experimental uncertainty estimation on the effective capture cross sections measured in the profil...
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PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable FutureThe Westin Miyako, Kyoto, Japan, September 28 - October 3, 2014, on CD-ROM (2014)
EXPERIMENTAL UNCERTAINTY ESTIMATION ON THE EFFECTIVECAPTURE CROSS SECTIONS MEASURED IN THE PROFIL
EXPERIMENTS IN PHENIX
E. Privas, G. Noguere C. De Saint Jean, J. Tommasi, P. ArchierCEA, DEN, Cadarache, SPRC, F-13108 Saint-Paul-lez-Durance, France
[email protected]@cea.fr
[email protected]@[email protected]
ABSTRACT
A desire of increasing nuclear system safety and fuel depletion is directly translated by abetter knowledge on nuclear data. PROFIL and PROFIL-2 experiments give integral infor-mation on capture and (n,2n) cross sections and cumulative fission yields for several isotopes(95Mo, 97Mo, 101Pd, 105Pd, 133Cs, 143Nd, 144Nd, 145Nd, 147Sm, 149Sm, 151Eu, 233U, 234U,235U, 238Pu, 239Pu, 240Pu, 241Pu, 242Pu, 244Cm ...). Interpretation have been done many timesin the past but without experimental uncertainty estimation. The cross section library JEFF-3.1.1, the covariance data base COMAC and the code system ERANOS-2.2 are used for thisupdated interpretation. This study is focusing on the uncertainty estimation on experimentalvalues sensitive to capture cross sections. Three steps are required: the fluence scaling, theuncertainty propagation on the fluence and finally the uncertainty estimation on ratio variationof interest. This work is done with CONRAD using Bayesian adjustment and marginalizationmethod. Mean C/E results and conclusions are identical to the previous interpretation. A flu-ence uncertainty of 1.4% is found for the two experimental pins of PROFIL-2 and 1.9% forPROFIL. Propagating this new information on the fluence to ratio variation of interest givesexperimental uncertainties between 1% to 2.5% for the isotopes present in the experimentalpins. One of the main results are for 238Pu, 239Pu, 240Pu, 241Pu and 242Pu capture cross sec-tions: C/E are respectively equal to 1.03, 0.98, 0.97, 1.08 and 1.14 with an uncertainty lowerthan 2.5%. All the results will provide feedback on variance-covariance matrices for furtherworks.
Key Words: “PROFIL”, “NUCLEAR DATA”, “UNCERTAINTY ESTIMATION”,“PHENIX”, “CONRAD”, “ERANOS”
1. INTRODUCTION
For core calculation with determinist code, uncertainties of neutronic cross sections are sometimesdefined group per group and correlations are taking into account either between groups, nuclearreactions and isotopes. Covariance matrices are produced by the evaluator using two types of in-formation: microscopic experiments (such as transmission, capture and fission reaction yields ...)and integral experiments (such as ICSBEP benchmarks [1]).
E. Privas, G. Noguere, C. De Saint Jean, J. Tommasi
PROFIL and PROFIL-2 experiments give integral information on capture, fission and (n,2n) crosssections and cumulative fission yields for several isotopes. A work has already been performedon neodymium cumulative fission yields of 235U and will not be presented here [2]. The last in-terpretation was done by J. Tommasi [3] using JEFF-3.1 cross section library and the code systemERANOS-2.1 [4]. Results given in previous studies presented small experimental uncertainties(from mass spectrometry). The work presented in this paper is performed with the cross sectionlibrary JEFF-3.1.1 and ERANOS-2.2. A new interpretation method with uncertainty propagationwas used. The mean values between JEFF-3.1 and JEFF-3.1.1 are the same (no major changesbetween these two libraries for fast reactor).
The first chapter is dedicated to the general presentation of the PROFIL and PROFIL-2 experi-ments. After understanding the purpose of the experiments and the fluence scaling issue, adjust-ment method is presented. Marginalization method is then explained. The uncertainty on the flu-ence obtained with CONRAD depends on the choice of the ratio variation and two study cases arepresented. The results with uncertainties on C/E mainly sensitive to capture cross sections and thefluence are presented in the last section with their uncertainty estimation.
