Cross Section Generation 

Guidelines for TRACEPARCS

Office of Nuclear Regulatory Research

NUREG/CR-7164

ORNL/TM-2012/518
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Cross Section Generation 

Guidelines for TRACEPARCS 

Manuscript Completed: November 2012

Date Published: June 2013

Prepared by: 

D. Wang, B. J. Ade, and A. M. Ward

Oak Ridge National Laboratory

P. O. Box 2008

Oak Ridge, TN 37831-6010

C. G. Thurston, NRC Project Manager

NRC Job V6233

Office of Nuclear Regulatory Research

 

NUREG/CR-7164

ORNL/TM-2012/518

iii

ABSTRACT

This report documents a comprehensive comparison of cross sections calculated using different 

methodologies and codes, including CASMO, HELIOS, and TRITON XS. The conclusions from 

this study have resulted in this guidance document on how to choose cross section histories 

and branches for boiling water reactor (BWR) analysis, and the methodology to collapse the 

fine-energy and -space fluxes calculated by the detailed lattice calculation. The guidance herein 

is applicable to all BWR designs.

For BWR steady-state and transient analysis, the PARCS code uses two-energy-group cross 

sections for each computational node in the 3-dimensional grid. The PARCS cross sections are 

tabulated as a function of four instantaneous state variables: (1) control rod insertion, (2) fuel 

temperature, (3) coolant density, and (4) soluble poison concentration. The cross section values 

also depend on the isotopic mixture (i.e., concentration of 235U, 239Pu ...), which is characterized 

as a function of control rod and moderator density history variables. 

In a typical calculation, the fuel temperature, moderator density, and soluble boron 

concentration are calculated by the TRACE code for a coupled TRACE/PARCS analysis. The 

instantaneous control rod insertion is provided by the user in the input deck. The historic control 

rod and moderator density values are provided by a steady-state core-follow simulator, which 

has followed the core operation since the initial loading up to the time of the transient to be 

calculated. All these parameters are taken into account to estimate the instantaneous cross 

section based on the tabulated values. This report documents the expected error in evaluating 

the instantaneous cross section as a function of the data table structure.

As a result of this study, guidelines for BWR cross section generation have been generated.

The recommendations are the use of four instantaneous moderator density values at 0, 40, 70, 

and 90% void fraction at three different fuel temperatures of 500, 950, and 1500 K. For the 

history effect, three moderator density values at 0, 40, and 70% at a single 950 K fuel 

temperature provide sufficiently accurate results. For all cases, the user must generate these 

branches for controlled and uncontrolled bundles.

In addition, a coolant density branch with a moderator density of 1000 kg/m3 is needed to 

accurately model cold depressurized conditions. If boron injection is modeled in a BWR 

transient analysis, boron branches should be included.

v

TABLE OF CONTENTS

Section Page

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

TABLE OF CONTENTS................................................................................................................v

LIST OF FIGURES ..................................................................................................................... vii

LIST OF TABLES......................................................................................................................... ix

ACRONYMS AND ABBREVIATIONS.......................................................................................... xi

1.0 INTRODUCTION ...................................................................................................................1

2.0 FUEL SPECIFICATIONS FOR THIS STUDY........................................................................3

2.1. SPECIFICATIONS OF PEACH BOTTOM TYPE 3C FUEL............................................3

2.2. SPECIFICATIONS OF PMAXS STRUCTURE...............................................................4

2.3. SPECIFICATIONS OF LATTICE CODES......................................................................4

3.0 COMPARISONS OF CROSS SECTION SETS.....................................................................7

3.1. COMPARISONS OF CROSS SECTIONS......................................................................7

3.2. BENCHMARK BETWEEN TRITON AND KENO..........................................................11

3.3. COMPARISONS OF KINETICS PARAMETERS .........................................................11

3.3.1. 1/v..........................................................................................................................11

3.3.2. βeff ..........................................................................................................................12

4.0 SENSITIVITY ANALYSIS OF CROSS SECTIONS.............................................................17

4.1. DC HISTORY EFFECTS..............................................................................................17

4.1.1. Results of Single CHAN Model .............................................................................21

4.1.2. Results of Oskarshamn Core-wide Stability ..........................................................22

4.1.3. Results of Ringhals Core-wide Stability ................................................................23

4.1.4. Conclusion of DC History Effects ..........................................................................24

4.2. DC BRANCH EFFECTS...............................................................................................24

4.2.1. Results for Single CHAN Model ............................................................................25

4.2.2. Results for Oskarshamn Core-wide Stability.........................................................27

4.2.3. Results for Ringhals Core-wide Stability ...............................................................28

4.2.4. 4.2.4 Conclusions of DC Branch Effects ...............................................................29

5.0 EFFECTIVE DELAYED NEUTRON FRACTION.................................................................31

5.1. DEFINITION OF βeff......................................................................................................31

5.2. CALCULATION OF βeff USING A SINGLE-ASSEMBLY LATTICE ..............................31

5.3. PHYSICAL DESCRIPTION OF THE PHENOMENA....................................................32

5.4. NUMERICAL SIMULATION .........................................................................................35

6.0 CODE-TO-CODE COMPARISON.......................................................................................39
vi

7.0 ISSUES WITH TWO-GROUP KINETICS EQUATIONS......................................................41

8.0 BWR PARCS CROSS SECTIONS GUIDANCE..................................................................43

8.1. HISTORY CROSS SECTION.......................................................................................44

8.2. BRANCH CROSS SECTION........................................................................................45

8.3. GUIDELINES................................................................................................................46

9.0 REFERENCES....................................................................................................................49
vii

LIST OF FIGURES

Figure Page

Figure 1. Type-3c fuel bundle. ....................................................................................................3

Figure 2. Control blade detail......................................................................................................3

Figure 3. Ref PMAXS histories (HDC) and instantaneous coolant density branches (DC). .......5

Figure 4. Cross Section Comparisons At BOL............................................................................8

Figure 5. HDC Cross Section Comparisons at EOL. ..................................................................9

Figure 6. Branch Cross Section at EOL (40% void history CR = 0)..........................................10

Figure 7. Benchmark between TRITON and KENO. ................................................................11

Figure 8. Fission-neutron-weighted β........................................................................................14

Figure 9. Fission-weighted (6.1) vs neutron-fission-weighted (6.1.1) beta effective.................15

Figure 10. CASMO fission-neutron-weighted β and adjoint infinite flux–weighted β...................15

Figure 11. HELIOS fission-neutron-weighted β and K-ratio–weighted β.....................................15

Figure 12. Σa1 and νΣf1 at BOL and EOL.................................................................................18

Figure 13. PMAXS HDC breakdown...........................................................................................19

Figure 14. PMAXS HDC breakdown (cont.)................................................................................20

Figure 15. HDC and HCR input for single-CHAN model.............................................................21

Figure 16. Response to a pressure perturbation for single-CHAN model...................................21

Figure 17. HDC and HCR input for Oskarsham..........................................................................22

Figure 18. Response to a control rod perturbation. ....................................................................22

Figure 19. HDC and HCR input for Ringhals. .............................................................................23

Figure 20. Core flow stability for Ringhals. .................................................................................23

Figure 21. Σa1 and νΣf1 at BOL and EOL as function of branch void.......................................24

Figure 22. Axial coolant density for single-CHAN model. ...........................................................26

Figure 23. Response to a pressure perturbation for single-CHAN model...................................26

Figure 24. Core axial coolant density for Oskarsham. ................................................................27

Figure 25. Response to a control rod perturbation for Oskarsham.............................................27

Figure 26. Core axial coolant density for Ringhals. ....................................................................28

Figure 27. Response to a control rod perturbation for Ringhals. ................................................29

Figure 28. Two bundle sample configuration. .............................................................................32

Figure 29. Hypothetical neutron path born on left bundle and absorbed on the right one. .........33

Figure 30. Illustration of single-lattice approximation, where the neutron path is reflected 

 back into the bundle...................................................................................................33

Figure 31. Illustration of the effect of neutrons leaked from surrounding bundles. .....................34

Figure 32. Example colorset configuration..................................................................................35

Figure 33. Single channel results................................................................................................39

Figure 34. Oskarshamn results...................................................................................................40

Figure 35. Ringhals results. ........................................................................................................40

Figure 36. Typical PMAXS structure...........................................................................................44

Figure 37. Void history effect on Σa1 and νΣf1..........................................................................45

Figure 38. Brach void fraction effect on Σa1 and νΣf1...............................................................46

ix

LIST OF TABLES

Table Page

Table 1. Control blade design parameters..................................................................................4

Table 2. Code versions and Cross Section library......................................................................4

Table 3. TRITON delayed neutron yields [3].............................................................................13

