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Variance Reduction Factor of Nuclear Data for Integral Neutronics Parameters
Available online at www.sciencedirect.com
Nuclear Data Sheets 123 (2015) 62–67 www.elsevier.com/locate/nds
Variance Reduction Factor of Nuclear Data for Integral Neutronics Parameters G. Chiba,1, ∗ M. Tsuji,1 and T. Narabayashi1 1
Hokkaido University, Sapporo, Hokkaido 060-8628, Japan (Received 12 May 2014; revised received 24 July 2014; accepted 24 July 2014) We propose a new quantity, a variance reduction factor, to identify nuclear data for which further improvements are required to reduce uncertainties of target integral neutronics parameters. Important energy ranges can be also identified with this variance reduction factor. Variance reduction factors are calculated for several integral neutronics parameters. The usefulness of the variance reduction factors is demonstrated. I.
INTRODUCTION
Nuclear data uncertainty is becoming a dominant component of uncertainty in fission reactor integral neutronics parameters since accurate and precise numerical simulations for particle transport and fission chain reactions are being realized nowadays. Even though the quality of evaluated nuclear data files also has been significantly improved, their accuracy is not yet sufficient in some fields of nuclear engineering application, such as reactor core designs of advanced and future nuclear systems [1, 2]. When a ’target’ reactor core design is provided, one would quantify nuclear data-induced uncertainties of integral parameters of this design. If the estimated uncertainties are larger than the target accuracy, further efforts should be made. One reasonable and well-known approach is to utilize measurement data of integral neutronics parameters obtained (or planned to be obtained) at mock-up nuclear facilities. This approach was initially proposed by Usachev et al. [3], and it has been applied by Palmiotti and Salvatores [4]. If reduced uncertainties do not satisfy the target even when integral measurement data are effectively utilized, one would attempt to know which nuclear data should be further improved. Lately, the Oak Ridge group has proposed the inverse sensitivity/uncertainty quantification (IS/UQ) method to quantify the required accuracy for the nuclear data to achieve the target accuracy of the integral parameters [5]. In this method, variance reduction required to achieve the target accuracy are uniquely determined by the optimization process assuming cost functions for all the nuclear data, which quantify how much the microscopic nuclear data improvement costs. Although the IS/UQ method is quite a beneficial tool to know the target accuracy of the microscopic nuclear
∗
Corresponding author: go [email protected]
http://dx.doi.org/10.1016/j.nds.2014.12.011 0090-3752/© 2014 Elsevier Inc. All rights reserved.
data, there is one defect; it assumes that correlations between nuclear data are invariant. It is natural to consider that covariance matrices including correlations are altered when new measurement data are obtained, so the assumption of invariant correlations would be unrealistic. In the present paper, we propose a new quantity to identify which nuclear data should be further improved to reduce the variance of the target integral parameters. With the proposed quantity we cannot quantify the target accuracy for the microscopic nuclear data like the IS/UQ method, but this quantity properly considers correlations between the nuclear data. Thus, the proposed method would be complementary to the IS/UQ method. II.
THEORY
Here, let us consider a nuclear data vector T and its covariance matrix M. In the cross section adjustment procedure with integral measurement data [6],one considers several dozens/hundreds of integral data and prepares numerically-predicted values for these integral data using T. Through the cross section adjustment, we get a posterior nuclear data vector T and its covariance matrix M which is given as −1 M = M − MGT GMGT + Ve + Vm GM, (1) where G is the sensitivity matrix of integral data with respect to T, and the matrices Ve and Vm are covariance matrices of integral measurement data and the numerically-predicted values, respectively. The uncertainty considered in Vm comes only from uncertainties due to employed numerical methods: statistical uncertainties in Monte Carlo calculations, for example. Next, let us suppose that new experimental data for the kth microscopic nuclear data with measurement uncertainty (variance) Vk is obtained, and T is adjusted with
Variance Reduction Factor . . .
NUCLEAR DATA SHEETS
ˆ k GT = the numerator of the VRF is written as Gt M t 2 Mkk (Gt,k ) , where Gt,k is the kth entry of Gt . It should be noted that a similar method for determining the contribution of each nuclear data to target integral parameters has been already proposed by Muir [7]. The basis of the theories of Muir’s method and ours is almost identical except for the uncertainty treatment for new measurement. In Muir’s method, the new measurement is done in an ideal condition and measurement uncertainty is assumed to be zero while in our method the measurement uncertainty is expressed by Eq. (3). It should be emphasized that the present study presents rich numerical examples in the following section to show performance of our method, whereas Muir’s study has not shown any application results of his method.
this new data. In this case, Vm is zero since the integral parameter is the microscopic nuclear data itself, and the sensitivity matrix G becomes a vector ek in which the kth entry is unity and others are zero. Thus, the posterior covariance matrix M(k) for the adjusted nuclear data T(k) is written as follows: M(k) = M − M∗k [Mkk + Vk ]
−1
¯ (k) . Mk∗ = M − M (2)
Here, we express the uncertainty of the new measurement Vk as Vk = αMkk .
