~ ' , and . N~~ f o r the pl u ton i un i so topes d f mass 239, 240, 241 and 242. he cross sect ion values are f l u x and volume averaged values and are. def ined

9

as =

Div i s i on o f a l l equations (1) by

ax-

e l im ina tes the f? ux 4 and- time:, t-avari~ables except i n the small terms

which descr ibe t h e decay. $The r e s u l t i s another s e t o f d i f f e r e n t i a l

equations i n v o l v ing cross sec t i on and concent ra t ion r a t i o s ,

Equation (3) w i l l descr ibe t h e 2 3 5 U changes i n the case o f a UOp o r

U02-Pu02 system and 239Pu changes f o r a Pu system. I n t h i s way t h e 2 3 5 U

and 239Pu concent ra t ions a re chosen as burnup i n d i cators. The i r

concent ra t ion changes are the o n l y ones which. a re descr ibed 'as exponent ia l

f unc t i ons o f t h e exposure $ t . . .

Two separate methods, each maki ng ex tens i ve use . o f speci a1 l y developed

l e a s t squares f i t t i n g programs have' been developed t o o b t a i n r a t i o s o f ' & . I '

, f lux and vo i mie. :f n t e g r a t i d cross scctians .T,c;li t q i t a i i ~ i i i 3 j . ,. .

I n . one method ( d i f f e r e n t i a l ) , the several burnup equations descr ibed

by equations (4) a re so lved by separa t ion o f the va r iab les and i n t e g r a t i o n

over a ' l ~ - ~ at/b-on change i n ~ j . Equations o f the form

are ob ta ined where the o j and oi a i -a considered t o br! cons tant over the

smal l r eg ion o f ir. i tegrati;on. Equations (5 ) a re "then 'successively s o l ved

f o r t he cross s e c t i o n rat i .os which bes t f i t the experimental da ta i n the

l e a s t squares sense.

Regression equat ions o f i nve rse polynomial form

a r e used t o rep resen t t he data. The values o f Ak which y i e l d the bes t

f i t t o the data together w i t h t h e i r u n c e r t a i n t i e s a re used t o determine

cross s e c t i o n r a t i o s by i n t e g r a t i o n over an app rop r ia te burnup increment.

S ince

then .

j i j j i j ANi = N ~ ' f (Nb) - Na. f (Na)

where a and b a r e t h e i n t e g r a t i o n l i m i t s o f the burnup increment. The

cross s e c t i o n r a t i o s a r e ob ta ined by s o l v i n g equat ions (5) us ing equat ions

I n another method ( i n t e g r a l ) , a n a l y t i c a l s o l u t i o n s can be obta ined f o r

equat ions (4) by proper choices o f i n t e g r a t i n g f a k t o r s i f the cross sec t ions

G~ r e l a t i v e t o ~j remain cons tant as a f u n c t i o n o f exposure. I n the cases /

s t u d i e d t h i s ik apprbximate ly t r u e f o r a l l cross sec t i ons except 6:O. An

.,,, acceptable e m p i r i c a l form has been found f o r t h e 240Pu r a t i o which s t i l l a l lows

a n a l y t i c a l s o l u t i o n s t o be ob ta ined f o r e$a,tions (4). This ex tens ion

makes i t p o s s i b l e ' t o o b t a i n a s e t of i n t e g r a t e d equat ions which a r e

amenable t o - b e i n g . . f i t t o exper imenta l da ta by the method o f l e a s t squares.

I

-_ A . l e a s t squares f i t o f . the anal-ytical sol at.ions; to:.:the:-experimental . I . -

data i s 'considerably more complex than i s ordinarily encountered. F i r s t ,

only variables N' are' considered as opposed t o a dependent variable and

one or more independent variables. The uncertainties associated with

each data point must be described by a matrix as opposed to the simpler

s i tua t ion wh.ere only the e r ror i n the dependent variable i s required.

Lastly, a s e r i e s . o f related equations must be f i t simtil t a n e o u s l ~ to a . .

s e t of. experimental data, Before this method could be used, i t was

necessary - to develop a l e a s t squares f i t t i n g routine with the required --

sophisti cation to solve these problems.

Each method has advantages and disadvantages to the extent tha t J-

the two methods complement each other. 1,ni t i a l l y , . the experimental -

., . ...

data ,!/as analyzed in each way, T h , e f h s t v e t W ~!4!" ?dWfer::ti.al . "

equations has the advantage of yielding cross section ra t ios as a function

o f exposure. The requi rements o f . assuming a val i e : for:one o f the,. ra t ios

and of neglecting correlated uncertainties between u . r ious N~ fo r a

given sample a re disadvantages. On the other hand, the second method

using integrated equations simul taneously f i t s 2.1 1 the data and correctly

accounts for correlated uncertainties. The integrated equation approach

also provides a rigorous s t a t i s t i c a l analysis of the data and required, no

' a ' p r i o r i values to be assumed for the cross section ra t ios . The dis-

advantage of the 111ethod l i e s i n the r e s t r i c t i v e form of the cross section

- ratiosfiwhich are necessdry t o obtain. >analytical solut ionss~ t.lowever, . i n . * . .

most cases . . . . empirical forms 'can be found which 'dd$quately meet the . . . . .

requirements of the analysis.

I ' - 8- . .....

..... . . ., . .

-.- A . s.&si . . . ti v e assessinen t of the measurement accuracy i s provided ..,+ . . . .

. , by t h e e x t e n t d f the agreement of the resu l t s of the two methods compared 3

7. ih with the s t a t i s t i c a l uncertainty. In addition, the to ta l f iss ions can

be calculated a f t e r the cross-section ra t ios a re obtained by e i the r

method. Comparison of calculated values of the total f iss ions o r

h . percent depletion with those obtained experimentally by 3 7 ~ s o r other

analysis provides fur ther assessment of measurement accuracy.

TRANSMUTATION MEASUREMENTS

Fuel composition and concentrati.ons have been measured as a

function of exposure and used t n correlattons w i t h analytical resu l t s .

The measurements include data from 19-rod c lus te rs trrad+ated in the

PRTR and individual rods i r radiated i n the Experimental Boiling Water

Reactor (EBWR) . (1'?-j The correlations include data from the Saxton

~eactor '?) and Yankee ~eactor 'g) t n addition to tha t from PRTR and

the EBWR. . .-

The types of fuels t h a t are being used and the i r exposures are

l i s t e d i n Table I . All exposures are given i n MWd/MTU except fo r the

Al-Pu systems for which exposures a re given as atom percent depletion

of . the P u . .. . .. ,

The quant i t ies determined by destructive analysis of th&'fuels

a re sample weights and dissolution volumes, l 37Cs a c t i v i t j , [I , P u , .:-

and I 4 8 ~ d contents, and U and P u isotopic. compositions. ..* . ..,.

. . -. .- 0 . I . .

/

(859) UOp (Na tu ra l ) - -

TABLE I ,

FUEL TYPES AND EXPOSURES

PRTR (D20)

I n i t i a l 2 4 0 P ~ (Wt%)

EBWR (BWR)

8 and 26

Maximum Exposure ( a )

(12,13) . . UO, . (6%. Enriched) - -

(14y15) uo, - -

8, 20 and 26 6800

Y an'kee Reactor (PWR)

Sax ton (PIIR)-Core I I - (16,171 -. .- U 0 2 (Natura l )-Pu02 - - -.

(a ) A1 1 exposures a r e i n FIWd/PlTIJ except f o r the A1 -Pu systems fo r which exposures a r e i n atom pe rcen t d e p l e t i o n o f the PIJ,.

Radiometal 1 urgy Procedures . . - , . - - ->

After approximately two months cooling, an e n t i r e c lu s t e r o r

rod i s shipped t o the Radiometallurgy Laboratory f o r cu t t ing and

dissolut ion. After a rod i s sawed in to segments, two samples of fuel

and cladding each approximately 1/2 i n . long a re cu t by an abrasive

wheel from a segment and weighed on an analyt ical balance. . .

For A1-Pu fue l s , one sample i s added to 150 ma solut ion consis t -

ing of 1 - M HN03, 0.01 5 - M Hg(E103)2, and 8 x - M CsN03 and heated t o

60°C. The CS' ions minfmize the exchange of 37Cs for mater ia ls i n

t he glassware. The, H ~ ~ ' ions a c t as a ca ta lys t . Heating i s continued

un t l l a vigorous reaction s t a r t s . Then 85 ma of a solut ion containing

'15.6 - M HN03 and heated t o 60°C i s added. The solut ion removes the

c a r s izatei-ia7 : eavi ng the Zi r ti - - ' p , l p : b j cladci'incj pract icai i y unattackeci.

,The weight of the cTean cladding i s obtained and used t o determine the

fuel weight. A standard cladding sanple weighing 5.886 gm was found

t o lose 40 mg a f t e r being submitted t o the dissolut ion proGess. After

removing the cladding, the solut ion of fuel i s d i lu ted to 250 ,mn, w i t h .

2M - HN03. The vo'iu:ne, and temperature a r e measured and the density

(gm fuellma so lu t ion) of the fuel in solut ion i s calculated.' The other

sample i s s tored f o r l a t e r analys is if needed.

For uranium oxide o r mixed oxide fue l s one sanple i s dissolved i n

117 ma of solut ' ion containing 13.5 - M HN03, 8 x - ?I CsH03 and 0.023 - M

HF. In order t o keep the HF from at tacking the g lass containers,A1NO3

. . i s added. , The so lu t ion i s heated but kept below boi1,inq u n t i l there i s

no more react ion taking place. Af te r disso1utio.n of the fuel.. material the

*I . . - .

I

. . . .,. . . -11- :...

. .

: . cl adciing. i s 'removed ani weighted. !%nother 10 me of water 'containing . . .

, . . . . . ..

-.;: a - ' 1-2; drops:.of HF i s added. to the fuel-solution i f needed for complete

, . j . dissolution. The solution i s reheated for 2 hours to assure tha t

a l l the fuel i s dissolved. After cooling, enough 2 - M HNGgis added

to bring the to ta l to 250 ma. The volume and temperature a re measured

I and the density of the fuel i n solution i s calculated. - . .

Analytical Chemistry Procedures

samples of unirradiated fuel . and .; 15 m t solutions of the i r radiated

fuel a re sent to the Analytical Laboratory for analyses. The 3 7 ~ s . .

content i s measured by gamma-ray spectrometry. The f i ss ion

product 1 4 8 ~ d content i s determined by the isotopic d i l ution-mass

spectrometry technique. (g) The plutonium and uranium 'contents and

! c c t o ~ i c abzndanczs ;re deteririced by i 5 ~ t c p - i ~ dilution arid mass

spectrometry. The plutonium i s also measured by alpha counting, and

the plutonium and uranium a re measured by control led-potential

coulometric t i t ra t ion . -'- 20) The isotopic di 1 uti on-mass spectrometry

method for plutonium and uranium provides the best accuracy. The resu l t s

from the coulometri c ti t ra t ion procedure for pl utoni urn and urani urn, and

the alpha counting procedure fo r pl utoni urn provide independent

verification of resu l t s obtained by isotopic-di 1 ution mass spectrometry. 1, .

I

Coulometric t i t r a t i o n of plutonium i s accomplished us'ing a platinum

electrode ce l l with an e lec t ro ly te of 1M - HC1 containing 1 b grams per

l i t e r ~ 1 ' ~ and a small quantity of urea ('50 pa of saturateid urea

solution i s added t o 5 me of e lec t ro ly te) . he ~ 1 ' ~ e l ininates

- J.LcL.. . ' . . . . i.nte.r!ference . . from fluoride ions and: the urea eliminates interference . ~ . G. L . .

