Nuclear Science

ISBN 978-92-64-99007-4

OECD/NEA Nuclear Science Committee
Working Party on the Physics of Plutonium Fuels and Innovative Fuel Cycles

Physics of Plutonium Recycling
Volume VIII
Results of a Benchmark Consideringa High-temperature Reactor (HTR)
Fuelled with Reactor-grade Plutonium

J.G. Hosking, T.D. Newton
Serco Assurance
A Report Produced for Nexia Solutions© OECD 2007
NEA No. 6200
NUCLEAR ENERGY AGENCY
ORGANISATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENTORGANISATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENTThe OECD is a unique forum where the governments of 30 democracies work together to address the economic, socialand environmental challenges of globalisation. The OECD is also at the forefront of efforts to understand and to help
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***This work is published on the responsibility of the Secretary-General of the OECD. The opinions expressed andarguments employed herein do not necessarily reflect the official views of the Organisation or of the governments of its member
countries.

NUCLEAR ENERGY AGENCYThe OECD Nuclear Energy Agency (NEA) was established on 1st February 1958 under the name of the OEEC European
Nuclear Energy Agency. It received its present designation on 20th April 1972, when Japan became its first non-European full
member. NEA membership today consists of 28 OECD member countries: Australia, Austria, Belgium, Canada, the Czech
Republic, Denmark, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Japan, Luxembourg, Mexico, theNetherlands, Norway, Portugal, Republic of Korea, the Slovak Republic, Spain, Sweden, Switzerland, Turkey, the United
Kingdom and the United States. The Commission of the European Communities also takes part in the work of the Agency.
The mission of the NEA is:

to assist its member countries in maintaining and further developing, through international co-operation, thescientific, technological and legal bases required for a safe, environmentally friendly and economical use of nuclear
energy for peaceful purposes, as well as
to provide authoritative assessments and to forge common understandings on key issues, as input to government
decisions on nuclear energy policy and to broader OECD policy analyses in areas such as energy and sustainable
development.

Specific areas of competence of the NEA include safety and regulation of nuclear activities, radioactive waste
management, radiological protection, nuclear science, economic and technical analyses of the nuclear fuel cycle, nuclear law andliability, and public information. The NEA Data Bank provides nuclear data and computer program services for participating
countries.
In these and related tasks, the NEA works in close collaboration with the International Atomic Energy Agency in Vienna,
with which it has a Co-operation Agreement, as well as with other international organisations in the nuclear field.
© OECD 2007
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Cover credit: PBMR, South Africa.

FOREWORD

This report contains the results submitted for the five phases of the OECD/NEA benchmark
concerning a high-temperature reactor (HTR) fuelled with reactor-grade plutonium. The benchmark
has been organised under the auspices of the OECD/NEA Working Party on the Physics of Plutonium
Fuels and Innovative Fuel Cycles. Specifications for these benchmark phases are reproduced inAppendix A. Although the benchmark has been specifically designed to provide intercomparisons forplutonium and thorium fuels, phases of calculations for uranium fuel have also been included. The
purpose of these phases is to identify any increased uncertainties, relative to uranium fuel, that areassociated with plutonium and thorium fuel.
The five phases of the benchmark cover:x

Phase 1: An infinite array of fuel pebbles containing uranium fuel.

x

Phase 2: An infinite array of fuel pebbles containing plutonium fuel.

x

Phase 3: A whole-core calculation for a plutonium-fuelled system.

x

Phase 4: A whole-core calculation for a uranium-fuelled system.

x

Phase 5: An infinite array of fuel pebbles containing thorium/uranium fuel.

Acknowledgements
Serco Assurance would like to thank all contributors to the benchmark.
The support and sponsorship of Nexia Solutions (formerly BNFL R&T) to the contribution madeby Serco Assurance is gratefully acknowledged.

3

TABLE OF CONTENTS

Foreword ............................................................................................................................................

3

Chapter 1

INTRODUCTION........................................................................................................

7

Chapter 2

METHODS AND CODES USED BY THE PARTICIPANTS ................................ 9
2.1 Deterministic calculation details ............................................................................ 9
2.1.1 Resonance shielding..................................................................................... 9
2.1.2 Double heterogeneity ................................................................................... 10
2.1.3 Number of energy groups ............................................................................ 10

Chapter 3

RESULTS .....................................................................................................................
3.1 Phase 1 ...................................................................................................................
3.1.1 Phase 1a: Reactivities and reaction rates .....................................................
3.1.2 Phase 1b: Reactivities and reaction rates .....................................................
3.1.3 Spectra .........................................................................................................
3.1.4 Burn-up ........................................................................................................
3.2 Phase 2 ...................................................................................................................
3.2.1 Phase 2a: Reactivities and reaction rates .....................................................
3.2.2 Phase 2b: Reactivities and reaction rates .....................................................
3.2.3 Spectra .........................................................................................................
3.2.4 Burn-up ........................................................................................................
3.3 Phase 3 ...................................................................................................................
3.3.1 Core reactivity..............................................................................................
3.3.2 Spectra .........................................................................................................
3.3.3 Cross-sections ..............................................................................................
3.4 Phase 4 ...................................................................................................................
3.4.1 Core reactivity..............................................................................................
3.4.2 Spectra .........................................................................................................
3.4.3 Cross-sections ..............................................................................................
3.5 Phase 5 ...................................................................................................................
3.5.1 Phase 5a: Reactivities and reaction rates .....................................................
3.5.2 Phase 5b: Reactivities and reaction rates .....................................................
3.5.3 Spectra .........................................................................................................
3.5.4 Burn-up ........................................................................................................

Chapter 4

CONCLUSIONS .......................................................................................................... 53

13
13
13
15
17
17
28
28
30
32
32
38
38
39
39
42
42
43
44
45
45
47
48
49

References .......................................................................................................................................... 55
Figures ................................................................................................................................................ 57
Appendix A – Benchmark definition.................................................................................................. 75
Appendix B – Energy group schemes used in the deterministic calculations .................................... 95

5

Chapter 1
INTRODUCTION

This report contains results submitted for the five phases of the OECD NEA benchmark for a
HTR fuelled with reactor-grade plutonium. The benchmark has been organised as part of the work by
the Working Party on the Physics of Plutonium Fuels and Innovative Fuel Cycles. Specifications ofthese benchmark phases are reproduced in Appendix A of this report. Although the benchmark has
been specifically designed to provide inter-comparisons for plutonium and thorium fuels, phasesof calculations for uranium fuel have also been included. The purpose of these phases is to identify
any increased uncertainties, relative to uranium fuel, that are associated with the plutonium andthorium fuels.
The five phases of the benchmark are defined as follows:x

Phase 1a: Uranium fuel: Cell calculation employing a spherical outer boundary.

x

Phase 1b: Uranium fuel: Cell calculation employing a cubic outer boundary.

x

Phase 2a: Plutonium fuel: Cell calculation employing a spherical outer boundary.

x

Phase 2b: Plutonium fuel: Cell calculation employing a cubic outer boundary.

x

Phase 3: Plutonium fuel: Whole-core calculations.

xPhase 4: Uranium fuel: Whole-core calculations.

xPhase 5a: Thorium/uranium fuel: Cell calculation employing a spherical outer boundary.

x

Phase 5b: Thorium/uranium fuel: Cell calculation employing a cubic outer boundary.

Participants have been invited to submit results from deterministic calculations and, whereappropriate, Monte Carlo calculations. The following participants have submitted results:
xParticipant A: J.G. Hosking and T.D. Newton, Serco Assurance, UK.

xParticipant B: O. Koberl, CEA, France.

x

Participant C: U. Colak, Nuclear Engineering Dept., Hacettepe University, Turkey.

xParticipant D: M. Tombakoglu, Nuclear Engineering Dept., Hacettepe University, Turkey.

xParticipant E: P. Morris, Nexia Solutions, UK (formerly BNFL R&T, UK).

x

Participant F: S. Goluoglu, Nuclear Science & Technology Division, ORNL, USA.