(3)
If the measurement is done with the same degree of uncertainty as the prior cross section uncertainty, α is unity. ¯ (k) can be written as Using Eq. (3), the entry of M Mik Mkj Cik Ckj σi σj Mkk = Mkk + Vk (1 + α)Mkk ˆ (k) M Cik Ckj σi σj ij = , = 1+α 1+α
¯ (k) = M ij
III.
(4)
1 ˆ (k) GT . Gt M t 1+α
A.
(5)
ˆ k GT Gt M t . Gt MGT t
Numerical Procedure
All the sensitivities of integral neutronics parameters to nuclear data are calculated with a general-purpose reactor physics calculation code system CBZ [8], which is being developed at Hokkaido university. Sensitivities of criticality and reactivity worth are calculated with the classical perturbation theory, and those of burnup-related integral parameters (nuclide number densities after burnup) are calculated with the generalized perturbation theory. Forward and (generalized) adjoint neutron fluxes are obtained from neutron transport calculations with the discrete ordinates method or the collision probability method. 70-group and 107-group energy-averaged cross sections based on JENDL-4.0 [9] are used for fast neutron systems and thermal neutron systems respectively. The covariance data given in JENDL-4.0 are also processed to those in a multi-group form. Note that correlations between different nuclides are ignored in the present study. Inter-energy and inter-reaction correlations are taken into account.
In the present paper, we define a variance reduction factor (VRF) of the kth nuclear data, fk , as fk =
RESULTS AND ANALYSES
In order to demonstrate the performance of our method, we calculate VRFs for several integral parameters. Some of them are measurement data obtained in previous critical experiments. In the actual application, such existing integral data would be used for the cross section adjustment,and then our method is used with the covariance data of the adjusted nuclear data for integral parameter of a target reactor core design such as an innovative fast reactor or an accelerator-driven subcritical system. In future study, we will apply the proposed method into such future nuclear systems.
where σi is the standard deviation of Ti and Cik is the correlation between Ti and Tk . Using this equation, it is easily shown that the correlation matrix for T (k) is not identical to that for T . Next, let us consider how much uncertainty of an integral parameter is reduced when we use the adjusted cross section set T(k) , i.e., when we obtain new measurement data for the kth nuclear data. If we write a sensitivity vector of the target integral parameter as Gt , variances of the integral parameter obtained using T and T(k) can be (k) T written as Gt MGT Gt , respectively. Thus, t and Gt M the variance reduction of the target integral parameter ΔVt is written as ¯ (k) GT ΔVt = Gt M − M(k) GT t = Gt M t =
G. Chiba et al.
(6)
With this VRF, we can know which nuclear data require further accuracy improvement in order to reduce the variance of the target integral parameter. It should be also emphasized that calculations of the proposed VRFs are quite easy; one needs sensitivity profiles for target neutronics parameters and covariance matrices of nuclear data. The proposed method can provide us useful information on microscopic nuclear data measurements. If correlations among different cross sections are igˆ (k) is written as nored in the adjustment process, M ij ˆ (k) = Mkk δik δjk , where δ is Kronecker’s delta, and M ij
63
Variance Reduction Factor . . . B.
NUCLEAR DATA SHEETS
Criticalities of Fast Neutron Systems
G. Chiba et al.
0.18 U-238 mu-bar U-235 inelastic U-238 elastic U-238 inelastic
0.16 Variance reduction factor
VRFs for criticalities of the following four fast critical assemblies are calculated. The geometrical specification and nuclide number densities of the compositions are taken from the ICSBEP handbook. • Jezebel: A bare sphere of plutonium (95at% 239 Pu). • Godiva: A bare sphere of high enriched (94wt%) uranium. • Flattop-25: A high enriched uranium sphere surrounded by a normal uranium reflector.
0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 103
• Big-Ten: A large mixed-uranium-metal cylindrical core with a depleted uranium reflector.
104
105 106 Neutron energy [eV]
107
FIG. 3. Variance reduction factor for criticality of Flattop-25.
Figures 1 to 4 show VRFs for criticalities of the above fast neutron systems.
