NUREG/CR-7249

ORNL/TM-2017/706

Overview of Nuclear Data 

Uncertainty in Scale and 

Application to Light Water 

Reactor Uncertainty Analysis 

Office of Nuclear Regulatory Research
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NUREG/CR-7249

ORNL/TM-2017/706

Overview of Nuclear Data 

Uncertainty in Scale and 

Application to Light Water 

Reactor Uncertainty Analysis

Office of Nuclear Regulatory Research

Manuscript Completed: December 2017

Date Published: December 2018

Prepared by:

W. Wieselquist

M. Williams

D. Wiarda

M. Pigni

U. Mertyurek

Oak Ridge National Laboratory

Bethel Valley Road

Oak Ridge, TN 37831

Mourad Aissa, NRC Project Manager

iii

ABSTRACT 

This report presents the current state of the art in SCALE for nuclear data uncertainty analysis 

capability, and it provides an overview of the uncertainty in multigroup cross sections, fission 

product yields, and decay data. The effect of nuclear data uncertainty is demonstrated for a 

typical light water reactor (LWR) depletion analysis problem involving a Combustion 

Engineering 14 × 14 assembly irradiated in Calvert Cliffs Unit 1. A single fuel rod from assembly 

D047, designated as MKP109, has been subjected to destructive radiochemical assay to 

measure the isotopic contents. 

The 95% range width (difference between the 97.5th and 2.5th percentiles) is used in this study 

to assess the calculation uncertainty. This approach uses the actual distribution of the data and 

does not make any assumptions about the normality of the distributions. If the distribution were 

normal, then the 95% range width would correspond to 4-sigma range, or +/- 2 sigma. 

The calculation uncertainty determined in nuclide concentrations for the MKP109 rod ranges 

from a few percent to 50%. The power factor for this fuel rod shows a very low uncertainty of 

less than 0.5%. 

Uncertainties in the macroscopic cross sections, reactivity, and power distributions are 

generally low, in the few percent range. The effective delayed neutron fraction, βeff, shows 

higher uncertainty of 20–100%. 

v

FOREWORD

Consideration of nuclear data uncertainty in calculations is an active area of research in data 

and software development. On the nuclear data side, uncertainty adds another dimension to an 

already complex database, and although the topic of uncertainty data is receiving attention—

each subsequent ENDF/B release has additional data uncertainty—there is still not widespread 

usage of the uncertainty data due in part to lack of software tools, but also due to 

disagreements within the nuclear engineering analysis community on how exactly it can be 

used. This report demonstrates usage of the uncertainty data in one of the areas where many 

code developers and analysts spend a lot of time: code and model validation. This report 

recommends error metrics that use uncertainty, additional avenues for interpreting validation 

results when calculation uncertainty is included, and avenues for additional usage of the 

uncertainty data. Ideally, data uncertainty should become standard in validation, leading to new 

pathways for data improvement.

vii 

TABLE OF CONTENTS 

ABSTRACT ................................................................................................................... iii

FOREWORD................................................................................................................... v

LIST OF FIGURES......................................................................................................... ix

LIST OF TABLES .......................................................................................................... xi

EXECUTIVE SUMMARY ............................................................................................. xiii

ABBREVIATIONS AND ACRONYMS.......................................................................... xv

1 INTRODUCTION ..................................................................................................... 1-1

2 DATA UNCERTAINTY ............................................................................................ 2-1

2.1 Cross Section Uncertainty ............................................................................................2-1

2.1.1 Background ....................................................................................................2-1

2.1.2 Implementation ...............................................................................................2-1

2.1.3 Data Characteristics .......................................................................................2-1

2.2 Fission Product Yield Uncertainty .................................................................................2-4

2.2.1 Background ....................................................................................................2-4

2.2.2 Implementation ...............................................................................................2-4

2.2.3 Characteristics ................................................................................................2-5

2.3 Decay Uncertainty ........................................................................................................2-6

2.3.1 Background ....................................................................................................2-6

2.3.2 Implementation ...............................................................................................2-6

2.3.3 Characteristics ................................................................................................2-6

3 LWR APPLICATIONS WITH UNCERTAINTY ........................................................ 3-1

3.1 Description ...................................................................................................................3-1

3.1.1 Operating History ............................................................................................3-1

3.1.2 Measurements ................................................................................................3-3

1.1 Model Equations ...........................................................................................................1-1

1.1.1 Transport Equation .........................................................................................1-2

1.1.2 Transmutation Equation ..................................................................................1-3

1.1.3 Multigroup Representation ..............................................................................1-4

1.2 Additional Models .........................................................................................................1-5

1.2.1 Few-Group Macroscopic Cross Sections ........................................................1-5

1.2.2 Activity and Decay Heat ..................................................................................1-5

1.2.3 Reactivity Coefficients ....................................................................................1-5

1.2.4 Kinetics Parameters........................................................................................1-6

1.2.5 Power Peaking Factors ...................................................................................1-6

1.3 Uncertainty Quantification Methods ..............................................................................1-7

1.3.1 Perturbation Theory ........................................................................................1-7

1.3.2 Sampling ........................................................................................................1-7
viii 

3.2 Calculation Results .......................................................................................................3-6

3.2.1 Macroscopic Cross Section Uncertainty..........................................................3-7

3.2.2 Reactivity Uncertainty ................................................................................... 3-13

3.2.3 Power Distribution Uncertainty ...................................................................... 3-16

3.2.4 Isotopic Uncertainties ................................................................................... 3-18

3.2.5 Source Terms ............................................................................................... 3-68

4 DISCUSSION........................................................................................................... 4-1

4.1 Importance of Non-Data Error ......................................................................................4-1

4.2 Calculation Uncertainty Is a Prediction .........................................................................4-1

4.3 Data Assimilation ..........................................................................................................4-2

4.4 Significant Bias Identification ........................................................................................4-2

5 CONCLUSIONS ...................................................................................................... 5-1

6 FUTURE WORK...................................................................................................... 6-1

7 REFERENCES ........................................................................................................ 7-1
ix

LIST OF FIGURES

Figure 1-1 SCALE/Sampler Work Flow ..............................................................................1-8

Figure 2-1 Uncertainty in Energy-Dependent 235U Fission Cross Section, σ 235U, fg, 

With Various Energy-Average Approximations ..................................................2-4

Figure 3-1 Configuration of Assembly D047 From CC-1 [7] ...............................................3-2

Figure 3-2 137Cs Gamma Scan Showing Sample S1 and S2 Locations [7] .........................3-3

Figure 3-3 Fast Macroscopic Absorption Cross Section .....................................................3-8

Figure 3-4 Thermal Macroscopic Absorption Cross Section ...............................................3-9

Figure 3-5 Fast Macroscopic Nu-Fission Cross Section ................................................... 3-10

Figure 3-6 Thermal Macroscopic Nu-Fission Cross Section ............................................. 3-11

Figure 3-7 Fast Macroscopic Scattering Cross Section .................................................... 3-12

Figure 3-8 Thermal Macroscopic Scattering Cross Section .............................................. 3-13

Figure 3-9 Eigenvalue Uncertainty ................................................................................... 3-14

Figure 3-10 Delayed Neutron Fraction Uncertainty ............................................................ 3-15

Figure 3-11 Reactivity Coefficient Uncertainty .................................................................... 3-16

Figure 3-12 Irradiation Sample Power Factor Uncertainty .................................................. 3-17

Figure 3-13 Flux Magnitude Uncertainty ............................................................................ 3-18

Figure 3-14 241Am Isotopic Uncertainty .............................................................................. 3-20

Figure 3-15 243Am Isotopic Uncertainty .............................................................................. 3-21

Figure 3-16 140Ce Isotopic Uncertainty ............................................................................... 3-22

Figure 3-17 142Ce Isotopic Uncertainty ............................................................................... 3-23

Figure 3-18 244Cm Isotopic Uncertainty .............................................................................. 3-24

Figure 3-19 133Cs Isotopic Uncertainty ............................................................................... 3-25

Figure 3-20 135Cs Isotopic Uncertainty ............................................................................... 3-26

Figure 3-21 137Cs Isotopic Uncertainty ............................................................................... 3-27

Figure 3-22 151Eu Isotopic Uncertainty ............................................................................... 3-28

Figure 3-23 152Eu Isotopic Uncertainty ............................................................................... 3-29

Figure 3-24 153Eu Isotopic Uncertainty ............................................................................... 3-30

Figure 3-25 154Eu Isotopic Uncertainty ............................................................................... 3-31

Figure 3-26 155Eu Isotopic Uncertainty ............................................................................... 3-32

Figure 3-27 152Gd Isotopics Uncertainty ............................................................................. 3-33

Figure 3-28 154Gd Isotopic Uncertainty ............................................................................... 3-34

Figure 3-29 155Gd Isotopic Uncertainty ............................................................................... 3-35

Figure 3-30 156Gd Isotopic Uncertainty ............................................................................... 3-36
x

Figure 3-31 157Gd Isotopic Uncertainty ............................................................................... 3-37

Figure 3-32 158Gd Isotopic Uncertainty ............................................................................... 3-38

Figure 3-33 160Gd Isotopic Uncertainty ............................................................................... 3-39

Figure 3-34 139La Isotopic Uncertainty ................................................................................ 3-40

Figure 3-35 95Mo Isotopic Uncertainty ................................................................................ 3-41

Figure 3-36 142Nd Isotopic Uncertainty ............................................................................... 3-42

Figure 3-37 143Nd Isotopic Uncertainty ............................................................................... 3-43

Figure 3-38 144Nd Isotopic Uncertainty ............................................................................... 3-44

Figure 3-39 145Nd Isotopic Uncertainty ............................................................................... 3-45

Figure 3-40 146Nd Isotopic Uncertainty ............................................................................... 3-46

Figure 3-41 148Nd Isotopic Uncertainty ............................................................................... 3-47

Figure 3-42 150Nd Isotopic Uncertainty ............................................................................... 3-48

Figure 3-43 237Np Isotopic Uncertainty ............................................................................... 3-49

Figure 3-44 238Pu Isotopic Uncertainty ............................................................................... 3-50

Figure 3-45 239Pu Isotopic Uncertainty ............................................................................... 3-51

Figure 3-46 240Pu Isotopic Uncertainty ............................................................................... 3-52

Figure 3-47 241P Isotopics Uncertainty ................................................................................ 3-53

Figure 3-48 242Pu Isotopic Uncertainty ............................................................................... 3-54

Figure 3-49 103Rh Isotopic Uncertainty ............................................................................... 3-55

Figure 3-50 101Ru Isotopic Uncertainty ............................................................................... 3-56

Figure 3-51 147Sm Isotopic Uncertainty .............................................................................. 3-57

Figure 3-52 148Sm Isotopic Uncertainty .............................................................................. 3-58

Figure 3-53 149Sm Isotopic Uncertainty .............................................................................. 3-59

Figure 3-54 150Sm Isotopic Uncertainty .............................................................................. 3-60

Figure 3-55 151Sm Isotopic Uncertainty .............................................................................. 3-61

Figure 3-56 152Sm Isotopic Uncertainty .............................................................................. 3-62

Figure 3-57 154Sm Isotopic Uncertainty .............................................................................. 3-63

Figure 3-58 90Sr Isotopic Uncertainty ................................................................................. 3-64

Figure 3-59 234U Isotopic Uncertainty ................................................................................. 3-65

Figure 3-60 235U Isotopic Uncertainty ................................................................................. 3-66

Figure 3-61 236U Isotopic Uncertainty ................................................................................. 3-67

Figure 3-62 238U Isotopic Uncertainty ................................................................................. 3-68

Figure 3-63 Activity Uncertainty ......................................................................................... 3-69
xi

LIST OF TABLES

Table 2-1 Key Cross Section Uncertainty Data in SCALE.................................................2-3

Table 2-2 Uncertainty in Key Fission Yield Data ...............................................................2-5

Table 2-3 Uncertainty in Key Decay Data .........................................................................2-7

Table 3-1 Measurement and Uncertainty Data for CC-1 MKP109 Samples S1 

and S2 ..............................................................................................................3-5

xiii

EXECUTIVE SUMMARY

This report presents the current state of the art in SCALE for nuclear data uncertainty analysis 

capability, and it provides an overview of the uncertainty in multigroup cross sections, fission 

product yields, and decay data. 

