Sensitivity and uncertainty in the effective delayed neutron fraction (βeff )

Nuclear Instruments and Methods in Physics Research A 715 (2013) 70–78

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Sensitivity and uncertainty in the effective delayed neutron fraction (βeff ) Ivan-Alexander Kodeli n Jožef Stefan Institute, Jamova 39, Ljubljana, Slovenia

art ic l e i nf o

a b s t r a c t

Article history: Received 8 January 2013 Received in revised form 14 March 2013 Accepted 15 March 2013 Available online 22 March 2013

Precise knowledge of the effective delayed neutron fraction (βeff ) and the corresponding uncertainty is important for nuclear reactor safety analysis. The interest in developing the methodology for estimating the uncertainty in βeff was expressed in the scope of the UAM project of the OECD/NEA. The sensitivity and uncertainty analysis of βeff performed using the standard first-order perturbation code SUSD3D is presented. The sensitivity coefficients of βeff with respect to the basic nuclear data were calculated by deriving Bretscher's k-ratio formula. The procedure was applied to several fast neutron benchmark experiments selected from the ICSBEP and IRPhE databases. According to the JENDL-4.0m covariance matrices and taking into account the uncertainties in the cross-sections and in the prompt and delayed fission spectra the total uncertainty in βeff was found to be in general around 3%, and up to ∼7% for the 233 U benchmarks. An approximation was applied to investigate the uncertainty due to the delayed fission neutron spectra. The βeff sensitivity and uncertainty analyses are furthermore demonstrated to be useful for the better understanding and interpretation of the physical phenomena involved. Due to their specific sensitivity profiles the βeff measurements are shown to provide valuable complementary information which could be used in combination with the criticality (keff ) measurements for the evaluation and validation of certain nuclear reaction data, such as for example the delayed (and prompt) fission neutron yields and interestingly also the 238U inelastic and elastic scattering cross-sections. & 2013 Elsevier B.V. All rights reserved.

Keywords: Nuclear safety Nuclear data sensitivity and uncertainty Benchmarks Covariance

1. Introduction The effective delayed neutron fraction is the key reactor safety parameter involved in the control rod worth calculations and reactor transient (reactivity feedbacks effect) studies. It is used to define a unit of reactivity known as the dollar representing the amount of reactivity necessary to make the reactor prompt critical. As such it plays an important role in reactivity accident analysis. Its accuracy should be therefore precisely understood and evaluated. Among the basic reactor parameters the kinetic parameters such as the effective delayed neutron fraction (βeff ) and the neutron generation lifetime (lambda) play, besides evidently the effective multiplication factor (keff ), a major role in the reactor safety and control analysis. Since the value of βeff varies from one isotope to the other, it is evident that the future reactor systems using a wider range of actinide isotopes in their fuel will have to face the problem of lower values of βeff due to the presence of Pu isotopes, making the reactor control of MOX fueled cores more challenging. The conception of the future GEN-IV reactors will have to rely more than in the past, when the design was supported by (expensive) experiments, on (cheaper) modelisation

n

Tel.: þ386 1 5885 412; fax: þ386 1 5885 412. E-mail addresses: [email protected], [email protected]

0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2013.03.020

using advanced computer codes. Precise knowledge of the basic reactor parameters as well as good understanding and reliable estimation of the associated uncertainties will be essential for new reactor design. The sensitivity and uncertainty methods provide powerful means to understand the most important physical processes involved in the calculation of the basic reactor parameters and to assess the modeling uncertainties. Within the Uncertainty Analysis in Modeling (UAM) project [1,2] of the OECD/Nuclear Energy Agency (NEA) the interest in the dynamic parameters and in the sensitivity and uncertainty analysis was expressed and the development of original sensitivity and uncertainty methods is being promoted. The UAM aims at developing the mathematical methods and computer codes for the propagation of different types of uncertainties throughout the entire reactor calculations, from the nuclear basic data, neutron transport and thermalhydraulics analysis. The project also identified the benchmark experiments, such as SNEAK-7A & 7B (Karlsruhe Fast Critical Facility) having a unique set of experimental data for βeff which can be used as an example for the βeff uncertainty calculations. SNEAK and several other experiments selected from the International Reactor Physics Benchmark Experiments (IRPhE) [3] and International Handbook of Evaluated Criticality Safety Benchmark Experiments (ICSBEP) [4] databases were used to investigate the βeff uncertainties. The uncertainty in the effective delayed neutron fraction βeff was already studied by Hammer [5], D'Angelo et al. [6,7] and

I.-A. Kodeli / Nuclear Instruments and Methods in Physics Research A 715 (2013) 70–78

Zukeran et al. [8]. Hammer [5] estimated the βeff uncertainty for power reactors to about 5%, which are decomposed to the uncertainty due to those in the direct and adjoint neutron flux (∼2%), uncertainty in the fission cross-section times fission neutron yield, νsf (∼1:5%) and in the delayed fission spectra, χ d (∼0:5%). A similar uncertainty of about 5% is reported for fast reactors in the paper by D'Angelo et al. [6]. The main sources were the uncertainty in the delayed neutron yield, ν d (∼2%) and the delayed fission spectra, χ d (∼0:5%). In Ref. [7] the βeff variations were calculated as the difference between the Conventional Perturbation Theory first-order solutions of the reference “promptplus-delayed-neutron” critical state and the altered “prompt-neutrononly” one-beta-subcritical state 〈Φ þ δBΦ〉 〈Φ þ P δBP ΦP 〉 − 〈Φ þ FΦ〉 〈Φ þ P F P ΦP 〉

δβeff ¼

ð1Þ

where brackets 〈〉 indicate integration over the space, angle and energy, F is the fission operator, δB is the Boltzmann operator perturbation, Φ and Φ þ the homogeneous direct and adjoint Boltzmann equation solutions and the index P indicates the “promptneutrons-only” status of the system. Zukeran et al. [12] used the generalised perturbation method to obtain the sensitivity coefficients and estimated the βeff uncertainty to about 4–5%. The principal components of this uncertainty were found to be the uncertainty in ν d (∼2:5%), sf especially of 238 U (∼1:6%) and χ d (∼0:5%).