0.7 U-235 chi U-238 inelastic U-238 capture
0.35 0.6 Variance reduction factor
Variance reduction factor
0.30
U-235 mu-bar U-235 elastic U-235 inelastic
0.25 0.20 0.15 0.10
0.5 0.4 0.3 0.2 0.1
0.05 0.00 3 10
4
10
5
10 10 Neutron energy [eV]
6
0.0 4 10
7
10
Variance reduction factor
Pu-239 fission Pu-239 chi Pu-239 inelastic
0.20 0.15 0.10 0.05
4
10
5
10 10 Neutron energy [eV]
6
7
10
take non-negligible values in a low-energy range in which there are few neutrons in the system. This is due to a storing positive correlation between high- and low-energy ranges in this cross section: reduction of the uncertainty in the low-energy range simultaneously results in reduction of the uncertainty in the high-energy range. For the Jezebel criticality, VRFs of 239 Pu fission cross sections are high. This is interesting because the high-enriched uranium system Godiva has much smaller VRFs of fission cross sections of principal fissile material than Jezebel. For the Flattop-25 criticality, VRFs of reflector materials are rather high in comparison with those of fuel materials. Big-Ten has high VRFs of 238 U inelastic cross sections around several MeV. The energy distribution of the VRFs is different from that for the Flattop-25 criticality.
0.30
0.00 3 10
6
FIG. 4. Variance reduction factor for criticality of Big-Ten.
FIG. 1. Variance reduction factor for criticality of Godiva.
0.25
5
10 10 Neutron energy [eV]
7
10
FIG. 2. Variance reduction factor for criticality of Jezebel. C.
Criticalities of Thermal Neutron Systems
235
For the Godiva criticality, VRFs of U inelastic cross sections around 1 MeV are high. It is one of specific features of VRFs that those of 235 U elastic cross sections
VRFs for criticalities of the following two thermal neutron systems are calculated. One is a mixed-oxide (MOX) 64
Variance Reduction Factor . . .
NUCLEAR DATA SHEETS
fuel-loaded critical assembly (9x9 reference core) constructed through the reactor physics experiment program FUBILA [10], and a high-enriched uranium solution system reflected by water, which is categorized as HEU-SOLTHERM-010 (case 1) in the ICSBEP handbook. Sensitivities of the FUBILA criticality are calculated in a radial two-dimension core model. Figures 5 and 6 show VRFs for the criticalities of these thermal systems.
D.
G. Chiba et al.
Sodium Void Reactivity Worth of Fast Critical Assembly ZPPR-10A
VRFs for the sodium void reactivity worth of the fast critical assembly ZPPR-10A are calculated. ZPPR-10A is one of fast critical assemblies constructed through the physical mock up experiments for large-size fast breeder reactors. VRFs for spectrum component-dominant sodium void reactivity and those for leakage componentdominant reactivity are shown in Figs. 7 and 8.
Pu-239 fission Pu-239 capture Pu-240 capture
0.5
0.30
0.4 Variance reduction factor
Variance reduction factor
0.6
0.3 0.2 0.1 0.0 -3 -2 -1 0 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 10 10 10 Neutron energy [eV]
0.25 0.20 0.15 0.10 0.05 0.00 3 10
FIG. 5. Variance reduction factor for criticality of FUBILA 9x9 reference core.
U-238 inelastic Na-23 elastic Na-23 inelastic
4
10
5
10 10 Neutron energy [eV]
6
7
10
FIG. 7. Variance reduction factor for spectrum componentdominant sodium void reactivity of ZPPR-10A. 0.9
0.7
U-235 capture U-235 nu-bar U-235 chi 0.30
0.6 0.5
Variance reduction factor
Variance reduction factor
0.8
0.4 0.3 0.2 0.1 0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 107 Neutron energy [eV]
0.25 0.20 0.15 0.10 0.05 0.00 103
FIG. 6. Variance reduction factor for criticality of HEU-SOLTHERM-010-1.
U-238 inelastic Na-23 elastic Na-23 inelastic
104
105 106 Neutron energy [eV]
107
FIG. 8. Variance reduction factor for leakage componentdominant sodium void reactivity of ZPPR-10A.
It is interesting to see that VRFs of thermal capture cross sections of 240 Pu are significantly high in the FUBILA core. For the HEU-SOL-THERM-010-1 criticality, VRFs of fission spectrum of 235 U are much higher than VRFs of other nuclear data. The fission spectrum uncertainty of 235 U has large negative correlation between high-energy and low-energy ranges, so new measurement of this nuclear data is desirable at arbitrary energy.