The nuclear data used in a typical SCALE calculation are processed from evaluated nuclear 

data files (ENDF/B) produced by the National Nuclear Data Center [1] through the AMPX code 

system distributed with SCALE. The ENDF/B data are based on a large set of nuclear data 

measurements that have been evaluated by the world’s top nuclear data experts and compiled 

into a cohesive, consistent database. Ideally, ENDF/B data should include an associated 

uncertainty (and correlations to other data) indicating the accuracy of the measurement, as well 

as any uncertainty introduced in the evaluation due to modeling fits, consistency adjustments, or 

other causes. This uncertainty in the nuclear data can be propagated to uncertainty in 

calculated quantities of interest for the nuclear analyst—whether it is keff for criticality safety 

applications or a void coefficient of reactivity for reactor physics applications. However, ENDF/B 

uncertainty information is not available for many nuclides or for some types of nuclear data 

relevant to nuclear engineering applications. To address this need, supplemental data have 

been developed within SCALE to provide complete uncertainty data sets for (1) fission yield 

data, (2) multigroup cross section data, and (3) decay data.

The effect of nuclear data uncertainty is demonstrated for a typical LWR depletion analysis 

problem involving a Combustion Engineering 14 × 14 assembly irradiated in Calvert Cliffs Unit 

1. A single fuel rod from assembly D047, designated as MKP109, has been subjected to 

destructive radiochemical assay to measure the isotopic contents. 

The 95% range width (difference between the 97.5th and 2.5th percentiles) is used in this study 

to assess the calculation uncertainty. This approach uses the actual distribution of the data and 

does not make any assumptions about the normality of the distributions. If the distribution were 

normal, then the 95% range width would correspond to 4-sigma range, or +/- 2 sigma. 

The calculation uncertainty determined in nuclide concentrations for the MKP109 rod ranges 

from a few percent to 50%. The power factor for this fuel rod shows a very low uncertainty of 

less than 0.5%. There is also a low uncertainty of less than 2% in activity. 

Uncertainties in the macroscopic cross sections, reactivity, and power distributions are generally 

low, in the few percent range. The effective delayed neutron fraction, βeff, shows an 

unexpectedly high uncertainty of 20–100%.

The considered application of the MKP109 fuel rod enabled a bias estimation and comparisons 

of calculated uncertainty in the isotopic concentrations to measurements of samples S1 and S2. 

In this case, the relative bias range was calculated to include both the calculation and 

measurement uncertainty. A bias range (2.5th–97.5th percentile) that includes zero is within 

expectation, whereas a bias which does not include zero is unexpected. Identifying unexpected 

biases can lead to improvements in the model or in nuclear data.

Section 1 introduces the relevant model equations used to explicitly define the data parameters, 

discusses the background of uncertainty quantification (UQ), and describes the sampling-based 

propagation technique as implemented in SCALE. Section 2 describes the uncertainty data in 

SCALE. Section 3 presents an application uncertainty problem in light water reactor (LWR) 

analysis. Section 4 provides general recommendations for interpreting results with uncertainty.

xv 

ABBREVIATIONS AND ACRONYMS 

AMPX Automation of MUG, POPOP4, and XLACS 

CC Calvert Cliffs 

CE Combustion Engineering (company) 

CE continuous energy 

COVERX cross section covariance data format 

ENDF evaluated nuclear data file 

GRS Gesellschaft für Anlagen- und Reaktorsicherheit laboratory 

JEFF joint evaluated fission and fusion file 

JENDL Japanese evaluated nuclear data library 

LWR light water reactor 

MEDUSA software developed by GRS for random sampling of covariances in XSUSA

ORIGEN Oak Ridge Isotope Generation code 

ORNL Oak Ridge National Laboratory 

PIE post-irradiation examination 

PWR pressurized water reactor 

sEL elastic scattering 

sIN inelastic scattering 

TENDL TALYS-based evaluated nuclear data library 

TRITON Transport Rigor Implemented with Time-Dependent Operation for Neutronic 

Depletion 

TSUNAMI Tools for Sensitivity and Uncertainty Analysis Methodology Implementation 

UQ uncertainty quantification 

XSUSA cross section software developed by GRS for uncertainty and sensitivity 

analyses 

1-1

1 INTRODUCTION 

This report summarizes the nuclear data uncertainty content available in version 6.2.2 of 

SCALE [2], presents the sampling-based uncertainty propagation methodology implemented in 

SCALE/Sampler which uses these data, and discusses some applications of the data. 

The nuclear data used in a typical SCALE calculation are processed from evaluated nuclear 

data files (ENDF/B) produced by the National Nuclear Data Center [1] through the AMPX code 

system distributed with SCALE. The ENDF/B data are based on a large set of nuclear data 

measurements that have been evaluated by the world’s top nuclear data experts and compiled 

into a cohesive, consistent database. Ideally, ENDF/B data should include an associated 

uncertainty (and correlations to other data) indicating the accuracy of the measurement, as well 

as any uncertainty introduced in the evaluation due to modeling fits, consistency adjustments, or 

other causes. This uncertainty in the nuclear data can be propagated to uncertainty in 

calculated quantities of interest for the nuclear analyst—whether it is keff for criticality safety 

applications or a void coefficient of reactivity for reactor physics applications. However, ENDF/B 

uncertainty information is not available for many nuclides or for some types of nuclear data 

relevant to nuclear engineering applications. To address this need, supplemental data have 

been developed within SCALE to provide complete uncertainty data sets for (1) fission yield 

data, (2) multigroup cross section data, and (3) decay data. 

The remainder of this document is organized as follows. Section 1.1 introduces the relevant 

model equations used to explicitly define the data parameters, and Section 1.2 discusses some 

background of uncertainty quantification (UQ). The sampling-based propagation technique as 

implemented in SCALE/Sampler is discussed in Section 1.3. Section 2 describes the uncertainty 

data in SCALE, Section 3 presents an application uncertainty problem in light water reactor 

(LWR) analysis, and Section 4 provides general recommendations for interpreting results with 

uncertainty. 

1.1 Model Equations 

Although the average SCALE user does not need to know details of the equations solved in 

SCALE calculations, the easiest way to discuss data uncertainty is to first show the relevant 

models in which the data parameters occur. This will provide a common reference point and 

basic understanding of which data parameters can affect which quantities of interest. The 

fundamental unknowns (those from which all other quantities of interest can be calculated) in 

SCALE are: 

 neutron angular flux, ψ(r⃗, E, Ω⃗⃗, t), and

 nuclide distribution, n(r⃗, t),

with the following independent variables: 

 r⃗ for the location in space,

 E for the neutron energy,
1-2

 Ω⃗⃗ for the neutron direction, and

 t for time.

Throughout this section, one or more of the independent variables may be suppressed for 

brevity. 

1.1.1 Transport Equation 

The neutron transport equation is framed as an eigenvalue problem with quasistatic 

representation of the time-dependent neutron angular flux:

Ω⃗⃗ ⋅ ∇⃗⃗ψl + Σt

(r⃗,E,tl) ψ(r⃗,E,Ω⃗⃗,tl) =

∫ dΩ

′

4π

∫ dE

′ Σs(r⃗,E′ → E,Ω⃗⃗′ → Ω⃗⃗,tl) ψ(r⃗,E′

,Ω⃗⃗′,tl)

∞

0

+

1

keff

∫ dE

′ νΣf

(r⃗, E

′ → E,tl

) φ(r⃗, E′,tl)

∞

0

,

(1)

where 

 l is a time index from 0 to L with ψl = ψ(tl);

 φl = φ(tl

) is the scalar flux at time tl

, defined as the angle-integrated angular flux, φl =

∫ dΩ′

 ψl(Ω⃗⃗′

)

4π ;

 Σt, Σs, and νΣf are macroscopic data corresponding to the total cross section, doubledifferential scattering cross section, multiple-neutron emission cross section, and fission

production cross section, respectively; and

 keff is the eigenvalue.

The macroscopic data are defined in terms of the fundamental microscopic nuclear data and the 

nuclide distributions. The total cross section is defined as 

Σt

(r⃗, E,tl) = ∑σj,t(E) nj(r⃗,tl)

j

,

(2)

where σj,t(E) is the total cross section for nuclide j at energy E, and nj(r⃗, tl) is the nuclide

number density for nuclide j at location r⃗ at time tl. The macroscopic double differential 

scattering cross section is similarly defined: 

Σs(r⃗, E

′ → E,Ω⃗⃗′ → Ω⃗⃗,tl) =

∑[σj,s(E

′ → E,Ω⃗⃗′ → Ω⃗⃗) + ∑ νσj,νn(E′ → E,Ω⃗⃗′ → Ω⃗⃗)

ν=2,3

] nj(r⃗,tl)

j

,

(3)
1-3

where σj,s(E

′ → E, Ω⃗⃗′ → Ω⃗⃗) is the microscopic double differential scattering cross section of

nuclide j, which describes scattering from energy E′ to energy E and direction Ω⃗⃗′ to direction Ω⃗⃗;

and σj,νn(E

′ → E, Ω⃗⃗′ → Ω⃗⃗) are microscopic double differential cross sections describing reactions

that emit an integer number of neutrons, ν. Typical nuclear engineering problems consider only 

ν = 2 and ν = 3. The microscopic scattering cross section is further divided into elastic and 

inelastic components, 

σj,s = σj,sEL + σj,sIN, (4)

where σj,sEL is the elastic component and σj,sIN is the inelastic component consisting of 

scattering from discrete inelastic levels and from the continuum range. The macroscopic fission 

production cross section is defined as 

υΣf

(r⃗, E

′ → E,tl

) = ∑χ̅j,f(E′ → E) ν̅j,f(E

′

) σj,f(E

′

) nj(r⃗,tl)

j

,

(5)

where 

 χ̅j,f(E′ → E) is the energy distribution of secondary neutrons emitted from fission at

energy E′,

 ν̅j,f(E

′

) is the average number of neutrons emitted per fission, and

 σj,f(E

′

) is the fission cross section for nuclide j.