2. Calculation of the effective delayed neutron fraction According to the conventional definition given in Ref. [9] the effective delayed neutron fraction (βeff ) for a mixture of fissionable isotopes, m, is given by βeff

∑m P m d ¼ ∑m P m

with Pm d ¼

Z Z V

Z 

E′,Ω′

E,Ω

ð2Þ

! Φ þ ð r ,E′,Ω′Þ∑αi,m χ i,m ðE′Þ dE′ dΩ′

Z Pm ¼

i

! ! ! ν d,m ðEÞΣ f ,m ð r ,EÞΦð r ,E,ΩÞ dE dΩ d r ,

E,E′,Ω,Ω′,V

ð3Þ

! ! Φð r ,E,ΩÞ dE dΩ dE′ dΩ′ d r

The total fission spectrum is defined as ν p,m ðEÞχ p,m ðE′,EÞ þ ν d,m ðEÞ∑αi,m χ i,m ðE′Þ

ð4Þ

where Φ and Φ þ symbolise the direct and the adjoint angular ! fluxes, Σ f ,m ð r ,EÞ represents the macroscopic fission cross-section of the fissile isotope m, χ i,m ðE′Þ, χ m ðE′,EÞ are the corresponding ithgroup delayed and total neutrons fission spectra, and ν d,m ðEÞ, ν m ðEÞ the delayed and total fission neutron yields (nu–bar–mean number of neutrons emitted during the fission reaction). αi represents the delayed-neutron fraction of the ith-group (normalised to 1).

i

χ m ðE′,EÞ ¼

ν p,m ðEÞ þν d,m ðEÞ

:

ð5Þ

Note that the total neutron fission spectrum in general depends on the incident neutron energy E, in particular at higher energies. On the other hand, the delayed spectra are assumed to be independent of E. The above terms (3) and (4) are very similar to the sensitivity of the keff with respect to the prompt and delayed fission neutron yields (ν p , ν d ) calculated using the first-order perturbation theory. In the cross-section sensitivity and uncertainty code SUSD3D [10,11] these sensitivity terms are computed by weighting the neutron yield with the prompt/delayed neutron spectra and the direct and adjoint neutron fluxes. In the multigroup approximation they are expressed for an energy group g and a fissile isotope m as Z 1 ! ! ! ! Skν p ,m,g ¼ ∑ d r Φg ð r ÞΦg′þ ð r Þχ p,m,g′ ν p,m,g sf ,m,g N m ð r Þ ð6Þ R g′ V Skν d ,m,g ¼

1 ∑ R g′,i

Z V

! ! ! ! d r Φg ð r ÞΦg′þ ð r Þχ i,m,g′ αi,m ν d,m,g sf ,m,g N m ð r Þ:

ð7Þ

R is the normalisation factor assuring that the sum of the total nu– bar (ν ¼ ν p þ ν d ) sensitivities of all involved fissile isotopes m adds ! up to 1, i.e. ∑g,m Skν,g,m ¼ 1. Nm ð r Þ symbolises the atomic number density of the isotope m. Comparing the above equations we see that the beta-effective can be easily calculated using the classical sensitivity–uncertainty codes (such as SUSD3D) by summing the delayed nu–bar (ν d ) sensitivities over all the energy groups and fissile isotopes βeff ¼

∑g,m Skν d ,m,g ∑g,m ðSkν p ,m,g þSkν d ,m,g Þ

¼ ∑ Skν d ,m,g : g,m

ð8Þ

Calculation of βeff by Eq. (2) (or (8)) is suitable for deterministic neutron transport codes. On the contrary, due to the complexity of the adjoint Monte Carlo (M/C) transport calculations an alternative formulation to Eq. (2) is often used to calculate the βeff in the M/C codes. The method is based on Bretscher's approximation, also called the prompt k-ratio method [12,13] βeff ¼ 1−

! ! Φ þ ð r ,E′,Ω′Þχ m ðE′,EÞν m ðEÞΣ f ,m ð r ,EÞ

71

kp k

ð9Þ

where kp is the effective multiplication factor (keff ) taking into account only prompt neutrons and k is the usual total (prompt plus delayed neutron) keff . To verify the equivalence between the βeff values calculated using the above Eqs. (8) and (9) several benchmark experiments selected from the International Reactor Physics Benchmark Experiments (IRPhE) [3] and International Handbook of Evaluated Criticality Safety Benchmark Experiments (ICSBEP) [4] databases were studied using deterministic and Monte Carlo codes. The

Table 1 List of benchmark experiments considered in this study. Name

ICSBEP/IRPhE ref.

Description

Jezebel Skidoo Popsy Topsy Flat-top 23 Bigten ZPPR-9 SNEAK-7A & -7B

PU-MET-FAST-001 U233-MET-FAST-001 PU-MET-FAST-006 HEU-MET-FAST-028 U233-MET-FAST-006 IEU-MET-FAST-007 ZPPR-LMFR-EXP-002 SNEAK-LMFR-EXP-001

Bare sphere of 95 at% 239Pu metal, 4.5 at% 240Pu, 6.385-cm radius Bare ∼98:1% 233U sphere, 5.983-cm radius ∼20-cm natural U reflected 94 wt% 239Pu sphere, 4.533-cm radius ∼20-cm natural U reflected 93 wt% 235U sphere, 6.116-cm radius ∼20-cm natural U reflected 98 at% 233U sphere, 4.2-cm radius Cylinder of 10% enriched U with depleted U-reflector, radius 41.91-cm, height 96.428-cm Cylindrical 2-zone, MOX core with Na cooling and depleted U blanket MOX fuel reflected by metallic depleted uranium

72

I.-A. Kodeli / Nuclear Instruments and Methods in Physics Research A 715 (2013) 70–78

benchmarks considered here are listed in Table 1 together with their main characteristics. The sensitivities of keff with respect to the delayed nu–bar (terms in Eq. (8)) were calculated using the SUSD3D [10] code. SUSD3D uses first-order perturbation theory to calculate sensitivity coefficients and standard deviations in the calculated detector responses or design parameters of interest (such as keff ) due to input cross-sections and their uncertainties. One-, two- and threedimensional shielding and criticality problems can be solved. Several types of uncertainties can be considered, i.e. those due to the neutron/gamma multigroup cross-sections, energy dependent response functions and secondary angular and energy distribution (e.g. scattering anisotropy, fission spectra) uncertainties. The direct and adjoint fluxes needed to calculate the sensitivities can be produced by different discrete ordinates codes, such as ANISN, DORT-TORT and DANTSYS. SUSD3D is available from the OECD/ NEA Data Bank and RSICC. For the need of the here presented sensitivity analysis the direct and adjoint neutron fluxes were calculated using one-, twoand three-dimensional discrete ordinates transport codes ONEDANT, TWODANT and THREEDANT included in the DANTSYS computer code package [14]. The Jezebel, Skidoo, Popsy, Topsy and Flattop 23 benchmarks were calculated in one-dimensional spherical geometry using ONEDANT, Big-ten and ZPPR-9 in twodimensional r–z geometry using TWODANT, whereas both TWODANT (two-dimensional r–z model) and THREEDANT (threedimensional x−y−z) were used for the two SNEAK-7 benchmarks. The cross-sections were taken from the ENDF/B-VII.0 evaluation [15] and processed into 33 energy groups by the NJOY-99 [16] Evaluated Nuclear Data File (ENDF) processing code. The TRANSX2 code was used to prepare the case-dependent multigroup selfshielded cross-sections. Table 2 proves the good equivalence between the βeff values calculated using the above Eqs. (8) and (9) as well as good agreement with the measured values. Note that although for the SNEAK-7A, Topsy and 23 Flattop benchmarks the C/E (calculation to measurement) ratio slightly exceeds 1s experimental uncertainty level, they are, as it will be shown later, well within the total (experimental plus experimental) 1s uncertainty.

