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TRANSCRIPT
LA-8333-MS
&3Informal Report .
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g.-
-WI I
CIC-14 REPORT COLLECTION
REPRODUCTION~n~v
A Comprehensive Neutron Cross-Section and
Secondary Energy Distribution Uncertainty
Analysis for a Fusion Reactor
L%- LOS ALAMOS SCIENTIFIC LABORATORYPost Office Box 1663 Los Alamos. New Mexico 87545
An AffumativeAction/EqualOpportunityEmploycr
ThisreportwasnoteditedbytheTechnicalInformationstaff.
ThisworkwassupportedbytheUS DepartmentofEnergy,OfficeofFusionEnergy.
This report was prepared as an account of work sponsoredby the United States Government. Neither the UnitedSIaIes nor the United States Department of Energy, norany of thek employees, makes any warranty, express or!mphed, or assumesany legal Iiab!lity or responsibility forthe accuracy. completcnc% or usefulness of any infor-mation, appar~tus, produc~ or process disclosed, or repre-sents that IIS US?would not infringe privwely owned rights.Reference herein to any specific commelcid product,process, or service by trade name, mark, manufacturer, orotherwise, does not necessarily constitute or imply itsendo,wncm, rccmnme”&tio”, or favoring by the UnitedSlates Government or any agency thereof. The views undopinions of authors expressed herein do not necessmilysfate or rcfkct those of the United StaIes Governmentor any agency thereof,
UNITED STATIS
DEPARTMENT OF ENERGY
CONTRACT W-740 S-CNG. 36
LA-8333-MSInformal Report
UC-20d, UC-34CIssued: May 1980
A Comprehensive Neutron Cross-Section and
Secondary Energy Distribution Uncertainty
Analysis for a Fusion Reactor
S.A. W. GerstlR. J. LaBauveP. G. Young
-- .. -,. ------- . . -,.“
---- . . .
—.. .—. , -.— .. . . . ----- ..- .-.
.
CONTENTS
ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...1
I.
II.
III.
IV.
v.
VI.
INTRODUCTION . . . .
THEORY . . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
CROSS-SECTION m SECONDARY ENERGY DISTRIBUTION (SED)
A. Cross-Section Covariance Data . . . . . . . . .
B. Secondary-Energy-DistributionCovariances . . .
. . . .
. . . .
. .
. .
. .
. . .
. . .
. . .
NUCLEONICS ANALYSIS FOR GA’S TNS REACTOR CONCEpT (PGFR) . .
QUANTITATIVE RESULTS . . . . . . . . . . . . . . . . . . .
.
.
.
.
A. Response Uncertainties caused by Reaction Cross-Section
Uncertainties . . . . . . . . . . . . . . . . . . . . .
B. Response Uncertainties caused by SED Uncertainties . . .
.
.
.
.
.
.
c. Total Response Uncertainties dueto all Data Uncertainties
CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
2
5
6
6
9
.11
.13
.15
.15
.18
REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
!
APPENDIXA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
APPENDIXB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
iv
A COMPREHENSIVE NEUTRON CROSS-SECTION AND SECONDARY
ENERGY DISTRIBUTION UNCERTAINTY ANALYSIS FOR
A FUSION REACTOR
by
S. A. W. Gerstl
R. J. LaBauve
P. G. Young
ABSTRACT
On the example of General Atomic’s well-documented PowerGenerating Fusion ~eactor (PGFR) design, this report exe;cises~ comprehensive neutron cross-section and secondary energydistribution (SED) uncertainty analysis. The LASL sensitivityand uncertainty analysis code SENSIT is used to calculate re- ,action cross-section sensitivity profiles and ’integralSEDsensitivity coefficients. These are then folded with covari-ance matrices and integral SED uncertainties to obtain theresulting uncertainties of three calculated neutronics designparameters: two critical radiation damage rates and a nuclearheating rate. The report documents the first sensitivity-baseddata uncertainty analysis,which incorporates a quantitative treat-ment of the effects of SED uncertainties. The results demonstratequantitatively that the ENDF/B-V cross-section data files for C,H,and O, including their SED data, are fully adequate for thisdesign application, while the data for Fe and Ni are at best mar-ginally adequate because they give rise to response uncertaintiesup to 25%. Much higher response uncertainties are caused bycross-section and SED data uncertainties in Cu (26 to 45%),tungsten (24 to 54%), and Cr (up to 98%). Specific recommen-dations are given for re-evaluations of certain reaction cross-sections, secondary energy distributions,and uncertainty estimates.
1
I. INTRODUCTION
One of the first steps in any new fusion reactor design is a neutronic
analysis to determine adequate tritium breeding, nuclear heating in blankets and
coils, acceptable radiation damage rates, etc. In an early phase it is usually
sufficient to perform radiation transport calculations with a one-dimensional
conceptual design model and allow for multi-dimensional streaming effects by
estimated “streaming factors” which assure the l-D.model to be conservative.
However, uncertainties in calculated neutronics design parameters due to cross-
section uncertainties will be present in both 1-D as well a~ multi-dimensional
analyses. These latter uncertainties can be conveniently estimated by performing
a cross-section sensitivity and uncertainty analysis(1)
based on the one-dimen-
sional model.