1.1.2 Transmutation Equation 

The nuclide distribution nj

(r⃗, tl

) is the solution of the transmutation equations for each nuclide j,

dnj

dt = ∑[(∑fi,x→j

(r⃗,t)σi,x(r⃗,t)

x

)φ(r⃗,t) +∑fi,d→j

d

λi,d] ni

(r⃗,t)

j

− [σj,tt

(r⃗,t) φ(r⃗,t) + λj

]nj

(r⃗,t)

for tl−1 ≤ t ≤ tl

(6)

where 

 fi,x→j(r⃗, t) is the fractional yield of nuclide j from reaction type x with nuclide i;

 σi,x(r⃗, t) is the energy-averaged cross section, σi,x =

∫ dE′σi,x(E

′

) φ(E

′

∞ )

∫ dE′

 φ(E′) ∞

, for reaction type x

of nuclide i; 

 φ(r⃗, t) is the energy-integrated scalar flux, φ = ∫ dE′φ(E′)

∞

0

; 

 fi,d→j is the branching ratio for creation of nuclide j from decay mode d of nuclide i;
1-4

 λi,d is the decay constant for decay mode d of decay of nuclide i;

 σj,tt(r⃗, t) is the energy-averaged transmutation cross section (i.e., absorption plus

discrete inelastic scatter to isomeric states) for nuclide j, σj,tt = ∑ σj,xx ; and

 λj is the total decay constant, λj = ∑ λj,dd .

The fractional yield term fi,x→j(r⃗, t) represents yields due to isomeric branching or fission. The

isomeric branching yield is defined as 

fi,x→j[m] =

∫ dE

′ fi,x→j[m]

(E

′

) σi,x(E

′

) φ(E

′

)

∞

0

∫ dE′ σi,x(E′)

∞

0

φ(E′)

, (7)

where fi,x→j[m](E

′

) is the energy-dependent isomeric branching ratio to nuclide j with isomeric

state m. For example, the (n, γ) reaction in U

234 can produce both U235 and U235m , with the 

energy-average yield of U

235m calculated as f U

234 ,γ→ U 235m =

∫ dE′f234U,γ→ U 235m (E

′

) σ234U,γ(E′

) ∞

∫ dE′

 σ234U,γ(E′) ∞

.

The fractional yield of fission product j from fissioning nuclide i in a particular system is usually 

obtained from the energy-dependent yield distribution, evaluated at the average fission energy 

for the system, 

fi,f→j = interp(fi,f→j

(E), E̅

i,f), (8)

where E̅

i,f is the average energy of fission calculated as 

E̅

i,f =

∫ dE

′ E′ σi,f(E′

)φ(E

′

) ∞

∫ dE′ σi,f(E′)φ(E′) ∞

. (9)

The transmutation cross section differs from the conventional total cross section by the omission 

of elastic scatter and discrete inelastic scatter to non-isomeric states, which has no effect on 

production or loss of nuclides appearing in Eq. (6). 

1.1.3 Multigroup Representation 

One standard solution technique for the neutron transport equation is to assume a multigroup 

representation in energy, which leads to multigroup angular fluxes defined as 

ψ

g = ∫ dE ψ(E)

E

g−1

Eg , (10)

and multigroup cross sections defined as 

σ

g =

∫ dE σ(E)w(E)

E

g−1

Eg

∫ dE w(E)

Eg−1

Eg

, (11)
1-5

where w(E) is a weighting function, and energies E

0

, E

1

,E

g

, and E

G are energy group 

boundaries in descending order. The energy group structure and number of groups are selected 

for a particular multigroup application. SCALE 6.2 includes two multigroup structures optimized 

for LWR applications: 56- and 252-groups. 

1.2 Additional Models

1.2.1 Few-Group Macroscopic Cross Sections 

SCALE is commonly used to generate data for nodal simulators such as the Purdue Advanced 

Reactor Core Simulator (PARCS). In this case, few-group macroscopic cross sections are one 

of the main outputs, with standard flux-volume weighting,

Σx

h(t) =

∑ ∑ φm

g

m (t)Vm ∑ σjm,x

g

(t)njm(t) g∈h j

∑ ∑ φm

g

g∈h m (t)Vm

, (12)

where h is the few-group energy index, m is a material index, Vm is the material volume, and φm

g

is the material-average multigroup scalar flux.

1.2.2 Activity and Decay Heat 

Activity and decay heat are key responses in spent fuel and source terms analysis. Activity is 

expressed as

A(t) = ∑λj nj

(t)

j

,

(13)

whereas decay heat is given as

H(t) = ∑Qj λj nj

(t)

j

,

(14)

where Qjis the nuclide-dependent energy release per decay. Decay heat responses are not 

addressed in this report.

1.2.3 Reactivity Coefficients 

Reactivity coefficients are simple coefficients that relate state changes to changes in reactivity, 

defined here as

αB =

∂keff

∂B , (15)

where B is a given state parameter such as fuel temperature. Reactivity coefficients are typically 

calculated from discrete state changes called branches. For example, a fuel temperature branch 

could be used to calculate the reactivity coefficient for fuel temperature as
1-6

αTfuel ≈

keff(Tfuel + ΔTfuel) − keff(Tfuel)

ΔTfuel

, (16)

where ΔTfuel is the branch temperature change. 

1.2.4 Kinetics Parameters 

SCALE does not have a time-dependent kinetics capability, but it supports that capability in 

nodal simulators through lattice physics outputs of domain-average delayed neutron fractions, 

βeff

p

, and decay constants, λ

p

, where p = 1, 2, ... ,6 in the traditional six-group precursor 

formalism. The six-group effective delayed neutron fractions are defined as 

βeff

p =

∑ nj

[∫ dE ∞

0

ν̅j,f(E) σj,f(E) φ(E)] [∫ dE

′ βj

p

(E

′

) χj

p

(E

′

) φ∗

(E

′

)

∞

0

] j

∑ nj

[∫ dE ∞

0

ν̅j,f(E) σj,f(E) φ(E)] [∫ dE′ χ̅j

(E′) φ∗(E′)

∞

0

] j

,

(17)

and six-group precursor decay constants are defined as 

λ

p =

∑ λi

p

i ni

∑j ni

, (18)

where 

 βj

p

(E) is the energy-dependent fraction of delayed neutrons into precursor group p,

from fission of nuclide i, emitted at energy E,

 λi

p

 is the decay constant corresponding to neutron emission in fission product nuclide

i,

 χj

p

(E

′

) is the energy spectrum of delayed neutrons produced by precursor group p,

and

 φ

∗

 is the adjoint flux.

λi

p

, χj

p

(E

′

), and βi

p

(E) are the fundamental nuclear data. The total effective delayed neutron

fraction is the sum over precursor groups 

βeff = ∑βeff

p

p

.

(19)

1.2.5 Power Peaking Factors 

One of the principal measures of the power distribution in simulations of nuclear 

reactors is the fission distribution. For use as a power distribution metric, the fission 

distribution is usually normalized as a region-to-average factor
1-7

pm =

∫ dE Σm,f

(E) φm(E)

∞

0

1

V

∑ Vm ∫ dE Σm,f(E) φm(E)

∞

m 0

,

where m is a material region index, Vm is a material region volume, and the total volume 

V = ∑ Vmm . These factors are used in safety analysis to obtain the peak local power,

Ppeak = P̅ max

m

pm, where P̅ is the core-average specific power. 

1.3 Uncertainty Quantification Methods 

SCALE currently includes two UQ methods: a perturbation theory–based method called Tools 

for Sensitivity and Uncertainty Analysis Methodology Implementation (TSUNAMI), and the 

sampling-based method known as Sampler. 

1.3.1 Perturbation Theory 

TSUNAMI implements a perturbation theory–based sensitivity analysis methodology that can 

produce first-order uncertainty estimates according to 

Cy = Sy/x

T Cx Sy/x, (21)

where Cx is a covariance matrix for inputs x, and Sy/x is a sensitivity matrix, computed from 

first-order perturbation theory, containing sensitivity coefficients for each response y in columns 

with respect to each input x in rows. 

Perturbation theory–based methods require solutions of adjoint equations specific to the forward 

equations and the desired response. In TSUNAMI, an adjoint scheme exists only for the 

eigenvalue transport equation. 

TSUNAMI’s original application in version 5 of SCALE was to determine appropriate criticality 

benchmarks for a particular user system, with the fundamental assumption being that cross 

section uncertainty is the primary uncertainty in a criticality benchmark. Therefore, the sensitivity 

matrix was originally for a keff response with respect to input cross sections. With SCALE 6.0, 

additional outputs were added for reaction rates, thus extending the applicability into reactor 

physics applications, but with the same limitation on the inputs that could be considered 

uncertain (only cross sections) and the types of problems which could be analyzed (only 

transport problems without depletion). With SCALE 6.2, TSUNAMI is available with continuous 

energy KENO, whereas before it was available only for multigroup transport methods. 

1.3.2 Sampling 

Sampler implements sampling-based uncertainty quantification, assuming that every input may 

have a probability distribution, pdf(x), which is propagated to determine the output probability 

distribution, pdf(y), 

pdf(y) = Θ[ pdf(x)], (22)

(20)
1-8

where Θ is an operator the effect of which can be approximated by sampling N realizations of x

from pdf(x), performing N realization calculations, and reconstructing pdf(y) from N

realizations of y. The perturbation theory approach calculates sensitivities for a single response 

per run and then approximates the uncertainties. The sampling-based approach calculates 

uncertainties directly and can only provide very limited sensitivity information. The 

SCALE/Sampler workflow is shown in Figure 1-1. 

Figure 1-1 SCALE/Sampler Work Flow 

Data uncertainty can be particularly complex because of the large amount of data and the 

potentially correlated nature. For this reason, SCALE/Sampler provides precalculated data 

samples. Sampler was initially released in SCALE 6.2. 

nominal 

SCALE model 

developed

uncertain 

data 

declared

uncertain

input

parameters 

defined

responses

defined

N

realizations 

created by 

random 

sampling

N 

independent 

realization 

calculations

responses 

scraped 

from output 

files

response

summary 

statistics 

evaluated
2-1

2 DATA UNCERTAINTY

The following sections discuss the data uncertainty available in SCALE 6.2.