In the both reactivity worth, we can observe high VRFs of 238 U inelastic scattering cross sections and sodium nuclear data. In the leakage component-dominant reactivity, VRFs of sodium nuclear data are slightly enhanced. 65
Variance Reduction Factor . . .
Nuclide Number Densities of 242 Cm and 244 Cm after Burnup in a Light Water Reactor Pincell
0.6 0.5 0.4 0.3 0.2
0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 107 Neutron energy [eV]
Pu-240 capture Pu-241 capture Am-241 capture
FIG. 11. Variance reduction factor for burnup of LWR-MOX pincell.
0.4
242
Cm inventory after
0.3 0.8 0.2
Pu-241 capture Pu-242 capture Am-243 capture
0.7 Variance reduction factor
Variance reduction factor
0.7
0.1
0.6
0.1 0.0 -3 -2 -1 0 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 10 10 10 Neutron energy [eV]
FIG. 9. Variance reduction factor for burnup of LWR-UO2 pincell.
242
Cm inventory after
0.5 0.4 0.3 0.2
0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 107 Neutron energy [eV]
Pu-241 capture Pu-242 capture Am-243 capture
0.7
0.6
0.1
0.8
Variance reduction factor
Pu-240 capture Pu-241 capture Am-241 capture
0.8
Nuclide number densities of 242 Cm and 244 Cm after burnup are very important because of their radioactivities and neutron generations by spontaneous fissions. VRFs for the nuclide number densities of these curium isotopes are calculated for PWR-represented pincells with UO2 fuel (235 U enrichment is 4.7%) and MOX fuel (Pu-fissile enrichment is 10%). The calculated VRFs are shown in Figs. 9 to 12.
0.5
G. Chiba et al.
0.9
Variance reduction factor
E.
NUCLEAR DATA SHEETS
FIG. 12. Variance reduction factor for burnup of LWR-MOX pincell.
0.6
244
Cm inventory after
0.5 0.4
UO2 pincell since initial number densities of plutonium and americium isotopes are zero. VRFs of 241 Pu capture cross sections for 244 Cm number densities are dominant in the UO2 pincell, and take quite high values below 100 eV. This is because inter-energy full positive correlations are given to this cross sections in this energy range. On the other hand, VRFs of 241 Pu capture cross sections are relatively low and those of 243 Am capture cross sections are rather high in the MOX cell.
0.3 0.2 0.1 0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 107 Neutron energy [eV]
FIG. 10. Variance reduction factor for burnup of LWR-UO2 pincell.
244
Cm inventory after IV.
242
CONCLUSIONS
244
Cm and Cm are generated via neutron capture reactions of their parent nuclides 241 Am and 243 Am respectively. VRFs of 241 Am capture cross sections for 242 Cm number densities after burnup are high in both the UO2 and MOX pincells, especially in a specific energy range from 0.1 eV to several eV. VRFs of capture cross sections of plutonium isotopes are relatively high in the
We have proposed a new quantity, a variance reduction factor, to identify nuclear data for which further improvements are required to reduce uncertainties of target integral neutronics parameters. Important energy ranges can be also identified with this variance reduction factor. Variance reduction factors have been calculated 66
Variance Reduction Factor . . .
NUCLEAR DATA SHEETS
for several integral neutronics parameters, such as criticalities of fast and thermal neutron systems, sodium void reactivity worth of a fast critical assembly and nuclide number densities after burnup in LWR pincells. The usefulness of the variance reduction factors has been
G. Chiba et al.
demonstrated. Acknowledgements: We thank Y. Kawamoto and T. Kajihara for their helpful comments on this manuscript. This work was supported by JSPS KAKENHI Grant Number 24561040.