. . . .,.

from n'it';.ite ions. The.pl utonium sample s i ze may range from 100

micrograms up to severdl milligrams. The best accuracy i s obtained with

the larger samples. The plutonium and iron i n the sample are reduced

to P U + ~ and ~ e + ~ , respectively, a t a potential of + 0.3 volts re la t ive

to a saturated calomel electrode. The potential i s raised to + 0.84

. . vol t s to oxidize the p l utoni um and iron to Pu+,' and ~ e + ~ . The pl utoni urn

i s then quant i ta t ively reduced to P U + ~ a t a potential of + 0.56 volts.

The plutonium content i s determined from the total change obtained by

integrating the reducti.on e lec t ro lys is current and subtracting background

currents during the l a s t reduction step. Also, a correction i s made

f o r the very imall fraction of the ~ e + ~ reduced during the reduction

a t + 0.56 volts. In the absence of contaminants which a f fec t the P u # -!

t i t ra t ion , accuracies ( g j o f s. 2 1 a re obtained from solutions having

greater than 50 m g / t Pu. , .

Uranium i s determined coulometri cal ly using a mercury el ectrode

ce l l and a lb1 - H2S04 electrolyte . The normal sample s i z e taken fo r

uranium i s 2 t o 5 mg. The uranium i s oxidized to u + ~ by adding 0.1N . . -

, .

c e r i c su l f a t e solution. he sample i s pre-reduced a t a potential of

+ 0.05 volts re la t ive to a saturated calomel electrode until a residual

current i s reached. This step removes constituents tha t reduce a t a I

more posit ive potential than uranium. The potential i s decreased to I

. I . - 0.30 volts and the uranium quant i ia t ively reduced frorn;~'~ t o u + ~ . 4 . 8

The uranium content i s determined from' t c e , total charge obtained by

integrating the reduction e lec t ro lys is current and subtracting Dack-

ground currents during the l a s t reduction step.

.. . . The r e s u l t s o f the coulometr i c ti t r a t i o n procedures p rov ide , . - .

independent v e r i f i c a t i o n s o f t he Pu and U contents obta ined from ' the

. i s o t o p i c d i 1 u t i o n mass spectrometry method. The v e r i f P c a t i on i s made

f o r 10% o f the ox ide samples and 25% o f the A1-Pu sarr~ples: The '

p lutonium from a measured p o r t i o n o f the sample i s removed by t h e n o y l t r i -

- fluo;roacetone(2Z! (TTA) ex t rac t i on . The amount o f p l u tbn i um i s d e t e r n i ned

t o an accuracy o f +_ 1% by count ing a p a r t i c l e s emitted, by the plutonium.

The t o t a l a l p h a - p a r t i c l e a c t i v i t y o f the p lutonium i s neasured w i t h

27~ alpha- Simpson p r o p o r t i o n a l counter c a l i b r a t e d w i t h a known Pu

standard. Energy ana lys is o f t he emi t ted p a r t i c l e s us ing a surface

b a r r i e r s i l i c o n d iode de tec to r ( o r fo rmer ly a F r i s c h g r i d i o n i z a t i o n

chamber) coupled t o a mu1 t ichanne l analyzer a l lows s u b t r a c t i o n o f the a

a c t i v i t y (23 )o f - 2 3 8 ~ u from the to ta l a c t i v i t y . The we ight of t h e P u ,

238 exc'l ud i ng Pu, i s c a l cu l a ted from the va l ue f o r the remai n i ng a c t i v i ty

u s i n g a weighted average s p e c i f i c a a c t i v i t y based on the i s o t o p i c

. abundances f o r 2 3 9 ~ u , 2 4 0 ~ u , 2 4 1 ~ u , and 2 4 2 ~ u determined by mass '

spectrometry.

The cesium i s removed from a measured p o r t i o n o f t he sample by

tet raphenyl bo ra te (TPB) e x t r a c t i o n . (E) The 622 KeV cja&na-ray a c t i v i t y

em i t ted from t h e cesium (137mBa daughter) i s counted i n a NaI w e l l

c r y s t a l . Cor rec t ions a r e made f o r 34Cs by energy ana lys i s and the '

r e s u l t i s normal ized t o a SRM-4233, Nat iona l Bureau o f Standards,

cesium standard t o o b t a i n the l 37Cs atorns/mg o f f u e l s o l u t i o n .

C. .-

A c o r r e c t i o n f o r I 37Cs decay i s made because the analyses a r e

...? 2 .....

... . . . . .

:.. comp1c:ted months a f t e r the i r radia t - ion was terminated. Tlie cesium . . . . . . .

. ; I a c t i v i t y , ( 1 3 7 ~ ~ ) I s converted t o 2 3 5 ~ and P u f i s s i ons using f i s s i on

., ,& y i e l d s ( a y c ) of 0.0622 2 0.0014, 0.0648 k 0.0019, and 0.0662 t 0.0033

2 G. . - f o r 2 3 5 ~ , 2 3 9 ~ u , and 241~!u, respect ively and a h a l f - l i f e ( z 1 of

. 29.68 k 0.10 years (a ) f o r 1 3 7 ~ s . The 14%d content i s a l so used t o

.-.= determine f i s s i ons espec ia l ly when p o s s i b i l i t i e s of cesi.um rnfgration - : -!.

are. present. In t h i s case, f i s s i on yields(-) of 0.0169 2 '0.0003,

0.0165 r 0.0005, and 0.0188 t 0.0009 a r e used f o r 2 3 5 ~ , 2 3 9 ~ u , and

. . Mass Spectrometry Procedures

Determi nations of the U and P u concentrat ions and i so top ic

compositions a r e i a d e f g ) using two mass spectrometers w i t h lSt order

angular focusing. Each mass spectrometer has a 12-in. radius , 60 degree

magnet s ec to r and uses a surface ionizat ion source. A s ing l e f i lament

o f carbonized rhenium o r a t r i p l e f i lament of rhenium metal produces

metal ions from samples adjusted to contain 10-50 ng of mater ia l .

The same instruments a r e used f o r the determination of f i s s i on

product 1 4 8 ~ d . A s ing l e pre-carbonized rhenium fi lament o r a s i ng l e

rhenium illelal f i lament i s loaded w i t h about 70 ng o f neodymi urn. . In the

case of the metal f i 1 ament, 10-20 m i crol i t e r s of .0.005 M sucrose sol ution

i s dried on the filament w i t h the sample. Metal ions a r e produced from

both types of filaments. I . .

.?. The ion beam c f constant energp of 12 k V i s swept pa3t the detector I . I . c.

ten times i n each di rect ion t o obtain the ' iesul t s . The de tec tor i s an

e lect ron mu1 t ip1 i e r operated a s an in tes ra ted cur ren t devi'ce. The g a i n

i s mass and r e l a t i v e i n t ens i t y dependent so the instrunerrt is calibrated

t

w i t h i s o t o p i c s t anda rds of' urani m and pl utoniurn from the Platioiial

- . . Bureau of . S tandards , and w i t h a n a t u r a l neodymi urn starid.ard f o r neodymi um. , ' -

Samples o f U and Pu a r e prepared f o r m'ass spec t romet ry by s e p a r a t i o n (R)

on a s i n g l e Dowex-1 X 4, 200-400 mesh 'anion-exchange column using a

s e q u e n t i a l e l u t i o n technique. ( I n an e a r l i e r procedure t h e Pu was .. .

sepa ra t ed and p u r i f i e d using a combination o f t r i - i s o - o c t y l ami ne and

thenoyl tri f l uoroacetone e x t r a c t i o n s . )

~ e f o r e t h e s e p a r a t i o n process and a f t e r being d i 1 uted a sample o f

f u e l s o l u t i o n from Radiometal l u rgy i s sp iked w i t h a s o l u t i o n con ta in ing

known amounts o f 2 3 3 ~ and 2 4 2 ~ u . A second sample is processed b u t w i t h o u t 0.

t h e s p i k e s o l u t i o n . The spike s o l u t i o n con ta in s a known amount of 2 4 2 ~ u

which has been c a l i b r a t e d w i th a m e t a l l i c 2 3 9 ~ u s t anda rd from the National

Bureau o f S tandards . I t a l s o con ta in s a known amount o f 2 3 3 ~ c a l i b r a t e d

wi th a na tu ra l uranium 's tandard from the Bureau o f S tandards . The

s e p a r i t e d s o l u t i o n s o f U , (U p l u s 2 3 3 ~ ) s p i t e , Pu, and ( P U p l u s 2 4 2 ~ u ) .

s p i k e a r e then analyzed s e p a r a t e l y f o r their i s o t o p i c composi t ions. The

uran i um and pl utoni urn concen t r a t i ons i n the sample a r e c a l c u l a t e d from

t h e known s p i k e volume and c o n c e n t r a t i o n s , the known sp iked sample

vol ume, t h e 'measured i s o t o p i c composi t ions , and t h e measured 2 3 8 ~ t o 233u

and 2 3 3 ~ u t o 2 4 2 ~ u atom r a t i o s i n sample, sp iked sample, and s p i k e

s o l u t ion . /

A Nd f r a c t i o n i s s e p a r a t e d f o r mass spec t romet ry using t h e anion-

exchange procedure o f Rider',' e t a1 . (!8) The same sample and spi bed' s imp le . -. ..

used f o r t h e U and Pu de t e rmina t ions i s used' f o r t h e Nd. In this c a s e

t he sp i ke s o l u t i o n conta ins a known amount o f I5ONd i n a d d i t i o n t o the

known amounts o f 2 3 3 ~ and ' 2 4 2 ~ u . The f i s s i o n 'produc:t "Nd concentrat ion '

i n t h e sample i s c a l c u l a t e d . from the known sp ike , volume and concent ra t ion ,

t he known sp iked sample volume, the known 5 0 ~ d t o 14'Nd and 1 4 8 ~ d t o 14'Nd

150 . atom r a t i o s i n n a t u r a l lid, and the 1 4 2 ~ d t o . Nd and 148#d t o 1 5 ' ~ d .

atom r a t i o s measured i n the sample, sp iked sample, and sp i ke s o l u t i o n . . . The . .. . . ..

I 4 * ~ d , a sh ie lded n u c l i d e which i s n o t produced d i r e c t l y i n f i s s i d n ,

i s used t o c o r r e c t f o r contaminat ion from n a t u r a l Nd. (Jg)

I t i s o f i n t e r e s t t o no te t h a t w i t h the use o f a " t r i p l e spike",

t h a t i s a s i n g l e s p i k e s o l u t i o n con ta in ing known amounts o f 2 3 3 1 ~ , 2 4 2 ~ u

and 150f.ld, t h a t t h e r a t i o s o f Pu t o U and 148~.~d t o U obta ined by the

i s o t o p i c d i 1 ution-mass spectrometry method a re independent o f t he s p i ke

and sample' volumes u,sed.

'Data Compi lat ion

Because o f the l a r g e volume o f data c o l l e c t e d i t i s convenient t o

compile t h e da ta f o r each' sample, perform subsequent c a l c u l a t i o n s and . (6) p l o t the r e s u l t s us ing computers. The Burnup Data Ana lys is . Code (ABC)--

amd t l i e HADES code(=) o r I s o t o p i c D i l u t i o n Code ( I S O D I L ) ( s ) a r e used t o

compile t h e da ta and determine the burnup parameters and t h e i r one

standard d e v i a t i o n u n c e r t a i n t i e s . The data which have been compiled are:

For A1-PU Samples ,

J e s c r i p t i v e and h i s t o r i c a l data . .<,-

Sample weight, c ladd ing weight, d i s s o l u t i o n vo l m e '

3 7 ~ s a c t i v i t y , alpha p a r t i c l e a c t i v i t y

238 % , P u a lpha a c t i v i t y

g /a ;Pu by coul omet r ic ti t r a t i o n .

~ f o m i c r a t i o s o f i s o t o p e s . . . For UOp and U02-Pu02 Samples . . .

Same a s f o r A1-PU excep t the a lpha p a r t i c l e and %238~u a lpha a c t i v i t y a r e n o t compiled when atom r a t i o s as te rmined from uranium and plutanium samples a r e used.