2.1 Cross Section Uncertainty

Uncertainty in cross sections σi,x

(E) for nuclide i, reaction type x, and energy E impacts all 

aspects of an analysis with SCALE, affecting both transport and depletion calculations.

2.1.1 Background

Cross section uncertainty has been a consideration in SCALE since the introduction of the 

TSUNAMI tools and their application to criticality safety in SCALE 5.0. Cross section data 

uncertainty is historically referred to as covariance data. The ENDF6 data format currently used 

by all evaluated data files (ENDF/B, joint evaluated fission and fusion file [JEFF], Japanese 

evaluated nuclear data library [JENDL], TALYS-based evaluated nuclear data library [TENDL]), 

imposes a normal distribution representation for the cross section probability distribution, and 

thus the uncertainty data are stored as a covariance matrix. The SCALE 6.2 covariance library 

is based on ENDF/B-VII.1 data for 187 nuclides, combined with previous SCALE 6.1

covariances from other sources for ~215 nuclides not available in ENDF/B-VII.1. Thus, SCALE6.2 has a complete set of uncertainties for important data of all nuclides in the multigroup cross 

sections [2].

2.1.2 Implementation

In SCALE 6.2, cross section covariance matrices are distributed in the COVERX multigroup 

format in both the 56- and 252-group structures that are processed through AMPX. The 

perturbation theory–based approach (TSUNAMI) operates directly on the covariance matrices, 

folding them with calculated sensitivity coefficients to arrive at uncertainty. In the samplingbased approach (Sampler), the multigroup covariance matrices are sampled using the 

XSUSA/MEDUSA [3] random sampling code developed by GRS (Gesellschaft für Anlagen- und 

Reaktorsicherheit) to produce 1,000 perturbed multigroup cross sections for each datum in the 

library,1 σ′

i,x

g

. A perturbed cross section is converted to a perturbation factor,

pi,x

g =

σ′

i,x

g

σi,x

g

. (23)

The perturbation factors are applied before the transport calculation to each self-shielded region 

in a multigroup problem [3]. 

2.1.3 Data Characteristics

Table 2-1 shows cross section uncertainty from the SCALE 6.2 library for some key nuclides in 

typical LWR analyses, with data condensed to approximate one-group cross sections, 

 

1 The number of samples (1,000) to ship with SCALE was chosen to exceed the needs of most analyses, for which

 100–300 samples have been deemed sufficient.
2-2

σi,x = ∑ σi,x

g w

g

g , (24)

where w

g

 is an LWR weighting spectra. Three different uncertainty measures are shown in the 

table. 

The first, denoted by the phrase with correlation, is the most accurate and uses the uncertainty 

propagation rule on the one-group cross section to approximate the variance in the one-group 

cross section as 

var(σi,x) ≈ ∑ ∑ w

g

′

cov

gg′

(σi,x) w

g

g′ g , (25)

where covgg′

(σi,x) is the covariance matrix for the group-wise data.

This is in contrast with the without correlation approach, which is simply 

var(σi,x) ≈ ∑ ∑ w

g

′

var(σi,x

g

) w

g

g′ g , (26)

where var(σi,x

g

) is the variance term on the diagonal of the covariance matrix. Comparing the 

first two approaches from Eqs. (25) and (26) allows one to estimate the importance of energy 

correlation, which is significant in most instances, with a higher uncertainty when correlation is 

included by a factor of two or three. This can be understood in the limit of a uniform cross 

section and weighting function. Without correlation, the result is an averaging procedure in 

which the uncertainty in the average will be a factor of 1

√N

 smaller than the uncertainty in the 

components, where N is the number of parameters or energy groups in this example. 

Introducing correlation basically limits the number of effective components to Ñ < N, with 

perfect correlation resulting in Ñ = 1 component, and the average having the same uncertainty 

as the components. 

The third weighted average is an erroneous approach that applies flux weighting to the standard 

deviations, 

std(σi,x) ≈ ∑ std(σi,x

g

) w

g

g , (27)

where std(σi,x

g

) = √var(σi,x

g

). This form is not valid since the total standard deviation is obtained 

from the square root of the summed of squares of the uncorrelated standard deviations; 

however, it is possibly what would be estimated by examining a plot of the standard deviation 

versus energy, as shown in the fission cross section uncertainty plot in Figure 2-1. In this plot, 

the energy-dependent relative uncertainty is shown, along with the three one-group estimates. 

Note that the incorrect weighted average is the only one that appears to represent an average 

of the data shown—it effectively multiplies the relative uncertainty times flux and cross section 

shapes and determines the average of that function. Due to the thermal spectrum and thermal 

cross section, the left-most region is by far the most important (note the shape functions are in a 

log scale), so the weighted average appears slightly above the left-most portion of the energy-
2-3

dependent curve. Comparing the with correlation (best) to without correlation shows that 

neglecting correlations is especially severe for this reaction, with an uncertainty that is lower by 

a factor of three. 

Table 2-1 Key Cross Section Uncertainty Data in SCALE 

Nuclide, 

i

Neutron reaction 

type, x

Cross 

section, 

σi,x

(barns)

Uncertainty 

with 

correlation 

(%)

Uncertainty 

without 

correlation 

(%)

Uncertainty 

weighted 

average 

(%)

235U fission (f) 3.73E+01 0.32 0.10 0.40

gamma (γ) 8.68E+00 1.26 0.48 2.15

ν̅ 2.54E+00 0.11 0.05 0.26

238U fission (f) 1.27E-01 0.52 0.29 0.64

gamma (γ) 2.12E+00 1.17 0.72 2.50

scattering (sIN) 1.24E+00 14.08 6.49 16.10

239Pu fission (f) 8.20E+01 0.77 0.31 1.10

gamma (γ) 4.42E+01 1.13 0.49 1.53

ν̅ 3.01E+00 0.07 0.05 0.22

240Pu gamma (γ) 2.16E+02 0.25 0.08 0.27

135I gamma (γ) 5.16E+00 3.65 3.65 13.24

135Xe gamma (γ) 1.63E+05 4.15 1.48 4.15

149Pm gamma (γ) 1.13E+02 16.14 5.53 21.43

149Sm gamma (γ) 4.44E+03 1.47 0.59 1.58

10B alpha (α) 2.50E+02 0.08 0.02 0.08

155Gd gamma (γ) 2.14E+03 3.27 1.76 4.95

90Zr gamma (γ) 9.53E-03 11.72 5.25 16.97

1H scattering (sEL) 1.21E+01 0.20 0.04 0.20

gamma (γ) 2.16E-02 1.07 0.29 1.08

2H scattering (sEL) 3.03E+00 1.67 0.40 1.97

gamma (γ) 3.63E-05 5.66 2.41 8.10

16O scattering (sEL) 3.41E+00 1.91 0.42 1.97

gamma (γ) 7.62E-05 32.22 10.70 34.84

232Th gamma (γ) 2.80E+00 1.64 1.59 4.07
2-4

Figure 2-1 Uncertainty in Energy-Dependent 235U Fission Cross Section, σ U

235 ,f

g

, With 

Various Energy-Average Approximations 

2.2 Fission Product Yield Uncertainty 

Uncertainty in fission product yields fi,f→j(E)—which is the number of atoms of fission product j

produced from fission of nuclide i by a neutron energy with E—impacts only simulations of 

depletion. 

2.2.1 Background 

SCALE 6.2 is the first version which includes uncertainties in yield data. SCALE 6.2 uses 

ENDF/B-VII.1 yields with approximately 30 fissionable nuclides i, 1,100 fission products j, with 

dependence on incident neutron energy causing fission, as represented by tabulation of the 

data at different energies. For important fissile nuclides, three energies are typically available: 

fast (14 MeV), intermediate (500 keV), and thermal (0.025 eV). For some fissionable nuclides, 

only a single yield energy may be available. Assuming a normal distribution, ENDF/B-VII.1 does 

contain standard deviations for the yield data, but it does not contain the inherent correlations of 

these data. 

2.2.2 Implementation 

Fission yield covariance matrices are constructed using a Bayesian update procedure described 

by Pigni et al. [5] based on knowledge of independent and cumulative yields and decay 

methods along a mass chain. In SCALE 6.2, these covariance matrices are available for 235U, 

238U, 239Pu, and 241Pu at energies relevant for LWR systems. One thousand random samples 

are drawn from these distributions and converted to yield perturbation factors, just as in the 

case of cross section perturbation factors. When Sampler is directed to consider uncertainty in 

0.0001

0.001

0.01

0.1

1

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03 1.0E+05 1.0E+07 Arbitrary shape units (-) Standard deviation (%)

Energy (eV)

energy-dependent with correlation without correlation

weighted average flux shape cross section shape
2-5

fission yield data, then the yield perturbation factors are applied to the nominal data and then 

renormalized. A particular nuclide with at least one energy set of yield perturbation factors will 

have those perturbation factors applied to all energy data of that nuclide. Yield data are not 

used by the TSUNAMI tools, which only support calculations with static nuclide compositions.

2.2.3 Characteristics

Table 2-2 shows the uncertainty in fission product yields from fissionable nuclides 235U, 238U, 

and 239Pu, for several key fission products in LWR systems: 135Te, 135I, 135Xe, 149Nd, 149Pm, and 

149Sm. The first three fission products are members of the A=135 mass chain, the main pathway 

for generating the most important neutron absorber for LWR reactor dynamics, 135Xe. The next 

three fission products are members of the A=149 mass chain, which includes stable 149Sm, an 

important neutron absorber in LWR systems. The buildup of 149Sm after a shutdown requires 

overcoming that negative reactivity penalty to return to power.

It is important to note that a fission product may have a high uncertainty (e.g., 149Sm yield has 

~50% uncertainty) in its yield, but the production of that isotope may be driven by different 

means, such as decay of precursors. Therefore, the effect of the cumulative uncertainty of 

actual isotopics predictions is significantly smaller than the yield uncertainty would indicate. 