2. PROFIL and PROFIL-2 experiments
PROFIL and PROFIL-2 experiments are based on irradiation of pure isotope samples in a wellcharacterized flux. They were performed four decades ago in the first cycles of Phenix (250 MWesodium-cooled fast reactor). A simplified radial view of the core is presented on Figure 3 andthe typical fast spectrum is given on Figure 4. PROFIL involved an experimental pin placed ina assembly inthe middle core with 46 separate samples and the PROFIL-2 experiment used twoexperimental pins labelled A and B with 42 separate samples each (see Figure 1 and 2). Samplescontents is described in Table I and Table II. Several information like neutron capture cross sectionscan be deduced from the concentration change during the irradiation. Mass spectrometry measure-ments performed for each sample give the ratio variation (RV) between two concentrations at thebeginning and at the end of the irradiation:
RVexp =N fIso1
N fIso2
− N iIso1
N iIso2
(1)
With N fIso1 and N i
Iso1 the final and initial concentration of an isotope in the measured sample andN fIso2 and N i
Iso2 the final and initial concentration of the second isotope. The uncertainties givenare small and stastistical.
More detailed analysis of these two experiments (experimental conditions and calculation schemes)can be found in a previous papers [3].
One of the most important step of the interpretation is the fluence scaling. Several hypotheses havebeen made in order to obtain a mean flux over time (four fuel cycles). The raw calculation resultsshow a discrepancy on samples containing the same isotope located in different position becauseof the hypothesis made on power and control rods height. The fluence scaling is done by makingsome experimental values equal to the calculated values.
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uncertainty estimation for PROFIL and PROFIL-2 experiments using JEFF-3.1.1 data library
5.5 mm
3.9 mm
4.5 mm 10 mmPure Isotope
Stainless Steel
Figure 1. Sample geometry used in both experiments containting as pure as possible isotope ofinterest.
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42
43 44 45 46 47 48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117
118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133
134 135 136 137 138 140 141 143 144 145 146 147 148
149 150 151 152 153 154 155 156 157 158 159 160 161 162
163 164 165 166 167 168 169 170 171 172 173 174 175
176 177 178 179 180 181 182 183 184 185 186 187
188 189 190 194192 193191 195 196 197 198
199 200 201 202 203 204 205 206 207 208
209 210 211 212 213 214 215 216 217
76
142Aig.
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42
43 44 45 46 47 48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117
118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133
134 135 136 137 138 139 140 141 143 144 145 146 147 148
149 150 151 152 153 154 155 156 157 158 159 160 161 162
163 164 165 166 167 168 169 170 171 172 173 174 175
176 177 178 179 180 181 182 183 184 185 186 187
188 189 190 194192 193191 195 196 197 198
199 200 201 202 203 204 205 206 207 208
209 210 211 212 213 214 215 216 217
Aig. A
Aig. B
Figure 2. PROFIL assembly on the left and PROFIL-2 assembly on the right.
Figure 3. Phenix radial core geometry.
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E. Privas, G. Noguere, C. De Saint Jean, J. Tommasi
Table I. Number of heavy element sample inPROFIL and PROFIL-2 pins.
Isotopes PROFIL PROFIL-2 (A+B)232Th - 1 + 2233U - 1 + 2234U - 2 + 1235U 6 7 + 7238U 3 3 + 3237Np - 2 + 1238Pu 2 2 + 1239Pu 3 2 + 2240Pu 3 2 + 2241Pu 3 -242Pu 3 2 + 2241Am 2 2 + 2243Am - 2 + 2244Cm - 2 + 2
Total NL 25 30 + 28
Table II. Number of light element sample inPROFIL and PROFIL-2 pins.
Isotopes PROFIL PROFIL-2 (A+B)92Zr - 2 + 195Mo 2 -97Mo 2 -101Ru 3 -105Pd 2 -106Pd - 2 + 2133Cs 2 -143Nd - 1 + 2144Nd - 1 + 2145Nd 2 -147Sm - 1 + 2149Sm 2 -151Sm - 1 + 2153Eu - 2 + 1
Total PF 15 10 + 12
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
Flu
x d
en
sit
y i
n le
tha
rgy
(arb
itra
ry u
nit
s, lo
ga
rith
mic
sc
ale
)
Energy (eV)
1968 groups
Figure 4. Typical neutron spectrum for PROFIL samples (logarithmic scale) obtained using theECCO lattice code of the ERANOS system with 1968 energy groups and a critical buckling.