The parameters determined from t h e d a t a a r e :

For A1-Pu Samples

Atom r a t i o s and atom pe rcen t s o f Pu i s o t o p e s ( 2 4 1 ~ u c o r r e c t e d f o r decay)

Pu atoms l e f t (from a counts o r coulometer r e s u l t s ) and Pu atoms d e p l e t e d (from 37Cs a c t i v i t y ) p e r m i 11 i 1 i t e r

Atoms 3 7 ~ s p e r m i 11 i l i t e r c o r r e c t e d f o r decay

% d e p l e t i o n (from Pu atoms dep le t ed and l e f t )

Pu atom d e n s i t i e s (from p e r c e n t d e p l e t i o n and i n i t i a l etom f r a c t i o n s Pu)

Comparison o f sample concen t r a t i on (weight and volume d a t a vs Pu atoms l e f t .and depl e t e d )

Rat io 1 3 7 ~ s t o Pu a p a r t i c l e a c t i v i t y

and

For U02 and U02-Pu02 Samples

Atom r a t i o s and atom pe rcen t s o f u and Pu I so topes ( 2 4 1 ~ u c o r r e c t e d for decay).

Pu/U r a t i o from Pu a count ing and atom r a t i o s and f u e l weight and composi t i o n of the sample; o r from i s o t o p i c d i 1 u t i o n technique where Pu/U r a t i o i s determined from atom r a t i o s o f t h e sp iked U and Pu s o l u t i o n s .

Beginning .o f 1 i fe (BOL) , end o f 1 i f e (EOL) , and n e t p roduct ion o f U and Pu i s o t o p e s i n gms pe r tn: ( o r .me t r i c t on ) o f beginning of l i f e uranium (determined from BOL a.nd EOL atom r a t i o s and Pu/U r a t i o s ) .

Burnu! = gms o f m a t e r i a l f i s s i o n e d f o r 2 3 5 ~ , 2 3 8 ~ ( f a s t ) , 2 3 9 ~ ~ and 24 P U p e r gm ( o r m e t r i c t on ) o f beginning o f l i f e U .

The burnups a r e a l s o conver ted i n t o MWd/MTU o f U and t o t a l e d .

235u - Fissions a r e determined from 3 7 ~ s (corrected f o r decay) and FiS -

236 - A u ~ ~ ~ - aU . and PuE4: = n P u 2 4 ~ / ~ b 1

P lo t s of the C '

241pu/239~u and 2 4 2 ~ u / 2 3 9 ~ u atom r a t i o vs. 240~i! /239~ 'u atom r a t i o

13%s t o alpha a c t i v i t y r a t i o Vs. 2 4 0 ~ u / 2 3 9 ~ u atom r a t i o s

Percent depletion vs. 2 4 0 ~ u / 2 3 9 ~ u atom r a t i o

2 3 6 ~ / 2 3 8 ~ atom r a t i o vs. 2 3 5 ~ / 2 3 8 ~ atom r a t i o . .

240pu/239~u, 241 and 2 4 2 ~ u / 2 3 9 ~ u atom r a t i o vs . 235 238u and 2 3 6 ~ / 2 3 8 ~ atom r a t i o s . u / 241pu/239pu and 2 4 2 ~ u 1 2 3 9 ~ u atom r a t i o s vs . 2 4 0 ~ u / 2 3 9 ~ u atom r a t i o -.--

235 2 3 8 ~ and 2 4 0 ~ u / 2 3 9 ~ u atom r a t i o s 13'cs t o alpha a c t i v i t y r a t i o vs. U / 236 238u and

Percent depletion vs. U / 2 4 0 ~ u / 2 3 9 ~ u atom r a t i o s

Atom densi ty of each isotope vs. percent .depletion

a r e included as output. Uncertainties corresponding to one standard

devia t i e 5 .:re 4 z t z x i ned f rcz e r r c r rqixtjszs dSr< , v.. ,,d from nz?al I*

procedures f o r propagation of errors.(;) I t i s assumed tha t no cor re la t ion

of e r ro r s of t he P u o r U atom r a t i o s ex i s t . However, uncer ta int ies due .

t o mass spectrometer b ias , sample handling, separa t ion procedures and

- 2 4 1 ~ h decay a r e included.

Analytical functions a r e f i t t o the data b j the method of l e a s t

squares. as a method to monitor the data and to. obtain more accurate

averaged data. For t h i s purpose the data i s represented as a function of

ver t i ca l posi t ion i n the reactor. The measurement quan t i t i e s 3 7 ~ s

(atons/me), fuel density (gmlma), and i: a c t i v i t y (dlmin-ma) r e s u l t i n 1 3 7 ~ s

(at/gm'), a a c t i v i t y (d/mi n-gia) and the 3 7 ~ ~ / a r a t i o (at-min/d) . The -. .-

measurements and data which devia te from the f i t t e d l i n e may do so because

of e i thc r inaccurate 1 3 7 ~ s o r a p a r t i c l e determination. I f the 1 3 7 ~ s

or LZ a c t i v i t y i s high the rx t l 'o would be l ~ ~ i y t ~ ur. lu\\r r-esprctively. I f

a low fuel density i s present the 1 3 7 ~ s (atlgm) and a a c t i v i t y (atlgm)

.. . . . . . .

; ' wil.1 . bo th be h ighe r than t h e f i t tcd"curve. Thus samples can be chosen : . C1 . ' , .. * .

. I : . .. .

, f o r reana lys is o r s p e i i F i c measurement techniques can be i d e n t i f i e d . .

whi ch .need improving.

. - STATI.STICAL ' ANALYSIS OF TRANSMUTATION ' MEAS!IREFIENTS . .

. . . . ... . .

F One of the approaches used i n analyz ing the da ta from t h e t ransmuta t ion ,

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

measurements i.s t o e l i m i n a t e the imprec i se l y known' exposure ( + t ) ' from .'

d i r e c t c o n s i d e r a t i on and t o determine parameters f o r equat ions which

descr ibe t h e data. Th i s i s accomplished by cas t i ng a n a l y t i ca'l s o l u t i o n s

t o equat ions (4 ) i n terms o f the i n t e r r e l a t e d ts.sotoptc concentrat ions.

These so lu t i ons , as i m p l i c i t f u n c t i o n o f exposure, c o n s i s t o f any non-

redundant s e t o f J-1 equat ions obta ined by a l g e b r a i c e l i m i n a t i o n o f the,

cxpns!!re . t frcrm the J.-R?d s e t o f i s o i ~ j j i i co r~cen t ra t i ons y. 'The i s o t & e - concen t ra t i on vec tors y as e x p l i c i t funct ions of exposure a re of m a t r i x - exponenti a1 form. (2)

y ( t ) = y ( o ) exp ( tH) + Q C exp ( tH) - 1 ] / t i . - - The theory parameters p (which a r e r e l a t e d t o the cross s e c t i o n r a t i o s ) - comprise i n i t i a l concent ra t ions ~ ( o ) , source s i n k vec tor Q, and t ransmuta t ion '

r a t e m a t r i x H. A l e a s t squares a n a l y s i s o f the exper imenta l da ta endeavors

t o eva lua te the parameters p- (which a re r e l a t e d t o t h e cross ' s e c t i o n r a t i o s ) ,, .

by op t ima l m i n i m i z a t i o n o f t he r e s i d u a l d i f fe rences ( i nve rse var iance - .

weighted) between the measured data and the theory equat ions (9). The

s t a t i s t i c a l ana lys i s problem haci t o b e 'solved before t h i s approach' cou ld . e .: . . be used w i t h the data because of the apparent absence i n t h e l i t e r a t u r e o f any

method having s u f f i c i e n t v e r s a t i l i t y f o r t he problem.' The s o l u t i o n o f

such a complex s t a t i s t i c a l probltllrl .is a r e s u l t o f t h e burnup program and

has an e f f e c t extending t o . most o t h e r general s t a t i s t i c a l problems.

-. . .. . t . .

. . * i h e ~ f o u n d a t i o n of t he ana lys is 1,ogic i s t h e minimdizatiion o f t he mean . s % , .

. square m i s f i t q u a n t i t y

by imposing ' t h e cond i t i ons

. - . . aQ/ap = 0 and a2Q[dp.p ap = p o s i t i v e d e f i n i t e . (11)

3 - - . . . . . - . - . . - -. - .

. . . The - f o l lowing subsect ions descr ibe t h e development o f ' t h e 1 eas t

squares m i s f i t q u a n t i t y , cons t ruc t i on o f t he non l i nea r theory equations, --.- -.)

. .

and the assignment o f s t a t i s t i c a l unce r ta in t i es . A l l a re requ i red f o r the , . . .

. .

s t a t i s t i c a l ana lys i s o f t h i s complex problem.

Mean Square l l i s f i t ,-*.

A t h ree dimensional p r o j e c t i o n o f a hypo the t i ca l f o u r dimensional

burnup study o f 2 3 9 P ~ i s shown i n Fig. 1 and provides a s imple i n t r o d u c t i o n

t o the complexi ty o f t h e problem. Each o f t he axes correspond t o one o f

t h e i s o t o p i c concentrat ions which descr ibe a data po i r r t : e i t h e r 2 3 9 ~ ~ ,

2 4 0 P ~ o r 2 4 1 P ~ . The f o u r t h a x i s y i = N(Pu-242) i s n o t shown b u t i s p resent

bo th i n measurement and theory. The data from a s i n g l e burnup sample

def ines a measurement vector , 1 and i s shown surrounded by i t s measurement

e r r o r e l 1 i pso id . The e l 1 i p s o i d i s a n a l y t i c a l l y descr ibed by a .measurement

var iance-covar i ance m a t r i x

U = . < (61 ) ($y )> . (12)

The: U can be cont ras ted w i t h t h e case fo r - t h e more fami 1 i a r 1 eas t

squares problem which has an e r r o r assigned t o o n l y one o f t h e var iab les .

The e l l i p s o i d o f t h e complex problem represents the one standard d e v i a t i o n

e r r o r reg ion i n the same sense as does the e r r o r f l a g f o r t h e s imple case.

239 .. : F i g u r e 1 . A T h r e e D i m e n s i o n a l P r o j e c t i o n o f the B u r n u p d?. . P u . -. .- 4 . 8 .

The sp i ra l 1 i ne represents the l eas t squares f i t 7 t o the data under c c :

the assumption of an appropriate analytical description of the bu rnup

process. The tube surrounding the sp i ra l represents . the s t a t i s t i c a l

uncertainty i n the f i t t e d sp i ra l as determined from t h p f i t t i n g process.

The construction of equation (10) can be verified by an inductive

l i n e of thought. F i rs t , when only one variable has an er ror associated

w i t h i t , the mean square mis f i t Q(p) i s the sum of the residuals divided - by the i r variance.

where m and T indicate measurement and theory quant i t ies respectively.

. . . In the more complex case 'dx i s a vector and U i s a matrix rather than

a number so tha t equation (13) i s composed of terms of the form

Equation '(14) reduces to the terms of equation (1 3) fb r the more

familiar l eas t squares problem i n whicli J = 1. In matrix form, U

includes both information about the errors on each of the variables

(diagonal terms) and information .about e r ro r correlations between

variables (off-diagonal terms). Transformed from the measurement co-

ordinates y to the misf i t coordinates q , t h i s expression becomes - -

. . . . . . . .. .$" ' :

. .

. . . . . . . . . . .. . 0 : . .... .. . . .

S ince the theory equat ions have been c a s t i n t h e i m p l i c i t form

q(y',p,)'.= 0 w i t h one l e s s q v a r i a b l e than y var iab le , t he m i s f i t - - - d i f f e r e n t i a l s

span o n l y the (J-1) dimensional subspace normal t o the theory curve.