Table 2-2 Uncertainty in Key Fission Yield Data

Fissionable 

nuclide

i

Fission 

product

j

fi,f→j

Thermal incident 

neutron energy 

Intermediate 

incident neutron 

energy

Fast incident 

neutron energy

Mean

(-)

Standard 

deviation

(%)

Mean

(-)

Standard 

deviation

(%)

Mean

(-)

Standard 

deviation

(%)

235U 135Te 3.23E-02 2.11 2.48E-02 9.93 1.05E-02 18.05

135I 2.92E-02 2.28 3.59E-02 6.58 3.13E-02 6.14

135Xe 7.97E-04 1.11 1.20E-03 10.86 4.54E-03 6.17

149Nd 6.87E-05 48.53 3.50E-05 48.39 6.98E-04 48.36

149Pm 3.93E-08 47.81 1.63E-08 49.60 1.31E-05 50.00

149Sm 1.75E-12 48.07 5.68E-13 48.57 2.67E-08 47.72

238U 135Te - - 4.61E-02 9.01 2.65E-02 9.18

135I - - 1.36E-02 28.92 2.59E-02 28.78

135Xe - - 1.23E-04 2.45 1.33E-03 2.54

149Nd - - 4.99E-06 47.71 6.30E-05 47.71

149Pm - - 1.19E-09 46.92 1.57E-07 46.92

149Sm - - 1.55E-14 50.35 2.27E-11 50.41

239Pu 135Te 2.19E-02 9.62 2.12E-02 28.11 6.87E-03 45.20

135I 4.28E-02 4.94 3.92E-02 9.97 3.25E-02 24.16

135Xe 3.14E-03 3.43 6.11E-03 6.14 8.54E-03 47.60

149Nd 4.99E-04 48.45 5.89E-04 49.10 2.37E-03 46.64

149Pm 2.38E-06 50.82 2.55E-06 48.70 1.39E-04 47.55

149Sm 8.22E-10 48.13 9.57E-10 48.52 9.81E-07 49.94
2-6

2.3 Decay Uncertainty 

Uncertainty in decay constants λi,d and branch ratios fi,d→j impact isotopic evolution in 

depletion or decay calculations. However, uncertainty in decay constants and decay heat Qi

impact any activity or decay heat response. This report only addresses activity responses. 

An analysis of decay heat uncertainty in this study identified errors in the uncertainties assigned 

to the Qi values in the uncertainty libraries used by Sampler, resulting in unrealistically large 

estimates of decay heat uncertainty. This error impacts versions up to and including SCALE 

6.2.2. Corrected libraries are being developed for distribution in parallel with the release of 

SCALE 6.2.3 through a separate distribution. This problem only impacts decay heat 

calculations. 

2.3.1 Background 

The SCALE decay data have only included uncertainties since the release of Sampler with 

SCALE 6.2. The decay data use ENDF/B VII.1 uncertainties and approximate correlations, 

which are described in the 2013 paper by Williams et al. [3]. 

2.3.2 Implementation 

Decay constant λi,d values are sampled according to independent normal probability 

distributions, with standard deviations given in ENDF/B-VII.1. The branch ratios fi,d→j are 

sampled from ENDF/B-VII.1 data, assuming that the largest two branching fractions are anticorrelated. Any smaller branch ratios are assumed to be independent. 

One thousand random samples of decay data are pre-generated and written to Oak Ridge 

Isotope Generation (ORIGEN)-formatted decay data files. When Sampler is directed to consider 

uncertainty in decay data, it substitutes one of these perturbed decay files in place of the 

nominal. Decay data are not used by the TSUNAMI tools, which only support calculations with 

fixed isotopics. 

2.3.3 Characteristics 

Table 2-3 shows uncertainty in total decay constants, λd, for nuclides that contribute to LWR 

fuel activity for decay times up to 40 years. 
2-7

Table 2-3 Uncertainty in Key Decay Data 

Radioactive 

nuclide i

λi

Mean

(1/s)

Standard 

deviation

(%)

238Pu 2.51E-10 0.11

106Rh 2.31E-02 1.14

140La 4.78E-06 0.01

144Pr 6.69E-04 0.29

90Sr 7.63E-10 0.21

137Cs 7.30E-10 0.30

95Zr 1.25E-07 0.01

137mBa 4.53E-03 0.04

134Cs 1.06E-08 0.02

90Y 3.01E-06 0.34

95Nb 2.29E-07 0.02

89Sr 1.59E-07 0.14

154Eu 2.55E-09 0.12

241Pu 1.54E-09 0.04

106Ru 2.16E-08 0.48

144Ce 2.82E-08 0.02

147Pm 8.37E-09 0.01

118mAg 3.50E-01 10.24

103mRh 2.06E-04 0.04

103Ru 2.04E-07 0.01

242Cm 4.92E-08 0.04

244Cm 1.21E-09 0.16

139I 3.04E-01 0.46

3H 1.78E-09 0.16

99Tc 1.04E-13 0.57

91Y 1.37E-07 0.10

243Cm 7.55E-10 0.35

240Pu 3.35E-12 0.10

243Am 2.98E-12 0.20

14C 3.85E-12 0.53

135Te 3.65E-02 1.07

135I 2.93E-05 0.31

135Xe 2.11E-05 0.22

149Nd 1.11E-04 0.06

149Pm 3.63E-06 0.09

3-1

3 LWR APPLICATIONS WITH UNCERTAINTY 

This section demonstrates the impact of nuclear data uncertainty on quantities of interest in 

typical LWR applications with SCALE, including 

 few-group macroscopic cross sections for nodal core simulators,

 reactivity coefficients,

 power distributions,

 spent fuel isotopic concentrations, and

 source terms.

This section also compares calculated isotopic concentrations to measurements, including 

uncertainty in both calculation and measurement, as seen in Williams et al. [3], but with 

expanded results. A significant portion of the following discussion is focused on the 

interpretation of these comparisons. 

3.1 Description 

The analysis case for this purpose is the Calvert Cliffs Unit 1 (CC-1) reactor assembly D047, a 

Combustion Engineering (CE) 14 ×14 design with known operating history and high-precision 

radiochemical assay data available for multiple axial locations of rod MKP109 in this assembly 

[6]. The assembly configuration and location of measured rod MKP109 are shown in Figure 1.]. 

Two samples analyzed by Oak Ridge National Laboratory (ORNL) in 2011 [7], identified as 

samples S1 and S2, are of particular interest. These samples were extracted from adjacent 

axial locations, as shown in Figure 2. High-precision measurements were made for 50 different 

isotopes. The similarity of the two samples provides a measurement cross check that allows an 

opportunity to investigate the reported measurement uncertainty in terms of the reproducibility of 

the measurement process. 

3.1.1 Operating History 

A simplified operating history was assumed to achieve the measured sample-average burnup of 

43.5 GWd/tU of the 3.038 wt% 235U fuel sample. The operating history assumptions are as 

follows: 

 constant material properties for soluble boron, temperatures, densities, etc.;

 4 cycles with 15 days of intermittent decay and sample-average powers of

o 34.21 MW/tU for 306 days,

o 29.69 MW/tU for 382 days,

o 24.61 MW/tU for 466 days, and

o 22.28 MW/tU for 461 days; and
3-2

 a final decay to 28.69 years when the measurements of samples S1 and S2 were

performed.

See Hu et al. [7] for additional details. 

Figure 3-1 Configuration of Assembly D047 From CC-1 [7] 

Fig. 3. Fuel rod and pellet dimensions of assembly D047 (Ref. [3], Fig. 3.2). 

Table	1.		Design	and	operating	data	for	assembly	D047	[4].	

Assembly and reactor data Value

Operating pressure, Pa 155 × 105

Core coolant inlet temperature, K 557

Core coolant outlet temperature, K 585

Lattice geometry 14 × 14

Number of fuel rods 176

Number of guide tubes 5

Number of burnable poison rods 0

Assembly pitchcm 2078

A

B

C

D

E

F

G

H

I

J

K

L

M

N

1 2 3 4 5 6 7 8 9 10 11 12 13 14

 Fuel rod Fuel rod MKP 109

 Guide tube 
3-3

Figure 3-2 137Cs Gamma Scan Showing Sample S1 and S2 Locations [7] 

3.1.2 Measurements 

The measured isotopic concentrations for samples S1 and S2 are shown in Table 3-1, along 

with four different measures of the uncertainty which are labeled Declared measurement 

uncertainty, Mean measurement uncertainty, Mean measurement standard error, and Effective 

measurement uncertainty. 

Declared measurement uncertainty 

The declared measurement uncertainty is the measurement standard deviation u(S) as reported 

for each isotope and sample [7], which is dependent on the measurement technique. 

Mean measurement uncertainty 

The Mean measurement uncertainty is calculated by applying uncertainty propagation to the 

sample mean of the two samples, 

S̅=

1

2

(S1 + S2),

(28)

	

Fig. 6. Axial 

137Cs gamma scans on segment G (of rod MKP109), from which the two ORNL samples (S1 and S2) were cut; the dashed

cutting lines also shown [7]. 
3-4

resulting in standard deviation of the mean, 

u(S̅) =

1

2

√v(S1) + v(S2) =

1

√2

u(S), (29)

where the variance is v(S) = [u(S)]

2

. 

Mean measurement standard error 

The Mean measurement standard error is the standard error of the mean for two samples, 

se(S̅),

se(S̅) =

1

√2

std(S). (30)

where std(S) is calculated from the sample standard deviation for two samples as

std(S) = √(S1 − S̅)

2 + (S2 − S̅)2. (31)

The true uncertainty is given by the standard error se(S̅), which is only applicable when there

are multiple equivalent measurements. These equivalent measurements allow for verification of 

the experimentalist’s uncertainty prediction. The only issue is when the standard error shows a 

lower uncertainty. In some cases, the two samples agree to the available three digits. In these 

cases, instead of using a very small standard error, the original uncertainty prediction is used. 

Effective measurement uncertainty 

The far right column shows this as the Effective measurement uncertainty, given as 

ueff(S̅) = max[u(S̅), se(S̅)]. (32)
3-5

Table 3-1 Measurement and Uncertainty Data for CC-1 MKP109 Samples S1 and S2

Nuclide S1

measurement

(g/gFuel)

S2

measurement

(g/gFuel)

Declared 

measurement

uncertainty

,

u(S) (%)

Mean

measurement

uncertainty

,

u(S̅) (%)

Mean

measurement

standard 

error, 

se(S̅) (%)

Effective 

measurement

uncertainty

, ueff(S̅)

(%)