Table 4. Steady-state parameters.............................................................................................21

Table 5. Steady-state parameters for Oskarshamn ..................................................................22

Table 6. Steady-state parameters for Ringhals ........................................................................23

Table 7. Steady-state parameters for single-CHAN model.......................................................25

Table 8. Steady-state parameters for Oskarshamn ..................................................................27

Table 9. Steady-state parameters for Ringhals ........................................................................28

Table 10. Three colorset burnup configurations studied.............................................................36

Table 11. Comparison of βeff for configuration C1—low burnup case.........................................36

Table 12. Comparison of βeff for configuration C2—medium burnup case .................................36

Table 13. Comparison of βeff for configuration C3 – high burnup case .......................................37

Table 14. Steady-state parameters.............................................................................................39

Table 15. Recommended history branches ................................................................................47

Table 16. Recommended instantaneous branches ....................................................................47

xi

ACRONYMS AND ABBREVIATIONS

BC boundary condition

BOL beginning of life

BWR boiling water reactor

CR control rod insertion

DC coolant density

DF discontinuity function

EOL end of life

GenPMAXS Generation of the Purdue Macroscopic XS Set

HCR control rod history

HDC history of coolant density

ORNL Oak Ridge National Laboratory

PARCS Purdue Advanced Reactor Core Simulator

PC poison concentration

pcm percent mil (or 10-5

)

RPV reactor pressure vessel

TF fuel temperature

TRACE TRAC/RELAP Advanced Computational Engine

XS cross sections

1

1.0 INTRODUCTION

In recent years, there has been increased application of TRACE/PARCS for light water reactor 

transient safety analysis, in particular for boiling water reactor (BWR) stability analysis. TRACE 

is a thermal-hydraulics system analysis code. PARCS solves time-dependent spatial kinetics 

equations based on the two-group diffusion nodal method.

A BWR core usually consists of hundreds of fuel assemblies. The coolant density (DC) changes 

significantly from the core inlet to the outlet. The coolant at the core inlet is subcooled and has a 

density of about 0.76 g/cm3

. The DC decreases from inlet to outlet as a result of boiling as the 

coolant flows through the core. At the core exit, DC decreases to approximately 0.2 g/cm3 on 

average with a void faction of about 75%. Therefore, for a BWR core operated with a wide 

distribution of DCs, historical effects of neutron spectrum on cross sections (XS) due to DC

should be considered in generating XS. In addition, the effects of control rod history (HCR) on 

XS should be considered.

A typical BWR fuel assembly is made up of several axial lattice segments (or zones) with 

different enrichments and burnable poisons. For each fuel segment, two-group XS are 

generated using a lattice physics code (e.g., TRITON, CASMO, or HELIOS). GenPMAXS is 

employed to process the two-group XS and kinetics data generated with the lattice code and 

store them in a PMAXS file, which can be read by PARCS. PMAXS provides all of the data 

necessary to perform core simulations for steady-state and transient applications, consisting of 

principal macroscopic XS, microscopic XS of Xe/Sm, group-wise form functions, and the 

kinetics data. 

The two-group XS for each computational node in the PARCS 3-D core model are typically 

parameterized as a function of five state variables: control rod insertion (CR), fuel temperature 

(TF), DC, and soluble poison concentration (PC), as well as control rod and moderator density 

history variables. Thermal-hydraulic variables such as fuel temperature, moderator density, and 

soluble boron concentration, are typically calculated by TRACE.

The main objective of this study of BWR cross sections is to answer the following questions:

1. How many histories and branches in the XS set are sufficient to have converged results 

for BWR analysis?

2. How sensitive are BWR analysis results to XS generated with different lattice codes 

(TRITON, CASMO, or HELIOS)?

A Peach Bottom fuel type with the 7×7 lattice was chosen for this study. Two-group XS and 

kinetics parameters for this fuel were generated with CASMO, HELIOS, and TRITON, 

respectively. Each of the XS sets has a very fine structure of histories and branches, which 

consists of 11 uncontrolled DC histories (HDCs) and 11 controlled HDCs; each history has 11 

uncontrolled DC branches and 11 controlled DC branches. 

Three BWR models employed to conduct this study are (1) Single-CHAN model, (2) 

Oskarshamn plant model, and (3) Ringhals plant model.

The results of the comprehensive comparison performed in this study on the three XS sets 

generated with CASMO, HELIOS, and TRITON show that, at beginning of life (BOL), TRITON 

slightly over predicts Σa

2 and νΣf

2 for high voids compared with CAMSO and HELIOS; overall, it 

has about 300 pcm of over prediction of Kinf. At the end of life (EOL) of each coolant history,

TRITON over predicts Kinf by 200—1500 pcm, and the over prediction increases as void 

fraction increases. In addition, TRITON slightly under predicts Σtr

1 by 2–8%. However, it should 
2

be noted that the discrepancies found with TRITON do not necessarily imply that TRITON 

predicts these XS poorly, because the predictability of lattice codes, including CASMO and 

HELIOS, for coolant with high void fraction (> 80%) needs further investigation. The TRITON 

calculations used SCALE version 6.1; Oak Ridge National Laboratory (ORNL) is in the process 

of performing bias assessments with Monte Carlo codes to improve the accuracy of future 

releases.

Both Σa

1 and νΣf

1 curves show more nonlinearity at high void fraction (>70%) than at low void 

fraction. Thus the extrapolation of the XS from the 70% void may overestimate the XS for high 

void faction. 

The sensitivity analysis performed on HDC effects shows that neither BWR steady-state nor

transient analysis is highly sensitive to HDC effects (nor HCR effects). It is found that three 

HDCs at 0%, 40%, and 70% (or 80%) voids are good enough for most BWR applications. 

The sensitivity analysis of DC instantaneous branch effects also shows that BWR steady-state 

results are not highly sensitive to DC branch effects. However, transients may be sensitive to 

the structure of DC branches, especially when the core upper region has void fractions that are 

outside the range of DC branches. The study has concluded that four DC branches at 0%, 40%, 

70%, and 90% voids provide sufficient accuracy for BWR transient analysis. The 100% void 

branch may be used to replace the 90% branch if XS at 100% void can be calculated 

accurately, but a Monte Carlo type calculation is required to confirm the accuracy of the lattice 

code. For controlled DC branches, it is suggested that the same density branch structure be 

used.
3

2.0 FUEL SPECIFICATIONS FOR THIS STUDY

2.1. SPECIFICATIONS OF PEACH BOTTOM TYPE 3C FUEL

Type-3c fuel was used from Peach Bottom Unit 2 Cycles 1 and 2 [1]. The type-3c fuel is a 7×7 

fuel lattice with fuel, gap, and cladding outer diameters of 0.477, 0.526, and 0.563 in. (1.211156, 

1.33604, and 1.43002 cm), respectively. The fuel pin pitch is 0.738 in. (1.87452 cm). The 

channel outer dimension is 5.438 in. (13.8152 cm) with a thickness of 0.08 in. (0.2032 cm) and 

a corner inner radius of 0.38 in. (0.9652 cm). Type 3c is the middle segment of the type-3 fuel 

bundle, as shown in Figure 1. Among the 49 fuel rods, 5 rods contain burnable poison Gd2O3. 

The enrichment of each fuel pin and the weight percentage of Gd2O3 are indicated in Figure 1. 

Detailed specifications for the control blade are given in Table 1. Figure 2 expands the control 

blade model in more detail.

Figure 1. Type-3c fuel bundle.

Figure 2. Control blade detail.
4

Table 1. Control blade design parameters

Parameter Specification

Control material B4C 

Material density (g/cc) 1.764

Control pin tube ID (cm) 0.35052

Control pin tube ID (cm) 0.47752

Control pin tube material SS-304 (7.94 g/cm3)

Control blade half span (cm) 12.3825

Control blade full thickness (cm) 0.79248

Control blade tip radius (cm) 0.39624

Control blade sheath material SS-304 (7.94 g/cm3)

Control blade sheath thickness (cm) 1.23698

Central structure wing length (cm) 1.98501

Total number of control pins 84

2.2. SPECIFICATIONS OF PMAXS STRUCTURE 

Three XS sets (PMAXS) for the PB type-3c fuel were generated with CASMO, HELIOS, and 

TRITON, respectively. For the CASMO and HELIOS XS sets, the PMAXS files consist of 12 

uncontrolled HDCs (CR = 0) and 12 controlled HDCs (CR = 1), each history having 12 DC 

branches. The PMAXS structure of histories and branches is shown in Figure 3. The DC ranges 

from 0.03591 g/cm3 (100% void) up to 0.73808 g/ cm3 (0% void) in 10% void spacing. The 

CASMO and HELIOS sets include one additional density at 1.0 g/ cm3 that represents cold core, 

depressurized conditions. The three XS sets have five TF branches at 500, 750, 100, 1250, and 

1500 K. 

2.3. SPECIFICATIONS OF LATTICE CODES

The version of each lattice code is given in Table 2. CAMSO and HELIOS use the ENDV-VI 

library, and TRITON uses the ENDF-VII library.