7

Chapter 2
METHODS AND CODES USED BY THE PARTICIPANTS

Table 2.1 gives an outline of the details of the calculational methods and codes used by each of
the benchmark participants.
2.1 Deterministic calculation details
This subsection contains details of some of the deterministic PBMR modelling methods
employed by the participants in order to obtain their submitted results.
2.1.1 Resonance shielding
Participant A has used the subgroup treatment within WIMS9 to calculate resonance shielding for
U and 238U in Phases 1 and 4 of the study, for 239Pu and 240Pu in Phases 2 and 3 of the study and for232
Th and 233U in Phase 5 of the study. All other shieldable nuclides have been treated using
equivalence theory. The use of the subgroup treatment allows the effects of the double heterogeneity
in the geometry to be accounted for directly in the resonance shielding of the cross-sections.
235

The subgroup method uses a set of library resonance integrals, interpolated to the correcttemperature, that are functions of nuclide, reaction, energy and the scattering power in the problem.
These are fitted by a set of subgroup cross-sections and weights for each reaction, energy group
and nuclide. An equation is then formed and solved for the subgroup fluxes and the final shieldedcross-sections formed from the subgroup cross-sections, weights and fluxes. To treat the double
heterogeneity of the particulate fuel a special formation is required for the solution of the subgroupflux equations. This is performed using collision probabilities, treating each energy group in turn.
Initially, collision probabilities are solved using the macroscopic subgroup total cross-sections, for theheterogeneous fuel particle with an additional outer layer representing the associated amount of carbon
support matrix material. A homogenised version of the particle is then considered and a homogenised
subgroup total cross-section is derived which has the same escape probability as the heterogeneousparticle. This homogenised subgroup total cross-section is then used in the central region of the pebble
to allow collision probabilities to be derived for all the pebble regions. The pebble and particle collision
probabilities are then combined to form a full set of collision probabilities for the complete pebble/
particle system. These are then used to solve for the subgroup fluxes used in deriving the shieldedcross-sections.
Participant B has generated self-shielded cross sections using the APOLLO2 TR model, which isa generalisation of the statistical model and is available for the whole resonance-energy range [2].
In Phases 2 and 3, Participant B has calculated self-shielded cross-sections for all plutonium isotopes.
Participant F has calculated self-shielded cross sections using the CENTRM/PMC/CHOPS
modules of the SCALE code system.
92.1.2 Double heterogeneity
Participant A’s treatment of double heterogeneity for deriving neutron fluxes is the same as that
described above for the derivation of the subgroup fluxes. Collision probabilities are solved using themacroscopic total cross-sections, for the heterogeneous fuel particle with an additional outer layer
representing the associated amount of carbon support matrix material. A homogenised version of theparticle is then considered and a homogenised total cross-section is derived which has the same escapeprobability as the heterogeneous particle. The homogenised total cross-section is then used in thecentral region of the pebble to allow collision probabilities to be derived for all the pebble regions. The
pebble and particle collision probabilities are then combined to form a full set of collision probabilities
for the complete pebble/particle system. These are then used to solve for the system fluxes.
The APOLLO2 double heterogeneity treatment used by Participant B is based on assumptions
made by J.R. Askew. The main principle of Participant B’s method is to apply the escape probability
for a single absorbing particle. An assumption is made that there is an isotropic incident flux uponeach particle. Slowing-down within the particles is neglected.
Participant F has treated double heterogeneity by first calculating the flux disadvantage factors forthe grains and then using these factors to create the homogenised grain/matrix mixture cross-sections.
The homogenised cross-sections are used on second pass to create the final resonance-shieldedcross-sections that represent the fuel pebbles.
2.1.3 Number of energy groups
Participant A has used a 172-energy-group scheme for Phases 1, 2 and 5 of the calculations. For
Phases 3 and 4, Participant A has used a 10-group energy scheme. The quoted Phase 3 and 4 resultsfrom Participant A have been corrected for the energy-group condensation effect of using a 10-groupscheme as opposed to the 172-group library scheme. The energy-group boundaries for both the
172-group and 10-group schemes used by Participant A are given in Appendix B.
Participant B has performed calculations using the 172-energy-group scheme provided inAppendix B for all phases of the benchmark.
Participant F has used a 238-group scheme, a description of which is also given in Appendix B.

1011None submitted
MCNP-4B + BURN-HUNEM
to calculate depletion

VSOP-99
None submitted
XSDRNPM within SCALE

KENO V.a within SCALE

Deterministic

Monte Carlo

Deterministic

Monte Carlo
Deterministic

Monte Carlo

Nexia Solutions

ORNL

None submitted
MCNP-4B

TRIPOLI4 [7]Monte Carlo

Deterministic
Monte Carlo

APOLLO2 [5]MONK9 [4]Monte Carlo

Deterministic

WIMS9 [3]Code used

Calculation
type
Deterministic

Hacettepe University
M. Tombakoglu
Hacettepe University
M. Tombakoglu

Hacettepe University
U. Colak

CEA

Serco Assurance

Participant

Data based on
ENDF/B-V libraries.

ENDF/B-VI.2 and V.
Results presented in this report wereobtained using a thermal S(D,E) treatment
for the graphite moderator.
Data required by BURN-HUNEM tocalculate depletion are taken from:
Decay constants – BNL website
Fission Q values – MCNP data
Fission yields – WIMSD-4 library
Data based on
ENDF/B-V and JEF1 libraries.
–
Data based on
ENDF/B-V libraries.

––
ENDF-B/VI continuous data.

JEF2.2 continuous data.

CEA93.V6 [6] – based on JEF2.2.

Development WIMS9 library based on
JEF2.2 cross-section set.
BINGO based on
JEF2.2 continuous data.

Data library usedNotes

XSDRNPM is a 1-D deterministic transport code using S8 quadrature.
Cell calculations performed with reflective boundary conditions at the edgeof a cube (which contains the pebble).

Cell calculations performed with white boundary conditions on the surface
of a pebble.

In all phases critical bucklings and migration areas are derived fromextrapolated lengths, with the extrapolated lengths being calculated using
MCNP surface tallies.

A cubic unit-cell model is used for Phases 1b and 2b. This provides apebble packing fraction of 51%.

A columnar hexagonal unit-cell model is used for Phases 1a, 2a and 3 toachieve a pebble packing fraction of 61%.

Phases 1a and 2a employ a collision probability solution.
Phases 1b and 2b employ a CACTUS3D characteristics transport solution.
As there is no facility to perform a critical-buckling search in the MONK
Monte Carlo code, critical bucklings must be determined from a series ofscoping calculations. As it was not practical to perform numerous Monte
Carlo calculations to find the exact critical buckling, the criterion was
relaxed. A buckling that gave a value of k-effective within ±250 pcm ofunity was found and taken to be the critical buckling. All particles in every
pebble were modelled in both cell and whole core calculations.
The deterministic whole-core calculations (which employed a transport
solution) did not include streaming effects.
In the Monte Carlo whole-core calculations, the outer graphite shell of thepebbles was homogenised with the helium coolant to avoid streaming
effects. This was done to enable comparisons to be made between thedeterministic and Monte Carlo whole-core calculations. The TRIPOLI4
Monte Carlo models included an exact treatment to model the pebbles.
–
MCNP-4B does not provide a S(D,E) treatment for graphite at atemperature of 1 000 K. Therefore, for the hot calculations, the graphite
temperature has been assumed to be 800 K.
–

Table 2.1. Inter-comparison of methods

Chapter 3
RESULTS

The submitted results are given in Tables 3.1 to 3.28. Spectra plots are given in Figures 3.1 to 3.16.
For brevity, calculation results are specified as originating from Participants A, B, C, D, E or F.
The letter code used for each of the participants is:
A. J.G. Hosking and T.D. Newton, Serco Assurance, UK.
B. O. Koberl, CEA, France.
C. U. Colak, Nuclear Engineering Department, Hacettepe University, Turkey.
D. M. Tombakoglu, Nuclear Engineering Department, Hacettepe University, Turkey.
E. P. Morris, Nexia Solutions, UK (formerly BNFL R&T, UK).
F. S. Goluoglu, Nuclear Science & Technology Division, ORNL, USA.
3.1 Phase 1
3.1.1 Phase 1a: Reactivities and reaction rates
Phase 1a considers a fuel pebble with a reflective boundary condition on the outer surface of thesphere. Results for Monte Carlo calculations are given in Table 3.1 and for deterministic calculationsin Table 3.2.
Participant A’s Monte Carlo result is for a sphere with an isotropic flux boundary condition at theouter surface, as there is no physical mechanism to model a reflective boundary condition for this typeof artificial spherical outer boundary geometry. This explains the artificially high k-effective forParticipant A. It is to be noted that, for Phase 1b (a reflected sphere in a box) Participant A’s results
using the Monte Carlo and deterministic treatments are in good agreement.
Participant C’s calculations in Phase 1a were performed for a single fuel element surrounded by aspherical shell containing helium. The boundary condition for Participant C’s Phase 1a calculation is
not known.
Participant D has modelled a columnar hexagonal unit-cell model to achieve a pebble packing
fraction of 61% in this phase of the benchmark. This avoids the difficulty in applying a reflective
boundary condition on the surface of a sphere, however, it does not directly correspond to the reflected
sphere benchmark.

13

Table 3.1. Phase 1a results – Monte Carlo calculations
Parameter
kinf with zerobuckling and at
T = 293.6 K
B2cr at T = 293.6 K
kinf(B2cr) at
T = 293.6 K
M2 at T = 293.6 K
kinf with zero buckling
and at T = 1 000 K
keff(B2cr) at
T = 1 000 K, using
B2cr at T = 239.6 K
B2cr at T = 1 000 K
kinf(B2cr) at
T = 1 000 K
M2 at T = 1 000 K
U238
G235
G238
C

Units
–

A
1.5364 r
100 pcm

B

C

cm–2
–
cm2
–

D
1.5108EFE
1.50600

F
1.50748

4.16E-04
1.44551056.61.4476 r
100 pcm–

0.92366cm2–

3.46E-04
1.3699

cm2
–
–
–
–

1032.99.26570.13490.00430.5212

U238 = 238Ucap(epithermal)/238Ucap(thermal): ratio of epithermal to thermal 238U captures.
G235 = 235Ufis(epithermal)/235Ufis(thermal): ratio of epithermal to thermal 235U fissions.
G238 = 238Ufis/235Ufis: ratio of fissions in 238U to fissions in 235U.
C = 238Ucap/235Ufis: ratio of captures in 238U to fissions in 235U.