241Am 8.47E-04 9.24E-04 5.0 3.5 4.3 4.3

243Am 1.69E-04 1.75E-04 5.0 3.5 1.7 3.5

140Ce 1.36E-03 1.35E-03 1.0 0.7 0.4 0.7

142Ce 1.23E-03 1.21E-03 1.0 0.7 0.8 0.8

244Cm 2.88E-05 2.99E-05 5.0 3.5 1.9 3.5

133Cs 1.20E-03 1.19E-03 1.0 0.7 0.4 0.7

135Cs 4.19E-04 4.17E-04 1.0 0.7 0.2 0.7

137Cs 6.70E-04 6.67E-04 1.0 0.7 0.2 0.7

151Eu 2.76E-06 2.26E-06 2.7 1.9 10.0 10.0

152Eu 2.64E-08 1.50E-07 15.0 10.6 70.1 70.1

153Eu 1.32E-04 1.32E-04 1.0 0.7 0.0 0.7

154Eu 2.58E-06 2.51E-06 2.7 1.9 1.4 1.9

155Eu 1.47E-07 1.63E-07 15.0 10.6 5.2 10.6

152Gd 4.92E-08 1.09E-07 15.0 10.6 37.8 37.8

154Gd 2.89E-05 2.88E-05 1.0 0.7 0.2 0.7

155Gd 1.02E-05 1.01E-05 1.0 0.7 0.5 0.7

156Gd 1.42E-04 1.43E-04 1.0 0.7 0.4 0.7

157Gd 3.55E-07 2.04E-07 15.0 10.6 27.0 27.0

158Gd 2.25E-05 2.21E-05 1.0 0.7 0.9 0.9

160Gd 1.78E-06 1.55E-06 2.7 1.9 6.9 6.9

139La 1.45E-03 1.52E-03 5.0 3.5 2.4 3.5

95Mo 8.35E-04 8.14E-04 5.0 3.5 1.3 3.5

142Nd 3.14E-05 3.24E-05 1.4 1.0 1.6 1.6

143Nd 7.81E-04 7.82E-04 1.0 0.7 0.1 0.7

144Nd 1.63E-03 1.62E-03 1.0 0.7 0.3 0.7

145Nd 7.30E-04 7.31E-04 1.0 0.7 0.1 0.7

146Nd 8.21E-04 8.21E-04 1.0 0.7 0.0 0.7

148Nd 4.28E-04 4.26E-04 1.0 0.7 0.2 0.7

150Nd 2.07E-04 2.06E-04 1.0 0.7 0.2 0.7

237Np 5.09E-04 5.40E-04 5.0 3.5 3.0 3.5

238Pu 1.94E-04 2.11E-04 1.0 0.7 4.2 4.2

239Pu 4.39E-03 4.57E-03 1.0 0.7 2.4 2.4

240Pu 2.49E-03 2.62E-03 1.0 0.7 2.5 2.5

241Pu 3.18E-04 3.38E-04 1.4 1.0 3.0 3.0

242Pu 7.82E-04 8.22E-04 1.4 1.0 2.5 2.5

103Rh 4.70E-04 4.85E-04 5.0 3.5 1.6 3.5

101Rh 8.25E-04 8.18E-04 5.0 3.5 0.4 3.5

147Sm 2.62E-04 2.60E-04 1.0 0.7 0.4 0.7

148Sm 1.91E-04 1.91E-04 1.0 0.7 0.0 0.7

149Sm 2.44E-06 2.34E-06 2.7 1.9 2.1 2.1

150Sm 3.16E-04 3.17E-04 1.0 0.7 0.2 0.7

151Sm 7.74E-06 7.72E-06 2.5 1.8 0.1 1.8
3-6

Nuclide S1

measurement

(g/gFuel)

S2

measurement

(g/gFuel)

Declared 

measurement

uncertainty

,

u(S) (%)

Mean

measurement

uncertainty

,

u(S̅) (%)

Mean

measurement

standard 

error, 

se(S̅) (%)

Effective 

measurement

uncertainty

, ueff(S̅)

(%)

152Sm 1.10E-04 1.09E-04 1.0 0.7 0.5 0.7

154Sm 4.43E-05 4.40E-05 1.0 0.7 0.3 0.7

90Sr 2.54E-04 2.30E-04 5.0 3.5 5.0 5.0

234U 1.64E-04 1.65E-04 5.0 3.5 0.3 3.5

235U 3.85E-03 3.79E-03 1.4 1.0 0.8 1.0

236U 3.64E-03 3.65E-03 1.4 1.0 0.1 1.0

238U 8.23E-01 8.23E-01 1.0 0.7 0.0 0.7

3.2 Calculation Results 

The results in this section demonstrate the uncertainty propagation capabilities of 

SCALE/Sampler, and they also provide a reference point for discussions on the implications or 

conclusions. The following results are provided: 

 two-group macroscopic cross sections, including

o absorption,

o nu-fission, and

o scattering;

 reactivity data, including

o eigenvalue,

o delayed neutron fraction, and

o fuel temperature coefficient of reactivity;

 power distribution data, including

o rod power fraction and

o flux magnitude uncertainty;

 isotopic concentrations for all measured nuclides; and

 source activity
3-7

In Figures 3–8, the quantity of interest is shown on the left y-axis in terms of mean—calc. 

mean—with bands for the standard deviation—calc. std. dev.—calculated according to standard 

sample statistics: 

std(C) = √

1

N − 1

∑(Cn − C̅)

2

n

, (33)

where C̅ is the calculation mean and Cn is the n-th sample. 

Through histograms, the 2.5th, 16th, 84th, 97.5th percentiles have also been estimated. These 

specific percentiles were chosen because if the distribution were normal, then the 84th and 16th

percentiles would coincide with the mean plus and minus one standard deviation (1-sigma), 

respectively, and the 97.5th and 2.5th percentiles would coincide plus and minus 2-sigma. 

Various error metrics are displayed in blue on the right y-axis. The calculation 95% range width 

(calc. 95% range width) and calculation 4-sigma (4 × calc. std. dev.) are provided for all results. 

The 95% range width is the 97.5th percentile minus the 2.5th percentile, and if the distribution 

were normal, then the 95% range width would be equivalent to four standard deviations 

(4-sigma). Thus, comparing these range widths indicates the normality of the distribution. 

Measurements are available for many nuclides considered in this study. On the left y-axis, the 

measurement is shown as a black circle (meas.) with 2-sigma error bars given by the 

measurement effective uncertainty defined in Eq. (32). The bias (bias) is also shown as a blue 

x, defined as 

bias =

C̅

S̅

− 1, (34)

with 2-sigma error bars shown, calculated as the (relative) standard deviation, 

rel. std(bias) = √ueff(S̅)

2 + std(C)2/C̅2. (35)

It is convenient in the following sections to refer to the uncertainty using a single number; we 

have chosen the 95% range width. The 95% probability range is a standard range of interest in 

engineering safety analysis, is valid for any distribution, and can be described by the 2.5th

percentile and 97.5th percentile values. To condense these two values to a single number that 

represents the uncertainty, the 95% range width is used here. In several instances, another 

measure has been used (e.g., 1-sigma measurement uncertainty). These instances are clearly 

marked. 

Across the different outputs of interest, uncertainty magnitudes were observed from fractions of 

a percent to 50% or more. Each set is discussed individually in the following subsections. 

3.2.1 Macroscopic Cross Section Uncertainty 

The uncertainties in scattering, absorption, and nu-fission macroscopic cross sections in the 

thermal energy range all show similar behavior with irradiation time (burnup), with values of 
3-8

approximately 2–4% over the course of the irradiation simulation. In the fast energy range, 

uncertainty for scattering and absorption ranges from 3–4%, and for fast nu-fission, it ranges 

from 2–7% with burnup. 

Figure 3-3 Fast Macroscopic Absorption Cross Section 
3-9

Figure 3-4 Thermal Macroscopic Absorption Cross Section
3-10

Figure 3-5 Fast Macroscopic Nu-Fission Cross Section 
3-11

Figure 3-6 Thermal Macroscopic Nu-Fission Cross Section
3-12

Figure 3-7 Fast Macroscopic Scattering Cross Section 
3-13

Figure 3-8 Thermal Macroscopic Scattering Cross Section

3.2.2 Reactivity Uncertainty

All reactivity parameters examined show increasing uncertainty with burnup, as illustrated in 

Figures 9 to 11. The eigenvalue, keff, shows 1.5–2% uncertainty. The total delayed neutron 

fraction, βeff, shows a remarkably high uncertainty, initially 20%, increasing with burnup to 

100%*. The temperature reactivity coefficient shows 5–7% uncertainty.

 

*

Initial investigations have shown that this is due to the delayed chi uncertainty.
3-14

Figure 3-9 Eigenvalue Uncertainty 
3-15

Figure 3-10 Delayed Neutron Fraction Uncertainty
3-16

Figure 3-11 Reactivity Coefficient Uncertainty

3.2.3 Power Distribution Uncertainty

As illustrated in Figure 12, the power factor for the MKP109 rod shows a low uncertainty, with a 

maximum value of only 0.25%. This is likely to the result of power normalization methods. For 

example, the uncertainty in an average power factor of 1.0 is by definition zero. The power of 

the MKP109 rod is slightly below the assembly average power, with a power factor ranging from 

0.93–0.97, which is very close to 1.0. As shown in Figure 13, the uncertainty in the flux is one of 

the few parameters where the uncertainty decreases with burnup, from an initial level of 3% 

down to 2%. The flux levels exhibit oscillations that occur when the flux distributions are 

recalculated by the transport calculation. This is performed in the middle of each depletion step 

corresponding to a reactor cycle in the Transport Rigor Implemented with Time-Dependent 

Operation for Neutronic Depletion (TRITON) depletion calculation. The flux decrease is caused 

by decreasing power in each subsequent cycle. During each cycle, power is assumed to be 

constant, and an expected gradual increase in flux is observed as the fissionable material is 

depleted.
3-17

Figure 3-12 Irradiation Sample Power Factor Uncertainty
3-18

Figure 3-13 Flux Magnitude Uncertainty 

3.2.4 Isotopic Uncertainties 

Figures 13–63 illustrate isotopic uncertainties. The isotopic calculations include the irradiation 

time and the decay time before measurement of the nuclide compositions. The calculated 

content and uncertainty in many isotopes change in behavior, moving from irradiation to decay. 

The calculation uncertainties range from a few percent to 50% or more. The availability of 

measurements gives an additional aspect to the analysis which will be used to organize the 

discussion into four groups representing the various combinations of large or small calculation, 

uncertainty, and expected or unexpected bias. Here, a 95% range width uncertainty of less than 

10% is considered small, and an unexpected bias is assumed when the bias 95% range does 

not include zero. For simplicity, the bias is assumed to be normally distributed. 

3.2.4.1 Small Calculation Uncertainty / Expected Bias 

These nuclides have less than 10% calculation uncertainty and an expected bias range that 

includes zero. It could be concluded that these nuclides are predicted accurately, within the 

expected uncertainties, and precisely, as the uncertainties are small. Nuclides in this category 

are 139La, 95Mo, 143,144,145,146Nd, 239,240,241Pu, 147,148,149,152Sm, and 234,238U, and they exhibit a wide 

range of notable behaviors. 

The uranium isotope 234U, present in the initial fuel, is observed to deplete during irradiation, and 

then it builds during decay due to production from 238Pu decay. 149Sm is an important neutron 
3-19

absorber; it builds to a maximum shortly after discharge from the decay of precursor 

149Pm (53 hours), depending on the power level reached during prior operation. It is well 

predicted, with a bias range of -0.6 to 10.6%, including a calculated uncertainty of ~6% and a 

measured uncertainty of ~8%. It can be reasonably concluded that, based on this bias, the 

amount of 149Sm may be predicted within 10% of the actual value. 

Plutonium isotopes 239Pu, Pu240,

 and 241Pu are predicted well within error bounds, with observed 

biases of only ~1%, while calculation uncertainties of ~5% are estimated from the data. 

Neodymium isotopes 143Nd, 

144Nd, 145 Nd, and 146Nd are predicted to within ~3%. 