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Two coefficients are introduced for each pin called Axial Adjustment Coefficient (XAAC) and Flu-ence Adjustment Coefficient (XFAC) in order to normalize the results. The first coefficient wasintroduced to account for the approximate modelling of control rods (only one average position percycle) and / or a bad positioning of the experimental samples at the pin filling stage. The secondcoefficient account for the inadequate knowledge on operation power during the different cycles.It is admitted that the axial shape of the flux is correctly calculated (well-characterized flux in theinner core). Finally, the fluence φtfitted is reconstructed as follow:
φtfitted = XFAC ∗ φttheoretical(z +XAAC) (2)
Where φt is the fluence, z the axial height corresponding to the samples positions and φttheoreticalthe fluence calculated by ERANOS.
3. ADJUSTMENT AND UNCERTAINTY PROPAGATION METHODS WITH CONRAD
3.1. Adjustment and Marginalization Methods
XFAC and XAAC are to be determined in order that selected experimental values matched the calcu-lation values. They are obtained using Bayes’ theorem [5] implemented in CONRAD [6] [7] andwith some assumptions on the prior and posterior probability distribution involved [8]. The devel-opment of the software tool CONRAD was initiated at CEA/Cadarache to give answers to variousproblems arising in the data analysis of nuclear reactions. The evaluation of posterior expecta-tion and covariances are done by finding the minimum of the following cost function (GeneralizedLeast-Square - GLS):
χ2GLS = (~x− ~xm)T M−1
x (~x− ~xm) + (~c− ~e)T M−1e (~c− ~e) (3)
Where
• ~x a vector containing the two fitted parameters XFAC and XAAC,
• ~xm and Mx the prior parameters and the prior covariance associated,
• −→e the experimental values of the ratio variation, RV = ∆(NIso1/NIso2) for a given isotopesamples as described in Equation (1),
• Me the experimental covariance matrix. It is diagonal and fill with the statistiacl experimentaluncertainty from mass spectrometry for RV ,
• ~c the calculated values for the same ratio RV . ~c is described as follow, with ~f definingERANOS calculation:
~c = ~f(φtfitted) = ~f(XAAC, XFAC) = ~f(~x) (4)
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E. Privas, G. Noguere, C. De Saint Jean, J. Tommasi
ERANOS
CONRAD
Sensitivities of selected C/E
to FAC and AAC
Initial FAC and AAC
Final Adjusted Coefficients
Convergence criteria
Figure 5. Fluence scaling method using CONRAD.
The scheme used to find the minimum of the previous cost function is described in Figure 5. Foreach iteration, ERANOS calculation has to be performed in order to have sensitivities Gx of theratio variationRV to theXFAC andXAAC. This methodology is simple and three iterations is enoughto converge thanks to the linearity of the problem. Once done, C/E(RV ) = 1 and others C/E areobtained. The covariance matrix on the ~x parameter is defined by M fitted
x . This step gives an twoinformation: small uncertainties and correlation on the two adjustment coefficients. But the selectedV R is also sensitive to nuclear data and it is important to take their uncertainties into account usinga marginalization method.
Let ~θ = {σ11, ..., σ
n11 , σ
12, ..., σ
n22 } be the multigroup cross sections of isotope 1 and isotope 2 which
C/E(RV ) are sensitive to. For this study, 33 groups cross sections are used. ~θ is called nuisanceparameters. The effect of other reactions from other isotope is considered negligible in this demon-stration (i.e. null sensitivities). The method used for doing this uncertainty estimation is imple-mented in CONRAD and is commonly called marginalization method [9]. It consists to propagatein a certain way the uncertainties from undesirable parameters to physical parameters (in that case,from ~θ toXFAC andXAAC). At the end of the marginalization, covariance matrices on marginalizedparameters is given by:
Mmargx = M fitted
x +(GTx ·Gx
)−1 ·GTx ·Gθ ·Mθ ·GT
θ ·Gx ·(GTx ·Gx
)−1 (5)
WithM statx the statistical covariance matrix find in the previous GLS adjustment,Mθ the covariance
matrix of the nuisance parameters and Gx the sensitivity vector of the selected RV to the adjustedparameters ~x.