Consequently, equat ion (14) t ransforms t o a degenerate form which ignores

t h e coord ina te t a n g e n t i a l t o t he f i t t e d curve. Th i s i s e x a c t l y what i s

r e q u i r e d f o r t he burnup ana lys is ; t rans format ion o f an imprec i se l y known

exposure ( 4 t ) t o an i gno rab le var iab le . Replacement o f dq by q(ym,p) and - - s m i n g the mean square m i s f i t s o f equat ion (15) over a l l measurements m,

. . . . . .

completes the construction o f the ! e r s t squzre m i s f i t , s f equ:tion,,,(?$).

. . - . . . . -. - . . - . .

Maximum L i ke l i hood N o n i i near Theory

The complexi ty o f t h e non l i nea r theory equat ions (9) i s l i m i t e d

o n l y t o the ex is tence o f t he minimal m i s f i t cond i t i ons o f equat ions

(11) and t h e m i s f i t q u a n t i t y equat ion (10) over t he domain o f

ap.p l icat ion. For example, they may be

e i m p l i c i t o r e x p l i c i t f unc t i ons o f t h e va r iab les y - 9 non l i nea r i n any o r a1 1 va r iab les y - e non l i nea r i n t he parameters p - . degenerate ( fewer f i e l d equat ions q = 0 than f i e l d va r iab les y ) . - -

The 'earch f o r t h e min imal n i s f i t ~ ( 6 ) ~ r o c e d e s v i a an' i t e r a t i ve -

process. "'The nonlinecr d i f fe rent ia l equations a Q / a p a re expanded in - truncated Taylor's se r ies

and solved for the parameter differences d p by matrix inversion -

These quant i t ies are then used in the algorithm

P = p - ( ~ Q ( P ) / ~ P ) - ( ~ ~ Q ( P ) / ~ P ~ P ) - ~ - - - - (1 6)

which, optimistically; provides parameters fo r an improved l e a s t squares

f i t ( i .e. , one i n which there i s a reduction in the mis f i t value Q ( ~ ) ) . - After each such parameter change, the mean square misf i t Q i s recomputed

from equation (10) and compared to i t s previous l e a s t value. A lower

mean square misf i t value, indicative of a convergent trend, closes the

' i t e r a t i v e loop by a recursive traverse of the refinement algo'ri t h m of

equation (16). A higher mean square misf i t value, indicative of a

divergent trend, branches the logic f i r s t to cubic-expansion of Q(p) i n

terms of the 1 ast. parameter change, and final ly to a reversing-and-

ha1 vi ng of the Tast. parameter change, i n order to provide a two-s tage

recovery e f fo r t . The occurrence of any s p i l l , division by zero, or

non-invertibl e matrix aborts the calculation. Complete agreement of

the mean square mis f i t w i t h i t s previous ' l e a s t value terminates the

. i t e r a t i v e process. O u t p u t incl udes the. normal principal :bl ades of the . ; misf i t second gradient a2Q/apap - - to provide decisive asses-sment of the

. .

posi t i ye-defi ni t e ,.requi.rement.. . ' C.

. . ,.

-25-

Max'imum Li'kel i hood PIonl i near E r r o r Ana lys is .

The t h i r d major ptiase o f t he s t a t i s t i c a l development i nvo l ves the

assignment o f e r r o r s t o .the parameters and o the r r e l a t e d q u a n t i t i e s .

Th is has been accomplished by a g e n e r a l i z a t i o n o f F i she r ' s maximum (33 )

li ke1 i hood s t a t i s t i c s t o the non-1 i n e a r l e a s t squares problem . The p r i n c i p a l r e s u l t o f t he g e n e r a l i z a t i o n i s embodied i n the parameter

var iance-covar i ance m a t r i x U. The m a t r i x U i s tw i ce t h e matrix'-- ' f 'r iverse

o f t h e mean square m i s f i t second grad ien t , a2Q(p)apap, t imes t h e - -- measurement var iance expected i f one begins over w i t h a new experiment.

The l a t t e r q u a n t i t y i s t h e mean square m i s f i . t Q(p) : d i v i d e d by the number - (LoM - K) o f degrees o f freedom f o r t he problem so

The L, M: and K are t h p t o t a l n l v b e r c f eq!!?t;cns q, data pc'nts, 2nd .

parameters p r e s p e c t i v e l y . The root-mean-square u n c e r t a i n t y i n each - theory parameter can be computed from t h e diagonal elements o f t h i s

Progrcms LI.Y.ELY and DBUFIT . .

. . " :. .

Thus a s e t of v e c t o r measurements &ln - + u:" a re f i - t t e d - b y a s e t

o f theory equat ions. 3 and a s e t o f parameters - D t o us ing a l e a s t squares

f i t t i n g method and a l l o w i n g a s e t o f parameters t o vary u n t i l a bes t

f i t i s obtained. Th i s i s accomplished us ing Program LIKELY and a theory

subrout ine. The theory subrout ine prov ides - _ t h e f u n c t i o n a(y,p) and . ?

..aq/ay, and" t h e i r f i r s t and second o r d e r 'par&metr i c cpadi en ts as/ - ae, a2q/ayag, a2q/aeae, and a3q/ayapap which a r e needed f o r t he f i t t i n g

process. The ou tpu t i nc1,udes an ex tens ive non1.-i near3.s'ta t i s t i c a l . anal.ysis

of t h e d i f f e r e n c e s between measurement and theor-y i n a d d i t i o n t o t h e para-

k meter r e s u l t s p + 6p . The s t a t i s t i c a l ana lys i s dec is ions a re based on a

nonl i n e a r general i z a t i o n o f t h e Student and F i she r the0r.y. Assessments

of t h e q u a l i t y of t h e measurements and the re levance o f t h e theory a r e

made i n a d d i t i o n t o the adequacy o f agreement between measurement and theory.

Graphical rep resen ta t i on o f enormous ou tpu t d e t a i l i s a l s o avail.ab.le i n a

m a g n i f i c a t i o n chosen by the user .

The general s t a ti s ti c a l ana lys i s capabi 1 i ty o f Program LIKELY has

been adap ted ' z ) t o t h e s o l u t i o n o f s p e c i f i c ana lys i s problems. One o f

th'ese DBUFIT-I , a double p r e c i s i o n burnup f i t t i n g code, has been .

s p e c i f i c a l l y developed f o r o b t a i n i n g the i n t e g r a l cross s e c t i o n - i n f o r m a t i o n

from . i so top i c t ransmuta t ion data. ~ h , DBUFIT-I code i s an improvement over

t h e DUBLIK c o d e ( 3 ) which had p r e v i o u s l y been used f o r t he l e a s t squares

.ana lys i s o f .burnup data.

. % APPLICATION USIFIG'A1-PU ELEMENTS

Samples o f A1-Pu f rom f u e l elements which have been i r r a d i a t e d i n

t he PRTR(%) have been d e s t r u c t i v e l y analyzed t o determine p lutonium

dep le t i ons and i s o t o p i c composit ions. The elements were 19-rod c l u s t e r s

which i n i t i a l l y conta ined two p lutonium composit ions. One type o f element

conta jned 88 i n . l ong r i d s _of an a l ' loy A1,-2 w t % N i - 1 . 8 2 wt% puJ6) Each

element conta ined 268 g p lutonium and thb'i. so top i c composi t ion before

i r r a d i a t i o n was 93.28/6.25/0.457/0.0178 a t % 2 3 9 ~ u / 2 4 0 ~ ~ / 2 4 ? . P ~ / 2 4 2 P ~ ,

respectively. The second! type contained an a1 loy Al-'2; w t % Ni - 2-.60 w t %

Pu which had an isotopic-composition before i r radiat ion of 81.06/16.45/

2.29/0.20 and 376 g plutonium per c lus te r . (7) The rods of both elements

were clad with 0.035 in . Zircaloy-2. .- -.

The PRTR (a verti c,al . pressure tube, heavy water moderated and cooled

reactor) was operated a t a thermal power of 70 MW. The i n l e t and . i ... .; . ..

o u t l e t temperatures were 470 and 530°F o r 125 and 150°F for the ioolant

o r moderator, respectively. The fuel elements were 1 ocated on an 8 in.

pitch w i t h fuel elements d i f fe rent from the experimental element. However,

the experimental elements were surrounded by 'other elements of the same

-'composition- and exposure history. . ,

The fuel elements were removed from the reactor a t approximatelv

equal increments o f exposure up to a maximum depletion of about 50% of the

i n i t i a l plutonium. Detailed analyses to obtain burnup data were carried

out for one rod from each of two rings and the center rod of the

c lus te rs . The data describes the variation .of the concentration of

plutonim isotopes as the plutonium i s being depleted. (6'L) The t o t a l .

+rradiation exposure i s computed from the depletion data and ef fec t ive

cross section ra t ios a re derived from the atom ra t ios and fuel depletion.

Data from samples having the same i n i t i a l fuel composition and

i r radiated in approximately the same nuclear surroundings. have been

analyzed together. This resu l t s in four'groups of data: . -..- . .

a 1.~82 wt9: P u in A1 (6.259: 240Pu) from rods of the outer ring of

the c l us t e r (LxO)'

0 1.82 w t % P u . i n 4;l (6.25"/140~u) from rods cf' the inner r ing

and cen te r rod of the c l u s t e r (LxMC) . .

c 2.6 wt% P u i n A1 (16.45% 2 4 0 P u ) from rods of the outer r ing . '

o f . the cTus t e r (tixO) . .

, 2.6 w t % P u i n A1 (16.45% 2 4 0 P ~ ) from rods of the inne; ring

and center rod of c l u s t e r (HxMC). ... . . .. . . . .. . ., . . - . - . . -- -. - -

1 . I f i t i s assumed t h a t the samples analyzed i n each proup have been i r r ad i a t ed

i n ne'utron spectra common to t h a t group, ther. i t can be concluded t h a t

. t h e fuel composi'tion depends only upon i n i t i a l fuel composition and the

exposure i t recei ved.

In many reac tors , the items which perturb the neutron spectrum in

the course of an extended i r r ad i a t i on a r e control rods, burnable poisons,

i s o t o p i c f u e l transformations, and leakage. Other f ac to r s , such as

temperature, . type and density of the moderator, and s t ruc tu r a l components

which a f f e c t the spectrum remain r e l a t i v e l y constant during the

i r r ad i a t i on . In the absence o f burnable poisons and i n fuel 1oca.tions . . '

su f f i c i en t l y removed from the control rods, only the fuel composition

a l t e r s the local spectrum and hence the f lux averaged cross Sections of

the fuel . These i n t u r n govern the incremental fuel changes and sugqest

t h a t the fuel composition evolves i n a manner amenable t o the d i f f e r e n t i a l -

. o r in tegrated equation method of analysis .

-. I he unique re la t ionsh ip between fuel ',composition and exposure cmbined

. . w i t h the f a c t t h a t the exposure var ies along a rod has the advantage of

allowing many data points t o be gathered from the same rod. , In t h i s

-., '

way the ,number of rods and reactor shutdowns ,required sto obtain a 'given -, . >;

. .

number:tsf samples i s reduced. Samples cut from the ends of the rod have

bee:n omitted from analysis because the perturbation of. ;the flux spectrum a t

these positions by flux leakage effects i s suspected to cause significant effects.

Samples from rods in similar positions of diyfererit elements have

been analyzed in one group since they also have been irradiated i n common

spectra. Even though some adjacent PRTR positions were loaded w i t h

different fuel elements they do not greatly influence the spectrum of the experi- ----

mental elements since. neutrons originatinq in one element are thermal i zed

by the time they reach another. Again, fuel composition changes of each , .

element are solely dependent upon in i t i a l fuel composition and exposure

of that el ement.

,Expansion of Equations (5) for the plutonium system results in

three equations containing five unknown ratios of effective cross

' sections. Solving for the cross section ratios a1 low three of them to

be determined from the experi,mental data and equations (6) and (8). T h s ' .

... . - . . . . effective cross section ratios -

(0i1/;i9)(1 + h4'/;:'4) ,. C ; and (20

.. . ' :..,'., ; .. , .