3.2.4.2 Large Calculation Uncertainty / Expected Bias 

These nuclides have 10% or greater calculation uncertainty and an expected bias range that 

includes zero. In contrast to the previous group of nuclides with small calculation uncertainty, it 

can be concluded that these nuclides are predicted within the expected uncertainty range, but 

the range exceeds a threshold of 10%. These nuclides are 243Am, 244Cm, 135Cs, 151,153,155Eu, 

154,155,160Gd, 142Nd, 237Np, and 151Sm, and they are candidates for data improvements. One 

possible reason for the large uncertainty is that the evaluation process has overestimated the 

uncertainty. 

244Cm has a very large calculation uncertainty of ~40%, but it only has an observed bias of 2% 

and an effective measurement uncertainty of ~14%. This is another candidate for improvement, 

as the uncertainties in the data significantly overestimate the biases observed when comparing 

calculations with measurements. 

151Eu has linear buildup during decay, with an uncertainty of ~20% during irradiation, rising to 

~40% at time of measurement. The effective measurement uncertainty in this case is also 

approximately 40%, based on observed differences in the two samples, although the reported 

measurement uncertainty for each sample was only 10%. 

155Gd, an important neutron absorber, is often added to fuel as a neutron-absorbing dopant, but 

here it is added as a fission product. The calculation uncertainty is ~60%, yet a bias of ~1% with 

a measurement uncertainty of ~1% was achieved. There may be overestimated data 

uncertainty that could be targeted for correction. 

3.2.4.3 Small Calculation Uncertainty / Unexpected Bias 

These nuclides have less than 10% calculation uncertainty and an unexpected bias range that 

does not include zero. In this group, the calculation is indicated to be precise, which implies that 

it should be accurate, but according to the comparison to measurement, it is actually inaccurate. 

Nuclides in this category are 241Am, 140Ce, 142Ce, 133,137Cs, 152,156,158Gd, 148,150Nd, 103Rh, 101Ru, 

154Sm, 90Sr, and 235,236U. Each of these nuclides contains an error that has not been taken into 

account. These errors could be due to issues in the model, the data, or the actual 

measurements. 

Consider 137Cs, an important radiological gamma-emitting nuclide used for determining axial 

and radial burnup profiles through gamma scanning. This nuclide is measured with an effective 

uncertainty of only ~1.4%, has an expected calculation uncertainty of only ~1.4%, and yet has a 

bias in the range of 7–9%. 
3-20

235U has a calculation uncertainty of 7% in this case, with 3% measurement uncertainty, but with 

a bias of 14%. According to Hu et al. [7], the observed bias in the amount of 235U was found to 

be only ~1%. The most likely cause is the simplified model used here for the depletion 

calculations. For example, the soluble boron variations were not modeled explicitly, and a 

simplified operating history was used. Because 235U is initially the main fissile nuclide in the 

system, and because the power is fixed in this type of model, there is a correlation between the 

major fissile isotopes of 235U and 239Pu. Thus, any changes made to the model which would 

impact 239Pu production would also influence the amount of 235U depleted. 

3.2.4.4 Large Calculation Uncertainty / Unexpected Bias 

The nuclides in this list include those with 10% or greater calculation uncertainty and an 

unexpected bias range that does not include zero. In this group, the calculation indicates 

significant uncertainty, but the result is still less accurate than expected. Nuclides in this 

category are 107Ag, 152,154Eu, 157Gd, and 238,242Pu. Historically, 238Pu has been difficult to predict, 

but recent updates in ENDF/B-VII.1 cross sections available with SCALE 6.2 indicate reduced 

bias in other comparisons to measurement [7]. Here, the calculation uncertainty in 238Pu is 16%, 

with an effective measurement uncertainty of 8%, but with a bias of 16%. In Hu et al. [7], a bias 

of 29% is reported, which indicates that the simplified model used here similarly predicts 238Pu. 

However, the same study found a bias of only 1.5% for a different, similar sample P that was 

measured at a different laboratory. This indicates that there may be a systematic measurement 

error in samples S1 and S2. 

Figure 3-14 241Am Isotopic Uncertainty 
3-21

Figure 3-15 243Am Isotopic Uncertainty 
3-22

Figure 3-16 140Ce Isotopic Uncertainty 
3-23

Figure 3-17 142Ce Isotopic Uncertainty
3-24

Figure 3-18 244Cm Isotopic Uncertainty
3-25

Figure 3-19 133Cs Isotopic Uncertainty
3-26

Figure 3-20 135Cs Isotopic Uncertainty 
3-27

Figure 3-21 137Cs Isotopic Uncertainty
3-28

Figure 3-22 151Eu Isotopic Uncertainty 
3-29

Figure 3-23 152Eu Isotopic Uncertainty
3-30

Figure 3-24 153Eu Isotopic Uncertainty
3-31

Figure 3-25 154Eu Isotopic Uncertainty
3-32

Figure 3-26 155Eu Isotopic Uncertainty 
3-33

Figure 3-27 152Gd Isotopics Uncertainty
3-34

Figure 3-28 154Gd Isotopic Uncertainty 
3-35

Figure 3-29 155Gd Isotopic Uncertainty
3-36

Figure 3-30 156Gd Isotopic Uncertainty
3-37

Figure 3-31 157Gd Isotopic Uncertainty
3-38

Figure 3-32 158Gd Isotopic Uncertainty 
3-39

Figure 3-33 160Gd Isotopic Uncertainty 
3-40

Figure 3-34 139La Isotopic Uncertainty 
3-41

Figure 3-35 95Mo Isotopic Uncertainty
3-42

Figure 3-36 142Nd Isotopic Uncertainty
3-43

Figure 3-37 143Nd Isotopic Uncertainty 
3-44

Figure 3-38 144Nd Isotopic Uncertainty 
3-45

Figure 3-39 145Nd Isotopic Uncertainty
3-46

Figure 3-40 146Nd Isotopic Uncertainty
3-47

Figure 3-41 148Nd Isotopic Uncertainty
3-48

Figure 3-42 150Nd Isotopic Uncertainty
3-49

Figure 3-43 237Np Isotopic Uncertainty
3-50

Figure 3-44 238Pu Isotopic Uncertainty 
3-51

Figure 3-45 239Pu Isotopic Uncertainty
3-52

Figure 3-46 240Pu Isotopic Uncertainty
3-53

Figure 3-47 241P Isotopics Uncertainty
3-54

Figure 3-48 242Pu Isotopic Uncertainty
3-55

Figure 3-49 103Rh Isotopic Uncertainty 
3-56

Figure 3-50 101Ru Isotopic Uncertainty 
3-57

Figure 3-51 147Sm Isotopic Uncertainty
3-58

Figure 3-52 148Sm Isotopic Uncertainty
3-59

Figure 3-53 149Sm Isotopic Uncertainty
3-60

Figure 3-54 150Sm Isotopic Uncertainty
3-61

Figure 3-55 151Sm Isotopic Uncertainty
3-62

Figure 3-56 152Sm Isotopic Uncertainty 
3-63

Figure 3-57 154Sm Isotopic Uncertainty
3-64

Figure 3-58 90Sr Isotopic Uncertainty
3-65

Figure 3-59 234U Isotopic Uncertainty
3-66

Figure 3-60 235U Isotopic Uncertainty 
3-67

Figure 3-61 236U Isotopic Uncertainty
3-68

Figure 3-62 238U Isotopic Uncertainty

3.2.5 Source Terms

Figure 64 shows the decrease in total activity after discharge, with an initial uncertainty of 1.6%, 

decreasing gradually to 1.4% after decay. These uncertainties reflect those in the predicted 

nuclide concentrations at the time of discharge, as well as the uncertainties in the decay data 

after discharge. 
3-69

Figure 3-63 Activity Uncertainty 

4-1

4 DISCUSSION 

In the previous section, uncertainties in the various quantities of interest are presented for an 

example application modeling the MKP109 fuel rod Calvert Cliffs fuel assembly D047. The 

previous section highlights the comparisons to destructive assay measurements for two similar 

samples, S1 and S2, from adjacent axial positions of the same fuel rod. 

This study used a simplified model as compared to the high fidelity model used for Calvert Cliffs 

assembly D047 described in Hu et al. [7]. Using the simplified model for this study reduced the 

run time by an order of magnitude. This study was only intended to investigate the various 

properties of the data uncertainty and how that uncertainty impacts calculations, while the 

Calvert Cliffs assembly was chosen for modeling because the available measurements facilitate 

discussion of the complex interrelationship and interpretation of calculation uncertainty, 

measurement uncertainty, and bias when using experimental data. 

This section provides general guidance on interpreting calculation uncertainties. 

4.1 Importance of Non-Data Error 

Depending on the specific scenario being modeled, error from sources other than nuclear data 

may significantly impact the total error and cannot be neglected. Examples can include 

manufacturing tolerances, geometry variations in the fuel during irradiation caused by fuel 

swelling and other physical changes, associated changes in the fuel temperature, bowing of fuel 

rods, and uncertainties in the moderator temperature and void. On the experimental side, 

uncertainties in measurement dates and measured burnup can also contribute to overall 

uncertainty when comparing calculations to measurements. 

Uncertainty propagation is not necessary when there is a strong validation suite for the 

application of interest. The distribution of biases from validation indicates the actual error. 

Uncertainty propagation methods described here only give an expectation for the error due to 

nuclear data. In the future, as data uncertainty becomes more routinely used and reliable, it will 

be useful to determine the minimum expected uncertainty present in quantities of interest for 

which validation data are not available. These methods can also be used to support the 

extrapolation of uncertainties beyond the validated range. 

4.2 Calculation Uncertainty Is a Prediction 

Uncertainty in calculated quantities, like calculated quantities themselves, must be regarded as 

a prediction. Confidence in each prediction is developed in different ways. 

Non-data sources of uncertainty can be minimized so that isotopic comparisons to the 

measurements presented may be used to gain further confidence in calculated data uncertainty. 

For example, it is not certain that including data uncertainty provides a better understanding of 

the bias. For example, the 243Am nuclide concentration has a bias of 7% compared to the 

experimental uncertainty of ~7% (2-sigma). However, the large data uncertainty of ~15% (2-

sigma) provides some additional information indicating that a bias of 7% is well within 

expectations. In some cases, the calculation uncertainty may be much greater than the 

observed bias. Both the fundamental data and the data uncertainty should undergo constant 

revision and refinement to better match observations. This is discussed in the next section on 

data assimilation. 
4-2

Just as calculation uncertainty is a prediction, so is measurement uncertainty. The 

measurement of CC-1 fuel was made with two nearly identical samples, S1 and S2. A 

measurement uncertainty was provided for each measurement, but it has been shown that 

when samples S1 and S2 are compared to provide a more robust estimate of the true error 

(more specifically the reproducibility component), the estimates do not hold for all samples. For 

example, for 152Eu, 15% (1-sigma) uncertainty has been claimed, but when the two samples 

(2.64E-8 and 1.50E-7 g/gFuel) are compared, there is a standard deviation of 70% when using 

sample statistics with two samples. This is identified in the work by Hu et al. as a likely 

underestimate of measurement uncertainty caused by the extremely low isotopic concentration 

of this nuclide that is near the detection limit. With only two samples, the likelihood of one of the 

measurements being an outlier is non-negligible. However, because measurement uncertainty 

is a prediction, additional cross checks and independent multi-laboratory measurements are 

useful to gain confidence in these uncertainty estimates. 