The sensitivity analyses allow a better understanding of the physical processes involved. Multiplying the sensitivity profiles with the covariance matrices using the “sandwich” equation furthermore allows estimating the uncertainty in the target quantity. However, βeff being itself the derivative with respect to the delayed neutron yields the calculation of the βeff sensitivity Table 2 Measured values of βeff compared with those calculated using Eqs. (8) and (9). Measured (pcm)

Calculated (pcm) Eq. (8) SUSD3D

SNEAK 7A SNEAK 7B Jezebel Skidoo Popsy Topsy 23 Flattop Big-ten ZPPR-9

395 7 20 4137 25 1947 10 290 7 10 2767 7 665 7 13 360 7 9 7207 7 N/A

373 419 185 296 277 688 374 720 360

or alternatively s ∂βeff ð1−βeff Þ k kp ¼ ðSs −Ss Þ: βeff ∂s βeff

ð11Þ

The two terms Sks and Sksp correspond to the sensitivities of k and kp which can be obtained using the standard linear perturbation theory. Due to a small difference between the two terms a high accuracy is required for the corresponding sensitivity calculations. Note that the sensitivity with respect to the delayed nu–bar (s ¼ ν d ) is particularly easily evaluated since naturally the sensitivity term Sksp ¼ 0. Eq. (11) is similar to Eq. (1) used in Ref. [7], differing only by a factor of ð1−βeff Þ which originates from the derivation of the denominator in Eq. (8). For example, in the case s ¼ νm d,g we can easily verify that β

Sν eff ¼ d ,m,g ¼

k Skν ,m,g ð1−∑g,m Skν d ,m,g Þ ν d,m,g ∂βeff ν d,m,g ∑g,m ∂Sν d ,m,g ¼ ¼ d βeff ∂ν d,m,g βeff ∂ν d,m,g βeff

1−βeff k S : βeff ν d ,m,g

ð12Þ

The first results obtained using this method were demonstrated at the UAM-5 Meeting in April 2011 and later (see Refs. [17–19]). An alternative procedure is being developed in the scope of the UAM project by Ivanov et al. [2] and Ivanov and Kodeli [20]. In January 2012 a similar method was independently proposed by Chiba [21]. 3.1. Application of the sensitivity method to fast benchmark analysis

3. Sensitivity of beta-effective to nuclear data

Benchmark

would actually require the evaluation of the second derivatives and application of the generalised sensitivity method. The method used here is on the other hand based on the derivation of Bretscher's afore mentioned prompt k-ratio method [12,13] and allows detailed analysis of components of βeff uncertainty using the existing first-order perturbation method. From the above expression (Eq. (9)) the sensitivities can be readily obtained as a (properly weighted) difference between two standard sensitivity terms     ∂kp kp ∂k kp s∂kp s∂k s ∂βeff s Sβseff ¼ ¼ þ 2 þ ¼ − − βeff ∂s βeff k∂s k ∂s k−kp kp ∂s k∂s kp ¼ ðSk −Skp Þ ð10Þ k−kp s s

Prompt k-ratio (Eq. (9)) DANTSYS

MCNP

379 429 186 297 278 690 375 734 362

369 415 186 284

As mentioned above, the k and kp sensitivity calculations were performed using the SUSD3D code based on the direct and adjoint neutron fluxes calculated by the DANTSYS package. For the specific needs of the βeff sensitivity analysis the accuracy of the SUSD3D sensitivity calculations was increased due to the small difference between the two sensitivity terms in Eq. (11). Several sources of covariance matrices needed to obtain the βeff uncertainties by error propagation (“sandwich”) formula were considered. However, among the available covariances only the JENDL-4.0m [22] evaluation includes covariance data relative to delayed fission neutron yields, which are of particular interest for the present studies. JENDL-4.0 also covers most of the relevant neutron reaction data and includes both cross-section as well as prompt neutron fission spectra covariance matrices for the main fissile isotopes. The COMMARA-2 covariance evaluation [23], otherwise the natural choice due to the use of the ENDF/B-VII cross-sections in the neutron transport calculations, includes the covariances of delayed nu–bar only for 233U. Nevertheless the COMMARA-2 covariances [23] provided valuable comparison and validation for reactions such as inelastic, elastic, fission, total neutron yield etc. JENDL-4.0m covariance matrices, both for crosssections and fission spectra (MF33 and MF35 data), were processed by the NJOY/ERRORR [16] code system. COMMARA-2 covariances are

I.-A. Kodeli / Nuclear Instruments and Methods in Physics Research A 715 (2013) 70–78

already available in the 33 energy groups, and were only reformatted using the ANGELO2 code [24,25]. 3.2. Covariances of delayed neutron energy spectra As stated already by Keepin [9] the energy distribution of the delayed neutrons (χ d ) is probably the poorest known of all βeff input parameters, and this may still be true nowadays. He believed that since the energy of delayed neutrons determines their effectiveness, the uncertainty in χ d could have an important contribution to the final βeff uncertainty. However, no nuclear data evaluation includes covariances of the delayed neutron fission spectra nor could this information be found in the literature. Keepin estimated the impact of this uncertainty by comparing results obtained based on two different delayed spectra evaluations taken as representative of the maximum uncertainty in delayed-neutron emission spectra. In the absence of more reliable data approximate “two-block” covariance matrices were constructed based on a simple common sense assumption of an energy-uniform standard deviation of 15% and a complete anti-correlation between the energies above and below the mean delayed neutron energy for each of the six delayed groups. Conservative assumption of the complete correlation between the six individual groups was adopted. To test the validity of this method a similar procedure, except assuming a uniform 4% standard deviation instead of 15%, was Table 3 Fission spectra uncertainties in keff and βeff calculated using the approximate “twoblock” prompt fission spectra covariances (i.e. assuming flat anti-correlated 4% standard deviation) compared to those based on covariances from JENDL-4.0 and SCALE-6.0.

Twoblock

JENDL4.0

SCALE6.0

Twoblock

JENDL4.0

SCALE6.0

U U Pu Total

22 71 261 271

27 99 288 305

20 78 264 276

53 49 572 577

50 25 523 526

36 17 414 416

235

U U 239 Pu Total

41 109 335 354

49 150 377 409

37 119 343 364

78 36 551 557

71 46 489 496

51 18 377 381

Jezebel

239

292

367

343

637

820

774

Skidoo Jez-23

233

106

121

97

212

106

91

Popsy Flat-Pu

235

6

8

6

28

30

22

238

U 239 Pu Total

47 302 306

68 371 377

54 348 352

105 100 147

94 172 199

79 45 93

235

220

290

229

195

374

289

238 U Total

44 224

64 279

50 234

47 201

92 385

70 297

233

U U 238 U Total

167 5 41 171

180 7 58 189

156 5 46 163

218 16 53 225

304 17 45 308

227 13 36 231

235

U U Total

456 189 493

575 273 637

441 217 491

43 218 218

200 400 448

132 307 334

235

6 76 331 340

7 103 371 385

5 81 332 342

14 45 706 708

13 16 639 639

9 11 520 521

235 238 239

SNEAK 7B

238

Pu

U U

applied to the prompt fission neutron spectra (PFNS), where comparison with detailed covariance matrices available in some nuclear data evaluation (JENDL-4.0, SCALE-6, etc.) was possible. JENDL-4.0 [22] contains PFNS covariance matrices for the main fissile isotopes such as 235U, 238U, 239Pu and 240Pu at several incident neutron energies. The code ERRORR of the NJOY [16] system was used to process these covariance data (Files 35 in ENDF terminology). The fission spectra covariance matrices for a large number of fissile isotopes (233U, 235U, 238U, 239Pu, 240Pu, 241 Pu, 242Pu, 232Th and 252Cf) are also included in the SCALE-5.1 and -6.0 packages [26] for use by the sensitivity/uncertainty code TSUNAMI. They were obtained assuming typical standard

Fig. 1. Sensitivity of βeff and keff with respect to the prompt fission spectra.

