Impact of Nuclear Data Uncertainties on Transmutation of Actinides

38 Pages·2003·1.22 MB·English

NUCLEARSCIENCEANDENGINEERING:146,13–50~2004!
Impact of Nuclear Data Uncertainties on Transmutation of Actinides
in Accelerator-Driven Assemblies
G.Aliberti, G. Palmiotti, M. Salvatores,*† and C. G. Stenberg
ArgonneNationalLaboratory
NuclearEngineeringDivision,Building208
9700SouthCassAvenue
Argonne,Illinois60439
ReceivedOctober15,2002
AcceptedMay8,2003
Abstract–Thepotentialimpactofnucleardatauncertaintiesonalargenumberofperformanceparam-
eters of reactor cores dedicated to the transmutation of radioactive wastes is discussed. An uncertainty
analysishasbeenperformedbasedonsensitivitytheory,whichunderlinesthecrosssections,theenergy
range,andtheisotopesthatareresponsibleforthemostsignificantuncertainties.
Toprovideguidelinesonprioritiesfornewevaluationsorvalidationexperiments,requiredaccura-
ciesonspecificnucleardatahavebeenderived,accountingfortargetaccuraciesonmajordesignparam-
eters. The required accuracies (mostly in the energy region below 20 MeV), in particular for minor
actinidedata,areofthesameorderofmagnitudeoftheachievedaccuraciesonmajoractinides.Specific
requirements also concern the improvement of minor actinide data related to decay heat and effective
delayed-neutronfractionassessment.
I.INTRODUCTION to the standard fuel components, or the use of reactor
coresdedicatedtotransmutationinaseparatestratumof
Amongthestrategiesforradioactivewastemanage- the fuel cycle.3,4 In the latter case, the dedicated reactor
ment,theso-calledpartitioningandtransmutation~P0T! coreshouldbeloadedwithMA-dominatedfuel,andboth
strategy has attracted considerable interest in the last critical and subcritical @i.e., accelerator-driven system
decade,andrelevantstudieshavebeenperformedinsev- ~ADS!# versions of such cores have been the subject of
eralleadinglaboratories,sometimesunderthecoordina- several studies.5
tion of international organizations ~see, for example, Althoughthemajorchallengesofthededicatedcores
Ref.1!.Mostofthestudieshavepointedouttheroleof aretobefoundintheappropriatefueldevelopmentand
minoractinide~MA!transmutationtoreducethesource inthedemonstrationoftheviabilityoftheADSconcept,
ofpotentialradiotoxicityindeepgeologicalstorageand one aspect of particular relevance is the uncertainty as-
of long-lived fission product transmutation in order to sessmentofthenominalpredictedcharacteristicsofsuch
eventually reduce the so-called residual risk.1 In both cores. A first partial intercomparison exercise was per-
cases, the transmutation should be performed in a neu- formedundertheauspicesoftheNuclearEnergyAgency
tron field, preferably with a fast neutron spectrum.2 ~NEA! of the Organization for Economic Cooperation
Among the different scenarios of implementation of the and Development6 ~OECD!. The published results did
P0T strategy, there has been a remarkable convergence show large discrepancies among the different param-
ontwomajoroptions,3,4namely,theuseofstandardcrit- eters, most probably to be attributed to nuclear data un-
ical fast reactors, where, for example, MAs are mixed certainties. Some other studies7–9 have been performed
thatexaminespecificaspects,butnocomprehensiveanaly-
*E-mail:[email protected] sis has been performed until now. Also, these studies
†Present address: DEN0DIR Building 101, CEA0 only partially address the issue of the impact of high-
Cadarache,F.13108St.-Paul-Lez-DuranceCedex,France energy ~E . 20 MeV! data on ADS core performance
13
14 ALIBERTIetal.
assessment.Theimpactofuncertaintiescanbeverysig- where the sensitivity coefficients S are formally given
j
nificant,bothonthesafetyassessmentandtheeconomic by
evaluation of a dedicated core. In fact, as an example,
]Q s
uncertaintiesonthesubcriticalitylevelofanADSdedi- S 5 { j . ~4!
cated to transmutation induce the need to define design j ]s Q
j
margins, which in turn can result in a proton beam-
The variance of Q can then be obtained as
power requirement that calls for an accelerator able to
deliver up to twice as much current of what is needed J
accordingtothenominaldesignvalueofthesubcritical- var~Q!5(S S c . ~5!