4.3 Data Assimilation 

Data assimilation (or data adjustment) is the step in which the underlying data are adjusted 

within uncertainty to better agree with observations: in effect, to reduce bias. Propagating data 

uncertainty is necessary for this procedure. Data assimilation is a rigorous procedure to improve 

data quality using measurements of integral quantities. Although the techniques are well 

known [8], the process is complex, and the nuclear data libraries are large, so a necessarily 

automated solution continues to be beyond reach. 

4.4 Significant Bias Identification 

Data uncertainty propagation allows for the capability to separate biases from a comparison to 

measurement into two categories: expected and unexpected (or significant) biases. Large 

unexpected biases may be the result of a source of error that has not been taken into account. 

This can lead to the need to refine models by including extra uncertainty in some variables or 

providing feedback to the nuclear data community/measurement laboratory about potential 

issues with the data/measurement. 
5-1

5 CONCLUSIONS

This report presents the current state of the art in SCALE for nuclear data uncertainty analysis 

capability, and it provides an overview of the uncertainty in multigroup cross sections, fission 

product yields, and decay data. The effect of nuclear data uncertainty is demonstrated for a 

typical LWR depletion analysis problem involving a Combustion Engineering 14 × 14 assembly 

irradiated in Calvert Cliffs Unit 1. A single fuel rod from assembly D047, designated MKP109, 

has been subjected to destructive radiochemical assay to measure the isotopic contents. 

The 95% range width (difference between the 97.5th and 2.5th percentiles) is used in this study 

to assess the calculation uncertainty. This approach uses the actual distribution of the data and 

does not make any assumptions about the normality of the distributions. If the distribution were 

normal, then the 95% range width would correspond to 4-sigma range, i.e., +/- 2 sigma. 

The calculation uncertainty determined in nuclide concentrations for the MKP109 rod range from 

a few percent to 50%. The power factor for this fuel rod shows a very low uncertainty of less 

than 0.5%. There is also a low uncertainty of less than 2% in activity. 

Uncertainties in the macroscopic cross sections, reactivity, and power distributions are generally 

low in the few percent range. The effective delayed neutron fraction, βeff, shows an unexpected 

high uncertainty of 20–100%.

The considered application of the MKP109 fuel rod enabled a bias estimation and comparisons 

of calculated uncertainty in the isotopic concentrations to measurements of samples S1 and S2. 

In this case, the relative bias range was calculated to include both the calculation and 

measurement uncertainty. A bias range (2.5th–97.5th percentile) that includes zero is within 

expectation, whereas a bias which does not include zero is unexpected. Identifying unexpected 

biases can lead to improvements in the model or in nuclear data.

6-1

6 FUTURE WORK

Numerous uncertainty-related improvements will be included in the SCALE 6.3 release. First, 

continuous energy perturbations will be available, which will allow the continuous energy Monte 

Carlo methods to be used with the Sampler code. Continuous perturbations are applied to the 

master library once at the beginning of a continuous energy calculation,

σi,x

′

(E) = pi,x

g

σi,x

(E) for E

g ≤ E ≤ Eg−1

. (36)

This introduces artificial discontinuities in the continuous energy cross sections at group 

boundaries. Future work will seek to limit this artifact. Additionally, the fission product 

uncertainty data have been expanded to include all available nuclides. Corrections to the decay 

heat uncertainty data (discussed in Section 2.3) are being implemented for distribution at the 

time that SCALE 6.2.3 is released. Other future work includes the following:

 Kinetics parameter uncertainty should be investigated, as it is of paramount importance 

to reactor transient and safety analysis.

 Cross section uncertainty should be extended to the so-called activation cross section 

library used by the ORIGEN depletion code. This multigroup cross section library is 

based on the JEFF/3.1-A activation library and is used to supplement reactions available 

in ENDF/B. The reactions available in ENDF/B have traditionally focused on neutron 

interactions relevant to neutron transport calculations, and they do not include such data 

as isomeric branching cross sections. Currently, zero uncertainty is assumed in all data 

originating from JEFF/3.1-A.

 Data uncertainty must be imposed consistently for any remaining data in SCALE. For 

example, alpha particle interaction probabilities are used in ORIGEN neutron emission 

models. These data effectively have no uncertainty. 

 Additional tools and processes should be implemented to make data uncertainty a 

standard part of the analysis process. Necessary improvements will occur only when 

such data are routinely used and required to be reliable.

 Data assimilation should be pursued in a rigorous, automated framework based on 

validation results comparing calculation to measurement for a wide array of 

measurements. 

7-1

7 REFERENCES

[1] M. B. Chadwick, M. Herman, P. Obložinský, M.E. Dunn, Y. Danon, A. C. Kahler, D. L. 

Smith, B. Pritychenko, G. Arbanas, R. Arcilla, R. Brewer, D. A. Brown, R. Capote, A.

D. Carlson, Y. S. Cho, H. Derrien, K. Guber, G. M. Hale, S. Hoblit, S. Holloway, T. D. 

Johnson, T. Kawano, B. C. Kiedrowski, H. Kim, S. Kunieda, N. M. Larson, L. Leal, J.

P. Lestone, R. C. Little, E. A. McCutchan, R. E. MacFarlane, M. MacInnes, C. M. 

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Palumbo, M. T. Pigni, V. G. Pronyaev, R. O. Sayer, A. A. Sonzogni, N. C. Summers, 

P. Talou, I. J. Thompson, A. Trkov, R. L. Vogt, S. C. van der Marck, A. Wallner, M. C. 

White, D. Wiarda, and P. G. Young, “ENDF/B-VII.1 Nuclear Data for Science and 

Technology: Cross Sections, Covariances, Fission Product Yields and Decay Data,”

In Nuclear Data Sheets, Volume 112, Issue 12, 2011, Pages 2887–2996, ISSN 0090-

3752, 

https://doi.org/10.1016/j.nds.2011.11.002.(http://www.sciencedirect.com/science/articl

e/pii/S009037521100113X).

[2] SCALE: A Comprehensive Modeling and Simulation Suite for Nuclear Safety Analysis 

and Design, ORNL/TM-2005/39, Version 6.1, Oak Ridge National Laboratory, Oak 

Ridge, Tennessee, June, 2011. Available from Radiation Safety Information 

Computational Center at Oak Ridge National Laboratory as CCC-785.

[3] M. L. Williams, D. Wiarda, G. Ilas, W. J. Marshall, and B. T. Rearden, “Covariance 

Applications in Criticality Safety, Light Water Reactor Analysis, and Spent Fuel 

Characterization,” Nuclear Data Sheets, Volume 123, January 2015, Pages 92–96, 

ISSN 0090-3752, https://doi.org/10.1016/j.nds.2014.12.016.

[4] M. L. Williams, G. Ilas, M. A. Jessee, B. T. Rearden, D. Wiarda, W. Zwermann, L. 

Gallner, M. Klein, B. Krzykacz-Hausmann, and A. Pautz, “A Statistical Sampling 

Method For Uncertainty Analysis With SCALE And XSUSA,” Nuclear Technology

183,3, p515–527 (Sept. 2013).

[5] M.T. Pigni, M. W. Francis, and I. C. Gauld, “Investigation of Inconsistent ENDF/B-VII.1 

Independent and Cumulative Fission Product Yields with Proposed Revisions,”

Nuclear Data Sheets, Volume 123, January 2015, Pages 231–236, ISSN 0090-3752, 

https://doi.org/10.1016/j.nds.2014.12.040.

[6] R. J. Guenther et al., Characterization of LWR Spent Fuel MCC-Approved Testing 

Material ATM-104, PNL-5109-104, 1991.

[7] J. Hu, J. M. Giaquinto, I. C. Gauld, G. Ilas, and T. J. Keever, “Analysis of new 

measurements of Calvert Cliffs spent fuel samples using SCALE 6.2,” Annals of 

Nuclear Energy, Volume 106, August 2017, Pages 221–234, ISSN 0306-4549, 

https://doi.org/10.1016/j.anucene.2017.04.005.

[8] G. Palmiotti, M. Salvatores, and G. Aliberti, “A-priori and A-posteriori Covariance Data 

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https://doi.org/10.1016/j.nds.2014.12.008. 

NUREG/CR-7249 

ORNL/TM-2017/706

William Wieselquist, Mark L. Williams, Dorothea Wiarda, Marco Pigni, 

Ugur Mertyurek

Oak Ridge National Laboratory

Managed by UT-Battelle, LLC

Oak Ridge, TN 37831-6170

Division of System Analysis and Regulatory Effectiveness

Office of Nuclear Regulatory Research

U. S. Nuclear Regulatory Commission

Washington, DC 20555–0001

This report presents the current state of the art in SCALE for nuclear data uncertainty analysis capability, and it 

provides an overview of the uncertainty in multigroup cross sections, fission product yields, and decay data. The 

effect of nuclear data uncertainty is demonstrated for a typical LWR depletion analysis problem involving a 

Combustion Engineering assembly irradiated in Calvert Cliffs Unit 1. A single fuel rod from assembly D047 has been 

subjected to destructive radiochemical assay to measure the isotopic contents. 

The 95% range width (difference between the 97.5th and 2.5th percentiles) is used in this study to assess the 

calculation uncertainty. This approach uses the actual distribution of the data and does not make any assumptions 

about the normality of the distributions. If the distribution were normal, then the 95% range width would correspond to 

4-sigma range, i.e., +/- 2 sigma. 

The calculation uncertainty determined in nuclide concentrations for the MKP109 rod range from a few percent to 

50%. The power factor for this fuel rod shows a very low uncertainty of less than 0.5%.

Uncertainties in the macroscopic cross sections, reactivity, and power distributions are generally low in the few 

percent range. The effective delayed neutron fraction shows an unexpected high uncertainty of 20–100%.

uncertainty quantification

calculation uncertainty

nuclear data

SCALE

spent nuclear fuel

validation

December 2018

Technical

Overview of Nuclear Data Uncertainty in Scale and Application to Light 

Water Reactor Uncertainty Analysis

 

NRC JCN V6449



NUREG/CR-7249 

ORNL/TM-2017/706

Overview of Nuclear Data Uncertainty in SCALE and Application 

to Light Water Reactor Uncertainty Analysis

December 2018