Table 3.2. Phase 1a results – deterministic calculations
Parameter
kinf with zerobuckling and at
T = 293.6 K
B2cr at T = 293.6 K
kinf(B2cr) at
T = 293.6 K
M2 at T = 293.6 K
kinf with zero buckling
and at T = 1 000 K
keff(B2cr) at
T = 1 000 K, using
B2cr at T = 239.6 K
B2cr at T = 1 000 K
kinf(B2cr) at
T = 1 000 K
M2 at T = 1 000 K
U238
G235
G238
C

Units
–

A
1.51759B
1.52232

C

D

cm–2
–

4.408E-04
1.460644.512E-4
1.46511cm2–

1045.01.426291030.788–

0.900360.90130

0.89483cm2–

3.369E-04
1.36483

3.455E-4
1.37096

3.088E-04
1.352

cm2
–
–
–
–

1082.99.334387
0.132525
0.004091
0.528724

1073.6
9.198128
0.132625
0.004208
0.520845

1139.89.519364
0.129738
0.003992
0.535737

4.130E-04
1.455

1.43126

14

1101.61.40773

Participant A has used collision probabilities with a white outer boundary condition to model thesphere in the deterministic treatment. Participants B and E have also used a white boundary conditionin these calculations.
Deterministic solutions for k-infinity at room temperature for Participants A and B are in very
reasonable agreement (473 pcm) with the APOLLO2 result slightly higher, as has been observed inother benchmarks. The reaction rate ratios are also in reasonable agreement Both participants haveused JEF2.2 nuclear data.
Participants E and F also show close agreement between their results (148 pcm) at room
temperature. Here both participants have used ENDF/B-V nuclear data
The relatively large difference between the cell reactivities calculated by Participants E and F andthose calculated by Participants A and B, about 1 320 pcm, is probably at least partly due to thedifferent nuclear data set employed.
There is a noticeable, unexplained, difference between Participants E and F results at 1 000 K.
Participants A and B have used the following expressions to calculate kinf(B2cr) and the migrationarea:
kinf(B2cr) = P/A
where P is the productions in the solution to which a critical buckling has been applied and A is theabsorptions in the solution to which a critical buckling has been applied:M2 = (kinf(B2cr) – 1)/B2cr.
Using these expressions, the migration area, buckling and kinf(B2cr) are consistent.
3.1.2 Phase 1b: Reactivities and reaction rates
Participant A is not able to give Monte Carlo results for reaction rates in cases where there is anapplied buckling. In these cases the MONK code solves the k-infinity calculation and reports reaction
rates for k-infinity, not k-effective. In addition, a prediction of k-effective with the applied buckling isgiven.
The geometry in Participant C’s Phase 1b model consists of a cubic array assembled from cubic
unit cells. The critical size and buckling were evaluated by performing a series of calculations in order
to ascertain the critical configuration.
The deterministic reaction rate ratios from Participant A for Phases 1a and 1b are very similar; theonly major difference in the two benchmark phases is the outer boundary condition. Given the very
similar deterministic results from Participants A and B for Phase 1a, it can be concluded that these twoparticipants are in good agreement for Phase 1 of the benchmark.
The Monte Carlo and deterministic calculations from Participant A are in good agreement. For
k-infinity at room temperature the difference is (1.5222 – 1.51785) 435 pcm. At temperature, using thecritical buckling at 293.6 K the difference is (0.9013 – 0.90040) 90 pcm. This also suggests that theMonte Carlo and deterministic evaluations of the temperature coefficient are in good agreement.

15

Table 3.3. Phase 1b results – Monte Carlo calculations
Parameter
kinf with zerobuckling and at
T = 293.6 K
B2cr at T = 293.6 K
kinf(B2cr) at
T = 293.6 K
M2 at T = 293.6 K
kinf with zero buckling
and at T = 1 000 K
keff(B2cr) at
T = 1 000 K, using
B2cr at T = 239.6 K
B2cr at T = 1 000 K
kinf(B2cr) at
T = 1 000 K
M2 at T = 1 000 K
U238
G235
G238
C

Units
–

A
1.5222 r
100 pcmcm–2–

2.64E-04

cm2
–
–
cm2
–

B

C
1.50697D
1.5107EF
1.50770 r
43 pcm

2.56E-04
1.4504
1706.71.4329 r
100 pcm
0.9013 r
100 pcm

1.40889 r
47 pcm
0.92372

2.03E-4

2.05E-04
1.3702

cm2
–
–
–
–

1766.29.17510.13250.00420.5178

Table 3.4. Phase 1b results – deterministic calculations
Parameter
kinf with zerobuckling and at
T = 293.6 K
B2cr at T = 293.6 K
kinf(B2cr) at
T = 293.6 K
M2 at T = 293.6 K
kinf with zero buckling
and at T = 1 000 K
keff(B2cr) at
T = 1 000 K, using
B2cr at T = 239.6 K
B2cr at T = 1 000 K
kinf(B2cr) at
T = 1 000 K
M2 at T = 1 000 K
U238
G235
G238
C

Units
–

A
1.51785cm–2–

3.245E–04
1.48131cm2–

1483.2
1.42668–0.90040cm2–

2.480E-04
1.38343

cm2
–
–
–
–

1546.19.323949
0.132374
0.004086
0.528246

B

C

16

D

E

F

The k-infinity (at room temperature) from the Monte Carlo calculation of Participant D is about
1 150 pcm lower than that of Participant A and Participant F’s is about 1 450 pcm lower. Participant A
used JEF2.2 nuclear data while Participants D and F used ENDF/B-V data. Participant C’s Monte Carloestimate of k-infinity (at room temperature) is 1 523 pcm lower, ENDF/B-VI data was used. The reaction
rate ratios, while being in reasonable agreement for Participants A and D, are significantly different
for Participant C, suggesting large differences between ENDF/B-V and ENDF/B-VI data.
There is also a large difference between Participant F’s Monte Carlo evaluation of the cell
reactivity at 1 000 K and the equivalent reactivity from Participant A.
3.1.3 Spectra
The spectrum for the Phase 1a Monte Carlo calculation submitted by Participant D is given in
Figure 3.1. The form of this spectrum appears to be in reasonable agreement with the Phase 1 spectrafrom other participants (Figures 3.2 to 3.4).
Figure 3.2 shows the spectra for deterministic calculations for Phase 1a from Participants A, B
and F.
Figure 3.3 compares the Phase 1b Monte Carlo spectra provided by Participants C and D.
However, it is not stated as to which material or averaged materials Participant C’s spectrum applies.
Figure 3.4 shows the deterministic solution spectra from Participant A for both Phases 1a and 1b
averaged over the matrix and fuel particles. It illustrates that the spectra from the two benchmark
phases are almost identical.
3.1.4 Burn-up
Participant A’s results from the burn-up calculation of Phase 1 (Tables 3.5, 3.9, 3.12 and 3.14)
indicate that irradiation in a critical spectrum, as opposed to a case where no buckling is applied,
increases 238U losses through capture and enhances the build-up of plutonium isotopes.
The results from Participant A for Phase 1a and Phase 1b are essentially identical.
For the majority of nuclides, the results from Participant A and B agree to within ~5%. However,there are significantly larger differences between their results for 237Np, 238Pu, 234U and 237U (althoughthese nuclides are only present in trace quantities).
The 239Pu content at 80 GWd/Te from Participant E indicates a higher 238U capture cross-section
in ENDF/B-V than JEF2.2, Participant A’s calculation. The results are also consistent with a lower
239
Pu capture cross-section in ENDF/B-V than JEF2.2.