j i ij
ity. Moreover, a sound uncertainty analysis can help to j:i
define new priority measurements of specific cross sec-
ToexploitEq.~5!oneneedstoobtainexplicitlythe
tionsinwell-definedenergydomains,togetherwithtar-
S coefficients and to establish an appropriate variance-
j
getaccuracies.Inthiswork,wehaveperformedsuchan
covariance matrix. For a set of integral parameters Q
n
analysis for a representative ADS-dedicated core with ~n51...N!,theassessmentofthevariancesasgivenby
U-free,MA-dominatedfuel.Wehaveaddressedbothstan-
Eq. ~5! is of course relevant, but it is also relevant to
dard core-related parameters ~which will be applicable
assess the inverse problem, i.e., what are the required
both to critical or subcritical versions of the core! and
data uncertainties to meet specific target accuracies on
high-energy-related parameters ~like damages and gas
the Q parameter.
n
productioninthestructures!potentiallysensitivetodata
The unknown uncertainty data requirements d
i
atenergiesE.20MeV.Anattempthasalsobeenmade
can be obtained solving the following minimization
to define target accuracies and to point out major data
problem12:
needs.
(l 0d25min , i51...I ~6!
i i
i
II.UNCERTAINTYANALYSIS
with the following constraints:
II.A. Theoretical Background (S2d2,QT , n51...N , ~7!
ni i n
i
The principles of uncertainty analysis and its appli-
cations to the fission reactor field are well document- where S are the sensitivity coefficients for the integral
ni
ed.10 We will simply recall here that we can represent a parameterQ ,andQTarethetargetaccuraciesontheN
n n
generic integral reactor parameter Q ~such as keff, or a integralparameters.Thecostparametersliarerelatedto
reactivity coefficient, or even a reaction rate like the each s and should give a relative figure of merit of the
i
neutron-induced damage in the structures! as a function difficultyofimprovingthatparameter~e.g.,reducingun-
of cross sections: certainties with an appropriate experiment!.
Q5f~s1,s2,...,sJ! , ~1! II.B. Sensitivity Coefficients and
Perturbation Theories
wheres ,s ,...,s representcrosssectionsbyisotope,
1 2 J
typeofreaction,andenergyrange~orenergygroup,ina For practical purposes, we will distinguish the ex-
multigrouprepresentation!.Theuncertaintiesassociated plicit dependence from some cross sections ~e.g., se!
i
withthecrosssectioncanberepresentedintheformofa andtheimplicitdependencefromsomeothercrosssec-
variance-covariance matrix: tions~e.g.,sim!inthegeneralexpressionofanyintegral
j
1 2 parameter Q:
c c J c
11 12 1J
C 5 c12 c22 J c2J , ~2! Q5f~sjim,sie! . ~8!
s J J J J As an example, we consider a reaction rate
c1J c2J J cJJ R5^st e,Fu & ~9!
where the elements cij represent the expected values re- where brackets ^,& indicate integration over the phase
lated to the parameters sj and si. space. Note that in the present analysis, Fu is the inho-
The variations of the integral parameter Q due to mogeneous flux driven by the external source. It would
variationsofscanbeexpressedusingperturbationtheo- bethehomogeneousfluxinthecaseofcriticalcorestud-
ries11 to evaluate sensitivity coefficients S: ies. Instead, the adjoint flux that appears later in the
papercorrespondstothehomogeneouscalculationinall
ds
dQ0Q5(S j , ~3! cases.InEq.~9!, st e canbeanenergy-dependentdetec-
j j sj torcrosssection;Risexplicitlydependentonthest e and
NUCLEARSCIENCEANDENGINEERING VOL.146 JAN.2004
NUCLEARDATAUNCERTAINTIESINACCELERATOR-DRIVENASSEMBLIES 15
implicitlydependentonthecrosssectionsthatcharacter- II.B.2. Reaction Rates
izethesystem,describedbythefluxFu .Inotherterms,R
dependsonthesystemcrosssectionsviaFu .Equation~3! The classical formulations found in Ref. 11, for ex-
can be rewritten as follows: ample, can be applied to the case of damage rate or He
S D production in the structures or to the power peak factor
dQ0Q5(S dsjim 1 ]Q {se {dse , ~10! in the core:
j sim ]se Q se
j j R5^Fu ,Su & . ~15!