[1] G. Aliberti et al., “Nuclear Data Sensitivity, Uncertainty and Target Accuracy Assessment for Future Nuclear Systems,” Ann. Nucl. Energy 33, 700 (2006). [2] G. Palmiotti et al., “Nuclear Data Target Accuracies for Generation-IV Systems Based on the Use of New Covariance Data,” J. Korean Phys. Soc. 59, 1264 (2011). [3] L.N. Usachev et al., Proc. Int. Conf. on Neutron Physics and Other Applied Physics, Harwell, Sep. 25-29, 1978, CONF-780921, p.181, International Atomic Energy Agency (1978). [4] G. Palmiotti, M. Salvatores, “Use of Integral Experiments in the Assessment of Large Liquid-Metal Fast Breeder Reactor Basic Design Parameters,” Nucl. Sci. Eng. 87, 333 (1984). [5] G. Aranas et al., “Integral Benchmark Experiments in the Inverse Sensitivity/Uncertainty Computations,” Trans. Am. Nucl. Soc. 109, 808 (2013). [6] J.B. Dragt et al., “Methods of Adjustment and Error Propagation of Neutron Capture Cross Sections; Appli-
[7]
[8]
[9]
[10]
67
cation to Fission Product Nuclides,” Nucl. Sci. Eng. 62, 117 (1977). D.W. Muir, “The Contribution of Individual Correlated Parameters to the Uncertainty of Integral Quantities,” Nucl. Inst. Methods in Phys. Res. A 644, 55 (2011). G. Chiba et al., “Uncertainty Quantification of Neutronic Parameters of Light Water Reactor Fuel Cells with JENDL-4.0 Covariance Data,” J. Nucl. Sci. Technol. 50, 751 (2013). K. Shibata et al., “JENDL-4.0: a New Library for Nuclear Science and Engineering,” J. Nucl. Sci. Technol. 48, 1 (2011). T. Yamamoto et al., “Neutronics Analysis of Full MOX BWR Core Simulation Experiments FUBILA,” J. Nucl. Sci. Technol. 48, 398 (2011).
Nuclear Data Sheets 123 (2015) 62–67 www.elsevier.com/locate/nds
Variance Reduction Factor of Nuclear Data for Integral Neutronics Parameters G. Chiba,1, ∗ M. Tsuji,1 and T. Narabayashi1 1
Hokkaido University, Sapporo, Hokkaido 060-8628, Japan (Received 12 May 2014; revised received 24 July 2014; accepted 24 July 2014) We propose a new quantity, a variance reduction factor, to identify nuclear data for which further improvements are required to reduce uncertainties of target integral neutronics parameters. Important energy ranges can be also identified with this variance reduction factor. Variance reduction factors are calculated for several integral neutronics parameters. The usefulness of the variance reduction factors is demonstrated. I.
INTRODUCTION
Nuclear data uncertainty is becoming a dominant component of uncertainty in fission reactor integral neutronics parameters since accurate and precise numerical simulations for particle transport and fission chain reactions are being realized nowadays. Even though the quality of evaluated nuclear data files also has been significantly improved, their accuracy is not yet sufficient in some fields of nuclear engineering application, such as reactor core designs of advanced and future nuclear systems [1, 2]. When a ’target’ reactor core design is provided, one would quantify nuclear data-induced uncertainties of integral parameters of this design. If the estimated uncertainties are larger than the target accuracy, further efforts should be made. One reasonable and well-known approach is to utilize measurement data of integral neutronics parameters obtained (or planned to be obtained) at mock-up nuclear facilities. This approach was initially proposed by Usachev et al. [3], and it has been applied by Palmiotti and Salvatores [4]. If reduced uncertainties do not satisfy the target even when integral measurement data are effectively utilized, one would attempt to know which nuclear data should be further improved. Lately, the Oak Ridge group has proposed the inverse sensitivity/uncertainty quantification (IS/UQ) method to quantify the required accuracy for the nuclear data to achieve the target accuracy of the integral parameters [5]. In this method, variance reduction required to achieve the target accuracy are uniquely determined by the optimization process assuming cost functions for all the nuclear data, which quantify how much the microscopic nuclear data improvement costs. Although the IS/UQ method is quite a beneficial tool to know the target accuracy of the microscopic nuclear
∗
Corresponding author: go [email protected]
http://dx.doi.org/10.1016/j.nds.2014.12.011 0090-3752/© 2014 Elsevier Inc. All rights reserved.
data, there is one defect; it assumes that correlations between nuclear data are invariant. It is natural to consider that covariance matrices including correlations are altered when new measurement data are obtained, so the assumption of invariant correlations would be unrealistic. In the present paper, we propose a new quantity to identify which nuclear data should be further improved to reduce the variance of the target integral parameters. With the proposed quantity we cannot quantify the target accuracy for the microscopic nuclear data like the IS/UQ method, but this quantity properly considers correlations between the nuclear data. Thus, the proposed method would be complementary to the IS/UQ method. II.
THEORY
Here, let us consider a nuclear data vector T and its covariance matrix M. In the cross section adjustment procedure with integral measurement data [6],one considers several dozens/hundreds of integral data and prepares numerically-predicted values for these integral data using T. Through the cross section adjustment, we get a posterior nuclear data vector T and its covariance matrix M which is given as −1 M = M − MGT GMGT + Ve + Vm GM, (1) where G is the sensitivity matrix of integral data with respect to T, and the matrices Ve and Vm are covariance matrices of integral measurement data and the numerically-predicted values, respectively. The uncertainty considered in Vm comes only from uncertainties due to employed numerical methods: statistical uncertainties in Monte Carlo calculations, for example. Next, let us suppose that new experimental data for the kth microscopic nuclear data with measurement uncertainty (variance) Vk is obtained, and T is adjusted with
Variance Reduction Factor . . .