Such quantitative data assessments have been performed in the past for var-
ious different fusion reactor designs, cf. e.g. Refs. (1) through (3). However,
none of these analyses considers the effects of all nuclear data uncertainties
simultaneously. Specifically, the effects of uncertainties in secondary energy
and angular distributions have not been
to the lack of a consistent methodology
secondary distributions. Only recently
and relevant uncertainty data are being
This uncertainty analysis includes
incorporated in the past, primarily due
and the lack of uncertainty data for
such methodology has been developed(4)
made available.
the effects of uncertainties in all
neutron reaction cross-sections relevant to the model, including correlations,
and the effects of estimated uncertainties in the neutron ~econdary Energy
Distributions (SED’S). Effects of uncertainties in secondary angular distribu-
tions are not incorporated for the lack of uncertainty data. Also, any uncer-
tainties in gamma ray cross sections are neglected for two reasons: (1) generally,
gamma ray interaction cross-sections are known to a much higher degree of accuracy
(at least an order of magnitude) than neutron interaction cross-sections,and (2)
only one of the three critical nuclear design parameters in our design depends at
all on the gamma ray
II. THEORY
The theoretical
uncertainty analysis
2
distribution.
expressions underlying any sensitivity-basedcross-section
have been developed previously, cf. e.g. Refs. (1) and (2),
and are given here only for completeness. The variance of any calculated inte-
gral design parameter Rk due to correlated uncertainties in given multigroup
cross-section sets {Z~] and {Z~] can be calculated from
g,g’
(1)
r-
neutron interaction cross-section for reaction i in energy
group g,
relative covariance matrix element for the multigroup cross-
sections Z! and Z;’,
cross-section sensitivity profile for Z? with respect to
the integral response Rk.
All cross-section uncertainty information about reaction cross-sections is con-
tained in the relative covariance matrix which is independent of the specific
reactor design. All sensitivity information about reaction cross-sections enters
Eq. (1) through the product of the sensitivity profiles which, of course, are
highly problem-dependent and specific for a particular reactor design and for
the particular design parameter considered.
If the total response uncertainty due to many cross-section uncertainties
is desired, then the relative standard deviation of the response Rk due to all
reaction cross-section uncertainties considered for a particular material is
given by
(2)
3
assuming that the variances due to individual (partial) cross-section uncertain-
ties are uncorrelated.
The theory for the consistent incorporation of the effects of uncertainties(5)
in secondary energy distributions (SED’S) has only recently been developed .
The concept of hot/cold integral SED-sensitivities,.which requires the specifica-
tion of integral SED-uncertainty parameters, is adopted and applied here. /LS
derived in Ref. 5, the relative standard deviation of an integral response
Rk due to SED-uncertainties for a specific reaction, !2,that generates secon-
daries, is given by
(-)‘k =
‘k ~
where
‘~,g’ =
SSED!?,g’ =
xl SSED!2,g’“fQ,g’ ‘
g’(3)
integral SED-uncertainty for neutron interaction ,Qat the in-
cident energy group g!; f is also called the spectral shape
uncertainty parameter,
integral SED-sensitivity for the
at the incident energy group g’.
As noted in Ref. 5 the integral SED-sensitivity
neutron interaction !2
may be positive or negative, in-
dicating whether the response Rk is more sensitive to the hot portion of the
SED or its cold part:
SSED = SSED SEDHOT - ‘COLD ‘
(4)
where the hot and cold portions of the integral SED-sensitivity are defined with
respect to the median energy of the secondary distribution, cf. Ref. 5.
Equation (3) is valid for the sum of all SED-uncertaintiespertaining to a
single type of neutron interaction, identified by the subscript !2. If the effects
of SED-uncertainties from all possible neutron interactions with one material are
to be considered, then we may assume that their effects on the responses Rk are
uncorrelated. With this assumption, the relative standard deviation of the res-
ponse Rk due to all SED-uncertainties considered for a particular material is
given by
EL-SE, =@ (2))~ (5)
Under the same assumption of independent, and therefore uncorrelated, effects
on R due to both all SED uncertainties and all reaction cross-section uncertain-k
ties per material, the total relative standard deviation of Rk per material is
given by
(&T=J((D , (6)
which results from the quadratic sum of Eqs. (2) and (5). However, before any of
the relative standard deviations defined in Eqs. (2), (3), (5), and (6) may be
evaluated quantitatively, the data uncertainties in the form of covariance
matrices and integral SED uncertainties, as well as the sensitivity profiles and
integral SED-sensitivities must be quantified. In section III we describe how
the required data uncertainties were obtained. In order to obtain the required
sensitivity information a complete neutronics design analysis of the reactor sys-
tem must be performed which is described in section IV.
III. CROSS-SECTION AND SECONDARY ENERGY DISTRIBUTION (SED) UNCERTAINTY DATA
One of the more important aspects of nuclear data is that the uncertainties
tend to be highly correlated through the measurement processes and the correc-
tions made to the observable quantities to obtain the microscopic cross sections.
In many applications, the correlations of the uncertainties in the nuclear data
play a crucial role in uncertainties in calculated results.
5
A. Cross-Section Covariance Data
Several versions of the reference cross-section data library known as ENDF/
B (EvaluatedNeutron Data Files-B) have been issued over the past 13 years, but
only the latest version, ENDF/B-V, contains formats14 and sufficient covariance
data for an application such as is described in this report. Covariance data
are given for 25 important nuclides in ENDF/B-V, which includes data for all
nuclides needed in this analysis except Cu and W. For these elements we used
the covariance files from Fe and Pb, respectively, assuming that the cross =c-
tions of Cu are as well known as those for Fe and the cross sections of W are
as well known as those of Pb. Note that both these assumptions are probably op-
timistic. It is planned, however, that covariance data for Cu and W will be in-
cluded in ENDF in the future, and the present calculationswill be repeated when
such data become available.