R
wherewehavethehypothesisofanexplicitdependence
of Q on only one se. If we drop the index im, The sensitivity coefficients are given by
S D
dQ0Q5(S dsj 1 ]Q {se {dse 5I1D , SjR5^CtR*,sjFu & , ~16!
j s ]se Q se
j j
where Fu is as defined previously, Ct * is the solution of
~11! R
where the term I is generally called the indirect effect, M*Ct *5Su , ~17!
R R
and the term D is called the direct effect. While the di-
rect effects can be obtained with explicit expressions of and M* is the adjoint of the operator M.
thederivativesofQ,theindirecteffect~i.e.,thesensitiv-
itycoefficientsS!canbeobtainedwithperturbationex-
II.B.3. Nuclide Transmutation
pression, most frequently at the first order.11
In what follows, we will recall in a simplified way ThegenericnuclideKtransmutationduringirradia-
theformulationsofthesensitivitycoefficientsatthefirst tion can be represented as the nuclide density variation
order for the indirect effects related to reactivity coeffi- betweentimet andt .IfwedenotenKthefinaldensity,
0 F F
cients,13reactionrates,11andnuclidetransmutation~i.e., the appropriate sensitivity coefficient is given by
evolution in time14!. Reactivity loss during irradiation
E
wtioilnlaolfsodebleayteredanteedutarsownselalnadsothfethceasdeescoayfehfefeact.tivefrac- SK5 ]nFK{sj 5 1 tFnt *s nt dt , ~18!
j ]s nK nK j
j F F t0
II.B.1. Reactivity Coefficients
where the time-dependent equations to obtain nt * and nt ,
Areactivitycoefficient~liketheDopplereffect!can together with their boundary conditions, are defined in
beexpressedasavariationofthereactivityoftheunper- Ref. 14.
turbed system ~characterized by a value K of the multi- Themethodpreviouslydescribeddoesnottakeinto
plication factor, a Boltzmann operator M, a flux Fu , and account the coupling with the flux field,15,16 neglecting
an adjoint flux Fu *!: inthiswaythefeedbackfromfluxandspectrumchanges
S D S D
during irradiation time. We show in Sec. IV.F that this
1 1 1 1
Dr5 12 2 12 5 2 , ~12! approximation is acceptable in the cases under study
Kp K K Kp and that the time dependence of the flux spectrum is
negligible.
where K corresponds to a variation of the Boltzmann
p
operator such that
II.B.4. Reactivity Loss During
M rMp~5M1dMp! Fu rFu p~5Fu 1dFu p! Irradiation, Drcycle
Fu * rFu *~5Fu *1dFu *! KrK ~5K1dK ! .
p p p p At the first order, and neglecting the cross-section
~13! variationduringirradiation~whichisagoodapproxima-
tion for fast neutron systems!, we can write
Thesensitivitycoefficients ~atfirstorder!forDrto
variations of the sj arHe given as in Ref. 13: J Drcycle 5(DnKr , ~19!
K
]~Dr! s 1 1 K
SRO5 { j 5 ^Fu *,s Fu &2 ^Fu *,s Fu & ,
j ]sj Dr Ifp p j p If j where
~14!
DnK 5nK2nK ~20!
F 0
where I 5 ^Fu *,FFu & and Ip 5 ^Fu *,FFu &, F being the
f f p p
neutron fission production part of the M ~5 F 2 A! and r is the reactivity per unit mass associated with
K
operator. isotope K.
NUCLEARSCIENCEANDENGINEERING VOL.146 JAN.2004
16 ALIBERTIetal.
The related sensitivity coefficients Scycle associated whereCt * andCt ~generalizedimportancefunctions!are
j
with the variation of an s are given by the solution of the following equations:
j
Scycle5 sj ]Drcycle M*Ct * 52nSf~r,E!^Fu *,xo & 1 nSf~r,E! ~26!
j Drcycle ]s ^Fu *,FFu & ^FFu &
j
S D
and
s ]nK ]r
5 Drcjycle (K ]sj{rK1(K DnK ]sKj . ~21! MCt 5 S~r,E! 2 x~E!^nSu fFu & , ~27!
^Fu *S& ^Fu *,FFu &
Using the formulations of Secs. II.B.1 and II.B.3,
we obtain where we have explicitly introduced the energy- and
E space-dependent form of the fission operator, and
Scycle5( rK tFnt *s nt dt nSf~E,r! ~component of the vector nSu f! is the macro-
j Drcycle j scopicfissioncrosssectionmultipliedbythepromptneu-
K t0 tron fraction at energy E and space point r, and x~E!