NUCLEAR DATA SHEETS
ˆ k GT = the numerator of the VRF is written as Gt M t 2 Mkk (Gt,k ) , where Gt,k is the kth entry of Gt . It should be noted that a similar method for determining the contribution of each nuclear data to target integral parameters has been already proposed by Muir [7]. The basis of the theories of Muir’s method and ours is almost identical except for the uncertainty treatment for new measurement. In Muir’s method, the new measurement is done in an ideal condition and measurement uncertainty is assumed to be zero while in our method the measurement uncertainty is expressed by Eq. (3). It should be emphasized that the present study presents rich numerical examples in the following section to show performance of our method, whereas Muir’s study has not shown any application results of his method.
this new data. In this case, Vm is zero since the integral parameter is the microscopic nuclear data itself, and the sensitivity matrix G becomes a vector ek in which the kth entry is unity and others are zero. Thus, the posterior covariance matrix M(k) for the adjusted nuclear data T(k) is written as follows: M(k) = M − M∗k [Mkk + Vk ]
−1
¯ (k) . Mk∗ = M − M (2)
Here, we express the uncertainty of the new measurement Vk as Vk = αMkk .
(3)
If the measurement is done with the same degree of uncertainty as the prior cross section uncertainty, α is unity. ¯ (k) can be written as Using Eq. (3), the entry of M Mik Mkj Cik Ckj σi σj Mkk = Mkk + Vk (1 + α)Mkk ˆ (k) M Cik Ckj σi σj ij = , = 1+α 1+α
¯ (k) = M ij
III.
(4)
1 ˆ (k) GT . Gt M t 1+α
A.
(5)
ˆ k GT Gt M t . Gt MGT t
Numerical Procedure
All the sensitivities of integral neutronics parameters to nuclear data are calculated with a general-purpose reactor physics calculation code system CBZ [8], which is being developed at Hokkaido university. Sensitivities of criticality and reactivity worth are calculated with the classical perturbation theory, and those of burnup-related integral parameters (nuclide number densities after burnup) are calculated with the generalized perturbation theory. Forward and (generalized) adjoint neutron fluxes are obtained from neutron transport calculations with the discrete ordinates method or the collision probability method. 70-group and 107-group energy-averaged cross sections based on JENDL-4.0 [9] are used for fast neutron systems and thermal neutron systems respectively. The covariance data given in JENDL-4.0 are also processed to those in a multi-group form. Note that correlations between different nuclides are ignored in the present study. Inter-energy and inter-reaction correlations are taken into account.
In the present paper, we define a variance reduction factor (VRF) of the kth nuclear data, fk , as fk =
RESULTS AND ANALYSES
In order to demonstrate the performance of our method, we calculate VRFs for several integral parameters. Some of them are measurement data obtained in previous critical experiments. In the actual application, such existing integral data would be used for the cross section adjustment,and then our method is used with the covariance data of the adjusted nuclear data for integral parameter of a target reactor core design such as an innovative fast reactor or an accelerator-driven subcritical system. In future study, we will apply the proposed method into such future nuclear systems.
where σi is the standard deviation of Ti and Cik is the correlation between Ti and Tk . Using this equation, it is easily shown that the correlation matrix for T (k) is not identical to that for T . Next, let us consider how much uncertainty of an integral parameter is reduced when we use the adjusted cross section set T(k) , i.e., when we obtain new measurement data for the kth nuclear data. If we write a sensitivity vector of the target integral parameter as Gt , variances of the integral parameter obtained using T and T(k) can be (k) T written as Gt MGT Gt , respectively. Thus, t and Gt M the variance reduction of the target integral parameter ΔVt is written as ¯ (k) GT ΔVt = Gt M − M(k) GT t = Gt M t =
G. Chiba et al.
(6)
With this VRF, we can know which nuclear data require further accuracy improvement in order to reduce the variance of the target integral parameter. It should be also emphasized that calculations of the proposed VRFs are quite easy; one needs sensitivity profiles for target neutronics parameters and covariance matrices of nuclear data. The proposed method can provide us useful information on microscopic nuclear data measurements. If correlations among different cross sections are igˆ (k) is written as nored in the adjustment process, M ij ˆ (k) = Mkk δik δjk , where δ is Kronecker’s delta, and M ij
63
Variance Reduction Factor . . . B.