The covariance data in ENDF/B-V were processed with the NJOY code15 to
transform the data into the 30-group multigroup format needed in this study.
Data from the various runs with the NJOY code were collected to form a 30-group
covariance data library. The contents of this library, which is-in an ENDF-like
format, are shown in Table I.
B. Secondary-Energy-Distribution Covariances
It should be noted that ENDF/B-V does not contain data uncertainties and
their correlations for secondary energy distribution (SED) data, and at the time
of this writing, formats have yet to be specified. Hence, for the purpose of
estimating the magnitude of these effects, we have generated SED covariance ma-
trices using the very simple “hot-cold” concept outlined in Ref. 5.
The angle-averagedmedian energies of the composite elastic plus nonelastic
neutron emission spectra were calculated for each material as functions of inci-
dent neutron energy. Relative errors were then estimated for the portions of the
emission spectra lying above and below the median energies. If & designates the
integrated neutron emission cross section for a given incident energy and ‘H =
;/2 is the integrated spectrum for E’ > E~edian, then the relative uncertainties
of the hot (denoted by subscript “H”) and cold (subscript “C”) regions can be
specified by the quantity f = ACJH/8= - A(Jc/~. For the purposes of this study,
we have assumed no correlation in the SED uncertainties with incident neutron
energy.
Table II lists the median energies and the relative errors assumed in the
6
,
TABLE I
ENDF/B-V COVARIANCE DATA (MF=33) PROCESSED WITH NJOY CODE
MAT Iiuclide Ref.
1301 H-1 16
1305 B-10 17
1306 C 18
1324 Cr 19 ~
1326 Fe 20
1328 Ni 21
1326 Cu(Fe) 20
1382 W(Pb) 22
MT-Nos, Processed
1,2
1,2,107,780,781
1,2,4,51-68,91,102,104,107
1,2,3,4,16,17,22,28,102,103,104,105,106
Reaction Cross Sections
Total, elastic
Total, elastic, (n,a), (n,ao),
and (n,al)
Total, elastic, teal inelastic,inelastic levels 1-18, inelasticcontinuum, (n,y), (n,d), (n,a)
Total, elastic, nonelastic, totalinelastic, (n,2n), (n,3n), (n,n’cy),(n,n’p), (n,y), (n,p), (n,t), (n,d),
(n,3He), (n,u)
1,2,3,4,16,22,28,102, Total, elastic, nonelastic, total103,104,105,106,107 inelastic, (n,2n), (n,n’ci),(n,n’p),
(n,y), (n,p), (n,d), (n,t), (n,3He),(n,~)
1,2,4,16,22,28,51-76, Total, elastic, total inelastic,91,102,103,104,107, (n,2n), (n,n’U), (n,n’p), inelastic111 lefels 1-26, inelastic continuum,
(n,y), (n,p), (n,d), (n,a), (n,2p)
1,2,3,4,16,17,22,28, Total, elastic, nonelastic, total102,103,104,106,107 inelastic, (n,2n), (n,3n), (n,n’a),
(n,n’p), (n,y), (n,p), (n,d), (n,3He),(n,CY,)
1,2,3,4,16,17,51,52, Total, elastic, nonelastic, total64,102 inelastic, (n,2n), (n,3n), inelastic
levels 1, 2, and 14, (n,y)
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8
total neutron emission spectra for each of the elements present. The relative
errors were determined by adding in quadrature separate error components due
to elastic and nonelastic neutron reactions.
The elastic scattering components are based upon the evaluated errors given
in the ENDF/B-V files for12C 16
* O, and Fe, and upon our estimates of these
errors for the other materials. Typically, the elastic uncertainty is approxi-
mately 8% near 14 MeV and gradually reduces to a few percent near the inelastic
threshold. These errors are significantly smaller in the case of12C, which is
used as a standard in neutron scattering experiments.
The nonelastic neutron spectrum errors were determined by combining quadra-
tically a 15% component assumed to exist for all materials at all energies and
a second component based on comparisons with the 14-MeV spectrum measurements of
Hermsdorf et al23 24and Clayeux and Voignier. This latter component was included
to roughly account for variations in the accuracy of the individual evaluations.
Only in the cases of Cr and Ni did the addition of the second component signifi-
cantly change the total SED uncertainty. It should be mentioned that significant
differences also exist between the measured and calculated spectra for W, as has
been shown by Hetrick et al.25
In averaging over the “hot” (or “cold”) portions
of the 14-MeV spectrum for W, however, a significant fraction of this spectrum
difference disappears. This cancellation indicates one of the problems jnherent
in using such a coarse representation of SED errors.