H J
~componentofthevectorxo !isthefractionofthefission
1 1
1 ^Fu *,s Fu &2 ^Fu *,s Fu & , ~22! spectrum at energy E; the brackets ^,& indicate integra-
Ip p j p I j
f f tion over energy and space.
where the index p refers to the core state at t5t .
F II.B.6. Decay Heat
Also in this case, the time-dependent variation of
thefluxspectrumduringirradiationissupposedtobeof The decay heat is defined as
negligible impact on the sensitivity coefficients for
Drcycle ~see Sec. IV.F!. H~t!5(lKQKnK~t! , ~28!
K
II.B.5.The w* Parameter whereforeachisotopeK,l arethedecayconstants,Q
K K
is the heat released in decay reaction, and n ~t! are the
K
Thew* parameterisdefinedforanexternalsource- nuclide densities at time t. The equations for n ~t! are
K
drivensystemastheratiooftheaverageexternalsource the classical ones:
importance to averaged fission neutron importance:
Y S DYS D dnK~t!
w* 5 ^Fu *S& ^Fu *,FFu & 5 1 21 1 21 , dt 5(F gK,ftf 1(j nKj~t!tjbjrK
^S& ^FFu & k K
eff S
1(ni~t!libirK2tKnK~t!2lKnK~t! ,
~23! i
~29!
where
or in a more compact form,
^Fu *,FFu &
keff 5 ^Fu *,AFu & dnk~t! 5b 1K(21C nj~t!2C nk~t! , ~30!
dt k kj kk
^FFu & j51
KS 5 ^AFu & where
g 5fission yields for fissionable isotope f
Fu 5solutionoftheinhomogeneousequationwith K,f
external source S: t5microscopic reaction rates
AFu 5FFu 1S . ~24! bjrk5branching ratios.
This is an inhomogeneous Bateman-type equation that
Equation ~23! is a special case of a real and adjoint
definestheappropriatenuclidefield.Theuncertaintyon
fluxfunctionalratioI forwhichageneralizedperturba-
S H~t! is obtained by combining the appropriate deriva-
tion theory ~GPT! has also been established.9
tivesofHwithrespecttol,Q,andnandaccountingfor
For that case, the sensitivity coefficients are given
possiblecorrelations.Asfarasvariationsofthen terms,
by K
they can be evaluated using the perturbation techniques
indicatedinSec.II.B.3.Aspecificfeatureisrepresented
]w* s s
Sw* 5 j 5 j $^Ct *,s Fu &1^Ct ,s Fu *&% , ~25! by the variation of the fission yields g, i.e., by the vari-
j ]sj w* w* j j ation of the source term bK in Eq. ~30!.
NUCLEARSCIENCEANDENGINEERING VOL.146 JAN.2004
NUCLEARDATAUNCERTAINTIESINACCELERATOR-DRIVENASSEMBLIES 17
The relative sensitivity coefficients corresponding and
to the decay heat at t5t are given by S D
x 1 @b ndS Fu #xd~E! @nS Fu #x
Sg 5t ]ntK5tx{gK,f 5 tf Etxnt *g dt . ~31! A2 K F Ct 5 ^xiidFu *,bf indSifFu & 2 ^Fu *,fFFu & .
K,f f ]gK,f ntK5tx ntK5tx 0 K,f ~36!
II.B.7.The Effective Fraction II.C. Calculational Tools and
of Delayed Neutrons Basic Data Library
The effective fraction of delayed neutrons, bZ , is Allthesensitivitycalculationshavebeenperformed
eff
defined by the following equation: with the ERANOS code system,17 which allows us to
calculatehomogeneousandinhomogeneoussolutionsof
bZ 5(bZ m , ~32! the Boltzmann equation, generalized importance func-
eff eff
tions,andtoperformperturbationanduncertaintyanaly-
m
sis.ThediscreteordinatemoduleBISTRO~Ref.18!has
where bZemff is the effective delayed-neutron fraction of been used to perform flux and generalized importance
fissile material m. For each fissile material m, bZeff 5 function calculations.An S4P1approximation in RZ ge-
( bZ , where bZ , the effective fraction for the precursor ometryhasbeenprovedaccurateenoughforthistypeof
i i i
group i, is expressed as follows: calculation.