NUCLEAR DATA SHEETS
Criticalities of Fast Neutron Systems
G. Chiba et al.
0.18 U-238 mu-bar U-235 inelastic U-238 elastic U-238 inelastic
0.16 Variance reduction factor
VRFs for criticalities of the following four fast critical assemblies are calculated. The geometrical specification and nuclide number densities of the compositions are taken from the ICSBEP handbook. • Jezebel: A bare sphere of plutonium (95at% 239 Pu). • Godiva: A bare sphere of high enriched (94wt%) uranium. • Flattop-25: A high enriched uranium sphere surrounded by a normal uranium reflector.
0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 103
• Big-Ten: A large mixed-uranium-metal cylindrical core with a depleted uranium reflector.
104
105 106 Neutron energy [eV]
107
FIG. 3. Variance reduction factor for criticality of Flattop-25.
Figures 1 to 4 show VRFs for criticalities of the above fast neutron systems.
0.7 U-235 chi U-238 inelastic U-238 capture
0.35 0.6 Variance reduction factor
Variance reduction factor
0.30
U-235 mu-bar U-235 elastic U-235 inelastic
0.25 0.20 0.15 0.10
0.5 0.4 0.3 0.2 0.1
0.05 0.00 3 10
4
10
5
10 10 Neutron energy [eV]
6
0.0 4 10
7
10
Variance reduction factor
Pu-239 fission Pu-239 chi Pu-239 inelastic
0.20 0.15 0.10 0.05
4
10
5
10 10 Neutron energy [eV]
6
7
10
take non-negligible values in a low-energy range in which there are few neutrons in the system. This is due to a storing positive correlation between high- and low-energy ranges in this cross section: reduction of the uncertainty in the low-energy range simultaneously results in reduction of the uncertainty in the high-energy range. For the Jezebel criticality, VRFs of 239 Pu fission cross sections are high. This is interesting because the high-enriched uranium system Godiva has much smaller VRFs of fission cross sections of principal fissile material than Jezebel. For the Flattop-25 criticality, VRFs of reflector materials are rather high in comparison with those of fuel materials. Big-Ten has high VRFs of 238 U inelastic cross sections around several MeV. The energy distribution of the VRFs is different from that for the Flattop-25 criticality.
0.30
0.00 3 10
6
FIG. 4. Variance reduction factor for criticality of Big-Ten.
FIG. 1. Variance reduction factor for criticality of Godiva.
0.25
5
10 10 Neutron energy [eV]
7
10
FIG. 2. Variance reduction factor for criticality of Jezebel. C.
Criticalities of Thermal Neutron Systems
235
For the Godiva criticality, VRFs of U inelastic cross sections around 1 MeV are high. It is one of specific features of VRFs that those of 235 U elastic cross sections
VRFs for criticalities of the following two thermal neutron systems are calculated. One is a mixed-oxide (MOX) 64
Variance Reduction Factor . . .
NUCLEAR DATA SHEETS
fuel-loaded critical assembly (9x9 reference core) constructed through the reactor physics experiment program FUBILA [10], and a high-enriched uranium solution system reflected by water, which is categorized as HEU-SOLTHERM-010 (case 1) in the ICSBEP handbook. Sensitivities of the FUBILA criticality are calculated in a radial two-dimension core model. Figures 5 and 6 show VRFs for the criticalities of these thermal systems.
D.
G. Chiba et al.
Sodium Void Reactivity Worth of Fast Critical Assembly ZPPR-10A
VRFs for the sodium void reactivity worth of the fast critical assembly ZPPR-10A are calculated. ZPPR-10A is one of fast critical assemblies constructed through the physical mock up experiments for large-size fast breeder reactors. VRFs for spectrum component-dominant sodium void reactivity and those for leakage componentdominant reactivity are shown in Figs. 7 and 8.
Pu-239 fission Pu-239 capture Pu-240 capture
0.5
0.30
0.4 Variance reduction factor
Variance reduction factor
0.6
0.3 0.2 0.1 0.0 -3 -2 -1 0 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 10 10 10 Neutron energy [eV]
0.25 0.20 0.15 0.10 0.05 0.00 3 10
FIG. 5. Variance reduction factor for criticality of FUBILA 9x9 reference core.
U-238 inelastic Na-23 elastic Na-23 inelastic
4
10
5
10 10 Neutron energy [eV]
6
7
10
FIG. 7. Variance reduction factor for spectrum componentdominant sodium void reactivity of ZPPR-10A. 0.9
0.7
U-235 capture U-235 nu-bar U-235 chi 0.30
0.6 0.5
Variance reduction factor
Variance reduction factor
0.8
0.4 0.3 0.2 0.1 0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 107 Neutron energy [eV]
0.25 0.20 0.15 0.10 0.05 0.00 103
FIG. 6. Variance reduction factor for criticality of HEU-SOLTHERM-010-1.