17

18

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.93E-03
0.00E+00
0.00E+00
2.13E-02

10 000
MWd/te9.01E-09
6.43E-11
8.01E-10
4.41E-10
7.24E-13
1.51E-11
6.64E-14
4.08E-07
1.17E-08
9.10E-05
1.00E-05
1.70E-06
5.99E-08
1.49E-13
2.05E-09
1.65E-03
4.71E-05
5.03E-08
2.12E-02

20 000
MWd/te1.03E-07
1.17E-09
2.20E-08
9.91E-09
3.41E-11
9.21E-10
9.38E-12
1.59E-06
9.76E-08
1.45E-04
2.85E-05
9.74E-06
7.30E-07
8.73E-13
3.83E-09
1.42E-03
8.69E-05
1.00E-07
2.10E-02

30 000
MWd/te
3.54E-07
5.00E-09
1.34E-07
5.05E-08
2.80E-10
9.16E-09
1.35E-10
3.28E-06
3.30E-07
1.78E-04
4.73E-05
2.24E-05
2.75E-06
1.26E-12
5.97E-09
1.21E-03
1.21E-04
1.36E-07
2.09E-02

40 000
MWd/te
7.61E-07
1.21E-08
4.58E-07
1.43E-07
1.10E-09
4.38E-08
8.81E-10
5.55E-06
7.83E-07
1.98E-04
6.37E-05
3.68E-05
6.54E-06
1.87E-12
8.94E-09
1.03E-03
1.51E-04
1.85E-07
2.08E-02

Units are u1024 atoms cm–3

50 000
MWd/te1.26E-06
2.15E-08
1.10E-06
2.92E-07
2.93E-09
1.43E-07
3.42E-09
8.22E-06
1.53E-06
2.08E-04
7.74E-05
5.04E-05
1.22E-05
2.79E-12
1.36E-08
8.70E-04
1.75E-04
2.25E-07
2.06E-02

60 000
MWd/te
1.79E-06
3.17E-08
2.21E-06
4.93E-07
6.10E-09
3.60E-07
1.04E-08
1.11E-05
2.63E-06
2.09E-04
8.74E-05
6.24E-05
1.97E-05
4.13E-12
2.09E-08
7.26E-04
1.97E-04
2.60E-07
2.05E-02

70 000
MWd/te2.27E-06
4.10E-08
3.86E-06
7.30E-07
1.08E-08
7.72E-07
2.57E-08
1.42E-05
4.09E-06
2.08E-04
9.53E-05
7.09E-05
2.86E-05
5.44E-12
3.17E-08
5.99E-04
2.14E-04
2.96E-07
2.03E-02

Table 3.5. Phase 1a – Monte Carlo calculations – results from Participant D – irradiation with a buckling included

80 000
MWd/te2.64E-06
4.80E-08
6.09E-06
9.78E-07
1.68E-08
1.46E-06
5.45E-08
1.73E-05
5.93E-06
2.04E-04
1.01E-04
7.68E-05
3.88E-05
6.96E-12
4.73E-08
4.87E-04
2.29E-04
3.22E-07
2.01E-02

19

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.9259E-03
1.0000E-19
1.0000E-19
2.1288E-02

10 000
MWd/te1.0698E-08
6.8437E-11
9.4636E-10
6.6307E-10
1.1398E-12
1.9097E-11
9.9221E-14
3.7784E-07
1.2780E-08
9.4500E-05
1.1547E-05
2.0356E-06
7.4614E-08
1.4148E-16
2.0000E-09
1.6584E-03
4.6823E-05
4.6237E-08
2.1153E-02

20 000
MWd/te1.1378E-07
1.1400E-09
2.3366E-08
1.3499E-08
4.9366E-11
1.0102E-09
1.0739E-11
1.3792E-06
9.8382E-08
1.4725E-04
3.2197E-05
1.0809E-05
8.5056E-07
5.2408E-16
3.8253E-09
1.4321E-03
8.6056E-05
8.3292E-08
2.1013E-02

30 000
MWd/te3.8085E-07
4.6793E-09
1.3683E-07
6.5549E-08
3.7924E-10
9.4126E-09
1.5171E-10
2.8115E-06
3.2236E-07
1.7653E-04
5.2328E-05
2.4506E-05
3.1262E-06
1.1199E-15
5.8570E-09
1.2341E-03
1.1984E-04
1.1473E-07
2.0868E-02

40 000
MWd/te8.0081E-07
1.1030E-08
4.4932E-07
1.7938E-07
1.4504E-09
4.3472E-08
9.3609E-10
4.5399E-06
7.5109E-07
1.9198E-04
6.9334E-05
3.9761E-05
7.3402E-06
1.9675E-15
8.6782E-09
1.0585E-03
1.4915E-04
1.4045E-07
2.0718E-02

Units are u1024 atoms cm–3

50 000
MWd/te1.3076E-06
1.9305E-08
1.0791E-06
3.6050E-07
3.7987E-09
1.3713E-07
3.6688E-09
6.4493E-06
1.4440E-06
1.9871E-04
8.2811E-05
5.4046E-05
1.3578E-05
3.1927E-15
1.3054E-08
9.0208E-04
1.7457E-04
1.6334E-07
2.0564E-02

60 000
MWd/te1.8184E-06
2.8030E-08
2.1319E-06
5.9777E-07
7.8427E-09
3.4057E-07
1.0782E-08
8.4671E-06
2.4444E-06
1.9959E-04
9.3076E-05
6.5891E-05
2.1692E-05
5.0167E-15
1.9874E-08
7.6245E-04
1.9653E-04
1.8424E-07
2.0406E-02

70 000
MWd/te2.2585E-06
3.5735E-08
3.6873E-06
8.6935E-07
1.3747E-08
7.1841E-07
2.5950E-08
1.0529E-05
3.7714E-06
1.9632E-04
1.0064E-04
7.4635E-05
3.1413E-05
7.7460E-15
3.0069E-08
6.3787E-04
2.1535E-04
2.0352E-07
2.0244E-02

80 000
MWd/te2.5740E-06
4.1310E-08
5.7942E-06
1.1492E-06
2.1371E-08
1.3467E-06
5.3883E-08
1.2575E-05
5.4153E-06
1.8997E-04
1.0601E-04
8.0148E-05
4.2425E-05
1.1740E-14
4.4498E-08
5.2695E-04
2.3130E-04
2.2146E-07
2.0077E-02

Table 3.6. Phase 1a – deterministic calculations – results from Participant A – irradiation with a buckling included

20

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
1.93E-03––
2.13E-02

10 000
MWd/te
1.11E-08
0.00E+00
1.04E-09
7.09E-10
1.24E-12
2.16E-11
1.17E-13
4.25E-07
1.36E-08
9.34E-05
1.18E-05
2.11E-06
8.00E-08
0.00E+00
2.18E-09
1.65E-03
4.77E-05
5.34E-08
2.12E-02

20 000
MWd/te
1.17E-07
0.00E+00
2.53E-08
1.42E-08
5.30E-11
1.12E-09
1.21E-11
1.63E-06
1.12E-07
1.45E-04
3.25E-05
1.11E-05
9.02E-07
0.00E+00
4.20E-09
1.42E-03
8.73E-05
1.03E-07
2.10E-02

30 000
MWd/te3.87E-07
0.00E+00
1.47E-07
6.84E-08
4.02E-10
1.03E-08
1.68E-10
3.46E-06
3.79E-07
1.72E-04
5.24E-05
2.49E-05
3.28E-06
0.00E+00
6.49E-09
1.22E-03
1.21E-04
1.48E-07
2.09E-02

40 000
MWd/te8.04E-07
0.00E+00
4.78E-07
1.85E-07
1.52E-09
4.71E-08
1.03E-09
5.77E-06
9.01E-07
1.86E-04
6.91E-05
3.99E-05
7.65E-06
0.00E+00
9.76E-09
1.05E-03
1.50E-04
1.90E-07
2.07E-02

Units are u1024 atoms cm–3

50 000
MWd/te1.30E-06
0.00E+00
1.14E-06
3.69E-07
3.95E-09
1.47E-07
3.98E-09
8.44E-06
1.76E-06
1.92E-04
8.22E-05
5.38E-05
1.41E-05
0.00E+00
1.50E-08
8.88E-04
1.74E-04
2.27E-07
2.06E-02

60 000
MWd/te
1.79E-06
0.00E+00
2.24E-06
6.07E-07
8.08E-09
3.63E-07
1.16E-08
1.13E-05
3.00E-06
1.92E-04
9.20E-05
6.51E-05
2.23E-05
0.00E+00
2.32E-08
7.48E-04
1.95E-04
2.61E-07
2.04E-02

70 000
MWd/te2.20E-06
0.00E+00
3.85E-06
8.75E-07
1.40E-08
7.61E-07
2.76E-08
1.43E-05
4.66E-06
1.89E-04
9.92E-05
7.32E-05
3.21E-05
0.00E+00
3.57E-08
6.24E-04
2.13E-04
2.92E-07
2.03E-02

80 000
MWd/te2.49E-06
0.00E+00
6.01E-06
1.15E-06
2.17E-08
1.42E-06
5.69E-08
1.74E-05
6.72E-06
1.82E-04
1.04E-04
7.81E-05
4.32E-05
0.00E+00
5.34E-08
5.13E-04
2.27E-04
3.20E-07
2.01E-02

Table 3.7. Phase 1a – deterministic calculations – results from Participant B – irradiation with a buckling included

21

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.93E-03
0.00E+00
0.00E+00
2.13E-02

10 000
MWd/te9.01E-09
6.43E-11
8.01E-10
4.41E-10
7.24E-13
1.51E-11
6.64E-14
4.08E-07
1.17E-08
9.10E-05
1.00E-05
1.70E-06
5.99E-08
1.49E-13
2.05E-09
1.65E-03
4.71E-05
5.03E-08
2.12E-02

20 000
MWd/te1.04E-07
1.18E-09
2.30E-08
9.96E-09
3.50E-11
9.73E-10
9.91E-12
1.69E-06
1.03E-07
1.51E-04
2.89E-05
9.86E-06
7.29E-07
1.70E-12
4.23E-09
1.42E-03
8.68E-05
1.08E-07
2.10E-02