IV. NUCLEONICS ANALYSIS FOR GA’S TNS REACTOR CONCEPT (PGFR)
General Atomic’s TNS (“The Next ~tep”) reactor design, also labelled Power(6)- -
—
Generating Fusion Reactor , has been selected as a representative model for all— — —
TNS reactor concepts; Fig. 1 shows a cut-away view of the PGFR. A nuclear analy-
sis for this reactor has been performed by General Atomic (GA) and is documented
in Ref. 7, which has been issued as Vol. IV of Ref. 6. In this analysis GA iden-
tifies as the three most critical nuclear design parameters (1) the radiation
damage to the superconducting TF-coil’s stabilizing matrix, (2) the radiation
damage to the alumina insulator in the F-coil, and (3) the nuclear heating in the
superconductingTF-coil. Two one-dimensional models have been employed in the GA
nucleonic analysis, an inboard and an outboard model. However, the inboard cal-
culations were sufficient to identify the above three most critical parameters.
Therefore, we selected for our data assessment task the PGFR inboard nucleonics
9
Fig. 1. General Atomic’s Bower ~enerating ~usion ~eactor (PGFR)
I I 1 I I I I I I I I I 1
n {cm
Fig. 2. PGFR inboard nucleonics model.
10
Imodel which is repr~duced from Ref. 7 and shown in Fig. 2. The symbol Q repre-
sents the 14-MeV neutron source at the plasma location while the RI ~ ~ indicate
the locations where the 3 critical responses are calculated for whi~h’the follow-
ing response functions were chosen:
RI =.=
R2 =.=
R3 =A=
CU dpa in the toroidal field coil (TF-coil)
radiation damage to the superconductingTF-coil’s stabilizing
Cu matrix,
Aluminum dpa in the field-shaping coil (F-coil)
radiation damage to the A1203 insulator in the T-coil,
Kerma in the TF-coil
nuclear heating in the superconducting TF-coil.
All data for the response functions as well as the multigroup cross-section sets
used
NJOY
tron
been
and :
base.
in the analysis (8,9) I(lo)
, were derived from ENDF/B-V and processed with the
code system into coupled neutron/gamma-ray multigroup sets with 30 neu-(11)and 12 gamma-ray groups . The resulting multigroup data library has
applied successfully at LASL to several other fusion nucleonics analyses
s therefore considered a well tested and reliable cross-section data
The nuclear analysis of the PGFR was performed with the LASL discrete-ordi-
nates code 0NETRAN(12) in S8 approximation and with P2 coupled neutron and gam-
ma-ray cross-sections. The angular flux distributions from the forward ONETRAN
run and the three adjoint runs (one for each of the above response functions)
were then used in the LASL sensitivity and uncertainty analysis code SENSIT(13)
to compute the relevant sensitivity profiles and response uncertainties.
V. QUANTITATIVE RESULTS
Uncertainties in the three critical design parameters
with SENSIT in two independent stages. First the response‘1,2,3
were calculated
uncertainties due to
uncertainties in neutron reaction cross-sections were calculated via Eqs. (1) and
(2), and then, in a second stage, the additional response uncertainties due to
estimated SED uncertainties were computed via Eqs. (3) through (5).
11
TRBLECROSS
ID-NO-----
1
232425262728293031323334353637383940
:;43444546474849505152535455565758
TABLE III
FOR DEFINITIONOF ID-NOS IN TERMS OF SPECIFICATION OFSECTION COV9RIRNCES. NOTE IN THIS VERSION,MRT1=MGT2
M9T1----
130513051305130513051305130613061306130613061324132413241324132413241324132613261326132613261326132613261326132613261326132613281328132813281328132!3132913291329132913291329132913291323132!313821382138213821382138213821382130113011301
MF7T2 MTl---- ---
1385 113E15 113051385 :1305 21305 18713E16 1130613E16 :1386 413~6 la?1324 11324 11324 21324 21324 41324 41324 1821326 11326 113261326 ;1326 21326 21326 41326 41326 41326 41326 1~21326 1(331326 1B713281328 ;1328 41328 IE121328 1E131329 113291329 i1329 21329 41329 41329 41329 41329 l@21329 1E131329 1071382 11382 113821382 :1382 21382 41382 41382 l@213E11 1130113B1 :
---
1
10?
10;107
;2
10?