Table 2. Code versions and Cross Section library

Version Cross Section Library

CASMO 4E ENDF-VI, 70G neutron + 18G gamma

HELIOS 1.10 ENDF-VI, 47G neutron + 17G gamma

TRITON SCALE 6.1 ENDF-VII, 49G neutron
5

Figure 3. Ref PMAXS histories (HDC) and instantaneous coolant density

branches (DC).

HDC: 12 + 12 DC: 12 + 12

7

3.0 COMPARISONS OF CROSS SECTION SETS 

Each of the three XS sets consists of two-group principal macroscopic XS; microscopic XS for 

Xe/Sm; assembly discontinuity factors; and kinetics parameters, including inverse neutron 

velocities (1/v), delayed neutron fractions (β), and decay constants of delayed neutrons (λ). 

3.1. COMPARISONS OF CROSS SECTIONS

Two-group macroscopic XS generated with CASMO, HELIOS, and TRITON, as a function of 

void fraction are shown in Figure 4, Figure 5, and Figure 6, along with the calculated Kinf 

values. 

Figure 4 shows XS comparisons for BOL. BOL fresh fuel has not undergone depletion. 

Therefore there are no fission products and other isotopes in the fuel. 

Figure 5 shows XS as a function of “history void fraction” at EOL (49.57 MWd/kg). The plotted 

XS are XS at the end of each void history depletion calculation. The horizontal axis is the history 

void fraction. There are a total of 11 void history cases (0%, 10%, 20%, 30%, 40%, 50%, 60%, 

70%, 80%, 90%, and 100%). 

Figure 6 shows XS as a function of “branch void fraction” at EOL (49.57 MWd/kg) for the 40% 

void history. The plotted XS are branch instantaneous XS at the end of the 40% void history 

depletion. The horizontal axis is the branch void fraction.

Observations on comparisons of XS are summarized below.

1. At BOL, Figure 4 shows that there are about 300 pcm differences in Kinf between 

TRITON and CASMO. TRITON slightly over predicts Σa

2 and νΣf

2 at high voids, compared

with CAMSO and HELIOS. 

2. For history XS comparison, Figure 5 shows that TRITON over predicts Kinf by 200–1500 

pcm compared with CASMO or HELIOS. Note that the difference increases as the void 

fraction increases. 

3. The literature reports that the approximations used to speed up CASMO and HELIOS 

calculations may not be applicable for void fractions higher than 80% (see for example

νΣf

1 on Figure 5).

4. For branch XS comparisons, overall TRITON over predicts Kinf by 500 pcm compared 

with CASMO or HELIOS at EOL. 

5. TRITON slightly under predicts Σtr

1 by 2–8%. An investigation indicates that the 

difference might be due to the collapsing method of transport XS. TRITON currently 

employs inverse-energy collapse to calculate the two-group transport XS using inverseenergy collapsing with B1-critical spectrum correction; however, both CASMO and 

HELIOS use direct energy collapsing. 

6. The values Σa

1 and νΣf

1 are the two key parameters for the study of XS sensitivity. Both 

XS curves show more curvature at high void than low void. 
8

Figure 4. Cross Section Comparisons at BOL.
9

Figure 5. HDC Cross Section Comparisons at EOL.
10

Figure 6. Branch Cross Section at EOL (40% void history CR = 0).
11

3.2. BENCHMARK BETWEEN TRITON AND KENO

In general, as the void fraction increases, the neutron spectrum becomes harder; so XS

databases plus solution methods may not be applicable. Therefore, there is some concern 

about the predictability of lattice codes for fuel regions with very high void. Our study found that 

CASMO and HELIOS have convergence difficulties for the void fraction >80%. To investigate 

how well TRITON performs at high void, a benchmark was performed between TRITON and 

KENO for a 7×7 BWR bundle similar to the PB type-3c, as shown in Figure 7. On the left plot of 

the figure, the green line is the results of the TRITON 49 group XS library, the red line is the 

results of the TRITON 238 group XS library, and the KENO results are shown with the blue line. 

The right plot shows relative differences between TRITON and KENO. The 49G TRITON results 

diverge at 100% void; the 238G results do not.

Figure 7. Benchmark between TRITON and KENO.

3.3. COMPARISONS OF KINETICS PARAMETERS

Kinetics data, together with two-group XS, are generated with lattice calculations. The kinetics 

data used by PARCS include two-group inverse neutron velocities (1/v), delayed neutron 

fractions (β), and decay constants of delayed neutrons (λ). 

3.3.1. 1/v

In CASMO, the assembly average value of the inverse neutron velocity is calculated by three 

different collapsing schemes:

�

1

v

�

G = ∑g∈G φg(1/v)g

∑g∈G φg

(3.1)

�

1

v

�

G

∗

= ∑ φg

∗ g∈G (1/v)g

∑ φg

∗ g∈G

(3.2)

�

1

v

�

G

′

= ∑ φg

∗ g∈G (1/v)gφg

∑ φg

∗ g∈G

(3.3)
12

In TRITON and HELIOS, Eq. (3.1) is employed to collapse two-group inverse neutron velocities. 

A comparison of inverse neutron velocities between TRITON and CASMO shows that the 

agreement is poor for high void histories; for example, the relative difference may be up to 30% 

for the fast group. HELIOS values have very good agreement with CAMSO for the fast group 

but are more than 20% lower than the CAMSO values for the thermal group.

3.3.2. βeff

Traditionally, βeff is calculated using bilinear weighting as below.

βeff, i = ∑ ∫ dr ∫ dEχd,i

m (E)φ∗�r, E� ∫ dE′ m βimν ∑ φ�r, E′� m

f

∑ ∫ dr ∫ dEχt m m(E)φ∗�r, E� ∫ dE′ν ∑ φ�r, E′� m

f

=

∑ βi

m ∫ dr ∫ dEχd,im (E)φ∗�r, E� ∫ dE′ m ν ∑ φ�r, E′� m

f

∑ ∫ dr ∫ dEχt m m(E)φ∗�r, E� ∫ dE′ν ∑ φ�r, E′� m

f

, i = 1, . . ,6 (3.4)

where

m = the mth isotope

φ∗(r, E) = adjoint flux

φ(r, E) = neutron flux

βi = delayed neutron fraction of the ith group

χd,i

m (E) = delayed neutron spectrum of the ith group

χt

m(E) = total fission spectrum (prompt + delayed)

ν ∑ φ(r, E′ f ) = total fission neutrons (prompt + delayed)

If φ∗�r, E� = constant (or 1), we obtain fission weighted β as

βi = ∑ βi

m ∫ dr ∫ dE′ m ν ∑ φ�r, E′� m

f

∑m ∫ dr ∫ dE′ ν ∑ φ�r, E′� m

f

, i = 1, ... ,6 (3.5)

In HELIOS, βeff is calculated with K-ratio weighting [2], which is calculated as

βeff = 〈χdυd〉

〈χυ〉 ≈ 1 − 〈χpυp〉

〈χυ〉 = 1 − kp

k , (3.6)

where kp is calculated based on the prompt neutron spectrum. It is found that βeff calculated 

using K-ratio weighting can produce nearly identical results to those calculated using the adjoint 

weighting method [2].

Delayed neutron yields for major isotopes used in TRITON are given in Table 3. The values 

used by CASMO and HELIOS are slightly different, varying between 2% to 10% differences for 

different isotopes.
13

Table 3. TRITON delayed neutron yields [3]

TRITON

235U 238U 239Pu 240Pu 241Pu

Group 

1

Group 

2

Group 

1

Group 

2

Group 

1

Group 

2

Group 

1

Group 

2

Group 

1

Group 

2

βm 0.0064 0.0067 0.0164 0 0.002 0.0022 0.0029 0 0 0.0054

αm,i

1 0.038 0.033 0.013 0.038 0.035 0.028 0.01

2 0.213 0.219 0.137 0.28 0.298 0.273 0.229

3 0.188 0.196 0.162 0.216 0.211 0.192 0.173

4 0.407 0.395 0.388 0.328 0.326 0.35 0.39

5 0.128 0.115 0.225 0.103 0.086 0.128 0.182

6 0.026 0.042 0.075 0.035 0.044 0.029 0.016

Figure 8 shows comparisons of fission-neutron-weighted β (calculated with Eq. [3.3]). Poor

agreement between TRITON and CASMO (or HELIOS) occurs because these calculations were 

performed with V6.1 of SCALE, and TRITON did not include the ν term in Eq. (3.3) when 

calculating fission-weighted β. In other words, V6.1 TRITON calculated the “fission-weighted” 

beta, while CASMO calculates the “fission-neutron-weighted” beta. As a result of this work, 

TRITON now includes an option to include the ν term, which is the recommended setting. A 

new option has been added in SCALE V6.1.1, and TRITON can also compute the fissionneutron-weighted beta, which accounts for the differences observed between the codes. Figure 

9 shows an example TRITON calculation.