30 000
MWd/te
3.66E-07
5.17E-09
1.39E-07
5.15E-08
2.81E-10
9.32E-09
1.43E-10
3.63E-06
3.54E-07
1.82E-04
4.77E-05
2.34E-05
2.81E-06
3.02E-12
6.58E-09
1.21E-03
1.20E-04
1.55E-07
2.09E-02

40 000
MWd/te8.01E-07
1.27E-08
4.80E-07
1.47E-07
1.15E-09
4.75E-08
9.80E-10
5.96E-06
8.50E-07
2.08E-04
6.33E-05
3.90E-05
6.70E-06
3.53E-12
9.96E-09
1.04E-03
1.50E-04
1.90E-07
2.07E-02

Units are u1024 atoms cm–3

50 000
MWd/te
1.35E-06
2.32E-08
1.18E-06
3.02E-07
3.15E-09
1.59E-07
4.38E-09
8.78E-06
1.66E-06
2.30E-04
7.66E-05
5.43E-05
1.24E-05
5.73E-12
1.52E-08
8.79E-04
1.74E-04
2.38E-07
2.06E-02

60 000
MWd/te1.96E-06
3.52E-08
2.39E-06
5.10E-07
6.62E-09
4.03E-07
1.31E-08
1.19E-05
2.84E-06
2.36E-04
8.88E-05
6.72E-05
1.98E-05
6.80E-12
2.31E-08
7.42E-04
1.95E-04
2.81E-07
2.04E-02

70 000
MWd/te2.53E-06
4.76E-08
4.23E-06
7.62E-07
1.17E-08
8.80E-07
3.37E-08
1.56E-05
4.46E-06
2.41E-04
9.63E-05
7.95E-05
2.85E-05
1.15E-11
3.52E-08
6.21E-04
2.11E-04
3.33E-07
2.02E-02

Table 3.8. Phase 1a – Monte Carlo calculations – results from Participant D – irradiation with a buckling included

80 000
MWd/te3.05E-06
5.81E-08
6.65E-06
1.03E-06
1.85E-08
1.67E-06
6.92E-08
1.87E-05
6.50E-06
2.45E-04
1.03E-04
8.78E-05
3.85E-05
1.32E-11
5.24E-08
5.16E-04
2.25E-04
3.34E-07
2.00E-02

22

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.9259E-03
1.0000E-19
1.0000E-19
2.1288E-02

10 000
MWd/te8.9324E-09
5.6583E-11
7.3494E-10
5.5376E-10
8.7861E-13
1.3625E-11
6.3452E-14
3.3204E-07
1.0816E-08
8.3734E-05
1.0531E-05
1.7137E-06
6.3895E-08
8.8739E-17
1.5874E-09
1.6564E-03
4.6321E-05
4.1166E-08
2.1168E-02

20 000
MWd/te9.7320E-08
9.6480E-10
1.9121E-08
1.1678E-08
3.9993E-11
7.6810E-10
7.4094E-12
1.2423E-06
8.5968E-08
1.3124E-04
3.0007E-05
9.3778E-06
7.5672E-07
3.4425E-16
3.1092E-09
1.4252E-03
8.5768E-05
7.6553E-08
2.1040E-02

30 000
MWd/te3.3087E-07
4.0179E-09
1.1672E-07
5.8185E-08
3.1959E-10
7.5433E-09
1.1162E-10
2.5822E-06
2.8871E-07
1.5830E-04
4.9456E-05
2.1713E-05
2.8596E-06
7.7086E-16
4.8841E-09
1.2213E-03
1.2007E-04
1.0821E-07
2.0904E-02

40 000
MWd/te7.0257E-07
9.5569E-09
3.9643E-07
1.6227E-07
1.2615E-09
3.6425E-08
7.2792E-10
4.2378E-06
6.8625E-07
1.7338E-04
6.6143E-05
3.5767E-05
6.8549E-06
1.4209E-15
7.4351E-09
1.0400E-03
1.4997E-04
1.3533E-07
2.0762E-02

Units are u1024 atoms cm–3

50 000
MWd/te1.1538E-06
1.6820E-08
9.7895E-07
3.3057E-07
3.3882E-09
1.1936E-07
2.9941E-09
6.1042E-06
1.3409E-06
1.8097E-04
7.9507E-05
4.9174E-05
1.2879E-05
2.4229E-15
1.1480E-08
8.7869E-04
1.7593E-04
1.6036E-07
2.0613E-02

60 000
MWd/te1.6096E-06
2.4510E-08
1.9792E-06
5.5326E-07
7.1352E-09
3.0640E-07
9.1853E-09
8.1111E-06
2.2999E-06
1.8362E-04
8.9765E-05
6.0518E-05
2.0816E-05
4.0017E-15
1.7879E-08
7.3538E-04
1.9826E-04
1.8392E-07
2.0457E-02

70 000
MWd/te2.0036E-06
3.1351E-08
3.4903E-06
8.0946E-07
1.2702E-08
6.6530E-07
2.2987E-08
1.0194E-05
3.5867E-06
1.8286E-04
9.7379E-05
6.9165E-05
3.0404E-05
6.4838E-15
2.7545E-08
6.0855E-04
2.1723E-04
2.0623E-07
2.0294E-02

80 000
MWd/te2.2902E-06
3.6416E-08
5.5752E-06
1.0738E-06
1.9982E-08
1.2794E-06
4.9511E-08
1.2293E-05
5.1960E-06
1.7965E-04
1.0283E-04
7.5016E-05
4.1311E-05
1.0281E-14
4.1338E-08
4.9699E-04
2.3308E-04
2.2751E-07
2.0125E-02

Table 3.9. Phase 1a – deterministic calculations – results from Participant A – irradiation with no buckling included

23

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
0.0000E+00
1.9259E-03
0.0000E+00
0.0000E+00
2.1288E-02

10 000
MWd/te4.25E-09
2.59E-11
2.66E-10
1.37E-10
2.59E-13
4.38E-12
0.00E+00
3.47E-07
8.35E-09
8.84E-05
5.84E-06
8.08E-07
2.19E-08
9.46E-13
1.73E-05
1.65E-03
4.76E-05
4.42E-08
2.12E-02

20 000
MWd/te
5.53E-08
5.48E-10
8.37E-09
3.53E-09
1.44E-11
3.02E-10
0.00E+00
1.44E-06
7.46E-08
1.54E-04
1.89E-05
5.29E-06
3.10E-07
1.86E-12
1.64E-05
1.40E-03
8.95E-05
9.35E-08
2.10E-02

30 000
MWd/te2.17E-07
2.66E-09
5.78E-08
2.06E-08
1.32E-10
3.34E-09
0.00E+00
3.19E-06
2.63E-07
2.00E-04
3.42E-05
1.38E-05
1.33E-06
2.78E-12
1.55E-05
1.17E-03
1.26E-04
1.41E-07
2.10E-02

40 000
MWd/te
5.21E-07
7.21E-09
2.17E-07
6.55E-08
5.87E-10
1.79E-08
0.00E+00
5.53E-06
6.38E-07
2.30E-04
4.89E-05
2.49E-05
3.51E-06
3.78E-12
1.46E-05
9.76E-04
1.58E-04
1.88E-07
2.08E-02

Units are u1024 atoms cm–3

50 000
MWd/te9.50E-07
1.41E-08
5.87E-07
1.50E-07
1.75E-09
6.44E-08
0.00E+00
8.33E-06
1.27E-06
2.47E-04
6.21E-05
3.70E-05
7.20E-06
4.85E-12
1.37E-05
7.99E-04
1.85E-04
2.34E-07
2.06E-02

60 000
MWd/te
1.46E-06
2.27E-08
1.28E-06
2.78E-07
4.02E-09
1.80E-07
0.00E+00
1.15E-05
2.21E-06
2.59E-04
7.31E-05
4.85E-05
1.24E-05
6.04E-12
1.28E-05
6.44E-04
2.08E-04
2.80E-07
2.04E-02

70 000
MWd/te1.96E-06
3.13E-08
2.42E-06
4.48E-07
7.76E-09
4.21E-07
0.00E+00
1.49E-05
3.47E-06
2.64E-04
8.20E-05
5.82E-05
1.93E-05
7.29E-12
1.20E-05
5.08E-04
2.25E-04
3.21E-07
2.04E-02

80 000
MWd/te2.44E-06
3.93E-08
4.12E-06
6.46E-07
1.31E-08
8.69E-07
0.00E+00
1.85E-05
5.10E-06
2.64E-04
8.88E-05
6.57E-05
2.74E-05
8.56E-12
1.11E-05
3.93E-04
2.40E-04
3.59E-07
2.02E-02

Table 3.10. Phase 1a – deterministic calculations – results from Participant E – irradiation with no buckling included

24

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.93E-03
0.00E+00
0.00E+00
2.13E-02