;24
10:102
1
10:2
10:
102103107102103107
12
10:103
;24
10:103107102103107
1
10:24
102102
:2
CROSS SECTION COV19RII?NCE--------------------- ---
BIEI TOTRL WITH BIEI TOTftLBla TOTfIL WITH BIEI ELfr3TICBl@ TOTRL WITH BIEI (N.I?LPHR)B1O ELRSTIC WITH B18 ELFISTICBIEI ELRSTIC WITH Bl~ (N,RLPHF?)BIEI (N>RLPHfl) WITH BIB (N,FILPHFI)C TOTFIL LJITH C TOTfILC TOTRL WITH C ELfr3TICC ELFISTIC WITH C ELf)STICC INELflSTIC WITH C INELr%TICC (N.RLPH(J) WITH C (N,RLPHR)CRCRCRCRCRCRCRFEFEFEFEFEFEFEFEFEFEFEFEFENININININICuCuCuCu
::CuCuCuCuCuPBPBPBPBPBP8
TOTfIL WITH CR TOTfILTOTRL WITH CR EL(Y3TICELfr3TIC WITH CR ELfXTICELr?3TIC WITH CR INELfISTICINELfY3TIC WITH CR INELflSTICINELF7STIC WITH CR CfIPTURECRPTURE WITH CR Cf)PTURETOTfIL WITH FE TOTRLTOTfIL WITH FE ELf)STICTOTfJL WITH FE CQPTUREELfr3TIC WITH FE ELfV3TICELFv3TIC WITH FE INELf2STICELfISTIC WITH FE CfIPTUREINELfKTIC WITH FE INELrX3TICINELfT3TIC WITH FE CFIPTUREINELRSTIC WITH FE (N,P)INELflSTIC WITH FE (N,fILPH9)CfIPTURE WITH FE CRPTURE(N,P) WITH FE (N,P)(N,FILPHFI) WITH FE (N,9LPHR)TOTfIL WITH NI TOTRLELF3STIC WITH NI EL(X3TICINELf%3TIC WITH NI INELRSTICCflPTURE WITH NI CRPTURE(N,P) WITH NI (N,P)TOTflL WITH CU TOTf+LTOTfIL WITH CU ELfKTICELfr3TIC WITH CU EL%TICELRSTIC WITH CU INELfISTICINELfISTIC WITH CU INELr9STICINELr?STIC WITH CU C9PTUREINELFISTIC WITH CU (N,P)INELI%3TIC WITH CU (N,f9LPHf))C9PTURE WITH CU CRPTURE(N,P) WITH CU (N,P)(N.fILPHf?l WITH CU (N.9LPHfl)TOTfIL WITH PB TOTRLTOTfiL WITH PB ELf13TICTOT9L WITH PB CfIPTUREELf13TIC WITH PB ELFV3TICELRSTIC WITH PB INELr?STICINELr?STIC WITH PB INELRSTIC
PB INELRSTIC WITH PB CRPTUREPB CRPTURE WITH PB C9PTUREH TOTRL WITH H TOTQLH TOTRL WITH H ELRSTICH ELFISTIC WITH H ELFISTIC
12
A. Response Uncertainties caused by Reaction Cross-Section Uncertainties
A total of 24 SENSIT runs were performed to compute ARk/Rk cause by reac-
tion cross-section uncertainties which are given as relative covariance matrices
for pairs of partial reaction cross-sections. Table III gives a listing of these
covariance matrices and identifies the relevant pairs of partial cross-sections
by ID-numbers. Partial cross-sections for one material were only paired with
partials for the same material because it is assumed that the cross-section un-
certainties for one material are uncorrelated with those for another material.
However, certain materials are present in more than one spatial zone of the
PGFR design; cf. Fig. 2. For example, chromium is present in the first wall, in
all stainless steel structural walls and in the TF and E coils, but with a dif-
ferent number density in each zone. Therefore, to simplify the interpretation
of our results, we report contributions to response uncertainties due to cross-
section uncertainties by material zones as def<ned by their operational function,
like Cr in the first wall, or in all SS walls, or in all coil structures, etc.
Appendix A gives detailed results of our cross-section uncertainty analysis
for those response uncertainty components which exceed a 10% standard deviation.
The sensitivity profiles for each of the two partial cross-sections of each pair
are also given in these tables. A summary of all calculated response uncer-
tainties due to all cross-section uncertainties is given in Table IV. All
standard deviations in Table IV per material (i) and zone (j) are obtained by
quadratically summing the standard deviations caused by all partial cross-
sections for that material according to Eq. (2). The overall response uncer-
tainties ‘or ‘1,2,3due to all reaction cross-section uncertainties are given in
the bottom line of Table IV. Large standard deviations of 71.9% and 125.2% are
predicted for the responses RI and R3, respectively. In both cases the largest
contributions originate from cross-section uncertainties in Cr, W, Cu, and Ni.
However, due to the unavailability of evaluated cross-section uncertainty data
for Cu and W we substituted the covariance data for these materials with those
for Fe and Pb, as was explained in Section 111A. These substitutions are thought
to be optimistic in the sense that the Fe and Pb covariances are probably
generally smaller than such data would be for Cu and W. Therefore, we feel these
substitutions do not seriously weaken our conclusions that the Cu and W cross
section data are serious sources of uncertainty in the RI and R3 responses.
13
TABLE IV
PREDICTED RESPONSE UNCERTAINTIES (STANDARDDEVIATION)DUE TO ESTIMATED CROSS-SECTION UNCERTAINTIES
CROSS SECT. Res onse Uncertainties in ercentMAT. ZONE(i)
‘j) g~; (..1 (+~x Qj ~’:
c TILES 1.69 1.69 2.05 2.05 1.70
Cr FIRST WALL 0.38 0.47 0.38
SS WALLS 0.13 28.7 0.04 0.47 5.3
TF+E COILS 28.7 6x10-10 98.2
Ni FIRST WALL 1.79 2.21 1.79
SS WALLS 1.81 6.91 0.78 2.34 1.5
TF+E COILS 6.43 9X10-9 24.8
Fe FIRST WALL 0.18 0.22 “ 0.18
SS WALLS 2.61 14.1 1.67 1.68 2.46
TF+E COILS 13.9 2X10-8 15.8
Cu F-COIL 3.68 4.8o 3.6633.0 4.80
TF+E COILS 32.8 8X10-9 45.1
w SHIELD 54.0 54.0 0.196 0.196 54.3
H FIRST WALL 0.07 1:11 0.073
H20 COOLT. 0.49 5.74 0.76 0.77 0.50
SHIELD 5.72 0.066 8.7
0 FIRST WALL 0.25 0.28 0’.25
H20 COOLT. 1.51 7.67 1.01 1.05 1.50
SHIELD 7.52 0.014 7.5
4LL* 71.9 6.12
:
H‘3 *
‘3 i
1.70
98.3
24.9
15.99
45.3
54.3
8.71
7.65
125.1
f:) quadratic S~S
14
B. Response Uncertainties caused by SED Uncertainties
In a second series of 24 computer runs with SENSIT (in SED anaylsis mode
ITYP = 3)13, the additional response uncertainties ARk/Rk due to estimated uncer-
tainties in secondary energy distributions were evaluated according to Eqs. (3)
and (5). Integral SED uncertainties f!?,g’
between 0.5% and 17% were used as input,
as specified in Table II, for incident neutron energies between 16 and 0.13 MeV.