Decay heat calculations have been performed with
bZ 5 ^xidFu *,bindSfFu & the ORIGEN code.19
i ^Fu *,FFu & Cross-section data have been processed to the re-
E quired multigroup structure, starting from the JEF-2.2
b @xd~E!Fu *~r,E,V!#@nd~E’!S ~r,E’!Fu ~r,E’,V’!#dr datafiles.20 Homogeneouscrosssectionshavebeencal-
i i f
5 , culated because heterogeneity effects on the cross sec-
^Fu *,FFu & tions are rather small in these hard neutron spectra.
Delayed-neutron data were also taken from the JEF-2.2
~33!
files.
The basic multigroup structure ~33 energy groups,
where
see,forexample,TableXI!hasanupperenergylimitat
nd 5numberofdelayedneutronsemittedbyfission 19.64 MeV.
Toinvestigatehigh-energy~E.20MeV!effectsin
xid5delayed-neutron spectrum for the group i asubcriticalsystemdrivenbyaspallationneutronsource
inducedbyhigh-energyprotons~E ’1GeV!,themulti-
b 5fractionofdelayedneutronsfromthegroupi. p
i groupdatahavebeenextendedupto150MeVusingthe
Using the GPT, the sensitivity coefficients for bZ , data provided in Ref. 21. For that purpose, ten energy
eff groups with a lethargy width of 0.2 have been added to
including both the direct ~i.e., related to the delayed-
the basic 33-energy-group structure to cover the energy
neutronparameters!andtheindirecteffect,aregivenby
range from 19.64 to 150 MeV.
]bZ b ]bZ xd ]bZ s
SbZ 5 eff i 1 eff i 1 eff j
j ]b bZ ]xd bZ ]s bZ
i eff i eff j eff III.THEREFERENCEDEDICATEDSYSTEM
]bZ b ]bZ xd s FORTHEUNCERTAINTYANALYSIS
5 eff i 1 eff i 1 j
]b bZ ]xd bZ bZ III.A. The Reference System
i eff i eff eff
3$^Ct *,s Fu &1^Ct ,s Fu *&% , ~34! The methodology outlined in Sec. II has been ap-
j j
plied to a dedicated system that has some general fea-
whereCt * andCt ~generalizedimportancefunctions!are tures ~e.g., the mass ratio between plutonium and MA,
the solutions of the following equations: the americium-to-curium ratio, etc.! that are representa-
tive of the class of MAtransmuters with a fast neutron
b @Fu *xd#ndS ~r,E! spectrum and a fertile-free fuel, as proposed, for exam-
~A*2F*!Ct * 5 i i f ple,intheframeworkoftheOMEGAprojectinJapan,as
^xdFu *,b ndS Fu &
i i f studied at Commissariat à l’Energie Atomique, France,
or examined in the United States.
@Fu *x#nS ~r,E!
2 f ~35! The target and the coolant material of the core are
^F*,FFu & the Pb-Bi eutectic. This is a more specific choice, in
NUCLEARSCIENCEANDENGINEERING VOL.146 JAN.2004
18 ALIBERTIetal.
particular in terms of coolant, which in principle, how-
ever,doesnotaffectmuchtheoveralluncertaintyanaly-
sisfeaturesbecausethesearemorerelatedtothetypeof
neutronspectrum~i.e.,fastversusthermalneutronspec-
trum!. Finally, the system that we have chosen is very
closetothesubcriticalcore,whichhasbeenanalyzedin
theframeworkofanOECD-NEAbenchmark.6Thegeo-
metric model and compositions are given in Fig. 1 and
Table I, respectively.
The spallation source in space and energy has been
generatedusingtheMCNPXcode,assumingabeamra-
dius of 10 cm of protons with an energy of 1 GeV. For
the successive propagation of neutrons using the deter-
ministic code system indicated in Sec. II.C, a cut-off
energyof20MeVhasbeenchosen.Asfarasthespectral
distribution,14%ofthespallationneutronsourceisabove
20 MeV. Figures 2 and 3 show the axial and radial dis-
tributions, respectively, of the neutron source split into
four energy bins: 0 to 6.1, 6.1 to 19.6, 19.6 to 55.2, and
55.2 to 150 MeV.