10 000
MWd/te9.10E-09
6.50E-11
8.36E-10
4.46E-10
7.19E-13
1.57E-11
7.24E-14
3.95E-07
1.12E-08
9.09E-05
1.00E-05
1.71E-06
6.05E-08
1.22E-12
2.01E-09
1.65E-03
4.71E-05
4.87E-08
2.12E-02

20 000
MWd/te1.03E-07
1.17E-09
2.22E-08
9.88E-09
3.46E-11
9.29E-10
9.01E-12
1.56E-06
9.54E-08
1.47E-04
2.85E-05
9.71E-06
7.28E-07
2.20E-12
4.02E-09
1.42E-03
8.69E-05
9.84E-08
2.10E-02

30 000
MWd/te
7.77E-07
1.22E-08
4.55E-07
1.44E-07
1.11E-09
4.40E-08
8.73E-10
5.60E-06
7.85E-07
1.97E-04
6.22E-05
3.80E-05
6.64E-06
3.52E-12
8.99E-09
1.03E-03
1.51E-04
1.85E-07
2.08E-02

40 000
MWd/te1.30E-06
2.21E-08
1.13E-06
2.98E-07
2.97E-09
1.44E-07
3.55E-09
8.34E-06
1.55E-06
2.07E-04
7.46E-05
5.19E-05
1.24E-05
5.21E-12
1.38E-08
8.71E-04
1.75E-04
2.31E-07
2.06E-02

Units are u1024 atoms cm–3

50 000
MWd/te
1.83E-06
3.23E-08
2.27E-06
5.04E-07
6.19E-09
3.65E-07
1.07E-08
1.12E-05
2.65E-06
2.09E-04
8.56E-05
6.24E-05
2.00E-05
6.16E-12
2.10E-08
7.27E-04
1.97E-04
2.55E-07
2.05E-02

60 000
MWd/te2.29E-06
4.15E-08
3.97E-06
7.41E-07
1.09E-08
7.83E-07
2.62E-08
1.43E-05
4.13E-06
2.09E-04
9.44E-05
7.04E-05
2.88E-05
6.95E-12
3.20E-08
6.00E-04
2.14E-04
2.94E-07
2.03E-02

70 000
MWd/te2.64E-06
4.81E-08
6.25E-06
9.85E-07
1.69E-08
1.49E-06
5.48E-08
1.73E-05
5.96E-06
2.04E-04
1.01E-04
7.62E-05
3.88E-05
8.62E-12
4.76E-08
4.88E-04
2.28E-04
3.23E-07
2.01E-02

80 000
MWd/te2.85E-06
5.22E-08
9.12E-06
1.22E-06
2.39E-08
2.59E-06
1.03E-07
2.04E-05
8.13E-06
1.99E-04
1.06E-04
7.92E-05
4.96E-05
9.39E-12
6.85E-08
3.90E-04
2.39E-04
3.53E-07
2.00E-02

Table 3.11. Phase 1b – Monte Carlo calculations – results from Participant D – irradiation with a buckling included

25

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.9259E-03
1.0000E-19
1.0000E-19
2.1288E-02

10 000
MWd/te1.0700E-08
6.8449E-11
9.4648E-10
6.6317E-10
1.1400E-12
1.9099E-11
9.9236E-14
3.7785E-07
1.2780E-08
9.4526E-05
1.1550E-05
2.0361E-06
7.4627E-08
1.4127E-16
1.9985E-09
1.6584E-03
4.6824E-05
4.6239E-08
2.1153E-02

20 000
MWd/te1.1381E-07
1.1403E-09
2.3369E-08
1.3501E-08
4.9376E-11
1.0103E-09
1.0741E-11
1.3793E-06
9.8381E-08
1.4730E-04
3.2202E-05
1.0812E-05
8.5072E-07
5.2334E-16
3.8227E-09
1.4322E-03
8.6056E-05
8.3296E-08
2.1013E-02

30 000
MWd/te3.8098E-07
4.6808E-09
1.3685E-07
6.5562E-08
3.7932E-10
9.4138E-09
1.5174E-10
2.8116E-06
3.2236E-07
1.7659E-04
5.2335E-05
2.4514E-05
3.1268E-06
1.1184E-15
5.8536E-09
1.2341E-03
1.1984E-04
1.1474E-07
2.0868E-02

40 000
MWd/te8.0114E-07
1.1034E-08
4.4939E-07
1.7942E-07
1.4507E-09
4.3477E-08
9.3629E-10
4.5402E-06
7.5109E-07
1.9206E-04
6.9339E-05
3.9777E-05
7.3416E-06
1.9652E-15
8.6743E-09
1.0586E-03
1.4914E-04
1.4046E-07
2.0718E-02

Units are u1024 atoms cm–3

50 000
MWd/te1.3083E-06
1.9314E-08
1.0793E-06
3.6059E-07
3.7996E-09
1.3714E-07
3.6697E-09
6.4498E-06
1.4440E-06
1.9880E-04
8.2813E-05
5.4070E-05
1.3580E-05
3.1896E-15
1.3050E-08
9.0215E-04
1.7456E-04
1.6334E-07
2.0564E-02

60 000
MWd/te1.8194E-06
2.8046E-08
2.1323E-06
5.9792E-07
7.8445E-09
3.4061E-07
1.0785E-08
8.4677E-06
2.4444E-06
1.9969E-04
9.3075E-05
6.5922E-05
2.1696E-05
5.0127E-15
1.9870E-08
7.6255E-04
1.9652E-04
1.8425E-07
2.0406E-02

70 000
MWd/te2.2599E-06
3.5760E-08
3.6879E-06
8.6960E-07
1.3751E-08
7.1848E-07
2.5957E-08
1.0530E-05
3.7715E-06
1.9644E-04
1.0064E-04
7.4674E-05
3.1418E-05
7.7411E-15
3.0065E-08
6.3799E-04
2.1534E-04
2.0353E-07
2.0243E-02

80 000
MWd/te2.5760E-06
4.1344E-08
5.7950E-06
1.1496E-06
2.1377E-08
1.3468E-06
5.3899E-08
1.2576E-05
5.4155E-06
1.9009E-04
1.0601E-04
8.0194E-05
4.2431E-05
1.1735E-14
4.4495E-08
5.2709E-04
2.3129E-04
2.2146E-07
2.0077E-02

Table 3.12. Phase 1b – deterministic calculations – results from Participant A – irradiation with a buckling included

26

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.93E-03
0.00E+00
0.00E+00
2.13E-02

10 000
MWd/te9.10E-09
6.50E-11
8.36E-10
4.46E-10
7.19E-13
1.57E-11
7.24E-14
3.95E-07
1.12E-08
9.09E-05
1.00E-05
1.71E-06
6.05E-08
1.22E-12
2.01E-09
1.65E-03
4.71E-05
4.87E-08
2.12E-02

20 000
MWd/te9.96E-08
1.12E-09
2.10E-08
9.64E-09
3.24E-11
8.35E-10
7.66E-12
1.50E-06
8.99E-08
1.39E-04
2.83E-05
9.34E-06
7.18E-07
2.18E-12
3.77E-09
1.42E-03
8.66E-05
9.29E-08
2.11E-02

30 000
MWd/te
3.49E-07
4.93E-09
1.35E-07
4.96E-08
2.79E-10
9.23E-09
1.40E-10
3.37E-06
3.24E-07
1.79E-04
4.62E-05
2.24E-05
2.72E-06
3.22E-12
6.06E-09
1.21E-03
1.21E-04
1.51E-07
2.09E-02

40 000
MWd/te
7.67E-07
1.22E-08
4.69E-07
1.42E-07
1.13E-09
4.61E-08
9.73E-10
5.84E-06
8.05E-07
2.06E-04
6.25E-05
3.77E-05
6.51E-06
3.63E-12
9.19E-09
1.03E-03
1.51E-04
2.00E-07
2.08E-02

Units are u1024 atoms cm–3

50 000
MWd/te1.30E-06
2.23E-08
1.17E-06
2.93E-07
3.06E-09
1.55E-07
4.07E-09
8.69E-06
1.60E-06
2.22E-04
7.64E-05
5.26E-05
1.21E-05
5.17E-12
1.41E-08
8.72E-04
1.75E-04
2.39E-07
2.06E-02

60 000
MWd/te
1.89E-06
3.40E-08
2.32E-06
4.98E-07
6.39E-09
3.96E-07
1.30E-08
1.18E-05
2.78E-06
2.33E-04
8.69E-05
6.65E-05
1.96E-05
7.47E-12
2.19E-08
7.33E-04
1.96E-04
2.80E-07
2.04E-02

70 000
MWd/te2.47E-06
4.58E-08
4.13E-06
7.45E-07
1.14E-08
8.57E-07
3.21E-08
1.52E-05
4.37E-06
2.39E-04
9.48E-05
7.80E-05
2.83E-05
8.11E-12
3.37E-08
6.12E-04
2.13E-04
3.11E-07
2.02E-02

80 000
MWd/te2.96E-06
5.63E-08
6.53E-06
1.01E-06
1.81E-08
1.63E-06
6.98E-08
1.86E-05
6.36E-06
2.41E-04
1.02E-04
8.64E-05
3.83E-05
9.65E-12
5.04E-08
5.07E-04
2.26E-04
3.51E-07
2.01E-02