Figure 10 shows CASMO fission-neutron-weighted β and adjoint infinite flux–weighted β. It is 

seen that adjoint flux–weighted β is about 10% less than fission-neutron-weighted β. 

In general, β decreases with burnup because of the depletion of 235U and the production of 

329Pu, which has a smaller β. At high voids, β increases because of enhanced 238U fission, which 

has higher β than 235U and 329Pu.

Figure 11 shows HELIOS fission-weighted β and K-ratio–weighted β. Generally, k-ratio–

weighted β can be up to 20% less than fission-neutron-weighted β for unrodded assemblies (CR 

=0). The right plot in Figure 11 shows that K-ratio–weighted β can be 20% less for CR =1 than 

for CR =0.
14

Figure 8. Fission-neutron-weighted β.
15

Figure 9. Fission-weighted (6.1) vs neutron-fission-weighted (6.1.1) beta effective.

Figure 10. CASMO fission-neutron-weighted β and adjoint infinite flux–weighted β.

Figure 11. HELIOS fission-neutron-weighted β and K-ratio–weighted β. 

17

4.0 SENSITIVITY ANALYSIS OF CROSS SECTIONS

Three TRACE/PARCS models employed to conduct the XS sensitivity study are 

1. Single CHAN model. The TRACE model consists of a CHAN, an inlet BREAK, and an 

outlet BREAK. The TRACE model is coupled with the PARCS core model. The flow is 

determined by the differential pressure between the inlet and outlet BREAKs. The 

transient is initiated by a pressure perturbation at the outlet. The power and flow of the 

single-CHAN model represent thermal-hydraulic conditions in a typical BWR. 

2. Oskarshamn plant model. The TRACE model is a detailed representation of the 

Oskarshamn plant, including the reactor pressure vessel (RPV), steam line, recirculation 

loop, and so on. The core consists of 444 fuel assemblies, modeled with 222 CHAN 

components in half-core symmetry. Each CHAN has 25 non-uniform powered axial 

nodes. The reactor has a power of 1723.8 MWth and a core flow of 3210 kg/s. The 

PARCS model has 444 radial nodes (node/assembly) and 25 axial powered nodes with 

uniform 12.42 cm spacing. A core-wide stability transient is induced by control rod 

perturbation. 

3. Ringhals plant model. The TRACE model is a detailed representation of the Ringhals 

plant, including the RPV, steam line, recirculation loop, and so on. The core consists of 

648 fuel assemblies, modeled with 325 CHAN components in half-core symmetry. Each 

CHAN has 25 powered axial nodes with uniform 14.72 cm spacing. The reactor has a 

power of 1770.6 MWth and a core flow of 3665 kg/s. The PARCS model has 648 radial 

nodes (node/assembly) and 25 axial powered nodes with uniform 14.72 cm spacing. A 

core-wide stability transient is induced by control rod perturbation.

The PMAXS data for PB type 3c fuel were used for all fuel types in these PARCS core models. 

Sensitivity analysis results for HDC effects and branch effects are presented in the following 

sections. Note that the results presented were calculated with the CASMO PMAXS data. 

4.1. DC HISTORY EFFECTS

The CASMO PMAXS data consist of 12 uncontrolled HDCs (CR = 0) and 12 controlled HDCs 

(CR = 1), each history having 12 DC branches as shown in Figure 3. The DC ranges from 

0.03591 g/ cm3 (100% void) up to 0.73808 g/ cm3 (0% void), with 10% spacing. One additional 

density at 1.0 g/ cm3 is for cold, unpressurized, core conditions. 

Macroscopic XS have a strong dependence on the history effects of DC and control rod 

position. Figure 12 shows Σa

1 and νΣf

1 as a function of history void fraction at BOL and EOL, 

respectively. The black dashed line on each figure is the second-order polynomial fitting of the 

XS at 0, 40, and 70% void. 

Note that both Σa

1 and νΣf

1 curves show more nonlinearity at high void fraction (>70%) than at 

low void fraction. Thus the extrapolation of the XS from the 70% void may overestimate the XS

at high void faction. 

For the study of HDC effects, the PMAXS file has been broken down into five PMAXS files as 

shown in Figure 13 and Figure 14. These small PMAXS files are described as follows. 

1. HDC 12+12: This is the reference case with 12 rodded and 12 unrodded HDCs. Each 

history has 12 rodded and 12 unrodded DC branches.
18

2. HDC 7+7: Seven rodded and seven unrodded HDCs (0%, 20%, 40%, 60%, 80%, 100% 

void, and 1 g/cc, for both CR = 0 and 1). Each history has 12 rodded and 12 unrodded

DC branches.

3. HDC 5+5: Five rodded and five unrodded HDCs (0%, 40%, 80%, 100% void, and 1 

g/cc). Each history has 12 rodded and 12 unrodded DC branches.

4. HDC 4+4: Four rodded and four unrodded HDCs (0%, 40%, 80% void, and 1 g/cc, for 

both CR = 0 and 1). Each history has 12 rodded and 12 unrodded DC branches.

5. HDC 3+3: Three rodded and 3 unrodded HDCs (0%, 40%, 80% void, for both CR = 0 

and 1). Each history has 12 rodded and 12 unrodded DC branches.

6. HDC 3+1: One rodded and three unrodded HDCs (0%, 40%, 80% void for CR = 0, and 

40% for CR =1). Each history has 12 rodded and 12 unrodded DC branches.

Figure 12. Σa

1 and νΣf

1 at BOL and EOL.
Figure 13. PMAXS HDC breakdown.

19
Figure 14. PMAXS HDC breakdown (cont.).

20
21

4.1.1. Results of Single CHAN Model 

Steady-state and transient results calculated for the three TRACE/PARCS models are 

presented in this section. All simulations were run with TRACE V5.509. 

HDC and HCR inputs for this case are shown in Figure 15. On the top of the channel, the 

coolant has an HDC of about 0.1 g/cc (i.e., 90% void). Steady-state results for each HDC case 

are summarized in Table 4, and they include core eigenvalue (Keff), 3-dimensional power 

peaking factor (Pxyz), core radial assembly power peaking factor (Pxy), and core axial power 

peaking factor (Pz). The value of Pxy = 1 because there is only one fuel assembly in the core. 

Figure 16 shows power transient oscillations induced by a pressure perturbation at the outlet 

pressure boundary. Overall, all history cases predict almost the same results for the steadystate and transient cases.

Figure 15. HDC and HCR input for single-CHAN model.

Table 4. Steady-state parameters

Steady-state

Parameters HDC: 12+12 HDC: 7+7 HDC: 5+5 HDC: 4+4 HDC: 3+3 HDC: 3+1

Keff 0.999154 0.999170 0.999218 0.999208 0.999208 0.999173

Pxyz 1.773 1.773 1.773 1.773 1.773 1.770

Pxy 1.0 1.0 1.0 1.0 1.0 1.0

Pz 1.773 1.773 1.773 1.773 1.773 1.770

Figure 16. Response to a pressure perturbation for single-CHAN model.
22

4.1.2. Results of Oskarshamn Core-wide Stability 

Figure 17 shows HDC and HCR inputs for this case for the different channels modeled. On the 

top of the channel, the coolant has an average HDC of about 0.2 g/cc (i.e., 75% void). Steadystate results for each history case are summarized in Table 5. There are no appreciable 

differences between HDC cases. However, the case of HDC 7+7 has the best agreement with 

the reference case HDC 12 + 12. Figure 18 shows core transient flow oscillations induced by 

control rod perturbation. Overall, all history cases predict almost the same results for the steadystate and transient.

Figure 17. HDC and HCR input for Oskarshamn

Table 5. Steady-state parameters for Oskarshamn

Steady-state

Parameters HDC: 12+12 HDC: 7+7 HDC: 5+5 HDC: 4+4 HDC: 3+3 HDC: 3+1

Keff 0.901037 0.901057 0.90119 0.901182 0.901181 0.900942

Pxyz 2.139 2.138 2.128 2.128 2.129 2.156

Pxy 1.482 1.482 1.482 1.481 1.481 1.481

Pz 1.315 1.314 1.307 1.307 1.308 1.333

Figure 18. Response to a control rod perturbation.