III.B. Main Parameters of the
Reference System
The main parameters of the reference system, ob-
Fig.1. GeometryofthereferenceADScore~R,Zmodel!. tained with the calculation route indicated in Secs. II.C
TABLE I
CompositionsoftheReferenceCore
Fuel Reflector Target0Buffer
Compositions Compositions Compositions Compositions
Isotope ~1024 at.0cm3! Isotope ~1024 at.0cm3! Isotope ~1024 at.0cm3! Isotope ~1024 at.0cm3!
237Np 4.377E204a 58Fe 4.386E205 54Fe 2.990E203 Pb 1.320E202
238Pu 4.226E205 50Cr 1.128E204 56Fe 4.560E202 Bi 1.632E202
239Pu 5.051E204 52Cr 2.096E203 57Fe 1.075E203
240Pu 2.321E204 53Cr 2.328E204 58Fe 1.344E204
241Pu 1.232E204 54Cr 5.682E205 50Cr 3.458E204
242Pu 9.102E205 58Ni 6.451E205 52Cr 6.422E203
241Am 8.084E204 60Ni 2.384E205 53Cr 7.134E204
242mAm 1.089E205 61Ni 1.015E206 54Cr 1.741E204
243Am 5.827E204 62Ni 3.173E206 58Ni 1.977E204
242Cm 4.079E208 64Ni 7.792E207 60Ni 7.305E205
243Cm 3.326E206 Mo 1.163E204 61Ni 3.111E206
244Cm 2.371E204 Mn 1.114E204 62Ni 9.724E206
245Cm 3.164E205 Pb 6.360E203 64Ni 2.388E206
246Cm 5.355E207 Bi 7.865E203 Mo 3.565E204
Zr 7.477E203 182W 6.984E206 Mn 3.412E204
15N 1.058E202 183W 3.770E206 Pb 4.075E203
54Fe 9.759E204 184W 8.045E206 Bi 5.039E203
56Fe 1.488E202 186W 7.439E206 182W 2.140E205
57Fe 3.507E204 183W 1.155E205
184W 2.465E205
186W 2.280E205
aReadas4.37731024.
NUCLEARSCIENCEANDENGINEERING VOL.146 JAN.2004
NUCLEARDATAUNCERTAINTIESINACCELERATOR-DRIVENASSEMBLIES 19
Fig.2. Axialdistributionoftheneutronsourcebyenergy Fig.3. Radial distribution of the neutron source by en-
domain~MCNPXcalculation!. ergydomain~MCNPXcalculation!.
TABLE II
MainParametersoftheReferenceSystem
Drcycle
bZ
eff
k ~pcm! DrDopplera Drvoid 1yrb 2yrb DecayHeatc PeakPowerd
eff
0.948164 185.4 20.00026 10.02906 20.01196 20.02158 25MW~thermal! 2.9
~Dn0n!cyclee @1024 at.0cm3#
238Pu 241Am 242mAm 243Am 242Cm 244Cm 245Cm
1.23 21.07E21f 7.66E21 28.99E22 6.57E12 9.62E22 4.74E22
aForDT5T2T 51773to980K.
Ref
bAtfullpower.
cAtdischarge.Nominalpowerofthecore:377MW~thermal!.
dSeetext.
e~nF2n0!0n0after1yrirradiation.
fReadas21.0731021.
TABLE III
MainParametersoftheReferenceSystem
Maximumdpaa MaximumHeProductiona MaximumHProductiona Maximum
w* ~s213cm23! ~s213cm23! ~s213cm23! ~Heproduction!0dpaa
1.18 2.58E116b 6.15E115 6.77E116 0.24
aSeetextfordescription.
bReadas2.5831016.
NUCLEARSCIENCEANDENGINEERING VOL.146 JAN.2004
20 ALIBERTIetal.
and III.Ausing the 33-energy-group structure, with up- of the displacements per atom ~dpa!, He production,
perenergyboundaryat19.64MeV,aregiveninTablesII and H production ~all in iron! at the spatial point where
and III. they reach their maximum value. The maximum value
In Table III, w* is the ratio of the average external oftheratio~Heproduction!0dpaiscalculatedatitsown
source importance to average fission neutron impor-
tance introduced previously.
The peak power is defined as the point maximum
power value normalized to the total power. Max dpa,
Max He production, Max H production are the values
Fig.4d. Maximum He production: ~R,Z! 5 ~20 cm,
Fig.4a. Referencecoordinatesforthefuelregion. 107.5cm!.