Table 3.13. Phase 1b – Monte Carlo calculations – results from Participant D – irradiation with no buckling included

27

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.9259E-03
1.0000E-19
1.0000E-19
2.1288E-02

10 000
MWd/te8.9348E-09
5.6597E-11
7.3507E-10
5.5388E-10
8.7883E-13
1.3627E-11
6.3467E-14
3.3222E-07
1.0821E-08
8.3756E-05
1.0533E-05
1.7142E-06
6.3910E-08
8.8685E-17
1.5869E-09
1.6564E-03
4.6321E-05
4.1187E-08
2.1168E-02

20 000
MWd/te9.7354E-08
9.6509E-10
1.9125E-08
1.1681E-08
4.0004E-11
7.6826E-10
7.4115E-12
1.2429E-06
8.6005E-08
1.3128E-04
3.0012E-05
9.3812E-06
7.5691E-07
3.4406E-16
3.1084E-09
1.4252E-03
8.5767E-05
7.6586E-08
2.1040E-02

30 000
MWd/te3.3101E-07
4.0194E-09
1.1675E-07
5.8201E-08
3.1968E-10
7.5450E-09
1.1165E-10
2.5832E-06
2.8882E-07
1.5836E-04
4.9461E-05
2.1722E-05
2.8603E-06
7.7046E-16
4.8832E-09
1.2213E-03
1.2007E-04
1.0824E-07
2.0904E-02

40 000
MWd/te7.0290E-07
9.5612E-09
3.9653E-07
1.6231E-07
1.2619E-09
3.6433E-08
7.2817E-10
4.2395E-06
6.8648E-07
1.7345E-04
6.6146E-05
3.5783E-05
6.8567E-06
1.4203E-15
7.4345E-09
1.0401E-03
1.4997E-04
1.3538E-07
2.0762E-02

Units are u1024 atoms cm–3

50 000
MWd/te1.1544E-06
1.6829E-08
9.7919E-07
3.3066E-07
3.3892E-09
1.1939E-07
2.9952E-09
6.1063E-06
1.3413E-06
1.8106E-04
7.9507E-05
4.9197E-05
1.2882E-05
2.4221E-15
1.1480E-08
8.7876E-04
1.7592E-04
1.6040E-07
2.0612E-02

60 000
MWd/te1.6106E-06
2.4526E-08
1.9797E-06
5.5342E-07
7.1373E-09
3.0647E-07
9.1886E-09
8.1136E-06
2.3005E-06
1.8372E-04
8.9762E-05
6.0548E-05
2.0821E-05
4.0007E-15
1.7881E-08
7.3548E-04
1.9825E-04
1.8395E-07
2.0456E-02

70 000
MWd/te2.0050E-06
3.1374E-08
3.4911E-06
8.0971E-07
1.2706E-08
6.6542E-07
2.2995E-08
1.0197E-05
3.5875E-06
1.8296E-04
9.7373E-05
6.9202E-05
3.0410E-05
6.4827E-15
2.7549E-08
6.0868E-04
2.1722E-04
2.0625E-07
2.0294E-02

80 000
MWd/te2.2920E-06
3.6447E-08
5.5763E-06
1.0742E-06
1.9988E-08
1.2795E-06
4.9529E-08
1.2296E-05
5.1972E-06
1.7977E-04
1.0283E-04
7.5060E-05
4.1317E-05
1.0280E-14
4.1346E-08
4.9713E-04
2.3306E-04
2.2752E-07
2.0125E-02

Table 3.14. Phase 1b – deterministic calculations – results from Participant A – irradiation with no buckling included3.2 Phase 2
3.2.1 Phase 2a: Reactivities and reaction rates
Again, Participant A’s k-infinity is too high due to the physical inability to model a reflective
boundary condition on the surface of a sphere in Monte Carlo calculations.
Table 3.15. Phase 2a results – Monte Carlo calculations
ParameterUnitskinf with zero buckling–
and at T = 293.6 K
B2cr at T = 293.6 K
cm–2
2
kinf(B cr) at T = 293.6 K–
M2 at T = 293.6 K
cm2
kinf with zero buckling–
and at T = 1 000 K
keff(B2cr) at T = 1 000 K,–
using B2cr at T = 239.6 K
B2cr at T = 1 000 K
cm2
2
kinf(B cr) at T = 1 000 K–
M2 at T = 1 000 K
cm2–
Reactivity at apebble irradiationof 1 500 MWd/te
135
Xe number density
u1024
at a pebble irradiation
atoms
of 1 500 MWd/te
cm–3
135
Xe absorption
barns
cross-section at apebble irradiationof 1 500 MWd/te
–
U239
–
G239
241
–
G
241
F
–
C8
–
C9
–
C0
–
C1
–
C2
–

A
1.4805 r
100 pcm

B

C

D
1.4573
4.50E-04
1.4334951.1

1.4068 r
100 pcm
0.97322
4.16E-04
1.3886925.3
0.98100.15250.1461
0.3616
0.24750.01210.59340.56980.0832
0.0565

U239 = 239Pucap(epithermal)/239Pucap(thermal): ratio of epithermal to thermal 239Pu captures.
G239 = 239Pufis(epithermal)/239Pufis(thermal): ratio of epithermal to thermal 239Pu fissions.
G241 = 241Pufis(epithermal)/241Pufis(thermal): ratio of epithermal to thermal 241Pu fissions.
F241 = 241Pufis/239Pufis: ratio of fissions in 241Pu to fissions in 239Pu.
C8 = 238Pucap/239Pufis: ratio of captures in 238Pu to fissions in 239Pu.
C9 = 239Pucap/239Pufis: ratio of captures in 239Pu to fissions in 239Pu.
C0 = 240Pucap/239Pufis: ratio of captures in 240Pu to fissions in 239Pu.
C1 = 241Pucap/239Pufis: ratio of captures in 241Pu to fissions in 239Pu.
C2 = 242Pucap/239Pufis: ratio of captures in 242Pu to fissions in 239Pu.

28

E

F

Table 3.16. Phase 2a results – deterministic calculations
Parameter
UnitsAkinf with zero buckling–
1.46154
and at T = 293.6 K
B2cr at T = 293.6 K
cm–2
3.49E-04
2
kinf(B cr) at T = 293.6 K–
1.44325
M2 at T = 293.6 K
cm2
1 271
kinf with zero buckling–
1.38884
and at T = 1 000 K
keff(B2cr) at T = 1 000 K,–
0.95562
using B2cr at T = 239.6 K
B2cr at T = 1 000 K
cm2
3.06E-04
2
kinf(B cr) at T = 1 000 K–
1.37491
M2 at T = 1 000 K
cm2
1 227
–
1.36772
Reactivity at apebble irradiationof 1 500 MWd/te
135
Xe number density at
u1024 3.4714E-07
a pebble irradiation of
atoms
1 500 MWd/te
cm–3
135
Xe absorption
barns 1.2025E+0
cross-section at a pebble
5
irradiation of
1 500 MWd/te–
0.14799
U239–
0.13904G239
241–
0.34356
G241
F–
0.23599C8–0.01070C9–0.60378
C0–0.57985
C1–
0.07887
C2–
0.05396B
1.46369

CD

E
1.43270F
1.476564.803E-4
1.44569927.9

4.428E-04
1.458
1 034.4
1.392410.954910.94877

4.194E-4
1.37629897.2
1.35605

3.809E-04
1.379
995.0

0.1482410.139550
0.291976
0.239237
0.010620
0.603985
0.576980
0.081506
0.046782

Again, good agreement is seen between Participant A and B for deterministic k-infinity
calculations and most of the reaction rate ratios at a temperature of 293.6 K. For this plutonium
system, the APOLLO2 k-infinity result is 215 pcm higher than WIMS9, a value of 473 pcm higher
was seen for the uranium system in Phase 1. The two reaction rate ratios that are in most significant
disagreement are 241Pufis(epithermal)/241Pufis(thermal) where Participant A’s result is about 25% higher
and 242Pucap/239Pufis where Participant A’s result is about 11% higher.
Participant D’s reaction ratio results are in reasonable agreement with those of Participants A and
B. Participant D’s estimate of the 241Pufis(epithermal)/241Pufis(thermal) ratio is in better agreement with
Participant A, while the 242Pucap/239Pufis ratio is in better agreement with Participant B.
The reactivities from Participants E and F at room temperature are significantly different
considering that both have used ENDF/B-V nuclear data. This is thought to be due to the inadequate
treatment of plutonium resonances in the VSOP code, Participant E’s result. Judging from the result ofParticipant F, ENDF/B-V predicts higher reactivities than JEF2.2 for plutonium systems. This is the
opposite of uranium systems in the Phase 1 calculations where reactivities from ENDF/B-V werelower than JEF2.2.
29