To run this method in CONRAD, the sensitivitiesGθ of the adjusted coefficients to the marginalizedcross sections and the covariance matrices of the nuisance parameters (ie. multigroup covariance ofnuclear reaction) are required. Finally, results are obtained for the scaling coefficients with higheruncertainties due to nuclear data.
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uncertainty estimation for PROFIL and PROFIL-2 experiments using JEFF-3.1.1 data library
3.2. Uncertainty Propagation on the Fluence of PROFIL-2A
The coefficients are calculated in order to match experimental and calculation values for some ratiovariations. The final uncertainties depend on this selected ratio variation and more precisely, to thesensitivity of this ratio to the nuclear reactions. This chapter is an example of how to use CONRADfor this problem.
3.2.1. Fluence Scaling on Ratio Variation 235U/238U
The last interpretations used the ratio variations C = 235U/238U in 235U samples for two main rea-sons:
• six samples of 235U are axially placed in each pins (PROFIL, PROFIL-2A and PROFIL-2B).Thus many measurements are available and can be used to fit two parameters.
• 235U/238U is mainly sensitive to 235U fission (0.754) and capture (0.217) (see Table III). 235Ubeing a standard, it is expected to calculate properly these ratio variations.
The results obtained depend on the cross sections variance-covariance. Taking COMAC-V0 [10]implies a high uncertainty of 6.0% on the XFAC while using JENDL-4 [11] covariances induce alower uncertainty 1.3% (see Table IV). The difference is due to the 235U capture cross section withuncertainties larger than 15% for energy higher than 1 keV with COMAC (see Figure 6). Even if Cis less sensitive to the 235U capture cross section than the 235U fission cross section, capture has aweight more important with its high uncertainties.
Thus, the previous interpretation gave very good mean value but uncertainty propagation showsthat is it not the best selected RV . In order to remove the sensitivity of this capture cross section,a new variation ratio is selected: (235U +236 U)/238U
3.2.2. Fluence Scaling on Ratio Variation (235U +236 U)/238U
New ratio variations for the fluence scaling are chosen:(235U +236 U)/238U. This new adjustmentmakes possible to decrease the influence of the 235U capture and increase the sensitivity due to the235U fission as shown in Table III. This new way means an adjustment on the fission standard of235U which different evaluations agree on. Applying the adjustment and marginalization method,uncertainty on the XFAC is 1.4% using COMAC and 1.2% using JENDL-4 (see Table IV). Lowuncertainties are obtained using the fluence scaling ratio variation (235U +236 U)/238U. This isonly possible for the experiment PROFIL-2A because some experimental issues appear for the twoothers experiments. However, PROFIL-2B is a pin placed in the same assembly than PROFIL-2Aand separate by few fuel pins. Thus, uncertainties find for PROFIL-2A on the fluence is the samefor PROFIL-2B. When adjusting with this ratio variation, the ratio 236U/235U sensitive to 235Ucapture cross section is equal to one. Doing a feedback in this nuclear reaction from PROFIL-2,
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E. Privas, G. Noguere, C. De Saint Jean, J. Tommasi
0
10
20
30
40
50
60
70
1.E-5 1.E-4 1.E-3 1.E-2 1.E-1 1.E+0 1.E+1
Re
lati
ve u
nce
rtai
nti
es
(%)
Energy (MeV)
COMAC
JENDL
Normalized flux
Figure 6. Comparison of the capture cross section uncertainties between COMAC-V.01 andJENDL-4. A normalized flux is represented to see the area of interest.
an uncertainty of 1.9% on the fluence of PROFIL is deduced using a fluence scaling with the ratio236U/235U. Finally, the three experiments are correlated and it implies that all C/E are correlated.
4. Experimental Uncertainty Estimation on Ratio Variation of Interest
The ratio variations of interest RVX presented in this study are the one sensitive to capture crosssection, meaning:
RVX = ∆
[n+1XnX
](6)
Where X is the isotope studied.