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

, . . . . . . ' .

,? . - :- , ~ l l radioact i ve decay o ' i t h e isotopes a r e considered negl i g i b l e except

f o r 241Pu ( i .e., A" # 0 ) and 0 4 0 = ;:O. An average value of C

.. A = 0.050 f 0.025 i s assigned t o each group of data on the bas i s

.. of experimental and calcula ted information. A l a r g e . uncertainty i s

. ' ' assigned t o the group average value s ince i t var ies widely from one data

po in t t o another.

Since the re a r e more unknowns than equations two other equations

must be es tabl ished before numerical values can be obtained f o r the r a t i o s .

The radio D i s r e l a t i v e l y small and could be s e t equal t o zero; however,

a ca lcula ted value of .D = 0.07 i s a' b e t t e r approximation. .4nother

ca lcula ted value C = 1.20 i s used. because of the g r e a t e r confidence

placed i n i t s calcula t ion because of the nature of the bas ic cross

sec t ions and the determination t h a t an i nco r r ec t value would have a

smaller e f f e c t on the remaining cross sect ion r a t i o s .

The r e l a t i ons between the r a t i o s and the data a r e

where

and

. . . . . . . . .

c ,. .. . The bi N are changes i n p lutonium concentrat ions i n atPb.~-on and E , E,

. .

C. D, a n d y a re assumed constant f o r , a p a r t i c u l a r b u i n u p i n t e r k a l (b-a).

The r a t i o s N ~ / N ~ ~ a r e represented by func t i ons o f t h e - f o r m g iven by

equat ion (6 ) . The parameters Ak o f these equations a r e determined from

t h e experimental data by t h e method o f l e a s t squares.b Th is i n t u r n

a l lows values o f t h e nFJi t o be obta ined from equat ion (8). F i n a l l y ,

t h e r a t i o s E , E and y a re determined as a f u n c t i o n o f p lutonium

d e p l e t i o n . I n the process the value o f lr9 = ,49/,49 c f and G ~ ~ = ; ~ ~ / G ~ ~ c f '

i s determined. The.,,

&49 = € / ( I - E ) and = y l / ( l - y ' ) -

where I, I . - I . 1 --:\ y' " . Y [ f + *"/a-"$).

' -The values obta ined f o r the r a t i o s o f ' e f f e c t i v e cross sec t ions are

summarized i n Table I 1 as a f u n c t i o n o f dep le t i on f o r rods o f t h e o u t e r

r i n g o f the c l u s t e r (LXO) and f o r rods o f the i n n e r r i n g and center r o d

o f t h e c l u s t e r (LxMC). Values o f t?9, or', and Gi0/G:9 were obta ined f o r f i v e

' equal sub in te rva l s between 1.5 and 48% dep le t ion . Values o f &49 and G 4 1

were constant w i t h i n the experimental u n c e r t a i n t y and the averages f o r

the t o t a l i n t e r v a l conf i rm the values obta ined by the i n t e g r a t e d equat ion

method (Table V). The values o f ;40/Z49 were n o t constant and agree

w i t h .the. values obta ined i n the i n t e g r a t e d equat ion ana lys i s and

corroborated the 'shape used f o r t h e r a t i 6 . : ' (F igure 4).

TABLE I 1

CROSS SECTION RATIO VALUES AND THEIR ONE STANDARD DEVIATIONS

FOR A1 -Pu CONTAIN-ING INITIALLY 6% 2 4 0 ~ u

Cros:s Sec t ion Percent Dep le t ion I n t e r v a l Ra t i o 1.5 t o 10.2 10.2 t o 19.5 - 19.5 t o 28.8 28.8 t o 38.2 38.2 t o 48.2

LxO 0 . 2 8 9 t 0 . 0 1 1 .13 .296 t0 .008 0 . 3 0 5 t 0 . 0 0 7 0 .307a0 .011 0 . 3 1 2 + 0 . 0 1 9 G4 1/;49

c a LxMC 0 . 2 9 6 t 0 . 0 1 0 0 . 2 9 5 t 0 . 0 0 8 9 . 3 0 2 t 0 . 0 0 8 0 . 3 1 4 t 0 . 0 0 7 . 9 .31140 .012

LxO 0.507 + 0,014 0.451 2 0.910 0.43'7 f 0.009 0.425 a 0.014 Q.391 + 0.024 G 4 0 / 6 4 9

a a LxMC , 0.456 + 0.012 0.404 t 0.009 0.390 t 0.008 0.382 2 0.010 0.350 + 0.013 ,.

LxO 0.300 + 0.009 0.305 a 0.008 ' 0.310 t 0.009 0.313 t 0.010 0.308 + 0.020 - ^49 /G49

3

LxMC 0.426 .a 0.018 0.430 t 0.017 0.438 t 0.018 . 0.449 5 0.020 0.468 5 0.032

. ,

. -33- : ..

I n t e g r a t e d Equat ions Method L --The same definition.^ at id 'assurr~ptions o 5 ; t h e d i - f f e r e n t i a l method

a r e used i n the expansiton o f equa t ion (4 ) and the i n t e g r a t e d equa t ion

method. The approximation t h a t 0i0/39 equa l s a c o n s t a n t E i s no longe r

vali-d s i n c e the burnrlp increment i s n o t smal l . 'Therefore ano the r r e l a t i o n -

s h i p is sought which w i l i s t i l l a l low a n a l y t i c a l s o l u t i o n s t o be ob t a ined

f o r the d i f f e r e n t i a l equa t ions . The cho ice o f

where A and ,B a r e c o n s t a n t s adequate ly meets the needs o f the, Al-Pu

a n a l y s i s . The o t h e r r a t i o s a r e assumed c o n s t a n t a s in , )equa t ions (19-21).

Under t h e s e c o n d i t i o n s , equa t ions ( 4 ) a.re i n t e g r a t e d by t h e

s t anda rd technique o f t r a n s forming t h e equa t ions t o e x a c t d i f f e r e n t i a l s

w i t h a p p r o p r i a t e . i n t e g r a t i n g f a c t o r s . The s o l u t i o n s a r e

. . . . . . . . ... . .-. . _ _ . . _ _ _ _ _ _ . .

. . . . The Ki a r e d e t e n i ned from equa t ions ( 2 7 ) , (2'8) , and (29) when t n e i ni t; a1 .

concen t r a t i ons N: a r e s u b s t i t u t e d f o r t h e N'. The o t h e r parameters a r e

def ined by equa t ions (19 ) , ( 2 0 ) , and (21) . Equations (27) , ( 2 8 ) , and

(29) a r e p r e c i s e l y the q ( y , p ) equa t ions r e q u i r e d by the l e a s t squa re s ---

. . f i t t i n g a n a l y s i s . The f i t t e d parameters p a r e t h e K i , E , A , - - C , y; and D whi l e the v a r i a b l e s a r e the i s o t o p i c concen t r a t i ons N ~ . The var iance-covar i ance ma t r ix ( e q u a t i o n 12) f o r each d a t a p o i n t i s

c a l c u l a t e d from random measurement e r r o r s using s t anda rd techniques o f

propagat ion of e r r o r s . The c r o s s s e c t i o n r a t i n D i s too small t o be

-34- :... . . .. . . .

.." ,. :;,- detennihod . . . from the experimental data. Therefore, the calculated val ue ..: j .*>.. saK2 ..

. , . .. . .

.;c; . of 0.07 i"s used as before. After a sat isfactory f i t has been obtained . ,,;

w i t h the equations, the fraction. B of plutonium atoms destroyed can be

calculated. Neglecting the e f fec ts of 2 4 2 P ~ neutron capture, B . : ~ s the

sum of 2 3 9 P ~ and 2 4 1 P ~ atoms fissioned and 2 4 1 P ~ atoms decayed divided

by the i n i t i a l plutonium content or

where the zero subscripts denote i n i t i a l values. In the actual analysis .

the more, complicated expression i s used which accounts fo r 2 4 2 ~ u neutron . . .

capture.

In principle, a generalized l e a s t squares f i t to the experimental -

dats is-ing eqtiations i : z i j , (281, and (29) will determine a l l of the unknown . .

parameters included in the equations. In practice, the vari.ous sets ' of .

data for the A1-Pu may yield more than one sol ution unless one of the cross

section ra t ios i s preset and held fixed during the i t e r a t i v e f i t t i n g process.

For th i s purpose 6:1/0:9 i s chosen as i n the d i f fe rent ia l method.

Graphical resu l t s of generalized l e a s t squares f i t s a re shown i n

Fig. 2 for the 1.8 w t % P u ; outer r i n g case (LxO). The three isotopic . .

curves are the specif ic projections of the generalized curve o.f F i g . 1 /

onto the appropriate planes. Because of the small distance between

one standard deviation l imits on each curve, the area between'them has

been shaded i n for purposes of c l a r i ty . The experimental 'data hqye . .: been . .

F i g u r e 2. G raph i ca l Resu l ts o f ~ e n e r a l i z e d Leas t Squares F i t s f o r 1.8 wt% Pu.

. ..... .. . . ..': p l o t t e d s.howi ng the one s t a r ~ d a r d d e v i a t i o n meas,urei~ent. e r r o r f l a g s .

. . .<. . : .: . . . * . .,. hi.: . E r r o r Y f l a g s for most p o i n t s f a i l t O show up because t h e y a re sma l l e r

than the mark i n d i c a t i n g the data p o i n t . The p l u t o n i wn d e p l e t i o n B

. ., p r e d i c t e d from the b e s t f i t t i n g parameters and equat ion (3 ) i s shown

i n F ig. 3. The. B va l ues determined experimental l y from 37Cs data are

a l s o p l o t t e d f o r comparison. The curves. f o r the o t h f r . three cases .. ,

(LxMC, HxO, and HxMC) are q u i t e s i m i l a r t o the ones shown and the goodness , ,

o f t he f i t s a r e about the same.

The cross s e c t i o n r a t i o s ob ta ined from the l e a s t squares f i t s a re

presented i n Tab1 e I I I for ,a1 1 t h e A1 -Pu data. A u n i f o r m i t y o f the . .

r a t i o s f o r t h e f o u r cases i s expected f o r i iU and B:1/G:9. I n the.

thermal energy reg ion a41 (E) changes l i t t l e w i t h neut ron ene,rgy and

spectrum changes should n o t change i t s . e f f e c t i v e value. The o i 9 ( E ) and '

, . - f r \

A , . , . A

i c , have s h i i a r shapes anu as a r e s u i t U - ~ ~ / U ~ i s r e l a t i v e i y . a a a

unaf fec ted by neutron spectrum changes. On the o t h e r hand, a49(E) does

vary apprec iab ly w i t h neutron energy i n the thermal r e g i o n and s ince

t h e exper imenta l values a r e approximate ly t he same i t i s concluded t h a t

t he thermal neut ron spec t ra o f t he f o u r cases a r e n o t apprec iab ly d i f f e ren t .

The 241Pu decay c o r r e c t i o n term h4l /u4 l ; r e q u i r e d t o determine

0:1/G:9 from t h e l e a s t squares value C has been c a l cu la ted from .. .

the heat re leased by the' element. Heat f low i s moni tored for . 'each

PRTR element and i s c a l c u l a t e d us inq the fonnula H(MI.ld): = kVdfiS$ ~ t . L

, The k = 3.91 x MWd/ f i~ and n t . i s t h e t ime i n seconds d u r i n g which i . 8 '

t he element was i r r a d i a t e d . The ffis ,anddF;" were computed va l ues.

. . . ' :,.,2. - ' .. ., :.. . . .,. ..., . .

- . . . .. . . , . ..,. . I .

. . . . .. . :.. . k i i . , . ... . .: . P L I ~ ' T o N I U M D E P L E T I O N FOR L X O F U E L . .