Wide Wide
23

4.1.3. Results of Ringhals Core-wide Stability 

HDC and HCR inputs for this case are shown in Figure 19. On the top of the channel, the 

coolant has an average HDC of about 0.2 g/cc (i.e., 75% void). Steady-state results for each 

history case are summarized in Table 6. Again, there are no appreciable differences between 

the history cases. The case of HDC 7+7 has best agreement with the reference case 

HDC 12+12. Figure 20 shows transient core flow stability induced by control rod perturbation. 

Again, all history cases predict almost the same results for the steady-state and transient.

Figure 19. HDC and HCR input for Ringhals.

Table 6. Steady-state parameters for Ringhals

Steady-state

Parameters HDC: 12+12 HDC: 7+7 HDC: 5+5 HDC: 4+4 HDC: 3+3 HDC: 3+1

Keff 0.925303 0.925328 0.925479 0.925469 0.925468 0.925368

Pxyz 2.388 2.386 2.373 2.374 2.375 2.378

Pxy 1.688 1.688 1.688 1.688 1.688 1.686

Pz 1.164 1.163 1.166 1.166 1.166 1.163

Figure 20. Core flow stability for Ringhals.

Wide Wide
24

4.1.4. Conclusion of DC History Effects 

Based on the above analysis of the HDC effects, the following conclusions can be reached:

1. BWR steady-state variables (i.e., Keff, Pxyz, Pxy, and Pz) are not very sensitive to the HDC 

and HCR inputs (history effects). Results of HDC 7+7 (20% void spacing) have the best 

agreement with the reference case HDC 12+12.

2. BWR transients are not highly sensitive to the HDC and HCR effects. Results of HDC 

7+7 (20% void apart) have best agreement with the reference case HDC 12+12.

3. Even the case of HDC 3+1 (3 uncontrolled HDC and 1 controlled HDC) does not show 

any significant deviation from the reference case.

4.2. DC BRANCH EFFECTS 

Figure 21 shows Σa

1 and νΣf

1 as a function of branch void faction at BOL and EOL. The black 

dashed line on each figure is the second-order polynomial fitting of the cross sections at 0, 40, 

and 70% void. At high void fraction (>70%) both Σa

1 and νΣf

1 curves show more nonlinearity (i.e., 

curvature) than those at low void fractions. Therefore the extrapolation of the branch XS from 

the 70% void may overestimate the XS at high void faction.

Figure 21. Σa

1 and νΣf

1 at BOL and EOL as function of branch void.
25

For the study of DC branch effects, the PMAXS file has been broken down into five PMAXS files

with less branches. These reduced PMAXS files are described as follows. 

1. DC 12+12: This is the reference case with seven rodded and seven unrodded HDCs 

(0%, 20%, 40%, 60%, 80%, 100% void, and 1 g/cm3, for both CR = 0 and 1). Each 

history has 12 rodded and 12 unrodded DC branches (0%, 10%, 20%, 30%, 40%, 50%, 

60%, 70%, 80%, 90%, 100% void, and 1 g/cc, for both CR = 0 and 1).

2. DC 7+7: Seven rodded and seven unrodded HDCs. Each history has seven rodded and 

seven unrodded DC branches (0%, 20%, 40%, 60%, 80%, 100% void, and 1 g/cm3

, for 

both CR = 0 and 1).

3. DC 5+5: Seven rodded and seven unrodded HDCs. Each history has five rodded and 

five unrodded DC branches (0%, 40%, 80%, 100% void, and 1 g/cm3, for both CR = 0 

and 1).

4. DC 4+4: Seven rodded and seven unrodded HDCs. Each history has four rodded and 

four unrodded DC branches (0%, 40%, 80% void, and 1 g/cc, for both CR = 0 and 1).

5. DC 3+3: Seven rodded and seven unrodded HDCs. Each history has 12 and 12 

unrodded DC branches (0%, 40%, 80% void, for both CR = 0 and 1).

Steady-state and transient results calculated for the three TRACE/PARCS models are 

presented in the following sections.

4.2.1. Results for Single CHAN Model 

Steady-state results for each DC branch case are summarized in Table 7. Figure 22 shows the 

steady-state core axial DC. The coolant in the top fuel channel is above 80% void. Figure 23

shows the power transient induced by an outlet pressure perturbation. It shows clearly that the 

cases of DC 4+4 and DC 3+3 differ significantly from the reference case. It indicates that the 

100% void branch is necessary to accurately interpolate XS for the core region with a void 

fraction greater than 80%. In other words, XS extrapolation from 80% cannot accurately 

generate XS for the core upper region with the void fraction > 80%.

In addition, the results of the case of DC 4+4 are almost identical to those of DC 3+3. For this 

simulation, the core inlet coolant is subcooled with the density of 0.77 g/cc, which is higher than 

the saturated density of 0.738 g/cm3 for 0% void. For the case DC 3+3, the XS for the inlet DC

has to be extrapolated from the branch of 0.738 g/cm3 (0% void). However, the case of DC 4+4 

has a XS branch at 1 g/cm3

, so the XS for the inlet DC is interpolated between 1 g/cm3 and 

0.73808 g/cm3 for this case. Good agreement between the cases DC 4+4 and 3+3 means that 

XS extrapolation is good at the high-density end (the branch XS curve at high density is more 

linear than at low density). 

Table 7. Steady-state parameters for single-CHAN model

Steady-state

Parameters DC: 12+12 DC: 7+7 DC: 5+5 DC: 4+4 DC: 3+3

Keff 0.99917 0.999342 0.999729 0.999021 0.999021

Pxyz 1.773 1.772 1.783 1.794 1.794

Pxy 1 1 1 1 1

Pz 1.773 1.772 1.783 1.794 1.794
26

Figure 22. Axial coolant density for single-CHAN model.

Figure 23. Response to a pressure perturbation for single-CHAN model.
27

4.2.2. Results for Oskarshamn Core-wide Stability

Steady-state results for each DC branch case are summarized in Table 8. Figure 24 shows 

steady-state core axial DC profiles. The core exit average DC is about 80%. The coolant in the 

top of a hot fuel channel can be up to 90% void. Figure 25 shows core flow oscillations induced 

by a control rod perturbation. Again, it shows clearly that the case of DC 4+4 differs significantly 

from the other cases when the 100% void branch is removed. 

Table 8. Steady-state parameters for Oskarshamn

Steady-state

Parameters DC: 12+12 DC: 7+7 DC: 5+5 DC: 4+4

Keff 0.901057 0.901236 0.90158 0.900782

Pxyz 2.138 2.132 2.12 2.144

Pxy 1.482 1.482 1.482 1.481

Pz 1.314 1.309 1.298 1.317

Figure 24. Core axial coolant density for Oskarshamn

Figure 25. Response to a control rod perturbation for Oskarshamn

Wide
28

4.2.3. Results for Ringhals Core-wide Stability

Steady-state results for each DC branch case are summarized in Table 9. Figure 26 shows 

steady-state core axial DC profiles. The core exit average DC is about 75%. For this case, the 

coolant in the top of a hot fuel channel can be up to 85% void. Figure 27 shows core flow 

oscillations induced by a control rod perturbation. The case of DC 5+5 shows some deviation 

from the reference case and the case of DC 7+7. The case of DC 4+4 differs significantly from 

the other cases when the 100% void branch is removed. 

Table 9. Steady-state parameters for Ringhals

Steady-state

Parameters DC: 12+12 DC: 7+7 DC: 5+5 DC: 4+4

Keff 0.925328 0.92542 0.92595 0.925837

Pxyz 2.386 2.382 2.357 2.361

Pxy 1.688 1.688 1.688 1.687

Pz 1.163 1.163 1.171 1.172

Figure 26. Core axial coolant density for Ringhals.
29

Figure 27. Response to a control rod perturbation for Ringhals.

4.2.4. 4.2.4 Conclusions of DC Branch Effects 

Based on the above analysis of the DC branch effects, the following conclusions are drawn.

1. BWR steady-state variables (i.e., Keff, Pxyz, Pxy, and Pz) are not highly sensitive to the DC 

branches. However, results of DC 7+7 (20% void spacing) have the best agreement with 

the reference case DC 12+12.

2. BWR transients show sensitivity to the DC branches, particularly in the core upper 

region where the void fraction is above the range of the DC branches.

Wide

31

5.0 EFFECTIVE DELAYED NEUTRON FRACTION 

5.1. DEFINITION OF βeff

The energy collapse of fine group XS data into few group constants requires proper weighting of 

the fine-energy and -space fluxes and reaction rates. Typically, XS are weighted by the local 

flux to maintain constant reaction rates. However, integral variables like the delayed neutron 

fraction, βeff, require weighting by both the forward and adjoint fluxes, also known as bilinear 

weighting. The following relation provides the optimal bilinear weighting to collapse the fineenergy and -space fluxes for βeff.