Appendix B gives the detailed results of our SED uncertainty analysis for
those calculated response uncertainties which exceed a 10% standard deviation..-SEDIntegral SED-sensitivity coefficiefitsS , together with S~T
ing to Eq. (4), are also given in Appendix B for each incident
calculated response uncertainties due to all SED uncertainties
Table V by material (i) and zone (j). It should be noted that
and S~D accord-
energy group. All
are summarized in
in this case no
substitutions of cross-section or uncertainty files were performed as was neces-
sary for the copper and tungsten reaction cross-section uncertainty analysis.
The largest response uncertainties due to SED uncertainties are calculated for
RI (33.1%) and R3(37.1%) which are both mainly due to Cu and W SED uncertainties,
each contributing between 23 and 28% in AR/R.
A note of caution must be added here, which should be taken into considera-
tion when the results of this SED uncertainty analysis are interpreted. As men-
tioned in Section 111.B, the integral SED uncertainties were estimated for the
composite secondary energy distributions of elastic and nonelastic reactions,
which resulted in the fairly low spectral shape uncertainty parameters fQ,g’
listed in Table II. In addition, such composite SED’S often exhibit two dis-
tinctly separated peaks, one due to the elastically scattered secondaries
peaking fairly close to the incident neutron energy, and the second at much
lower energies due to nonelastic emission neutrons. In these situations it is
questionable how ade’quatethe SED uncertainties can be realistically described
by the simple hot/cold concept which is the basis for our”anaylsis(5). Quite
possibly this concept may result in too coarse a representation and may there-
fore underestimate the real response uncertainties due to SED uncertainties.
This potential inadequacy could be remedied, however, if SED uncertainties were
treated separately for individual partial cross-sections.
c. Total Response Uncertainties due to all Data Uncertainties
Summary Tables IV and V give the calculated response uncertainties by mate-
rial as defined in Eqs. (2) and (5), respectively. In Table VI we compare the
15
TABLE V
PREDICTED RESPONSE UNCERTAINTIES (STANDARDDEVIATION)DUE TO ESTIMATED SED UNCERTAINTIES
CROSS SECT. Res onse Uncertainties in ercent IMAT. ZONE(i)
‘j)
(;$: p g~x~~’ ’100
+-k==-&SS WALLS I 0.33
SS WALLS I 0.31
TF+E COILS 6.50
Cu F-COIL 5.2
ITF+E COILS I 25.6
++=-=-&
/ )ARl~
FF’_i
0.41
1.17
0.86
6.57
26.1
23.5
0.99
1.05
33.1
*) quadratic sums
16
*
+
0.23 0.35
2’10-10
0.74
----10.08 0.75
1’10-10
0.07
0.51 0.52
6X10-1” iz10.1 10.11X10-9
0.37
0.07
0.36
0.01
0.10
0.46
0.02
0.37
0.37
0.47
10.2
H‘3R
3 i,j
0.41
0.24
0.31
0.76
0.79
0.13
0.28
0.06
0.84
4.50
5.2
27.5
23.9
0.09
0.39
0.95
0.11
0.46
1.03
/~3
+\R3 i
0.41
0.86
0.85
4.58
28.0
223.9
1.03
1.13
37.1
TABLE VI
TOTAL RESPONSE UNCERTAINTIES (STANDARDDEVIATION)DUE TO ALL DATA UNCERTAINTIES
Response Uncertainties in precent[nput Cross-Section Data AR, AR. I AR.
Jx 100‘1
111 C due to XS-Uncert. 1.69
due to SED-Uncert. 0.41
\ll Cr due to XS-Uncert. 28.7
due SED-Uncert. 1.17
\ll Ni due to XS-Uncert. 6.91
due to SED-Uncert. 0.86
\llFe due to XS-Uncert. 14.1
due to SED-Uncert. 6.57 “
ill Cu due to XS-Uncert. 33.0
due to SED-Uncert. 26:1
dl W due to XS-Uncert. 54.0
du& to SED-Uncert. 23.5
dl H due to XS-Uncert. 5.74
due to SED-Uncert. 0.99
~11O due to XS-Uncert. 7.67
due to SED-Uncert. 1.052..
Lll 79.2
~’)quadratic sums
-JX 100 I Jxloo‘2 ‘3
2.05 1.70
0.26 0.41
0.46 98.3
0.35 0.86
2.34 24.9
0.75 0.85
1.68 15.99
0.52 4.58
4.80 45.3
10.1 28.0
0.20 54.3
0.37 23.9
0.77 8.71
0.37 1.03
1.05 7.65
0.47 1.13
11.9 130.6
calculated response uncertainties due to cross-section uncertainties with those
due to SED uncertainties per material. We note that in almost all cases the
response uncertainties due to SED uncertainties are smaller than those due to
reaction cross-section uncertainties.