Figure 3. The Plutonium Depletion f o r 1.8 w t X P u as a Function of , 3 9 P ~ ~ Concentration.

TABLE I11

CROSS SECTION RATIOS FROM INTEGRATED EQUATION METHOD

The values a r e ob ta ined by keeping 31/G:9 = 1.1429, 0:2/39 = 0.07 . . . .&. ,.. .

a . . 8 - and ~ ~ ~ / 6 : ~ 5 = 0.05 k 0.025 du r i ng t h e f i t t i n g process. Numbers i n %, s., .. *.

parentheses a r e one s tandard d e v i a t i o n u n c e r t a i n t i e s ob ta ined from

t h e f i t t i n g process.

-- - .. . . -. . .

H ighes t Fuel 6 4 0 " 4 9 a /'a Dep le t i on

Fuel (%) ;4 9 A . . B i . 1 0 5 6 4 1 . .

LxO 50.4- 0.4391 (0.0032) 0.3097 0.2153 0.351 (0.013)

Lx?lC c8.C 2.435: (C.OC32) 0.2673 0.2102 0.355 (U .O ' i j )

HxO 51.2 0.4421 (0.0026) 0.2043 0.7305 0.342 (0.012)

HxPlC 48.1, 0.4465(0.0025) 0.1751 0.7133 0.339 (0.011)

- 1 i i ! 241 . F i n a i ly, = 1.655 x 10" sec . (T = 13.3 y r s f o r 2 Pu) completes 112.

the i n fo rmat ion necessary f o r t he correction.(^) Values of vary w ide ly

f o r i n d i v i d u a l elements as w e l l as f o r i n d i v i d u a l sainples taken from

- w i t h i n these elements. Also, rfiS and ;"change apprec iab ly du r ing the

i r r a d i a t i o n o f the f u e l . Consequently, t h e choice of a s i n g l e value

f o r A / f o r a l l data p o i n t s o f a g iven case i s q u i t e unce r ta in

and has been assigned an e r r o r o f 50%. Once ~ ~ ~ / o i ~ S has been est imated, . . , . - i t s e f f k t on the values o f G:1/04g and Pi c a n be computed.

a -- --

The values o f 6 i 0 / G i 9 are p l o t t e d i n F ig. 4 as a f u n c t i o n o f t h e

240Pu atom concent ra t ion w h i l e the 239Pu was depleted t o ;50% o f i t s

o r i g i n a l value. Also p l o t t e d i n the f i g u r e are values obta ined from the

d i f f e r e n t i a l methods o f ana lys is . The s o l i d curves are de f ined by t h e

A and, B values obta ined i n t h e simultaneous . l e a s t squares f i t and the

bounds o f the curves are the standard dev ia t i ons determined from the f i t s .

The crosses p l o t t e d i n the f i g u r e were obta ined by the. d i f f e r e n t i a l

ana lys i s method which a l lows t h e v a r i a t i o n o f a l l t he e f fbc t i v ; cross

s e c t i o n r a t i o s from the data. The agreement between t h e crosses and the

curves l end credence t o t h e shape .assumed . f o r t h e simultaneous fit. . . .

Figure 4. Values o f 8i0/G4,9 as a Func t i on o f 2 4 0 ~ u Atom Concentrat ion

I I

.. A In reactor neutronics calculations, the t n i t i a l s tep i s to calculate

the mul t i~roup neutron spectrum::.for the basic representative sub-region

or uni t cell of the reactor. The neutron spectrum within thi's uni t ce l l

, i s usually computed using d ig i ta l colllputer programs which solve the Boltzmann

equation for neutron transport. Mu1 tigroup ( the order of 100 groups o r more)

values of the neutron cross sections are averaged over th i s neutron spectrum

to reduce the mu1 tigroup values to few group values. Traditionally, the

calculation of the neutron spectrum i s divided in to non-the'rmal groups

( lo7 eV to thermal cutoff) and a thermal group (thermal cutoff t o 0 . 0 eV).

In th i s section we show how the cross section values obtained w i t h the

unit ce l l codes are used to obtain the one group values needed for compari.son

to the ra t ios deduced from the f i t to the isotopic data. We then i l l u s t r a t e

the use of the deduced cross section ra t ios and the calculated values i n

verification of burnup calculations. In the process the theoretical model

employed and the resu l t s obtained are described.

Ca1 cul ational Amroach

A n approach which i s simple i n practice and expected to be accurate i n

principle i s used to calculate the cross section ra t ios fo r comparison w i t h

the ,values deduced from the experimental data. The approach i s based upon - .

using the measured isotopic concentratians for the f i s s i l e and f e r t i l e . .

nuclides i n t he . fuel in calcul~t . ions off,:the neutron spectrurn2:5'n a ,uni.t cel l - . i . 8 . .-

of the reactor: Thus, i t i s assumed tha t the measured f e r t i l e .and f i s s i l e

nuclides characterize the neutron spectrum within a u n i t ce l l and tha t

. f iss ion products.do not perturb the spectrum signif icant ly.

. , ;. The spectrum averaged val ues , cal led here e f fec t ive val ues, are :defined l i . d . ,

according to equation (2)

where the spat ia l integration i s over the fuel volume V . of i n t e re s t . J

The neutron spectrum within a unit cell i s usually characterized into

thermal and non-thermal neutron energy groups. Thus, the mu1 tigroup val ues

for each isotope must be reduced to spectral average values for the thermal

and non-thermal energy groups. Separating the energy integrals in numerator

7 and denominator into non-thermal ( E c to 10 eV) and thermal (0.0 to Ec) terms

and assuming the spa t ia l integral of the nonthermal flux above Ec i s constant

allows equation (31) to be written as . . .

:. .

where subscript 1 refers t o the nonthenal neutrbn energy group and

Ec is'- the energy boundary, . . betGeen the thermal a n d non.therma1 groups. The

second term of the nmerator i s spatially constant and can be written

as the product of the average thermal cross section, o:'j and the total

thermal neutron flux 4; in the fuel region of interest. Subscript 2

refers t o the thermal ne,utron energy group. The second term in the

denominator i s just the total thermal neutron flux 4; in fuel region j. . . . . .

Thus, equation (32) becomes ... _.. .. . . ...

The fluxes 4: and 4; are normalized assuming t h a t al l neutrons scattered

from group 1 t o group 2 are absorbed i n group 2. Thus

where 4; and 4; are the t o t a l flux i n the cell for group 1 and 2. Now

m C i s related t o the total flux in the fuel by

.\ C - C 'mi = m; ( v ~ @ / ) / ( v and

where V refers again t o volume and a i s the average 'flux. Assuming

mi/< = 1, substituting equations (35) and (36) in equation (34) and

j j solving for the ratio 41/42 and then using this normalizing condition

i . in g u a t i o n (33), the expression f d the effective cross~~~ectionLbecomes ' C. .-

I

.- C The quant i t ies Z: and z , ~ are obtained from cel l cai chl ations using a

code which calculates the, slowing down of neutrons and a l l other quant i t ies

a re obtained from thermal ization cal cul ations. The ef fec t i ve cross

sections as cal cul ated w i t h equation (37) for each isotope are .. ... used

t o obtain the ra t ios needed f o r comparison t o the values deduced from

the f i t to the isotopic data.

Theory-Experiment Comparison

The cross section ra t ios which were deduced from the Al-Pu fuel

i r radiat ions i n the PRTR are used to i l l u s t r a t e how these data can help

verify the accuracy of hurnu? calculatinns. The zo2scred i s s t o p i c

concentrations a t pre-selected exposures are used as i n p u t t o cel l codes

for u n i t ce l l calculations to obtain ra t ios of effect ive cross sections.

These ra t ios a re compared to the values deduced from the experimental data

to provide a basis for verifying the accuracy of the calculated resu l t s .

Similar comparis[ms using resul ts from calculations which use calculated

isotopic concentrations i s an a1 ternate approach which can provide

additional informati.on.

Theoreti cal Methods I'

The theoretical methods used a t BNW for th i s study are based on w

approximate solutions of the .neutron transport equation. The neutron

dis t r ibut ions in . space and energy w i t h i n a s ingle cel l of the reactor

were computed using the multigroup t ranspor t theory codes, H R G ( ~ ~ ) and

. . . . . .

. ..

s,L:zw ,- THERMOS:~) : ~ l l basic neutron c rab ;~~ sect ions for these: ce l l codes were s i., .

i & % . obtai ped f r k t h e Ba t t e l l e-Northwes t Master Library (BFIWML) . (a) The \ ,;

,3. . . thermal neutron cross sections for the f i s s i l e nuclides were normalized ..*

to the 2200 m/sec values given in the 1965 International Atomic Energy

(IAEA) Eva1 uation cdnducted by Westcott, e t . a1 . (a)' a -

The behavior of neutrons slowing down to thermal energies i s

canputed using the HRG code. The time independent Bol tzmann equation

is solved w i t h HRG for isotropic sources of neutrons i n the B-1 o r P-1

approximations. The neutron flux and current spectra are computed

assuming 68 groups of neutrons w i t h p constant group w i d t h of lethargy =

0.25.. Corrections fo r heterogeneity, Doppler broadening, and leakage

are incl uded. The boundary di'vi'ding the thermal and non-thermal energy

groups (Ec of equation 32) was chosen as 0.683 eV. The multigroup cross

sections were reduced to a one group value fo r the energy region from

0.683 to l o 7 eV. .

The thermal neutron spectrum i s computed using the THERMOS code.

The version of th i s code i n use a t PNL fo r th i s study was f lex ib le as

to . the number of energy groups (up. t o 30), space points (up to 30) and

mixtures\(up to 8) such t h a t the code can be ta i lored to the spec i f ic

problem to be solved. The th i r ty group cross sections for each isotope

were spectral averaged to obtain one group values fo r the: energy range L

from 0.0 to 0.683 eV. . . C. - I . I .

Neutron Spectrum Computations

The neutron spectrum i s computed a t various exposures u t i 1 i z i ng

/ the experimental atom concentration of the plutonium isotopes. The

average physical ' temperatures of the fuel , coolLan t, 'a;n'd moderator were *

assumed to be 343, 260,, and 5g°C, respectively. These are the average

inferred fuel temperature d u r i n g irradiation and the average of the

observed i n 1 e t and out le t temperatures for the moderator and coolant. for

the positions i n which these A1-PU fuels were irradiated i n the PRTR.

The slowing down of neutrons i s computed using the HRG code. The \ .

. .

options utilized in the current study are the P-1 approximation'f&- zero

leakage, the 2 3 9 P ~ . fission spectra,, and Doppler broadening and spatial

se l f shielding corrections for 2 3 9 P ~ , 240Pu' , 'and 2 4 1 ~ ~ .

Spatial corrections are based upon the methods developed by

~ o r d h e i m . ( ~ ) The mean chord length ( i ) for the cluster was obtained from

'the relationship i = 4V/Seff where V i s the vol ume enclosed by a rubber

band stretched around the cl us isr and Seff = 1 . 2 3 x rubber band surtace -,

area. The admixed moderator scattering cross section per absorber atom was

computed f 0 r . a single rod .of the 19-rod cluster , ( i .e., A1 as the admixed

moderator in the fuel) . This procedure for obtaining the spatial correction

parameters evol ved empirical ly from analytical correl ations wi t h He1 1 strand's

experiman ti for I S-rod cl us ters . (42)

'The thermal neutron spectrum i s computed using the THERMOS code. b

Thirty neutron energy groups were used to describe events occuring below

0.683 eV. A concentric cylinder model of the cluster obtained by

preserving atom concentrations and volumes was used. The cell i s described

i n seven regions w i t h 30 space points. The Honeck-Nelkin kernel was

used for D20 with an approximate correction to \ the kernel to account

. - for an3 sotropy . ~ h k ~ s c a t t 6 r - t r q s f e r fron?sll o*her cel l ma@,ri a1 s . . .

were computed with the Brown-St. John Model. ( Reflecting cel l boundary

conditions were also u t i ! ized. , . . , . .