βeff, i = ∑ ∫ dr ∫ dEχd,i

m (E)φ∗�r, E� ∫ dE′ m βimν ∑ φ�r, E′� m

f

∑ ∫ dr ∫ dEχt m m(E)φ∗�r, E� ∫ dE′ν ∑ φ�r, E′� m

f

=

∑ βi

m ∫ dr ∫ dEχd,im (E)φ∗�r, E� ∫ dE′ m ν ∑ φ�r, E′� m

f

∑ ∫ dr ∫ dEχt m m(E)φ∗�r, E� ∫ dE′ν ∑ φ�r, E′� m

f

, i = 1, . . ,6 (5.1)

where

m = the mth isotope

φ∗(r, E) = adjoint flux

φ(r, E) = neutron flux

βi = delayed neutron fraction of the ith energy group for fissions of isotope m

χd,i

m (E) = delayed neutron spectrum of the ith energy group for fissions of isotope m

χt

m(E) = total fission energy spectrum (prompt + delayed)

ν ∑ φ(r, E′ f ) = total fission neutrons per fission (prompt + delayed)

If φ∗�r, E� is assumed constant, we obtain the “fission-weighted” βeff as in Eq. (5.2).

βeff,i = ∑ βi

m ∫ dr ∫ dE′ m ν ∑ φ�r, E′� m

f

∑m ∫ dr ∫ dE′ ν ∑ φ�r, E′� m

f

, i = 1, ... ,6 (5.2)

5.2. CALCULATION OF βeff USING A SINGLE-ASSEMBLY LATTICE

In practice, the problem is more complex because the fine-energy and -space fluxes are not 

known exactly. They must be approximated based on the results of single assembly lattice 

calculation. Lattice codes like TRITON, CASMO, or LANCER only calculate the fine-energy and 

-space fluxes for a single bundle, and they represent the core surrounding the bundle via 

boundary conditions (BCs). Two BCs are typically used:

1. Reflective BC (denoted Infinite BC). The bundle is assumed to be surrounded by a 

large number of similar bundles, so any neutron leakage that occurs will be 

compensated by neutrons that leak from surrounding bundles into the specified one. The 

BC used for this case is a pure reflective BC, where all the neutrons leaked are 
32

immediately returned to the bundle. This calculation produces an eigenvalue, which is 

appropriately labeled Kinf.

2. Reflective BC with Critical Spectrum (denoted Critical BC). In reality, cores are 

typically loaded in a checkerboard pattern, in which high-reactivity bundles are 

surrounded by low-reactivity bundles (once- or twice-burned); therefore, net leakage 

occurs from the high-reactivity to the low-reactivity bundles. To compensate for this 

effect, the critical BC adds the concept of a material buckling, Bm

2 , which acts as an 

absorber and effectively removes neutrons from the high-reactivity bundles and adds 

them to the low-reactivity ones. For low-reactivity bundles, the material buckling is 

negative and effectively reduces the fraction of neutrons absorbed in the bundle; for 

high-reactivity bundles, the material buckling is positive and increases the absorption 

fraction.

The critical BC physically represents more accurately the neutron balances in a real reactor, 

and it has been proved in practice that reaction rates calculated by the critical BC (i.e., using the 

material buckling approximation) are a better approximation than rates calculated with the 

infinite BC.

5.3. PHYSICAL DESCRIPTION OF THE PHENOMENA

The adjoint calculation, however, may be a bit counterintuitive. The adjoint, Φ+ (r, E), is not 

related to a production rate, but it is defined as the probability that a neutron of energy E at 

location r will produce an additional fission neutron. The following example illustrates the 

process. For this example, we have two bundles side by side, as shown in Figure 28.

Figure 28. Two bundle sample configuration.

Figure 29 represents a hypothetical neutron path in which a neutron is born on the left bundle 

but is leaked and interacts in the right bundle, where it produces either a capture or a fission. If 

the final interaction is a fission event, this path would contribute to the adjoint flux at the path 

originating point; if the neutron is absorbed, it does not.

Since lattice calculations are performed on a single lattice, we must model this type of neutron 

path using an approximation through a BC. In all cases (either infinite or critical approximations), 

lattice codes use reflective BCs, as illustrated in Figure 30.
33

Figure 29. Hypothetical neutron path born on left bundle and absorbed on the right one.

Figure 30. Illustration of single-lattice approximation, 

where the neutron path is reflected back into the bundle.

Both approximations (Infinite and critical) simulate the neutron path of Figure 29 by reflecting it 

as in Figure 30. In the case of the infinite approximation, the neutron is allowed to interact with 

the bundle. In the case of the critical approximation, the bundle absorption is modified with a 

material buckling, Bm

2 , that is adjusted to make the bundle critical. The question is which of the 

two approximations more closely reflects the real adjoint flux?

Let us assume a limiting case in which the control rod is inserted. In this case, the calculated 

eigenvalue for a single lattice will be significantly subcritical, and a large material buckling 

correction will be required for the critical approximation. Thus the control bundle scenario will 

result in the largest difference between the two approximations. We need to remember that the 

adjoint flux we are trying to estimate is the probability that at the end of the neutron path, there 

will be a fission-induced neutron or a capture.

If we follow the neutron path on the real reactor (Figure 29), we see that the neutron exits the 

left bundle through the left side of the control rod and then must cross the right half of the 
34

control rod before entering the right bundle. If the neutron is absorbed by either the left or the 

right side of the control rod, this event counts as a capture and the adjoint value is reduced. If 

the neutron makes it through the control rod, then the probability of fission in the bundle, which 

depends on its ratio of 235U to other absorbers, need not be corrected by the presence of the 

control rod because the rod was taken into account explicitly when the path was calculated.

These physical phenomena are more closely simulated by the infinite approximation, in which

the neutron path must cross the left side of the control rod twice (one to the right and once again 

to the left to enter the bundle); therefore, the bundle absorption and fission cross sections are 

unmodified.

If we use the critical approximation, we must decrease the absorption in the bundle using a 

negative material buckling (because K<1 in a controlled bundle). The net effect of this negative 

buckling on the critical approximation is to artificially increase the probability that the neutron 

path will create a fission neutron, which would bias the adjoint flux in the positive direction.

Figure 32 and Figure 31 illustrates the effect of neutron leakages from other bundles. In this 

particular example, we assume that the top and bottom neighbors have higher reactivity and 

leaked neutrons into our bundle, but the left and right bundles have lower reactivity and receive 

a net leakage. When we ask “what is the probability the red neutron path will generate a 

fission?” (i.e., what is the adjoint), the answer does not change, and we see that the leakage 

pattern is second order and does not impact the adjoint flux.

Figure 31. Illustration of the effect of neutrons leaked from surrounding bundles.
35

5.4. NUMERICAL SIMULATION

Calculations indicate that the βeff calculated with adjoint critical flux weighting (or K-ratio 

weighting) could be ~20% lower for rodded assemblies than for unrodded ones. However, βeff

calculated with adjoint infinite flux weighting for rodded assemblies is almost the same as for 

unrodded ones. This difference of 20% in βeff is very significant because it affects the reactivity 

directly when measured in dollars. For the same perturbation, if βeff is 20% lower, the reactivity 

and associated transient power response is 20% higher.

A numerical study was performed to confirm that the infinite approximation yields a better 

adjoint value for βeff weighting. For this study, a “mini core” in a 2×2 colorset configuration was 

used, as shown in Figure 32. The colorset contains two types of assemblies: Assembly A is a 

low-burnup assembly with high reactivity, and Assembly B is high-burnup with low reactivity; 

thus there is a net leak of neutrons from the A assemblies to the B assemblies. It is assumed 

that this colorset will repeat itself in the core for a typical checkerboard pattern. Since to the right 

side of the B assembly there would be an A assembly, the simulation used periodic BCs, which 

are more realistic for this colorset. For this calculation, all bundles are unrodded (the control rod 

is fully out).

Figure 32. Example colorset configuration.

In this study, all of the assemblies in the colorset are the same Peach Bottom type-3c fuel used 

in previous sections. Three different burnup and void configurations were considered for the 

study, as shown in Table 10. Separately, single-bundle lattice calculations were performed for 

each of the configurations and assembly types.