The total response uncertainties due to all data uncertainties were calcu-
lated using Eq. (6), which assumes all individual results summarized in Table VI
are uncorrelated and may be summed quadratically by column. The large total re-
sponse uncertainties of 79% for RI and 130% for R3 are reasons for concern in a
real design environment. In our conclusion we recommend, therefore, that certain
cross-section files including the SED data, and certain covariance files, be re-
evaluated. However, if such re-evaluations or re-measurements should result in
only insignificantly lower response uncertainties, then some additional conserva-
tism might have to be built into future blanket and shield designs.
VI. CONCLUSIONS AND RECOMMENDATIONS
In general, when a cross-section uncertainty analysis is performed with pre-
sently available codes and data, some care must be taken in the interpretation
of the results. Specifically, it must be recognized that considerable uncertain-
ty generally exists in the covariance files with the result that fairly large
errors are possible in the calculated response uncertainties themselves. Addition-
ally, any conclusions regarding the adequacy of nuclear data for a particular
application requires a statement as to what errors are tolerable in the calculated
responses. For the conclusions below we have assumed that response uncertainties
greater than - 25% are not acceptable.
With these qualifications in mind, the main conclusions drawn from this
PGFR cross-section and SED uncertainty analysis are the following:
1. The quadratic combination of the worst case response uncertainties (R3)
for H, C, and O results in a combined uncertainty of less than 13%.
Therefore, the existing ENDF/B-V neutron cross-section files for these
materials, including SED data, appear to be fully adequate for this
application.
2. The calculated R3 response uncertainties for Ni and Fe are 25% and 17%,
respectively,with the major components resulting from cross-section
18
uncertainties. The data for these materials therefore appear to be
marginally adequate for the present application, although some further
reduction in uncertainty would probably be desirable.
3. The W cross-section data are probably inadequate, as indicated by
a calculated response uncertainty of 54% for both RI and R3. This
conclusion must be qualified somewhat because Pb covariance data were
used in the W analysis. However, an examination of the Pb covariance
file suggests strongly that this qualitative conclusion would stand
even if W covariances were available. Additionally, the calculated
24% response uncertainties in RI and R due to tungsten SED uncertain-3
ties alone indicate a need for improved data. It is recommended,
therefore, that as a first step the cross-section and uncertainty files
for W be re-evaluated-to include the most recent experimental results.
4. For Cu the cross-section as well as the SED data appear inadequate
because they produce response uncertainties in RI and R3 between 26
and 45%. Specifically, large sensitivities are obtained for the
elastic and total copper cross-sections (see Appendix A) and SED’S
(see Appendix B) in the energy range from 1.3 keV to 1.3 MeV. These
cross-sections and secondary energy distributions are recommended for
re-evaluation and possibly re-measurement. Again, these conclusions
are subject to a similar qualification as was given for the W results
in that Fe covariance data were used in the absense of such data for
Cu. As with tungsten, however, we believe the qualitative conclusions
for Cu are valid.
5. The cross-section data for Cr appear grossly inadequate because they
produce an almost 100% uncertainty in R3, while the SED data are
found fully adequate. The main contribution to the 98% standard de-
viation is from the Cr total and elastic cross-sections (compare
Appendix A). While the sensitivity of R3 to the Cr total and elastic
cross-sections is roughly a factor of 10 lower than the sensitivity of
R3 to the copper cross-sections, the uncertainty estimates for Cr are
much larger than those for Cu. We recommend, therefore, first a re-
evaluation of the covariance files for the total and elastic chromium
cross-sections, and secondly, if the new uncertainty estimates are not
substantially lower, a re-evaluation of the Cr cross-sections them-
selves.
19
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
S. A. W. Gerstl, Donald J. Dudziak, and D. W. Muir, “Cross-Section Sensiti-vity and Uncertainty Analysis with Application to a Fusion Reactor,t’Nucl.Sci. Eng., ~, 137-156 (1977).
E. T. Cheng, “Generalized Variational Principles for Controlled Thermo-clear Reactor Neutronics Analysis”, General Atomic Company Report GA-A15378,June 1979.
J. M. Barnes, R. T. Santoro, and T. A. Gabriel, “The sensitivity of theFirst-Wall Radiation Damage to Fusion Reactor Blanket Composition”, ORNL/TM-6105, December 1977.
S. A. W. Gerstl, “Sensitivity Profiles for Secondary Energy and AngularDistributions”, Proc. Fifth Int. Conf. Reactor Shielding, held April 18-23,1977, in Knoxville, Tennessee, Proceedings edited by R. W. Roussin et.al.,Science Press, Princeton, 1978.
S. A. W. Gerstl, “Uncertainty Analysis for Secondary Energy Distributions”,Proc. Seminar-Workshop on Theory and Application of Sensitivity and Uncer-tainty Analysis, held Aug. 22-24, 1978, in oak Ridge, Tennessee, proceedings
edited by C. R. Weisbin et.al., ORNL/RSIC-42, Feb. 1979; see also D. W. Muirin “Applied Nuclear Data Research and Development, April l-June 30, 1977,”Los Alamos Scientific Laboratory report LA-6971-PRp. 30 (1977).
GA TNS Project Status Report for FY-78, Volumes I-VIII, GA-A151OO, October1978.