To simplify the calculation, i t was assumed t h a t the neutron spectrum

i n the fuel was, to the f i r s t order, sensi t ive to only the concentration

of f i s s i l e and f e r t i l e nuclides. That i s , the f iss ion product inventory

and the presence of neutron leakage a f fec t mainly the neutron popir"1ation

and the i r a f f e c t on the neutron spectrum i s negligible.

Comparison of Resul t s

A s e t , o f ra t ios obtained from resul t s of ce l l calculations and

equation (15) are compared in Table IV to values obtained from the l e a s t

: . squares analysis of experimental data using Program DUBLIK i n Table IV.

The calculated values of & 4 9 and G41are within one standard deviation

of the experimental values and lower by approximately 4-1/2 and 9%,

respectively. The values deduced i n the l eas t squares analysis show

s l i g h t variations between the .outer 12 rods and the inner s ix rods whereas

the calculated values do not. Because of the disagreement i n this

variation, i t appears tha t both the basic energy s e l f shielding and the

spa t ia l se l f shielding i s being calculated incorrectly. '

n

The calculated'values of are outside the standard deviations of - the experimental values. However, a discrepancy was expected in these ra t ios

.e: since &he calculation of resonance absorption in 2 4 0 P u was) made assuming a . 7 .-

homogeneous c e l l . The experimental data i t s e l f suggest t h a t spa t ia l e f fec ts

a re important i n calculating resonance absorption in 2 4 0 P . ~ as shown by the

TABLE I V -- COMPARISCN OF CALCULATED RATIOS OF EFFECTIVE CROSS SECTIONS

TO V!.LUES OBTAINED FROM LEAST SQUARES ANALYSIS OF EXPERIMENTAL DATA FOR LX Pu-A1 CLUSTERS

Cal cu l a ted .. Least Squares Ana lys is F rac t i ona l , Outer 12 Rods I n n e r 6 Rods

Burnup -- R a t i o .. .:'; . o f C l u s t e r - o f C l u s t e r Outer 12 Rods I n n e r 7'Rods - . . . .. .

0.351' 0.030 0.355 + 0.030

1.143 ( ~ i x e d ) ' 1.143 (F ixed)

0.581 + 0.016 0.531 + 0.016

0.07 (F ixed) 0.07 (F ixed)

variat ion of the ra t io ;,i.o/;>g.; between the outenll2 rods and the imner ..i . .

six rods of the cluster. .:Since the other three ratios are fixed i n

the least squares analysis no information could be obtained from thei r

comparison.

-...

. A simple way t o resolve the 0 4 0 / 2 + 9 discrepancjes would be t o force I . .

the data to agree via the spatial correction (mean chord length, :)..term 1

in the slowing 'down calculation of the HRG. code. For 'example, cal cul ati'6ris

of the rat io 0:0/6:9 for different mean chord lengths i resulted i n the

conclusion that a value of i between 2 and 3 instead of the value 6.95 which

was used would provide a cross section ra t io in reasonable agreement with ($6 ') experiment. The Bell - prescription i s found to give i = 2.9 when applied

as follows:

where Eo = diameter of fuel i n a rod of the cluster,

"u = vnl~nnt! of fuel i n the cluster,

V1 = vol me 'enclosed by a rubber band enriching t'he clirster, and a . 5 = effective epi thermal neutron cross section when the cladding

and D20 coolant are homogenized.

Hence, i f the criterion were t o match calculation and experiment for the ',. CT:~/;:~ the Be1 1 recipe would provide a reasonable match- for t h i ~ . . ~ e t of

. . . . . . . . . . . . . - . . . ... . . - -

experiments .

. * ,,' . t c : . Asmkntioned above,' the effects of neutron leakage and the presence

of f iss ion products (gaseous and s tab1 e nucl ides) have been neglected

2, . i n . the calculations. The impact of neglecting the leakage e f f e c t was

evaluated by performing a slowing down calculation (P-1 , approximation

i n the HRG code) using the geometrical buckl i n g , 8' = 5 m-', f o r , the 9

.; PRTR loading as the leakage factor i n the calculation (assuming leakage

does not a f fec t the thermal cross section averages). The cross sections

from th i s calculation are compared i n Table V to those obtained where

leakage was neglected completely. The effect ive absorption cross sections ,

for 2 3 9 P ~ and 2 4 0 P ~ change by s l igh t ly less than 2% b u t the ra t ios are

unaffected when leakage i s included i n the calculation. In order to observe

changes i n the r a t ios , buckling values larger than 50 ms2 are required.

Similar calculations of the e f fec t of including thermal neutron leakage

show t h a t the local buckl i ng of the fuel zone where the samples were

irradi3ted must be larger than the geometrical buckling of the reactor

to introduce errors i n the analysis suf f ic ient to account fo r the observed

discrepancy. A resul t of a two dimensional diffusion theory reactor

calculation f o r the loading i n which the data were collected shows tha t the ,

gradient across the sampling zone i s negligible. Thus, the assumption

tha t leakage does not a f fec t the thermal cross section averages appears ,

valid, . & . . ,

' E f f e c t i v e ' Cross Sections, ' G (barns) - Rat ios 4-9 i40 ;2 39 ;40/;49 a a a a a

F r a c t i o n a l N o No No .No Burnup Leakage Leakage Leakage Leakage - Leakage Leakage Leakage ' Leakage

. . %,*.' +. . . . : ' . : ?, ; ~.;.c,al culat-ion of the neutron spectrum was made with f iss ion rproducts - ;,j3

: . .: 2 . : : .. . .

b .

. . included i n the fuel o f t h e ce l l . The concentrations of f iss ion products

were taken from resul ts of an e a r l i e r burnup calculation(z) performed fo r

th i s same ce l l . The f iss ion products , n 1 3 5 ~ e , 149Srn, 151Sm, l S 5 E u , 155Gd,

and four pseudo elements were assumed, to be present. The cross

sections calculated using the spec t rm which include f iss ion products a re

-- compared in Table VI to those obtained when f iss ion products were neglected. YI

The ef fec t ive cross section for 2 3 9 P ~ changes by about 2-3% whereas the

value for 2 4 0 P ~ i s re la t ive ly unaffected indicating a change mainly i n

the thermal 'neutron energy component of G . The r a t i o ug i s insensi t ive

to the e f f e c t of f i ss ion products. The r a t i o s:0/6:9 increases b y about

1 to 4% between 25 and 0.3% depletion, respectively, when the f i ss ion

products are considered.

In summary, the ra t ios u 4 l , and 6;O/6i9 are a l l d i f fe rent from

the values deduced i n the f i t . However, & 4 9 and G41 are w i t h i n one standard

deviation. The neglect of the f iss ion products and neutron leakage i n

the spectrum calculation does, not introduce s igni f icant e r ro r in the

calculated resu l t s . Therefore, assuming tha t the method of deducing the

values from the isotopic data i s correct, the deviation between the

calculated and deduced values must be caused by the calculated resul ts

and occurs because of inaccurate theore ti cal models and basic neutron

crvss section data.

, Exami nation of Po t en t i a1 Causes:.of the Di screpanc.~ .-.r <_..

0. .- I . E

The .accuracy of calculating fuel b u r n u p .behavior i s usually eval uated

by using the i n i t i a l fuel concentration in a time dependent calculation

of the flux and spectr.urii averaye cross sections. In the evaluation a

. . . #

TABLE V I -

THE EFFECT OF F ISSION PRODUCTS ON EFFECTIVE CROSS SECTIONS AND RATIOS

A Effective Cross Sections, a (b. L Ratios

A49 G A 4.0 (3

A49 a A40,$+9 a a (T Fractional

Burnup No F.P? R P . No F.P. -- F.P. No F.P. F.P. No F.P. F.P.

* F.P. meaning f i s s i o n products.

\

determination i s made concerning ho\.;r we1 1 the i so topieyivari a t ions ,

w i t h exposure a r e reproduced. Each i so top ic concentration i s compared

separate ly and usual l y various degrees of accuracy a r e obtained.

Correlat ions of the e f f ec t i ve cross sect ion r a t i o s a r e believed t o be a

more s t r i ngen t t e s t than the i so top ic concentration. A s e t of r a t i o s

a r e obtained which res ul t i n an equal l y good f i t to a1 1 of the measured

i so top ic data. 1 n addi t ion, they allow a separate evaluat ion of cross

sect ions and theore t ica l models s ince the uncertainty of calcula t ing

the i so top ic concentrations can be removed by making a comparison

independent of the t,ransmutation calcula t ion. This i s accomplished by

incl uding the measured concentrations i n the calculation. . However, the

method of deducing the cross sect ion r a t i o s must be cor rec t i n order

t o obtain val id conclusions about the cslcu!ations. -

The r e s u l t s of a study by Page usiyg the same Al-PU data have

shown t h a t when agreement i s obtained between measured ,and cal culated

i so top ic concentrations the calcula ted and deduced cross sec t ion r a t i o s

were a l so i n agreement. Based upon the r e su l t s of t h a t study i t i s

expect.eTI t h a t . t h e values of deduced cross s e c t i m . r a t i o s i n Table IV

a r e valid.

The r e s u l t s from Reference 47 a l so val idate the cal cula t ional

approach presented i n t h i s paper. In addi t i on, the resul ti of a previous '

study based upon our calcula t ional appro ich@') agree w i t h those obtained - :50' 5lJ by Page when the basic neutron cross sect ionsh ' a r e iden t ica l i n

.? , *. - The cal cul ated val ues of i 4 9 andci41 obtained from2,each study.:were . - -, . *. -. )ti'

i n good agreement (t 1%) whereas the values of 3 0 / 0 : 9 .,.obtained by Page

were d i f fe rent from other values obtained in .our study. The principal

reason for the disagreement in th i s r a t io i s tha t Pags used a one-

dimensional model to calculate the neutron spectra in the vicini ty of the .

large 2 4 0 P ~ resonance a t 1.056 eV. This one-dimensicnal representation '

provides a bet ter ' calculation of the neutron spectra i n t h i s energy region

and therefore, should give be t te r values for the spec t rm averaged cross . .

section for 2 4 0 P ~ . -. .

Improvements have been made to the HRG and THERMOS ce l l codes and

' ( 3 9 3 j In' hew are reported as HRG-3 and BRT-I , respecti vely . ,

evaluations of cross 'sections were made available a f t e r these studies were

.cofip:etzd. Thz rise6 '- c~ --"".- I l l u ~ I ly the ccl? codes srew o u t s f thecc resu! t s , .

and the resu l t s of correlations of other calculated and measured reactor

parameters. ' (%), The advent of the Evaluated Nuclear Data File (EEIDF/B)

has provided improved estimates of - the neutron cross sections. The 1965

. . (55 ) and included in version I ! :nd the IAEA e ~ a l u a t i o n ' ~ has been updated - . ..

ENDF/B f i l e as the hazis of nnrmalization o f the f i s s i l e nuclides, (56) - . ., .

The modifications incorporated in HRG-3 a f fec t the calculations of the

resonance integral by adapting the method of Adler, Hinman, and Nordheim to

an in temedia te resonance formulation, by dis t r ibut ing the contribution of

broad resonances over several f ine groups, .and by correcting the epi-

thermal t ransfer cross sections i n water. tbAllow fo r upscattering by

hydrogen. These modifications resul t i n appreciable increases in the

calculated nonthermal cross sections of 2 4 0 P ~ and 2 4 2 P ~ b u t give only

. . . .. :.. , :,

, . . ,.

.'hi . smal l . {c,hhges i n the nonthermal crcs.s sec t ions o f 239P'u and 2 4 1 P ~ . , .