Fig.4e. Maximum H production: ~R,Z! 5 ~20 cm,
Fig.4b. Maximum peak power: ~R,Z! 5 ~20 cm,
107.5cm!.
102.5cm!.
Fig.4f. Maximum~HeProduction!0dpa:~R,Z!5~20cm,
Fig.4c. Maximumdpa:~R,Z!5~20cm,105cm!. 107.5cm!.
NUCLEARSCIENCEANDENGINEERING VOL.146 JAN.2004
NUCLEARDATAUNCERTAINTIESINACCELERATOR-DRIVENASSEMBLIES 21
maximum value position. In Figs. 4b through 4f, the
spatial distributions of the peak power, Max dpa, Max
He production, Max H production, and Max ~He
production!0dpa are shown ~Fig. 4a allows visualiza-
tion of the reference coordinate system!.
One can observe that the power peak is obtained
approximately at core midplane, where the flux reaches
its highest value. The Max dpa, He-, and H production
are located a few centimetres above the core midplane,
the location of the maximum of the higher energy neu-
tronscomingfromspallation.TheHe-andH-production
spatial distribution are peaked at the core0buffer inter-
face, while the dpa and power are more evenly distrib-
uted in the core.
Fig.6. Coreaverageadjointfluxspectrum.Forcompar-
In Table II, DrDoppler corresponds to the reactivity
ison, the adjoint flux spectrum for the typical fast reactor
induced by a jump in temperature between 980 and
PHENIXisalsoshown.
1773K. The Drcycle is the reactivity variation resulting
froma1-or2-yrirradiation.Amoredetailedanalysisof
these parameters is given later in this paper.
Finally,the~Dn0n!cycle,i.e.,therelativevariationof
a few selected major isotope nuclear densities ~238Pu,
241Am, 242mAm, 243Am, 242Cm, 244Cm, and 245Cm!, is
a measure of the effectiveness of the transmutation
process.
InFigs.5and6,theneutronfluxandadjointspectra
are also given. One can observe a harder neutron spec-
trum ~both real and adjoint! with respect to a standard
fastreactor~e.g.,forthePHENIXreactor!.Thiseffectis
relatedpartlytothecontributionofthehigh-energyneu-
trons coming from spallation, partly to the presence of
Pb-Biascoolant,andpartlytothehigherimportanceof
thehigh-energyfissionsinthesystem.Itcanbeofinter-
estinthisrespecttoinspecttheenergyshapeoftheh5 Fig.7. Theh5nsf0saenergyshapeforselectedactinides.
ns 0s parameterforseveralactinides.Infact,thesharp
f a
high-energy slope of the h of 241Am, 243Am, and 244Cm
~presentinhighpercentageinourreferencesystem!shows
aremarkabledifferencewithrespecttothatof 239Pu,for IV.UNCERTAINTYANALYSIS
example ~see Fig. 7!.
IV.A. Variance-Covariance Matrix
for Multigroup Data
Variance-covariancedataarestillscarceinallmajor
datafiles,inparticularforminoractinidesandmaterials
likePborBi,whichplayanimportantroleinourstudy.
Since a significant part of our work was based on the
JEF library data, we have used for major actinides and
some structural material ~Fe, Cr, Ni! uncertainty data
providedinRef.22.Formajoractinides,sincemosteval-
uationsinthemajordatafilesarebasedoncommonsets
of experimental data, significant variation of the uncer-
tainty values is not expected. For minor actinides, we
havedefineduncertaintiesbasedonacomparativeanaly-
sis among major data files performed in the framework
oftheNuclearScienceCommitteeoftheNEAofOECD
~Ref. 23!. For example, large uncertainties for thermal
Fig.5. Coreaveragefluxspectrum.Forcomparison,the andepithermaldataof241Amor243Amhavebeenpointed
fluxspectrumforthetypicalfastreactorPHENIXisalsoshown. outinthisstudy.ForstructuralmaterialslikePbandBi,