“Fusion Energy Flow”, GA-A151OO, Volume IV, October 1978.
ENDF/B-IV, Report BNL-17541 (ENDF-201), edited by D. Garber, available fromthe National Nuclear Data Center (NNDC), Brookhaven National Laboratory(BNL), Upton, N.Y. (October 1975).
“Summary Documentation of LASL Nuclear Data Evaluations for ENDF/B-V,”compiled by P. G. Young, LASL informal report LA-7663-MS, January 1979.
R. E. MacFarlane, R. J. Barrett, D. W. Muir, and R. M. Boicourt, “The NJOYNuclear Data Processing System: User’s Manual”, Los Alamos Scientific Lab-oratory report LA-7584-MS (ENDF-272),December 1978.
“MATXS, 30x12 Neutron, Photon, and Heating Library”, private communicationfrom R. E. MacFarlane and D. W. Muir, Los Alamos Scientific Laboratory,T-2, Feburary 18, 1977.
T. R. Hill, “ONETRAN: A Discrete Ordinates Finite Element Code for the Solu-tion of the One-Dimensional Multigroup Transport Equation”, Los AlamosScientific Laboratory report LA-5990-MS, June 1975.
S. A. W. Gerstl, “SENSIT: A Cross-Section and Design Sensitivity and Uncer-tainty Analysis Code,” Los Alamos Scientific Laboratory Report in prepara-tion.
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14.
15.
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18.
19.
20.
21.
22.
23.
24.
25.
F. G. Perey, “The Data Covariance Files for ENDF/B-V,” ORNL/TM-5938 (1977).
R. E. MacFarlane, R. J. Barrett, D. W. Muir, and R. M. Boicourt, “The NJOYNuclear Data Processing System: User’s Manual,” LA-7584-M (1978).
L. Stewart, R. J. LaBauve~ P. G. Young, and D. G. Foster, Jr., ENDF/B-VEvaluation MAT 1301, personal communication through the National NuclearDatalCenter (1979); D. G. Foster and P. G. Young, “Cross Section Covariancesfor H,” in “Summary Documentation of LASL Nuclear Data Evaluations forENDF/B-V,” LA-7663-MS (1979), p. 134.
G. M. Hale, L. Stewart, and P. G. Young, ENDF/B-V Evaluation MAT 1305,personal communication through the National uclear Data Center (1979);G. M. Hale, “Cross Section Covariances for 111
B,” in “Summary Documentationof LASL Nuclear Data Evaluations for ENDF/B-V,” LA-7663-MS (1979), p. 138.
C. Y. Fu and F. G. Perey, ENDF/B-V Evaluation MAT 1306, personal communica-tion through the National Nuclear Data Center (1979).
A Prince and T. W. Burrows, ENDF/B-V Evaluation MAT 1324, personal communi-cation through the National Nuclear Data Center (1979).
C. Y. Fu and F. G. Perey, ENDF/B-V Evaluation MAT 1326, personal communica-tion through the National Nuclear Data Center (1979).
M. Divadeenam, ENDF/B-V Evaluation MAT 1328? personal communication throughthe National Nuclear Data Center (1979).
C. Y. Fu and F. G. Perey, ENDF/B-V Evaluation MAT 1382, personal communica-tion through the National Nuclear Data Center (1979).
D. Hermsdorf, A. Meister, S. Sassonoff, D. Seeliger, K. Seidel, and F.Shalin, “DifferentielleNeutronenemissionsquerschnitteJ “ E; 8) bei~(Eo ,14.6 MeV Einschussenergie ffirdie Elemente Be, C, Na, Mg, A~, Si, P, S,Ca, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Se, Br, Zr, Nb, Cd, In, Sn,Sb, I, Ta, W, Au, Hg, Pb and Bi,” Zentralinstitut fiirKernforschungreport ZKF-277(U) (1975).
G. Clayeux and J. Voignier, “Diffusion Non Elastique de Neutrons de14 MeV sur Mg, Al?,Si, S, Ca, Sc, Fe, Ni, Cu, Au, Pb et Bi,” Commissariatsa Q’Energie Atomique report CEA-R-4279 (1972).
D. M. Hetrick, D. C. Larson, and C. Y. Fu, “Status of ENDF/B-V Neutron Emis-sion Spectra Induced by 14-MeV Neutrons,” ORNL/TM-6637 (1979).
21
APPENDIX A
In this appendix we reproduce 24 tables from the detailed SENSIT printouts for
those cases of our reaction cross-section uncertainty analysis where response un-
certainties greater than 10% have been obtained. The ID-numbers at the top of
the tables identify the cross-section pair for which correlated uncertainties in
the form of a covariance matrix have been used in the uncertainty analysis, as
listed in Table III. The sensitivity profiles for each of the two reaction cross-
sections of each pair are also given as PI(G) and P2(G), which are printed per
lethargy interval width DELTA-U.
22
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s
APPENDIX B
The following 11 tables are reproduced from the SENSIT printout of our SED
uncertainty analysis for those cases where the response uncertainties exieed 10%.
The title lines are self explanatory and the nomenclature coincides with that
used in the theory section (Sec. II) of the text. Only for the first case (Cu
in TF+F coils for response function RI) the detailed neutron cross-section and
SED sensitivity profiles are also reproduced. The gamma ray sensitivity pro-
files are all zero for this case because RI is a dpa cross-section which has
no gamma ray component.
47
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