βeff uncertainty (pcm)

Benchmark Isotope keff uncertainty (pcm)

SNEAK 7A

Table 4 Jezebel: sensitivity of βeff relative to nuclear data. MAT

239

Pu Pu Pu

240 241

MAT

Flattop 23

U

235

Big-ten

238

ZPPR-9

U U 239 Pu Total 238

Sensitivity (%/%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

νt

0.079 0.005 2  10−4

0.009 3  10−4 7  10−5

−0.014 −0.002 0.005

−0.022 −0.001 −7  10−5

0.948 0.043 0.007

−0.947 −0.049 −0.002

0.002 −0.007 0.005

Table 5 Jezebel: variance penalties and total uncertainty in βeff relative to nuclear data uncertainties, based on JENDL-4.0m covariances except for χ d .

239

Topsy Flat-25

73

Pu Pu 241 Pu Sum 240

Uncertainty (%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

χp

χd

Total

0.342 0.023 0.005 0.32

0.220 0.012 0.002 0.20

0.210 0.147 0.020 0.24

0.138 0.006 0.001 0.13

2.274 0.203 0.033 2.27

0.353 0.009 0.001 0.34

0.820

0.134

0.820

0.148

2.49 0.25 0.04 2.5

Table 6 23 Jezebel 23: sensitivity of βeff relative to nuclear data. MAT

233

U U U

234

238

Sensitivity (%/%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

νt

0.056 0.001 5  10−4

−0.005 −2  10−4 −2  10−4

−0.068 0.001 0.003

−0.021 −3  10−4 −1  10−4

0.980 0.012 0.005

−0.983 −0.012 −0.002

−0.003 1  10−4 0.003

74

I.-A. Kodeli / Nuclear Instruments and Methods in Physics Research A 715 (2013) 70–78

deviations for the Watt spectrum parameters a and b (e.g. for 235U of 1.2% and 5.9% for a and b, respectively). The data are provided in 44 energy groups and were converted in the 33-group structure used in this study by the ANGELO2 [24] code. Flat flux weighting is used in the conversion between the energy structures.

Table 12 SNEAK-7A: sensitivity of keff relative to nuclear data. MAT

235

Table 7 Jezebel 23: variance penalties and total uncertainty in βeff relative to nuclear data uncertainties. MAT

Uncertainty (%) Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

χp

χd

Total

JENDL-4.0m U 0.466 238 U 0.002 Sum 0.47

0.478 0.003 0.48

0.170 0.002 0.17

0.289 0.000 0.29

6.985 0.015 6.99

0.859 0.001 0.86

0.106 0.000 0.11

1.014 0.005 1.01

7.08 0.02 7.1

COMMARA-2 233 U 0.252 238 U 0.003 Sum 0.25

0.122 0.004 0.12

0.098 0.002 0.10

0.316 0.000 0.32

8.855 N/A 8.85

0.223 0.002 0.22

0.091 0.000 0.09

– – N/A

8.87 0.01 8.9

Elastic

233

U U 239 Pu 240 Pu 241 Pu 12 C 16 O Fe 238

MAT

MAT

U U 239 Pu 240 Pu 241 Pu 16 O Fe 238

234

U U 238 U 235

Sensitivity (%/%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

νt

1  10−5 0.001 0.009

−3  10−5 −0.005 −0.061

−0.001 0.016 0.046

−8  10−5 −0.012 −0.041

0.001 0.548 0.443

−0.003 −0.516 −0.473

−0.002 0.032 −0.030

Table 9 Big-ten: variance penalties and total uncertainty in βeff relative to nuclear data uncertainties, based on JENDL-4.0m covariances except for χ d . MAT

235

U U Sum

238

Uncertainty (%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

χp

χd

Total

0.010 0.086 0.09

0.066 0.613 0.62

0.046 0.038 0.06

0.049 0.162 0.17

1.857 1.457 2.36

0.129 0.277 0.31

0.200 0.400 0.45

0.109 0.148 0.18

1.88 1.67 2.5

Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

0.001 0.102 0.006 0.001 5  10−5 0.040 0.034 0.012

−2  10−4 −0.017 −0.001 −2  10−4 −2  10−5 −0.001 −4  10−4 −0.004

0.037 0.087 0.540 0.014 0.006 – – –

−0.005 −0.158 −0.058 −0.005 −4  10−4 −0:001ðn,αÞ −0:002ðn,αÞ −0.003

3  10−4 0.002 0.002 5  10−5 4  10−5 – – –

0.056 0.137 0.779 0.020 0.009 – – –

Table 13 SNEAK-7A: energy-integrated sensitivity of βeff relative to nuclear data.

235

Table 8 BIG-TEN: sensitivity of βeff relative to nuclear data.

Sensitivity (%/%)

Sensitivity (%/%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

νt

−2  10−4 −0.011 −0.002 −3  10−4 −2  10−5 −0.043 −0.009

−0.001 −0.151 −0.012 −0.001 −1  10−4 −1  10−4 −0.024

0.052 0.276 −0.252 −0.012 0.005 – –

−0.001 −0.017 −0.006 −4  10−4 −2  10−5 0:001n,α 1  10−4

0.080 0.488 0.402 0.014 0.011 – –

−0.025 −0.233 −0.700 −0.030 −0.007 – –

0.055 0.255 −0.298 −0.016 0.005 – –

The fission neutron spectra sensitivities, both for prompt and delayed neutrons, were derived using the constrained sensitivity method proposed in Refs. [27,28]. Use of this method was in particular necessary for the simplified “two-block” covariances, since the fission neutron spectra covariances prepared in this way naturally do not fulfill the so-called zero-sum rule, i.e. the sum of the elements of any row of the absolute matrices does not sum-up to zero. The constrained sensitivity coefficients of delayed neutron spectra for a delayed group i and a fissile isotope m were calculated from k S~ χ i,m ¼ M χ i,m  Skχ i,m