However, the resonance a l l o c a t i o n change increases the absorp t ion

..: whereas the i n te rmed ia te resonance fo rmu la t i on decreases the absorp t ion

r a t e w i t h the n e t r e s u l t t h a t these changes tend t o cancel. The

_ a d d i t i o n a l upsca t te r i ng by hydrogen i s n e g l i g i b l e f o r the D20 moderated

and cooled PRTR f u e l . The c a l c u l a t e d r a t i o o:b/;:9 and 642/G:9 t h e r e f o r e

i s expected t o increase by a few percent w i t h t'hese m o d i f i c a t i o n s , w h i l e

t he r a t i o s cr9 ,(141, and 0:1/39 would remain e s s e n t i a l l y unchanged. The

m o d i f i c a t i o n s incorpora ted i.n t he BRT-I code i n c l ude general i z e d boundary

cond i t ions , an improved . cu r ren t approximation, and more ' p r a c t i c a l e d i t s .

None o f these changes are expected t o s i g n i f i c a n t l y mod i fy the r e s u l t s

. o f t h i s study.

The resonance i n t e g r a l s f o r i n f i n i t e d i ' l u t i o n and the 2200 m/sec

values f o r t he cross sec t i ons used i n t h i s s tudy f o r the p lu ton ium

iso topes a r e summarized i n Table V I I I . A lso shown are the values

(9, conta ined i n Version I 1 o f the ENDFIB f i l e .

The d i f fe , rence i n the abso rp t i sn cross sec t i ons a t 2200. m/sec i s n o t

sign,E.fi can t enough t o cause a spectrum change. heref fore, the d i f f e rence

i n t h e e f f e c t i v e values would be p r 6 p o r t i o n a l ( a j t o those o f t he . .

2200 m/sec values f o r 2 3 9 ' P ~ and. 241Pu. Thus, t he d iscrepanc ies between

the c a l c u l a t e d and measured value o f &49 w i l l i nc rease from 4.5 percent A

t o Q 6.Z percent . Correspondingly, t h e c a l c u l a t e d value 6f (141 i s go ing

to.-decrease, by s 3.7 pe rcen t and be i n het'ter agreement w i t h t h e value

de r i ved from the burnup data. The c a l c u l a t e d r a t i o d:l/o:g w i l l decrease

Q 1.5 percent .

TABLE ' V I I -

SUMMARY OF THE CROSS SECTION DATA FOR Pu ISOTOPES

~esonance I n t e g r a l , . I _ . f o r I n f i n i t e D i l u t i o n - - - - - - -

I OC f o r a 0.41 4 eV C u t o f f (barns) , 2200 mlsec Values (barns)

F i s s i o n Capture Cross I so tope BNWML ENDFIB-I I BIiWML ENDFIB-I I Psotope Sec t ion BNWML ENDFIB-11 '

The s 3 percent difference in the 2200 m/sec caplure-cross section ,

\ .' showri i n .Table VIn, for 2 4 0 P u wi 11 propagate to an % 1 ?percent change in \

the calculated ef fec t ive value. The current best estjmate of the 2200

m/sec. cross section fo r 2 4 2 ~ ~ i s near, 18.5 b ra ther than the 30 b value

and this new value will be i n Version I11 of ENDF/B,. Thus, the calculated

r a t i o 6:2/6:9 i s expected t o decrease s ignif icant ly; +.The shape of the

cross section for 2 3 9 P u between the BNWML and ENDF/B-I1 are negligible; (58

therefore, no change i s expected --I i n the r a t io s considering th i s as

a source. . .

-

The resonance integral differences shown i n tab!^ VIII are not

expected to cause the conclusions of the comparisons to change. The

only cross sections which are expected to a f fec t the calculated ra t ios a re

the 2 4 0 P u .and 2 4 2 P ~ cross sections because the nonthermal absorption rates

i n the f i s s i l e isotopes a re negligible. However, the differences in the

2 4 0 P u . and 2 4 2 P ~ resonance integral val ues ar.e small.

Thus, very l i t t l e change ( l e s s than a few percent) would~occur i n the

resu l t s presented i n Table IV by subst i tut ing the improved versions of the

codes (HRG-3 and BRT-I) and the use of Version-Ii ENDF/B cross sections. .

The changes a re i n some cases a reduction. i n the 'discrepancies noted (e.g. ,

040/G49 and i 4 1 ) and i n other cases increase the discrepancy (e.g., 2"). a a Although the discrepancy between the calculated and measured values of & 4 9

i s < 2 standard deviations ,the thermal ization cal cul ation and/or the thermal

: neutron cross sections fo r . 2 3 9 ~ u might be judged 'inaccura,ie and ci'using .

0. .-

the noted discrepancy. Clearly, a homogeneous representation of the 19-rod

. . c lus ter i s inadequate for cal cuq ating the nonthermal absorption rates in

Z'tOPu and 2 4 2 P u and' i s the major cause o f the discrepancy i n the r a t i o

SUMMARY AN.D CONCLUS IOFIS I , . . . . . .

. "

Two methods for deducing r a t ios of effect ive crosscsections from

isotopic data have been presented. One method uses equations which are

expl i c i t functions of the concentrations and th i s method y ie l ds ra t ios

which are a function of exposure. The other method uses equations which

are imp1 i c i t functions of the concentrations and th i s method yields

ra t ios which are .constant or have a predetermined functional re1 ationship

with respect to exposure. Where applicable the preferred method i s the

l a t t e r or integrated equation approach. I t provides a more rigorous -- --.A

s t a t i s t i c a l analysis of the data and requires no a pr ior i values to be

assumed' for the cross section ra t ios . . , In th i s method, a1 1 the concentration

data can be f i t simul taneously and , the method correctly accounts fo r

correlated uncertainties. Both niethods uti 1 i ze specially developed programs

of l e a s t squares technique to f i t the transmutation re?ationships to the

isotopic data. The coeff ic ients of these f i t s a re the ra t ios of effecti.ve

cross sections.

Measurements of fuel isotopic composition have been obtained as

functions of exposure from fuels i r radiated in !!arious power reactors.

These data have been f i t t e d using these .methods to obtain ra t ios of

effect ive cross sections. The resu l t s of the f i t s obtained from A1-Pu fuels

i r radiated in PRTR have been shown to i l l u s t r a t e the application of the

- technique. These data were f i t w i t h good s t a t i s t i c a l precision basically

~.:.;beca.use~ the condi t ions under.!.whi ch these fuel s were i rradi ated wece we1 1 . .

' -, .-

known: and f ree from extraneous perturbations .to the neutron spectrum such

as those resulting from control rod ef fec ts .

Calculations were made to obtain resdl t s fo r coniparison w i t h the value

. deducedrfrom the experimental data. ' In order to -bri ngefthe calculated and > ra

deduced values of G 4 9 in agreement would require about 'a 4-5 percent

. . ..decrease i n the 2200 mlsec value of 6 for 2 3 P ~ from the value used in our

analysis. Correspondingly, the .cur rent value on ENDFIB.-I1 would have to be

decreased by % 6 percent to force agreement. A more l ikcly candidate in

the cross sections themselves of th i s di-screpancy is the variation of i 3 9 ~

with energy i n the thermal neutron energy range. However, the s t a tus of

knowledge-on the thermal cross sections fo r 2 3 9 ~ u has improved over recent

years to the point where the uncertainties associated with these data are not . ---- . .

. .

large enough to account for the discrepancy. ' Therefore, the more

logi cal candidate for the cause of th i s discrepancy' i s inadequacies

i n the THERMOS code. The resu l t s of other plutonium fueled reactor (59 ,ti ) +..A .--. -- + . s ~ p p a r t t h i s ccntention. -. 1.uu1 e3

The avai lab i l i ty of data on the ra t ios of effect ive cross sections

are important fo r tes t ing the accuracy of burnup cal culations. Where

these ra t ios can be derived with good s t a t i s t i c a l preci'sion, they are a

sens i t ive t e s t of the calculations and can be used as the c r i t e r i a fo r

judging the accuracy of u n i t ce l l calculations, as shown by the example

presented in th i s paper. For those data, where the ra t ios are l e s s

certain due to spectral perturbations during burnup, they cannot be used

singly as a c r i t e r i a for tes t ing the calcilations. @) ~lev&rtheless ,

these rati.os a re s t i l l a::useful adjunct to the isotopic data i n theory- i m

experiment correlations in tha t they can po.int to gross inadequacies in

the cal cul a t i ons .

. . . . . .

The authors thank W.. L. Purcell f o r assistan'ce ir; performing

the analyt ical ca lcu la t ions and E. B. Reppond f o r helping t o analyze

the data and combining the data compiling codes i n to an' e f f i c i e n t

analysis package. In addi t ion, !d. Y . Matsumoto provided a c r i t i c a l .: :. .

review and many useful suggestions f o r the sect ion on rans smut at ion Measurements. The enormous amount of iiork accomplished by personnel

of t he Radiometallurgy, Chemistry, and Mass'Spectroscopy Labora- . .

t o r i e s i s a l so appreciated. hanks . a re . , 'given espec ia l ly t o

W. Y: Matsumoto, A . C . Leaf, F. A . Scot t , C. R. Lagergren and

M. W . Goheen whose con ti nual e f f o r t s t o provide precise data res ul t ed

i n the s:cczssful car-' ,ztSon .sf t k . 2 sttidy.

LIST OF FIGURE CAPTICINS ::: . . . . . .

' ;Figure 1.. A Three Dimensiona1"Pro:jection of .theCBurnu!piof 2 3 9 ~ u . . -

Figure 2.' Graphical Resu1,ts of Generalized Least Squares Fits for 1.8 w t % P u .

. . . . .

-Figure 3 . h Plutonium Depletion for 1.8 w t % P U as a Function of J35~u concentration. . . . .

~ i g u r e 4. Values of 0 2 0 / 0 2 3 as a Function of 2 4 0 ~ u Atom Concentration.

. . LIST OF TABLE HEADINGS :;.

- . . . . . . . . .

Tab le I. Fuel Types and Exposures

Tab le 11. Cross S e c t i o n R a t i o Values and T h e i r One Standard 'i D e v i a t i o n s 5 o r A1-Pu Con ta in i ng I n i t i i t l ' l y : 6 % 2 4 0 ~ ~

Tab le 111. Cross S e e t i on Ra t i os ~ r o m I n t e g r a t e d Equat ion Method

Tab le I V . Compariso.n o f C a l c u l a t e d Ra t i os o f E f f e c t i v e Cross Sec t ions t o Values Obta ined f rom Leas t Squares Ana l ys i s of Exper imenta l Data f o r Lx PuAl C l u s t e r s '

. .

Tab le V. The E f f ec t of Leakage on ~ f f e c t i v e Cross ~ e c t i o n s ' ; h d Rati.os -

Tab le V I . The E f f e c t o f F i s s i o n Products on E f f e c t i v e Cross s e c t i o n s and Ra t i os

Tab le V I I . Summary o f t h e S ta tus o f Cross S e c t i o n Data f o r Pu Iso topes :

1.

i

FOOTNOTES

( a ) ~ o w the current best value for the ~s~~~ half l i f e appeais to

be 30.12 + 0.21 yrs . reported by S. A. Reynolds and E. I . Wyatt

i n the "Analytical Chemistry Division Annual Progress Report fo r

Period Ending October 31, 1966," ORNL-4039 (January 1967).

. . . . , (6) This f i t t i n g process involves the more familia'r l e a s t squares method

and i s accomplished using Program Learn. The analysis logic of

Proyam Learn i s a special case of Program Likely. I t i s obtained

by se t t ing the span of the vector f ie lds 9 and y_ to one ( i . e . ,

J = L = 1 ) and particularizing the imp1 i c i t vector theory g(1,g) = 0

to the exp l i c i t sca lar the0ry.y = y(A). ' .

A recent report: M. G. Cabell and M. Wilkins, "An Isometric

S ta te of 2 4 1 ~ u , " ~ . Iriorg. NUcl. Chem., - 33, 903 (!971), gives a

15..16 year half l i f e for 2 . 4 1 P ~ .

-65- . . Z # .

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