The TRITON code was used to calculate βeff for each assembly, using the neutron flux φ(r, E)

and its adjoint flux φ∗(r, E) directly calculated in the above colorset configuration. It was 

compared with the βeff calculated from single assembly lattice calculations using the two 

approximations based on the calculated adjoint critical flux and the adjoint infinite flux for the 

single assembly. The βeff results are summarized in Table 11 through Table 13 along with the 

eigenvalues for each of the configurations modeled.
36

Table 10. Three colorset burnup configurations studied

Configuration

A B

Burnup 

(GWd/MTU)

Coolant density

(g/cc)

Burnup

(GWd/MTU)

Coolant density

(g/cc)

C1 3.79 0.73808 (0% void) 24.70 0.73808

C2 7.25 0.45721 (40% void) 31.11 0.45721

C3 8.40 0.24657 (70% void) 27.50 0.24657

Table 11. Comparison of βeff for configuration C1—low burnup case

Group Single Assembly

Adjoint critical flux Colorset

weighting

Adjoint infinite flux 

weighting

A B A B A B

1 1.83E-04 1.12E-04 1.78E-04 1.17E-04 1.78E-04 1.17E-04

2 1.29E-03 9.61E-04 1.26E-03 1.00E-03 1.25E-03 1.01E-03

3 1.17E-03 8.24E-04 1.14E-03 8.60E-04 1.13E-03 8.64E-04

4 2.39E-03 1.65E-03 2.33E-03 1.72E-03 2.32E-03 1.73E-03

5 8.17E-04 6.36E-04 7.97E-04 6.63E-04 7.93E-04 6.67E-04

6 2.85E-04 2.02E-04 2.78E-04 2.11E-04 2.77E-04 2.12E-04

Total 6.13E-03 4.39E-03 5.98E-03 4.58E-03 5.95E-03 4.60E-03

Note: Kinf of A: 1.0590; Kinf of B: 0.9092; Kinf of colorset: 0.9854.

Table 12. Comparison of βeff for configuration C2—medium burnup case

Group Single assembly

Adjoint critical flux Colorset

weighting

Adjoint infinite flux 

weighting

A B A B A B

1 1.72E-04 1.04E-04 1.65E-04 1.09E-04 1.64E-04 1.10E-04

2 1.24E-03 9.29E-04 1.19E-03 9.70E-04 1.18E-03 9.79E-04

3 1.12E-03 7.99E-04 1.08E-03 8.34E-04 1.07E-03 8.41E-04

4 2.28E-03 1.61E-03 2.19E-03 1.68E-03 2.18E-03 1.70E-03

5 8.04E-04 6.47E-04 7.75E-04 6.75E-04 7.70E-04 6.80E-04

6 2.78E-04 2.03E-04 2.68E-04 2.12E-04 2.66E-04 2.13E-04

Total 5.89E-03 4.29E-03 5.67E-03 4.48E-03 5.63E-03 4.52E-03

Note: Kinf of A: 1.0952; Kinf of B: 0.9023; Kinf of colorset: 0.9987.
37

Table 13. Comparison of βeff for configuration C3 – high burnup case

Group Single assembly

Adjoint critical flux Colorset

weighting

Adjoint infinite flux 

weighting

A B A B A B

1 1.58E-04 1.03E-04 1.54E-04 1.07E-04 1.53E-04 1.08E-04

2 1.17E-03 9.18E-04 1.14E-03 9.55E-04 1.13E-03 9.64E-04

3 1.06E-03 8.00E-04 1.03E-03 8.33E-04 1.02E-03 8.39E-04

4 2.16E-03 1.63E-03 2.11E-03 1.69E-03 2.10E-03 1.70E-03

5 7.98E-04 6.73E-04 7.80E-04 7.00E-04 7.76E-04 7.04E-04

6 2.72E-04 2.12E-04 2.65E-04 2.21E-04 2.64E-04 2.22E-04

Total 5.62E-03 4.33E-03 5.49E-03 4.51E-03 5.45E-03 4.54E-03

Note: Kinf of A: 1.0590; Kinf of B: 0.9092; Kinf of colorset: 0.9818.

The colorset calculations indicate that, overall, the adjoint infinite flux weighted βeff gives more 

accurate results (~1% error) than the adjoint critical flux option (~5% error). Therefore, the 

recommendation from this study is to use the infinite approximation to estimate the adjoint flux 

for βeff weighting.

Differences in the actual implementation of the βeff weighting in lattice codes (CASMO, HELIOS, 

TRITON, and LANCER02) may result in measurable differences between the βeff values 

provided by different codes for the same conditions.

39

6.0 CODE-TO-CODE COMPARISON

Each of the three TRACE/PARCS models was run with the CASMO, HELIOS, and TRITON XS, 

respectively. Steady-state results are summarized in Table 14, and transient results are shown 

in Figure 33, Figure 34, and Figure 35. Note that delayed neutron fractions of each XS set used 

for this comparison analysis are fission-weighted.

Comparisons for the steady-state variables show good agreement between the XS sets. 

Differences in Keff are less than 200 pcm for the single-CHAN case, 150 pcm for the 

Oskarshamn, and 580 pcm for the Ringhals. In addition, the maximum relative difference in all 

power peaking factors is less than 3%. 

Comparisons for the transients show good agreement between the CASMO and HELIOS XS 

sets. Poor agreement of TRITON is believed to be due to the over predicted fission-weighted β

(because of a code bug in the fission weighting, see Figure 8). 

Table 14. Steady-state parameters

Steady-state

Parameters Single-CHAN Oskarshamn Ringhals

CASMO HELIOS TRITON CASMO HELIOS TRITON CASMO HELIOS TRITON

Keff 0.99915 1.0014 1.00102 0.901 0.90156 0.90231 0.9253 0.931071 0.92919

Pxyz 1.773 1.766 1.792 2.14 2.189 2.127 2.388 2.389 2.302

Pxy 1 1 1 1.482 1.488 1.482 1.688 1.683 1.685

Pz 1.773 1.766 1.792 1.315 1.345 1.295 1.164 1.195 1.167

Figure 33. Single channel results.
40

Figure 34. Oskarshamn results.

Figure 35. Ringhals results.

Wide
41

7.0 ISSUES WITH TWO-GROUP KINETICS EQUATIONS

There is a degree of inconsistency in the traditional formulation of two-group kinetics equations. 

In the traditional formulation, the two-group XS data are obtained by an energy collapse with 

forward flux weighting, while β is weighted with bilinear adjoint flux and forward flux weighting. 

This inconsistency in the traditional formulation of two-group kinetics equations is a well-known 

issue. One of the major problems with this formulation is that it has considered the relative

importance difference of delayed neutrons but has not considered the effects of scattering terms 

on the neutron importance. Some researchers [6, 7] have suggested that bilinear weighting of 

the collapsed XS be used to provide a consistent and rigorous mathematical foundation for the 

formulation of the two-group kinetics equations. Unlike the linear weighting method, bilinear 

weighting employs both the forward and adjoint spectra for group collapsing, including 1/v, XS, 

and β. 

Wade and Bucher [6] performed a detailed analysis of bilinear-weighted XS. They found that the 

errors in reactivity worth of scattering materials can be reduced significantly when XS are

weighted bilinearly (adjoint flux and forward flux). Lee et al. [7] performed an assessment study 

of the bilinear weighted two-group kinetics equations with a 1-dimensional, fine-mesh diffusion 

code. Their study shows that bilinear weighing of the two-group kinetics equations has good 

agreement with a 97-group formulation of kinetics equation.

For the calculation of βeff, our review of XS methods has discovered differences in the actual 

implementation of Eq. (3.4) in lattice codes (CASMO, HELIOS, TRITON, and LANCER02), 

which may result in measurable differences between the βeff values provided by different codes 

for the same conditions. For example, for some configurations, the βeff calculated with adjoint 

critical flux weighting or K-ratio weighting could be up to 20% lower for rodded assemblies than 

for unrodded ones. However, the βeff calculated with adjoint infinite flux weighting for rodded 

assemblies is almost the same as for unrodded ones. The difference of up to 20% in βeff can 

result in errors of up to 20% in transient power. 

Two-group spatial kinetics equations are extensively used for light water reactor transient 

analysis, and are the methodology used by TRACE-PARCS. The inconsistency between the 

methodologies for generating βeff and the traditional formulation of the two-group cross sections, 

may pose some concerns about the accuracy of transient analysis. However, the sensitivity 

analyses performed in Section 5.4 indicate that these inconsistency errors are minimized if the 

adjoint infinite flux weighting is used for βeff generation, which is the recommended methodology

used in SCALE/TRITON and CASMO. For the cases studied in in Section 5.4, this error is of 

the order of a few percent, but it affects mostly the controlled bundles, which contribute less to 

the transient power; thus we judge this error acceptable for confirmatory calculations.

43

8.0 BWR PARCS CROSS SECTIONS GUIDANCE 

This section provides guidance on how to choose XS histories and branches for BWR analysis. 

The guidance herein is applicable to all BWR designs. 