ð13Þ

where M χ i,m is the matrix with the components defined as [27] M g,g′ χ i,m ¼ δg,g′ −χ i,m,g Table 10 Topsy: sensitivity of βeff relative to nuclear data. MAT

234

U U 238 U 235

and Skχ i,m symbolises the unconstrained sensitivity vector with terms defined as R ! þ ! ! ! V d r Φg ð r Þχ i,m,g αi,m ∑g′ Φg′ ð r Þν d,m,g′ sf ,m,g′ N m ð r Þ Skχ i,m,g ¼ R ! ! ! þ ! V d r ∑g,i,m Φg ð r Þχ i,m,g αi,m ∑g′ Φg′ ð r Þν d,m,g′ sf ,m,g′ N m ð r Þ

Sensitivity (%/%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

νt

2  10−4 0.016 0.047

−2  10−4 −0.014 −0.051

−0.004 −0.059 0.028

−3  10−4 −0.033 −0.013

0.004 0.836 0.153

−0.010 −0.843 −0.140

−0.005 −0.007 0.013

Table 11 Topsy: variance penalties and total uncertainty in βeff relative to nuclear data uncertainties, based on JENDL-4.0m covariances except for χ d . MAT

235

U U Sum

238

ð14Þ

Uncertainty (%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

χp

χd

Total

0.157 0.210 0.26

0.320 0.552 0.64

0.102 0.023 0.10

0.169 0.043 0.17

2.403 0.510 2.46

0.358 0.083 0.32

0.374 0.092 0.39

0.773 0.036 0.77

2.60 0.76 2.7

ð15Þ where g′ and g symbolise the incident and the outgoing neutron energies. Note that for the “two-block” covariances the same uniform standard deviations (4% and 15% for the prompt and delayed fission spectra, respectively) were adopted for all fissionable nuclides, the only difference between isotopes were the differences in mean neutron energies. Table 3 compares the uncertainties in keff and βeff calculated using the above “two-block” PFNS covariances with those based on the PFNS covariances from JENDL-4.0 and SCALE-6.0. In spite of its simplicity the procedure is shown to predict similar uncertainties, both for keff and βeff uncertainties, as the more sophisticated methods used in the JENDL-4.0 and SCALE-6.0 covariance data evaluations. This good agreement can be explained by the relatively narrow-energy

I.-A. Kodeli / Nuclear Instruments and Methods in Physics Research A 715 (2013) 70–78

sensitivity of the keff and βeff to the fission spectra, as shown in Fig. 1. The agreement is particularly good for the “well-behaved” PFNS sensitivities of the keff , whereas the βeff sensitivities in some cases show more complex energy dependence leading to a larger

Table 14 SNEAK-7A: variance penalties [29] and total uncertainty in βeff relative to nuclear data uncertainties. Note that the ν d covariances are not available in the COMMARA2 evaluation. MAT

JENDL-4.0m U ∼0 238 U 0.051 239 Pu 0.008 240 Pu 0.001 241 Pu ∼0 16 O 0.118 Fe 0.039 Sum 0.13 235

ðn,γÞ

νd

νp

χp

χd

0.050 0.025 0.523

0.044 0.23 0.279 2.18 0.091 1.63 0.07 0.06 0.12 0.20 0.30 2.7

0.017 1.425 0.120 0.009 0.002 0.002 0.197 1.44

0.074 0.101 0.074 0.115 0.020 – – 0.19

0.004 0.066 0.018 0.002 0.001 0:014n,α 0.003 0.07

0.218 1.610 1.529 0.067 0.055 – – 2.23

0.006 0.138 0.122 0.006 0.003 – – 0.18

0.53

0.008 2.595 0.216 0.034 0.003 2.60

0.025 0.144 0.126 0.021 0.003 0.19

0.017 0.045 0.055 0.008 0.002 0.07

N/A N/A N/A N/A N/A N/A

0.002 0.270 0.051 0.077 0.002 0.28

0.036 0.017 0.341 0.038 0.005 0.35

Total

COMMARA-2 U U Pu 240 Pu 241 Pu Sum 238 239

0.008 2.515 0.169 0.029 ∼0 2.52

N/A N/A N/A –

0.05 2.62 0.43 0.10 0.01 2.6

N/A

Table 15 SNEAK-7B: sensitivity of βeff relative to nuclear data. MAT

235

U U 239 Pu 240 Pu 241 Pu 16 O Fe 238

spread of calculated uncertainties. Few exceptions, such as the sensitivity of βeff to 235U PFNS in the Big-ten benchmark and to 239 Pu PFNS in the Popsy benchmark, can be viewed as indication that a more sophisticated approach is needed in general to evaluate reliable delayed neutron fission spectra covariances. This relatively good agreement is nevertheless giving us justification for using the delayed spectra covariances and a reasonable confidence that the above approach is able to predict at least the correct order of magnitude of the delayed fission spectra uncertainty, provided the sensitivity profiles cover a sufficiently narrow energy range.

Uncertainty (%) Elastic Inelastic ðn,f Þ

235

75

3.3. Results of the βeff sensitivity and uncertainty analysis The nine benchmark experiments listed in Table 1 were analysed using the above described procedure and the results are presented in Tables 4–22. The individual reactions uncertainty contributions in Tables 5, 7, 9, 11, 14, 16, 18, 20 and 22 are characterised in terms of the variance penalty proposed in Ref. [29]. Variance penalty takes into account the variances due to specific reaction uncertainty as well as the correlations among the reactions. It is attractive since it guarantees a non-negative variance, but note that the total variance is in general not equal to the sum of partial variance penalty contributions. 3.3.1. Sensitivity and uncertainty analysis of the Jezebel, Skidoo, Bigten and Topsy benchmarks The energy-integrated sensitivity coefficients for the Jezebel, Skidoo, Big-ten and Topsy benchmarks are given in Tables 4, 6,

Table 17 Popsy: sensitivity of βeff relative to nuclear data.

Sensitivity (%/%)

MAT

Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

νt

−2  10−4 −0.019 −0.001 −1  10−4 −7  10−6 −0.040 −0.007

−0.002 −0.164 −0.008 −0.001 −7  10−5 2  10−4 −0.026

0.061 0.267 −0.233 −0.010 0.003 – –

−0.001 0.011 −0.001 5  10−5 1  10−5 0:002n,α 0.001

0.114 0.564 0.300 0.009 0.008 – –

−0.052 −0.334 −0.579 −0.022 −0.005 – –

0.061 0.230 −0.280 −0.013 0.003 – –

Sensitivity (%/%)

235

U U 239 Pu 240 Pu 241 Pu 238

Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

νt

0.001 0.103 −0.010 −3  10−4 −4  10−5

−0.001 −0.170 −0.042 −0.002 −1  10−4

0.027 0.261 −0.305 −0.015 0.002

−0.001 −0.050 −0.017 −0.001 −5  10−5

0.020 0.361 0.588 0.024 0.005

0.010 −0.083 −0.879 −0.043 −0.002

0.030 0.278 −0.292 −0.019 0.002

Table 16 SNEAK-7B: variance penalties [29] and total uncertainty in βeff relative to nuclear data uncertainties. Note that the ν d covariances are not available in the COMMARA-2 evaluation. MAT