NUCLEARSCIENCEANDENGINEERING VOL.146 JAN.2004
22 ALIBERTIetal.
we have intercompared data files and extracted an edu- files, and in particular data for MAs have not been
cated guess for uncertainties. assessed. However, it was considered of interest to al-
The diagonal values used, reduced to a 15-energy- low for some hypothesis, at least on the energy correla-
group substructure of the reference 33-group structure, tions,togainsomeinsightontheirpotentialimpact.We
are shown inTables IV,V, and VI. have chosen rather arbitrarily to introduce a full corre-
As far as correlations, most of our analysis has lationonselectedenergydomains,thesameforalltypes
been based on the hypothesis of no correlation among of cross sections and isotopes. These energy ranges are
uncertainties,inparticularofnoenergycorrelationfora 20 to 1 MeV, 1 MeV to 100 keV, 100 keV to 1 keV, and
specific reaction type. Since the present analysis is per- 1 keVdowntoepithermalenergy.Thepurposeofthese
formedat15energygroups,itimplicitlyallowsforafull correlations is essentially to impose energy shapes on
energy correlation for each reaction type within the en- thecrosssections,asobtained,forexample,frommodel
ergy range of each group. calculations.Werefertouncertaintiesobtainedwiththis
As mentioned previously, the variance-covariance hypothesis as having been obtained with partial energy
dataarerelativelyseldomassociatedwithevaluateddata correlations ~PECs!. The uncertainty analysis presented
TABLE IV
VarianceMatrixforMajorActinides*
Group ~MeV!a n s s s s s n s s s s s
f inel el capt n,2n f inel el capt n,2n
238Puand 240Pu 239Pu
1 19.6 0.012 0.05 0.15 ˆ 0.3 0.16 0.008 0.03 0.1 ˆ 0.1 0.13
2 6.07 0.014 0.05 0.15 0.3 0.0075 0.037 0.1 0.085 0.25
3 2.23 0.018 0.1 0.15 0.3 0.007 0.037 0.1 0.095
4 1.35 0.02 0.1 0.15 0.3 0.0065 0.065 0.15 0.13
5 4.98E21b 0.028 ˆ 0.2 0.25 0.0055 0.04 0.15 0.13
6 1.83E21 0.03 0.2 0.15 0.008 0.028 0.15 0.078
7 6.74E22 0.0312 0.2 0.1 0.1 0.015 0.03 0.2 0.039
8 2.48E22 0.0311 0.1 0.008 0.045 0.25 0.05 0.056
9 9.12E23 0.031 0.2 0.1 0.008 0.063 0.25 0.056
10 2.04E23 0.03 0.1 0.0051 0.02 0.065
11 4.54E24 0.029 0.1 0.005 0.025 0.065
12 2.26E25 0.028 0.08 0.003 0.025 0.065
13 4.00E26 0.027 ˇ ˇ 0.03 0.0024 0.025 0.039
14 5.40E27 0.026 0.5 0.05 0.005 0.0022 0.0025 0.008
15 1.00E27 0.019 0.5 0.05 0.005 0.002 0.0025 ˇ 0.008
241Pu 242Pu
1 19.6 0.01 0.125 0.15 ˆ 0.5 0.18 0.012 0.05 0.15 ˆ 0.3 0.25
2 6.07 0.0095 0.2 0.15 0.5 0.2 0.015 0.05 0.15 0.3
3 2.23 0.009 0.05 0.15 0.4 0.019 0.1 0.15 0.3
4 1.35 0.0085 0.05 0.15 0.3 0.02 0.1 0.15 0.3
5 4.98E21 0.008 0.06 0.2 0.2 0.03 ˆ 0.2 0.25
6 1.83E21 0.007 0.1 0.2 0.2 0.0317 0.2 0.15
7 6.74E22 0.0065 0.1 0.2 0.15 0.0316 0.2 0.1 0.1
8 2.48E22 0.006 0.08 0.1 0.15 0.0315 0.1
9 9.12E23 0.0055 0.08 0.1 0.031 0.2 0.1
10 2.04E23 0.005 0.03 0.1 0.03 0.1
11 4.54E24 0.0045 0.03 0.1 0.029 0.09
12 2.26E25 0.004 0.03 0.1 0.028 0.08
13 4.00E26 0.0035 0.03 0.1 0.027 ˇ ˇ 0.08
14 5.40E27 0.003 0.006 0.014 0.025 0.05 0.07 0.01
15 1.00E27 0.0024 0.006 ˇ 0.014 0.02 0.05 0.07 0.01
*Variancematrix~ds0s!.
aUpperenergyboundary.
bReadas4.931021.
NUCLEARSCIENCEANDENGINEERING VOL.146 JAN.2004