Those ratios are also sensitive to the fluence and the uncertainty propagation is done by usingequivalent method than Section 3. The results obtained are presented in the Figure 7. It is dividedinto two areas: one on the left represent the C/E relative to the capture of the fission products andone on the right C/E relative mainly to the capture of actinide and lowly to the fission for fissileisotopes.
For all the C/E presented, at least 2 measures have been taken into account (see Table I and Ta-ble II). The results give two information: a trend on the capture cross section and the associateduncertainty. Uncertainties are laying between 0.9% to 2.5% depending on the fluence sensitivity ofthe ratio variations considered and the origin of the data. Indeed, PROFIL experiment has a fluenceuncertainty of 1.9% higher than PROFIL-2 thus if a measure is taken from PROFIL only, a greateruncertainty would be found.
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Table III. Sensitivity of the C((235U +236 U)/238U) and C(235U/238U) values to the nuclear crosssections. Others reactions are negligible.
Selected RV 235U/238U (235U +236 U)/238U
Cross section σc235U σf235U σc238U σf238U σc235U σf235U σc238U σf238U
Sensitivity to C 0.217 0.754 -0.115 –0.019 -0.036 1.068 0.136 -0.02
Table IV. Results obtained with uncertainties for XAAC and XFAC using COMAC and JENDL-4covariance.
Selected RV 235U/238U (235U +236 U)/238U
Covariance database COMAC JENDL-4 COMAC JENDL-4
XFAC (%) 0.963± 6.0% 0.963± 1.2% 0.963± 1.4% 0.963± 1.2%
XAAC (%) 0.990± 0.2% 0.754± 0.2% 0.990± 0.2% 0.754± 0.1%
Correlation -0.06 0.05 0.53 0.27
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
Ratio C
/E
Pd
10
5
Cs1
33
Nd
14
4
Sm
14
9
Eu
15
3
Mo
97
Mo
95
Ru
101
Nd1
43
Sm
14
7
Nd
14
5
Sm
15
1
U2
34
U2
33 Pu
24
0
Pu
23
9
U2
35
Pu
23
8
Pu
241
Pu
242
Cm244
Fission Products Actinides
Figure 7. Results obtained for capture cross sections. In red final uncertainties at one sigma andin grey at two sigma.
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Some capture cross sections are to high. For example,151Sm and 244Cm fast capture cross sectionare overestimate by more than 20% in JEFF-3.1.1. Moreover, PROFIL and PROFIL-2 give smalluncertainties showing that a feedback on those cross section is recommended. It has been done for151Sm capture cross section [12]. 105Pd, 233U U234U, 143Nd, 145Nd, 147Sm, 241Pu and 242Pu captureare badly calculated by more than 10% and regarding the uncertainties, a change of the fast crosssection is recommended. For 242Pu, a work was realised by E. Rich using PROFIL and PROFIL-2experiments [13]. For all the other (C-E < 10%) small change may solve the trend obtained withJEFF-3.1.1. However, all the results can be used as a feedback to produce variance-covariancematrices. An example is detailed in [14] using PROFIL and PROFIL-2 results for a feedback on239Pu covariance matrices.
5. CONCLUSION
The analysis of PROFIL and PROFIL-2 experiments with integral data assimilation using CON-RAD provide nuclear data feedback on trends and uncertainties. The adjustment and marginaliza-tion methods were successfully used and give consistent results.
The scaling step enables the evaluator to adjust the level and the axial shape of the fluence inorder to have good C/E for the ratio variations (235U +236 U)/238U mainly sensitive to the 235Ufission cross section. Using COMAC as the variance-covariance database, an uncertainty of 1.4%is found for the two experimental pins in PROFIL-2 and 1.9% for the irradiated pin in PROFIL. Theuncertainty propagation of the fluence on the others ratio variations give important results for anintegral feedback on nuclear trend and covariance for fast reactor. A work on 239Pu has been done[14] using JEZEBEL and PROFIL experiments. The effect of those experiments are significant onthe variance-covariance matrice (anti-correlations are created for the capture cross section betweenthe resonance and the high energy domain). Further work on 239Pu and 238U in particular will bedone using this study for the capture cross section and others experiences from ICSBEP in order toproduce new variance-covariance matrices.
REFERENCES
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