Uncertainty (%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

χp

χd

Total

JENDL-4.0m 235 U 238 U 239 Pu 240 Pu 241 Pu 16 O Fe Sum

∼0 0.051 0.003 0.001 ∼0 0.149 0.067 0.17

0.023 1.701 0.086 0.006 0.002 0.001 0.235 1.72

0.079 0.112 0.055 0.106 0.021 – – 0.18

0.005 0.067 0.012 0.002 0.001 0.024n, α 0.005 0.07

0.329 1.848 1.162 0.045 0.040 – – 2.21

0.011 0.196 0.099 0.004 0.003 – – 0.22

0.071 0.046 0.489

0.086 0.469 0.195

0.50

0.52

0.35 2.57 1.29 0.04 0.04 0.15 0.24 2.9

COMMARA-2 235 U 238 U 239 Pu 240 Pu 241 Pu Sum

0.009 2.863 0.105 0.019 ∼0 2.87

0.010 2.991 0.148 0.024 0.002 2.99

0.028 0.139 0.108 0.018 0.002 0.18

0.011 0.046 0.070 0.006 0.002 0.06

N/A N/A N/A N/A N/A N/A

0.005 0.385 0.044 0.060 0.001 0.39

0.051 0.018 0.323 0.032 0.004 0.33

N/A N/A N/A – – N/A

0.06 3.01 0.38 0.07 0.01 3.0

76

I.-A. Kodeli / Nuclear Instruments and Methods in Physics Research A 715 (2013) 70–78

Table 18 Popsy: variance penalties and total uncertainty in βeff relative to nuclear data uncertainties. Note that the ν d covariances are not available in the COMMARA-2 evaluation. MAT

Uncertainty (%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

χp

χd

Total

JENDL-4.0m 235 U 238 U 239 Pu 240 Pu 241 Pu Sum

0.004 0.380 0.034 0.002 ∼0 0.38

0.011 1.712 0.414 0.017 0.003 1.76

0.100 0.029 0.184 0.163 0.055 0.27

0.002 0.103 0.107 0.005 ∼0 0.15

0.066 1.191 1.347 0.114 0.022 1.80

0.003 0.048 0.265 0.008 0.001 0.05

0.030 0.094 0.172

0.011 0.098 0.073

0.20

0.12

0.08 2.13 1.47 0.12 0.02 2.6

COMMARA-2 235 U 238 U 239 Pu 240 Pu 241 Pu Sum

0.009 1.645 0.622 0.054 ∼0 1.76

0.006 3.273 0.657 0.058 0.004 3.34

0.011 0.136 0.273 0.026 0.002 0.31

0.015 0.066 0.131 0.005 0.001 0.15

N/A N/A N/A N/A N/A N/A

0.001 0.096 0.102 0.101 0.001 0.17

0.022 0.079 0.045

N/A N/A –

0.09

–

Table 19 ZPPR-9: sensitivity of βeff relative to nuclear data. MAT

235

U U 239 Pu 240 Pu 241 Pu 16 O 55 Mn Fe 238

Table 21 Flattop 23: sensitivity of βeff relative to nuclear data.

Sensitivity (%/%)

MAT

Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

νt

−1  10−5 −0.009 −0.001 −8  10−5 −5  10−6 −0.023 −0.010 −0.014

−2  10−4 −0.116 −0.006 −0.001 −8  10−5 −6  10−5 −0.002 −0.069

0.013 0.293 −0.194 −0.012 0.008 – – –

1  10−4 0.038 0.004 0.001 7  10−5 0:002n,α 0.001 0.002

0.022 0.537 0.403 0.016 0.018 – – –

−0.010 −0.281 −0.660 −0.033 −0.010 – – –

0.012 0.256 −0.257 −0.017 0.007 – – –

233

U U 235 U 238 U 234

235

U U 239 Pu 240 Pu 241 Pu 16 O 55 Mn Fe Sum 238

Uncertainty (%) Elastic Inelastic ðn,f Þ

ðn,γÞ

νd

νp

χp

0 0.021 0.002 0.000 0. 0.110 0.090 0.220 0.26

0.001 0.075 0.028 0.005 0.001 0:017n,α 0.014 0.015 0.08

0.060 1.794 1.751 0.076 0.088 – – – 2.51

0.002 0.170 0.110 0.006 0.005 – – – 0.20

0.013 0.022 0.07 0.016 0.512 2.28 0.639 0.338 1.90 0.08 0.09 – – 0.11 – – 0.09 – – 0.65 0.64 0.62 3.0

0.002 1.285 0.071 0.008 0.002 0.001 0.012 0.638 1.44

0.028 0.121 0.146 0.109 0.025 – – – 0.17

χd

Total

8 and 10, respectively. The corresponding uncertainties are given in Tables 5, 7, 9 and 11. It can be seen that βeff is particularly sensitive to the delayed and prompt neutron yields. According to the JENDL-4.0m covariance data the uncertainties in the delayed fission neutron yields are by far the main sources of uncertainty, leading to the total uncertainty in βeff of around 2.5%, and up to ∼7% for 233U Jezebel 23 benchmark. For the latter benchmark the results could be verified against the COMMARA-2 based uncertainties of ∼9% and were found to be consistent (see Table 6). These benchmarks can be therefore considered as suitable above all for the validation of the delayed fission neutron yield data.

Sensitivity (%/%) Elastic

Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

νt

−0.005 2  10−5 0.001 0.075

−0.034 −0.001 −0.001 −0.129

−0.231 −0.001 0.015 0.167

−0.016 −2  10−4 −0.001 −0.033

0.700 0.007 0.015 0.274

−0.885 −0.009 0.002 −0.104

−0.185 −0.002 0.017 0.170

Table 22 Flattop 23: variance penalties and total uncertainty in βeff relative to nuclear data uncertainties. MAT

Table 20 ZPPR-9: variance penalties and total uncertainty in βeff relative to nuclear data uncertainties, based on JENDL-4.0m covariances except for χ d . MAT

0.02 3.30 0.73 0.12 0.004 3.4

Uncertainty (%) Inelastic

ðn,f Þ

ðn,γÞ

νd

νp

χp

χd

Total

JENDL-4.0m 233 U 0.089 235 U 0.004 238 U 0.318 Sum 0.33

0.584 0.009 1.293 1.42

0.209 0.063 0.103 0.24

0.217 0.001 0.073 0.23

5.097 0.048 0.916 5.18

0.657 0.002 0.061 0.81

0.304 0.017 0.045 0.31

0.762 0.001 0.110 0.77

5.27 0.06 1.63 5.5

COMMARA-2 233 U 0.298 235 U 0.003 238 U 1.770 Sum 1.77

0.432 0.004 2.430 2.47

0.195 0.006 0.088 0.21

0.227 0.010 0.046 0.23

6.444 N/A N/A 6.44

0.197 0.001 0.122 0.23

0.227 0.013 0.036 0.23

N/A N/A N/A N/A

6.47 0.01 2.44 6.9

Elastic

For the physical understanding it is interesting to notice a large compensation of the βeff sensitivity to the prompt and delayed 238 U fission neutron yields, leading possibly even to a small negative sensitivity to the total fission yield in the case of the Big-ten benchmark, and this in spite of the fact that the βeff of 233U (∼1480 pcm) is considerably higher than the one of the Big-ten benchmark (∼720 pcm). This can be explained by the high threshold of the 238U fission reaction as compared to the energy of delayed neutrons and different relative effectiveness of delayed and prompt neutrons in 235U and 238U (see Fig. 2). βeff of 238U is also decreasing at high energies contributing to the negative sensitivity above few MeV. The sensitivity to the fission neutron yield of 235U becomes positive below the 238U fission threshold being the sole reaction capable to provide fission neutrons for 238U fission.