For BWR steady-state and transient analysis, PARCS uses two-energy-group XS for each 

computational node in the 3-dimensional grid. The PARCS XS are parameterized as a function 

of four instantaneous state variables: CR, TF, DC, and PC. The XS also depend on the 

isotopics (i.e., concentration of 235U, 239Pu ...), which are parameterized as a function of control

rod and moderator density history variables. 

In a typical calculation, the fuel temperature, moderator density, and soluble boron 

concentration are calculated by TRACE for a coupled TRACE/PARCS analysis. The 

instantaneous CR is provided by the user in the input deck. The historic control rod and 

moderator density values are provided by a steady-state core-follow simulator, which has 

followed the core operation since the initial loading up to the time of the transient to be 

calculated with TRACE/PARCS. 

It is obvious that XS tabulation plays an important role in the coupled neutronics/thermalhydraulics analysis. A good XS tabulation can reduce the error caused by XS and therefore 

improve the accuracy of the analysis. The XS tabulation defines the XS data set structure, that 

is, what histories and branches are considered and how many of them should be included in the 

XS set. Generally speaking, XS should be developed based on the specific transient to be 

analyzed. For example, for a BWR analysis, HDC and branch XS should cover the whole range 

of core DC change under both the steady-state and the expected transient conditions to avoid 

the need for table extrapolation. 

In a typical BWR core, the DC changes significantly from the core inlet to the outlet. The coolant 

at the core inlet is subcooled with a density of ~760 kg/m3

, and the DC is reduced to ~200 kg/m3

on average at the top of the core with a void faction of ~75%. 

Therefore, for a BWR core operated under such a wide distribution of DC, history effects of DC

on XS due to the neutron spectrum difference should be considered when generating XS. In 

addition, the effects of HCR on XS should be considered as well.

A BWR core usually consists of hundreds of fuel assemblies. A fuel assembly typically is made 

up of several axial segments with different enrichment levels and burnable poisons. For each 

fuel segment, two-group PARCS XS including histories and instantaneous branches are 

generated using a lattice physics code (e.g., TRITON, CASMO, or HELIOS). A processing code, 

GenPMAXS, is employed to process the two-group XS and kinetics data generated with the 

lattice code calculations and store them in PMAXS format for each fuel segment; the PMAXS 

files can be read by PARCS directly. An XS file in PMAXS format provides all of the data

necessary to perform PARCS simulations for steady-state and transient applications. These 

PMAXS data consist of two-group macroscopic XS, microscopic XS of Xe/Sm, group-wise 

discontinuity functions (DF) for higher-order solution methods, and the kinetics data (e.g., 

delayed neutron parameters and inverse velocities). The lattice code process to generate XS is 

described in detail in Spriggs et al. [2, 8].

The structure of PMAXS for a sample BWR fuel type is given in Figure 36. This sample PMAXS 

file contains three HDCs at DCs corresponding to 0%, 40%, and 70% void at full pressure. Each 

HDC contains three instantaneous DC branches, three TF branches, and 3 soluble boron 

concentration branches. The PMAXS data structure shown in Figure 36 is repeated for control 

and uncontrolled bundles for each burnup step.
44

Figure 36. Typical PMAXS structure.

8.1. HISTORY XS

Macroscopic XS have a strong dependence on history effects of DC and control rod position. 

Figure 37 shows Σa

1 and νΣf

1 of a BWR assembly as a function of history void faction at BOL and 

EOL, respectively. The black dashed line on each figure is the second-order polynomial fitting of 

the XS data at 0, 40, and 70% void, and the blue solid line is the XS calculated by the lattice 

code at that void. 

It is seen that both Σa

1 and νΣf

1 curves show more nonlinearity at high void fraction (>70%) than 

at low void fraction, so the extrapolation of the XS above the 70% void may overestimate the XS

at high void faction (see Figure 36). However, a recent sensitivity study of HDC effects 

concludes that neither BWR steady-state nor transient analysis is highly sensitive to HDC and 

HCR effects. It is found that three HDCs at 0, 40, 70% (or 80%) voids are sufficiently accurate 

for most BWR applications. A history point at the 90% void would be recommended for highpower density reactors to improve the XS interpolation and/or extrapolation in the core upper 

region with a void fraction of above 80%. 
45

Figure 37. Void history effect on Σa

1 and νΣf

1.

8.2. BRANCH XS

At each time-step of a transient calculation, XS for each PARCS neutronic node are updated 

based on the instantaneous value of DC, CR, TF, and boron concentration at that node. The 

instantaneous XS are obtained by interpolation/extrapolation between branch XS tabulated in 

the PMAXS files. Figure 38 shows Σa

1 and νΣf

1 of a BWR assembly as a function of branch void 

at EOL [1]. The BOL Σa

1 and νΣf

1 are the same as those shown in Figure 2. The black dashed 

line on each figure is the second-order polynomial fitting of the XS at 0, 40, and 70% void. 
46

Figure 38. Brach void fraction effect on Σa

1 and νΣf

1.

At high void fraction (>70%) both Σa

1 and νΣf

1 curves show more nonlinearity than at low void 

fraction, so the extrapolation of the XS from the 70% void may overestimate the XS at high void 

fraction; however sensitivity studies show that BWR steady-state analysis is not very sensitive 

to the DC branch effects [1]. The same sensitivity analysis shows that transient results may be 

sensitive to the DC branches, especially when the DC of the core upper region is significantly 

larger than that of the upper branch void. The study shows that four DC branches at 0, 40, 70, 

and 90% voids are recommended for high-power-density BWR transient analysis. Most currentgeneration lattice codes cannot model the 100% void case accurately because of its extreme 

hard spectrum, but this result is dependent on the lattice methodology used. It is suggested that 

the same DC branches be used for the controlled DC branches.

For TF branches, three branches at 500, 950, and 1500 K are sufficient to calculate the TF

reactivity feedback. In addition, a DC branch with DC = 1000 kg/m3 is needed to model 

shutdown conditions accurately. If boron injection is modeled in a BWR transient analysis, XS 

boron branches should be included.

8.3. GUIDELINES

For most BWR applications, the XS branch structure of Table 15 and Table 16 is recommended; 

however, specific studies may require a different structure. The user must review the 

requirements and ensure the applicability of the XS set to the particular application. For 

anticipated-transient-without-scram calculations, an additional set of boron concentration 

branches are needed, but that topic was not part of this study.

For βeff calculations, it is recommended that the infinite adjoint approximation be used to weight 

the fine-energy and -space results.
47

Table 15. Recommended history branches

History coolant 

density 

(kg/m3)

Percentage 

void at 7 Mpa

(%)

Control 

rod

Fuel 

temperature 

(K)

738 0

0

950

457 40

247 70

738 0

457 40 1

247 70

Table 16. Recommended instantaneous branches

Instantaneous 

coolant density 

(kg/m3

)

Percentage

void at 7 Mpa

(%)

Control 

rod

Fuel 

temperature 

(K)

738 0

0

950

457 40

247 70

106 90

738 0

457 40 1

247 70

106 90

738 0

0

500

457 40

247 70

106 90

738 0

457 40 1

247 70

106 90

738 0

0

1500

457 40

247 70

106 90

738 0

457 40 1

247 70

106 90

1000

0

950

500

293

1

950

500

293

49

9.0 REFERENCES

1. Core Design and Operation Data for Cycles 1 and 2 of Peach Bottom 2, EPRI NP-563,

Electric Power Research Institute, June 1978.

2. G. D. Spriggs et al., “Calculation of the delayed neutron effectiveness factor using ratios 

of K-eigenvalues,” Annals of Nuclear Energy 28 (2001) 477–487.

3. TRITON source code, SCALE 6.1.

4. CASMO-4 User’s Manual, SSP-01/400 Rev 3. 

5. Y. Xu and T. Downar, GenPMAXS— Code for Generating the PARCS Cross Section 

Interface File PMAXS, PU/NE-00-20, Purdue University, April 2009.

6. D. C. Wade and R. G. Bucher, “Conservation of the adjoint neutron spectrum by use of 

bilinear-weighted cross sections and its effect on fast reactor calculations,” Nuclear 

Science and Engineering 64, 517–538 (1977).

7. C. Lee, T. J. Downar, K. O. Ott, “An assessment of consistent bilinear weighted twogroup spatial kinetics for MOX fuel applications,” in PHYSOR 2000 : 2000 ANS 

International Topical Meeting on Advances in Reactor Physics and Mathematics and 

Computation into the Next Millennium (2000).

8. B. Ade, SCALE/TRITON Primer: A Primer for Light Water Reactor Lattice Physics 

Calculations, NUREG/CR-7041/ ORNL/TM-2011/21, November 2012.




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