SPECTRUM AVEP AGED CROSS SECT1 ONS DEDUCED FROM …
Transcript of 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
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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
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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.
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..... . . ., . .
-.- 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 . . - .
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. . . .,. . . -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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