I.-A. Kodeli / Nuclear Instruments and Methods in Physics Research A 715 (2013) 70–78

77

for the Popsy and Sneak-7A benchmarks, and the energyintegrated sensitivities of keff and βeff can be compared on Tables 12 and 13. Due to the different shapes of the sensitivity profiles it is expected that the combined use of criticality and βeff measurements would provide a better insight and an efficient validation of these nuclear data in the energy range above ∼1 MeV.

4. Conclusions

Fig. 2. Sensitivity of βeff with respect to total fission neutron yields of and 238U.

235

U

Fig. 3. Comparison of βeff and keff sensitivities with respect to the inelastic and elastic cross-sections of 238U for the Popsy experiment.

3.3.2. Sensitivity and uncertainty analysis of the SNEAK-7A, -7B, Popsy, FLATTOP 23 and ZPPR-9 benchmarks The energy-integrated sensitivity coefficients for the SNEAK-7A, -7B, Popsy, ZPPR-9 and FLATTOP 23 benchmarks are given in Tables 13, 15, 17, 19 and 21, respectively. It can be seen that βeff is particularly sensitive to the prompt and delayed neutron yields, fission, inelastic and capture neutron cross-sections. According to the JENDL-4.0m covariance data the main sources of uncertainty in βeff are, in addition to the uncertainty in delayed neutron yields of 238U and 239Pu, also the inelastic scattering on 238 U and fission cross-sections (see Tables 14, 16, 18, 20 and 22). It is interesting to note that the uncertainty due to 238U inelastic cross-sections are even almost twice as large when using COMMARA-2 covariances (1.5–1.7% vs. 2.5–3%) which differ with respect to JENDL-4.0m in particular around and above 1 MeV (see Fig. 4). The total uncertainty in βeff is in general around 3% and up to 5.5% using JENDL-4.0m (∼7% using COMMARA-2) for Flattop 23 benchmark due to 233U uncertainties. The fact that an important part of the total uncertainty comes from the 238U inelastic cross-section uncertainty makes these benchmarks potentially suitable for the validation of these nuclear reaction data. An example of the βeff sensitivity to the 238U inelastic data compared to the keff sensitivity is shown on Fig. 3

First, βeff was shown to be equivalent to the energy and isotope integrated sensitivity of the keff with respect to the delayed fission neutron yield as calculated using the first order perturbation theory. Consequently βeff can be easily calculated using the existing nuclear data sensitivity and uncertainty codes (such as SUSD3D). Good agreement between βeff values obtained using SUSD3D and Bretscher's approximation was actually demonstrated on the series of benchmark experiments. Furthermore, the sensitivity coefficients of the effective delayed neutron fraction βeff were obtained by deriving Bretscher's (prompt k-ratio) βeff expression with respect to the basic nuclear data. JENDL-4.0m covariance data were found to include the most complete evaluated data for the βeff uncertainty evaluations. For verification, JENDL-4.0m based uncertainties were partly compared with those based on COMMARA-2.0 covariances for the available reactions. JENDL-4.0m covariance data evaluations cover most of the relevant reactions, except the uncertainties in the delayed neutron fission spectra. The latter were roughly estimated assuming some common sense approximations, i.e. “two-block” covariance matrix with constant 15% standard deviation, anticorrelated below and above the mean energy of the emitted neutrons. In the case of the prompt fission neutron spectra this simplified method (only assuming 4% instead of 15% uncertainty) was found to predict reasonably well the impact of the fission spectra uncertainties comparing to more sophisticated covariances available, both for keff and βeff uncertainties. The sensitivity and uncertainty method was successfully applied to the sensitivity and uncertainty analysis of several fast reactor benchmarks. The method was demonstrated to allow detailed analysis of various components of βeff uncertainty, including those due to the uncertainty in basic nuclear data and neutron fission spectra. The method is robust provided sufficiently high precision is used in the sensitivity calculations. In addition to providing an insight into the physical phenomena involved (e.g. effectiveness of 238U delayed/prompt neutrons vs. those of other fissile isotopes) the results of the sensitivity and uncertainty analysis would be valuable for nuclear data validation. According to the available covariance data the total uncertainty in βeff was found to be in general around 3%, but up to 7% in the 233U reactor systems. The βeff uncertainty is in most cases predominantly due to the uncertainties in delayed neutron yields, contributing in the bare sphere experiments Jezebel, Skidoo, as well as in Topsy, Big-ten and Flattop 23 benchmarks over 90% to the total uncertainty. In some cases (Popsy, SNEAK-7A, -7B and ZPPR-9) the inelastic and elastic scattering (contributing with few %), fission cross-sections (∼0:2%) and prompt neutron yields (∼0:2%), as well as the prompt and delayed fission spectra (roughly 0.5%) play an important role. Due to the high sensitivity and the specific sensitivity profiles of βeff the latter experiments can provide a complementary information to critical experiments for the validation of other quantities than νd . Inelastic and elastic scattering on 238 U is a particularly interesting example where βeff measurements could contribute to the improved nuclear data evaluations. Precise knowledge of the βeff uncertainty will be particularly important for the future reactor systems using a wider range of

78

I.-A. Kodeli / Nuclear Instruments and Methods in Physics Research A 715 (2013) 70–78

Δσ/σ vs. E for 238U(n,inel.)

Δσ/σ vs. E for 238U(n,inel.)

35

Ordinate scales are % relative

30

standard deviation and barns.

25 20

Abscissa scales are energy (eV).

15 10 5

Ordinate scales are % relative standard deviation and barns. Abscissa scales are energy (eV).

σ vs. E for U(n,inel.)

238

U(n,inel.)