U-238 inelastic Na-23 elastic Na-23 inelastic
104
105 106 Neutron energy [eV]
107
FIG. 8. Variance reduction factor for leakage componentdominant sodium void reactivity of ZPPR-10A.
It is interesting to see that VRFs of thermal capture cross sections of 240 Pu are significantly high in the FUBILA core. For the HEU-SOL-THERM-010-1 criticality, VRFs of fission spectrum of 235 U are much higher than VRFs of other nuclear data. The fission spectrum uncertainty of 235 U has large negative correlation between high-energy and low-energy ranges, so new measurement of this nuclear data is desirable at arbitrary energy.
In the both reactivity worth, we can observe high VRFs of 238 U inelastic scattering cross sections and sodium nuclear data. In the leakage component-dominant reactivity, VRFs of sodium nuclear data are slightly enhanced. 65
Variance Reduction Factor . . .
Nuclide Number Densities of 242 Cm and 244 Cm after Burnup in a Light Water Reactor Pincell
0.6 0.5 0.4 0.3 0.2
0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 107 Neutron energy [eV]
Pu-240 capture Pu-241 capture Am-241 capture
FIG. 11. Variance reduction factor for burnup of LWR-MOX pincell.
0.4
242
Cm inventory after
0.3 0.8 0.2
Pu-241 capture Pu-242 capture Am-243 capture
0.7 Variance reduction factor
Variance reduction factor
0.7
0.1
0.6
0.1 0.0 -3 -2 -1 0 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 10 10 10 Neutron energy [eV]
FIG. 9. Variance reduction factor for burnup of LWR-UO2 pincell.
242
Cm inventory after
0.5 0.4 0.3 0.2
0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 107 Neutron energy [eV]
Pu-241 capture Pu-242 capture Am-243 capture
0.7
0.6
0.1
0.8
Variance reduction factor
Pu-240 capture Pu-241 capture Am-241 capture
0.8
Nuclide number densities of 242 Cm and 244 Cm after burnup are very important because of their radioactivities and neutron generations by spontaneous fissions. VRFs for the nuclide number densities of these curium isotopes are calculated for PWR-represented pincells with UO2 fuel (235 U enrichment is 4.7%) and MOX fuel (Pu-fissile enrichment is 10%). The calculated VRFs are shown in Figs. 9 to 12.
0.5
G. Chiba et al.
0.9
Variance reduction factor
E.
NUCLEAR DATA SHEETS
FIG. 12. Variance reduction factor for burnup of LWR-MOX pincell.
0.6
244
Cm inventory after
0.5 0.4
UO2 pincell since initial number densities of plutonium and americium isotopes are zero. VRFs of 241 Pu capture cross sections for 244 Cm number densities are dominant in the UO2 pincell, and take quite high values below 100 eV. This is because inter-energy full positive correlations are given to this cross sections in this energy range. On the other hand, VRFs of 241 Pu capture cross sections are relatively low and those of 243 Am capture cross sections are rather high in the MOX cell.
0.3 0.2 0.1 0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 107 Neutron energy [eV]
FIG. 10. Variance reduction factor for burnup of LWR-UO2 pincell.
244
Cm inventory after IV.
242
CONCLUSIONS
244
Cm and Cm are generated via neutron capture reactions of their parent nuclides 241 Am and 243 Am respectively. VRFs of 241 Am capture cross sections for 242 Cm number densities after burnup are high in both the UO2 and MOX pincells, especially in a specific energy range from 0.1 eV to several eV. VRFs of capture cross sections of plutonium isotopes are relatively high in the
We have proposed a new quantity, a variance reduction factor, to identify nuclear data for which further improvements are required to reduce uncertainties of target integral neutronics parameters. Important energy ranges can be also identified with this variance reduction factor. Variance reduction factors have been calculated 66
Variance Reduction Factor . . .
NUCLEAR DATA SHEETS
for several integral neutronics parameters, such as criticalities of fast and thermal neutron systems, sodium void reactivity worth of a fast critical assembly and nuclide number densities after burnup in LWR pincells. The usefulness of the variance reduction factors has been
G. Chiba et al.
demonstrated. Acknowledgements: We thank Y. Kawamoto and T. Kajihara for their helpful comments on this manuscript. This work was supported by JSPS KAKENHI Grant Number 24561040.
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