106

238

106

107

107

Correlation Matrix

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

Fig. 4. Covariance matrices of

101

105

σ vs. E for

105

Correlation Matrix 1.0 0.8 0.6 0.4 0.2 0.0

107

100

106

10-1

105

10-2

107

101

106

100

105

10-1

0

18 16 14 12 10 8 6 4 2 0

1.0 0.8 0.6 0.4 0.2 0.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

238

U inelastic cross-sections from the JENDL-4.0m and COMMARA-2 evaluations.

actinide isotopes with lower values of βeff (Pu isotopes), making the reactor control of MOX fueled cores more challenging. References [1] K. Ivanov, M. Avramova, I. Kodeli, E. Sartori, Benchmark for uncertainty analysis in modelling (UAM) for design, operation and safety analysis of LWRs, in: Specification and Support Data for the Neutronics Cases (Phase I), vol. I, NEA/NSC/DOC(2007)23. [2] K. Ivanov, M. Avramova, S. Kamerow, I. Kodeli, E. Sartori, E. Ivanov, O. Cabellos, Benchmark for uncertainty analysis in modelling (UAM) for design, operation and safety analysis of LWRs, in: Specification and Support Data for the Neutronics Cases (Phase I), vol. I, Version 2.0, 2012. [3] International Reactor Physics Experiments Database Project (IRPhE), “International Handbook of Evaluated Reactor Physics Benchmark Experiments”, NEA/ NSC/DOC(2006)1, Published on DVD March 2011 Edition, OECD NEA, ISBN 978-92-64-99141-5. [4] International Handbook of Evaluated Criticality Safety Benchmark Experiments, OECD/NEA, NEA/NSC/DOC(95)03, Paris, Published on DVD, 2011. [5] P. Hammer, Requirements of delayed neutron data for the design, operation, dynamics and safety of fast breeder and thermal power reactors, in: Proceedings of the Consultants Meeting on Delayed Neutron Properties, Vienna, 26–30 March 1979, INDC (NDS)-107/Gþ Special. [6] A. D'Angelo, B. Vuillemin, J.C. Cabrillat, NEACRP-A-766, 1987. [7] A. D'Angelo, A total delayed neutron yields adjustment using “ZPR” and “SNEAK” effective-beta integral measurements, in: Proceedings of the PHYSOR'90 Conference, Marseille, 1990. [8] A. Zukeran, H. Hanaki, S. Sawada, T. Suzuki, Journal of Nuclear Science and Technology 36 (January (1)) (1999) 61. [9] G.R. Keepin, Physics of Nuclear Kinetics, Addison-Wesley, Reading, MA, 1965. [10] I. Kodeli, Nuclear Science and Engineering 138 (2001) 45. [11] I. Kodeli, The SUSD3D code for cross-section sensitivity and uncertainty analysis—recent development, in: Invited Paper, Transactions of the American Nuclear Society, vol. 104, Hollywood, FL, June 26–30, 2011. [12] M.M. Bretscher, Evaluation of reactor kinetics parameters without the need for perturbation codes, in: Proceedings of the International Meeting on Reduced Enrichment for Research and Test Reactors, Jackson Hole, Wyoming, USA, October 5–10, 1997. [13] R.K. Meulekamp, S.C. van der Marck, Nuclear Science and Engineering 152 (2006) 142. [14] R.E. Alcouffe, et al., DANTSYS 3.0—A Diffusion-Accelerated, Neutral-Particle Transport Code System, LA-12969-M, Los Alamos National Laboratory, 1995, RSICC Code Package CCC-0547/08. [15] M.B. Chadwick, P. Obložinsky, M. Herman, N.M. Greene, R.D. McKnight, D. L. Smith, P.G. Young, R.E. MacFarlane, G.M. Hale, S.C. Frankle, A.C. Kahler,

[16]

[17]

[18]

[19]

[20]

[21] [22] [23]

[24] [25]

[26]

[27]

[28] [29]

T. Kawano, R.C. Little, D.G. Madland, P. Moller, R.D. Mosteller, P.R. Page, P. Talou, H. Trellue, M.C. White, et al., Nuclear Data Sheets 107 (2006) 2931. R.E. MacFarlane, D.W. Muir, The NJOY Nuclear Data Processing System Version 99, LA-12740-M, Los Alamos National Laboratory. RSICC Code Package PSR368, 1999. I. Kodeli, Sensitivity and Uncertainty in the Effective Delayed Neutron Fraction βeff (Method & SNEAK-7A Example), in: Proceedings of the Fifth Workshop for the OECD Benchmark for Uncertainty Analysis in Best-Estimate Modelling (UAM) for Design, Operation and Safety Analysis of LWRs (UAM-5), Stockholm, April 13–15, 2011,, NEA-1769/04 Package, OECD-NEA Data Bank. I. Kodeli, Sensitivity and uncertainty in beta-effective—new results, in: Proceedings of the OECD Benchmark for Uncertainty Analysis in BestEstimate Modelling (UAM) for Design, Operation and Safety Analysis of LWRs—Sixth Workshop (UAM-6), Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany, May 9–11, 2012. I. Kodeli, Sensitivity and uncertainty in the effective delayed neutron fraction (βeff ), in: Proceedings of the PHYSOR 2012 – Advances in Reactor Physics – Linking Research, Industry, and Education, Knoxville, TN, USA, April 15–20, 2012, on CD-ROM, American Nuclear Society, LaGrange Park, IL. E. Ivanov, I. Kodeli, Kinetics parameters S/U deterministic analysis, in: Proceedings of the OECD Benchmark for Uncertainty Analysis in BestEstimate Modelling (UAM) for Design, Operation and Safety Analysis of LWRs—Sixth Workshop (UAM-6), Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany, May 9–11, 2012. G. Chiba, et al., Journal of Nuclear Science and Technology 48 (12) (2011) 1471. (Available Online 5 January 2012). K. Shibata, et al., Journal of Nuclear Science and Technology 48 (1) (2011) 1. M. Herman, P. Obložinsk ́ y, C.M. Mattoon, M. Pigni, S. Hoblit, S.F. Mughabghab, A. Sonzogni, P. Talou, M.B. Chadwick, G.M. Hale, A.C. Kahler, T. Kawano, R.C. Little, P.G. Young, COMMARA-2.0 Neutron Cross Section Covariance Library, BNL-94830-2011, March 2011. I. Kodeli, ANGELO-LAMBDA, Covariance Matrix Interpolation and Mathematical Verification, NEA-DB Computer Code Collection, NEA-1798/02, 2008. I. Kodeli, E. Sartori, Neutron cross-section covariance data in multigroup form and procedure for interpolation to users' group structures for uncertainty analysis applications, in: Proceedings of the PHYSOR'90 Conference, Marseille, 1990. SCALE: A Modular Code System for Performing Standardized Computer Analyses for Licensing Evaluations, ORNL/TM-2005/39, Version 5.1, vols. I–III, November 2006. I. Kodeli, A. Trkov, R. Capote, Y. Nagaya, V. Maslov, Nuclear Instruments and Methods in Physics Research A 610 (2009) 540, http://dx.doi.org/10.1016/j. nima.2009.08.076. Y. Nagaya, I. Kodeli, G. Chiba, M. Ishikawa, Nuclear Instruments and Methods in Physics Research A 603 (2009) 485. D.W. Muir, Nuclear Instruments and Methods in Physics Research A 644 (2011) 55.