3.2.2 Phase 2b: Reactivities and reaction rates
Table 3.17. Phase 2b – Monte Carlo calculations
Parameter
kinf with zerobuckling and at
T = 293.6 K
B2cr at T = 293.6 K
kinf(B2cr) at
T = 293.6 K
M2 at T = 293.6 K
kinf with zero buckling
and at T = 1 000 K
keff(B2cr) at
T = 1 000 K, using
B2cr at T = 239.6 K
B2cr at T = 1 000 K
kinf(B2cr) at
T = 1 000 K
M2 at T = 1 000 K
Reactivity at apebble irradiationof 1 500 MWd/te
135
Xe number density
at a pebble irradiationof 1 500 MWd/te
135
Xe absorption
cross-section at apebble irradiation of
1 500 MWd/te
U239
G239
G241
F241
C8
C9
C0
C1
C2

Units
–

A
1.4657 ±100pcmcm–2–

2.84E-4

cm2
–
–
cm2
–

B

C
1.44418D
1.4575EF
1.47717 r
34 pcm

2.77E-04
1.43491 574.9
1.39218 r
38 pcm1.3928 ±100 pcm
0.9571 r
100 pcm

0.96482

2.49E-4

2.49E-04
1.3884cm2–

1 531.0
0.98358

u1024
atoms
cm–3
barns

–
–
–
–
–
–
–
–
–

0.15090.14510.35720.24810.01210.59270.57140.083330

Table 3.18. Phase 2b – deterministic calculations
Parameter
kinf with zerobuckling and at
T = 293.6 K
B2cr at T = 293.6 K
kinf(B2cr) at
T = 293.6 K
M2 at T = 293.6 K
kinf with zero buckling
and at T = 1 000 K
keff(B2cr) at
T = 1 000 K, using
B2cr at T = 239.6 K
B2cr at T = 1 000 K
kinf(B2cr) at
T = 1 000 K
M2 at T = 1 000 K
Reactivity at apebble irradiationof 1 500 MWd/te
135
Xe number density
at a pebble irradiationof 1 500 MWd/te
135
Xe absorption
cross-section at apebble irradiation of
1 500 MWd/te
U239
G239
G241
F241
C8
C9
C0
C1
C2

Units
–

A
1.46154cm–2–

3.49E-04
1.44325

cm2
–

1272
1.38884–

0.95562cm2–

3.05E-04
1.37491

cm2
–

1227
1.36772

B

C

D

E

F

u1024 3.4716E-07
atoms
cm–3
barns 1.2044E+05

–
–
–
–
–
–
–
–
–

0.147990.13904
0.343560.235990.01070
0.60378
0.579890.07887
0.05396

Again, the deterministic results from Participant A for Phases 2a and 2b are very similar otherthan the effect of the outer boundary condition. It can again be concluded that Participants A and B are
in close agreement based on the Phase 2a results.
The Monte Carlo and deterministic calculations for Participant A are in good agreement. For
k-infinity the difference is (1.4657 – 1.46154) 416 pcm. At temperature, using the critical buckling at
293.6 K the difference is (0.9571 – 0.95622) 88 pcm. This also suggests the Monte Carlo anddeterministic evaluations of the temperature coefficient are in good agreement. It should be noted that,for the plutonium system, the temperature coefficient is only about one half of the value for the uranium
system in Phase 1.

31

Participants D and F, both using ENDF/B-V cross-section data, have very different results, by
about 2 000 pcm. Participant D is in reasonable agreement with the JEF2.2 result but Participant F issignificantly higher.
Xenon rapidly builds up in the reactor as a poison and reaches equilibrium concentrations aftera few days at constant power. In Phases 2a and 2b of the benchmark Participant D calculates thiseffect, in addition to the change in reactivity when going to 1 000 K, to be a reduction in reactivity of(1.4573 – 0.9810) 47 630 pcm for Phase 2a and (1.4575 – 0.98358) 47 392 pcm in Phase 2b.
Participant A calculates the effect as a reduction by (1.46154 – 1.36772) 9 382 pcm in Phase 2a and(1.46154 – 1.36772) 9 382 pcm in Phase 2b. Participant B calculates the effect as (1.46369 – 1.35605)
10 764 pcm for Phase 2a. There is significant disagreement between the results of Participant D andthose from the other participants in this part of the benchmark. There is a large difference between thepebble reactivity at 1 500 MWd/te calculated by Participant A and the equivalent value calculated byParticipant B. This result is surprising given the close agreement between Participant A and B’s
general results for other parts of the benchmark. This could be due to differences in xenon worth
and/or xenon number density at 1 500 MWd/te (perhaps resulting from differences in the employedburn-up chain or yield data).
3.2.3 Spectra
Figure 3.5 shows the Monte Carlo solution spectrum for Phase 2a from Participant D.
Figure 3.6 shows the spectra for Phase 2a submitted by Participants A, B and F, all using
deterministic solutions.
Figure 3.7 shows the Monte Carlo solution spectrum for Phase 2b from Participant D. Thereappears to be close agreement between this spectrum and that from Participant D for the Phase 2asolution.
No deterministic solutions have been shown for Phase 2b, as these are the same as Phase 2a.
3.2.4 Burn-up
The number densities at 0, 10 000, 20 000, 30 000, 40 000, 50 000, 60 000, 70 000, 80 000 MWd/te
supplied for Phase 2a or 2b are given in Tables 3.19 to 3.23. The results for Monte Carlo calculations
from Participant D for Phases 2a and 2b are essentially identical, Tables 3.19 and 3.22, as are thosefrom Participant A using deterministic methods, Tables 3.20 and 3.23.
The deterministic results from Participants A and B show reasonable agreement. There appears to
be a slightly larger accumulation of the higher actinides in Participant B’s calculation. The mostnoticeable differences are for 243Am, 245Cm, 237Np and 238U (although these only occur in trace
quantities). The rate of accumulation of 237Np and 238U between 10 000 and 80 000 MWd/te is farlarger in Participant B’s calculation.

32

33

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
6.01E-04
1.24E-02
5.45E-03
3.01E-03
1.55E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00

10 000
MWd/te7.12E-06
9.08E-09
1.02E-05
4.53E-08
1.38E-11
6.56E-08
8.67E-11
4.00E-10
5.98E-04
1.21E-02
5.45E-03
3.04E-03
1.55E-03
3.94E-15
2.32E-07
1.80E-08
2.83E-08
1.04E-10
1.42E-10

20 000
MWd/te
1.42E-05
3.58E-08
2.11E-05
1.90E-07
1.22E-10
2.88E-07
7.78E-10
1.49E-09
5.96E-04
1.19E-02
5.45E-03
3.08E-03
1.56E-03
1.49E-14
4.63E-07
3.66E-08
5.65E-08
1.55E-10
2.84E-1030 000
MWd/te2.12E-05
7.89E-08
3.20E-05
4.27E-07
4.22E-10
6.71E-07
2.73E-09
3.22E-09
5.93E-04
1.16E-02
5.45E-03
3.12E-03
1.56E-03
3.27E-14
6.90E-07
5.60E-08
8.47E-08
2.00E-10
4.26E-1040 000
MWd/te2.81E-05
1.37E-07
4.23E-05
7.54E-07
9.78E-10
1.22E-06
6.65E-09
5.57E-09
5.90E-04
1.13E-02
5.45E-03
3.16E-03
1.57E-03
5.95E-14
9.16E-07
7.61E-08
1.13E-07
2.42E-10
5.69E-10

Units are u1024 atoms cm–3

50 000
MWd/te
3.49E-05
2.08E-07
5.26E-05
1.16E-06
1.93E-09
1.89E-06
1.32E-08
8.51E-09
5.87E-04
1.10E-02
5.44E-03
3.19E-03
1.58E-03
9.64E-14
1.14E-06
9.70E-08
1.41E-07
2.76E-10
7.13E-1060 000
MWd/te
4.17E-05
2.91E-07
6.31E-05
1.64E-06
3.30E-09
2.71E-06
2.27E-08
1.20E-08
5.85E-04
1.07E-02
5.44E-03
3.22E-03
1.58E-03
1.34E-13
1.36E-06
1.19E-07
1.69E-07
3.29E-10
8.57E-10

Table 3.19. Phase 2a – Monte Carlo calculations – results from Participant D

70 000
MWd/te
4.84E-05
3.85E-07
7.34E-05
2.20E-06
5.12E-09
3.68E-06
3.70E-08
1.62E-08
5.82E-04
1.04E-02
5.43E-03
3.26E-03
1.59E-03
1.81E-13
1.58E-06
1.41E-07
1.97E-07
3.77E-10
1.00E-09

80 000
MWd/te5.50E-05
4.90E-07
8.39E-05
2.83E-06
7.61E-09
4.81E-06
5.50E-08
2.09E-08
5.79E-04
1.01E-02
5.42E-03
3.28E-03
1.60E-03
2.35E-13
1.79E-06
1.64E-07
2.25E-07
4.11E-10
1.15E-09

34

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
6.0118E-04
1.2447E-02
5.4460E-03
3.0097E-03
1.5454E-03
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19

10 000
MWd/te7.0773E-06
7.3501E-09
9.9436E-06
5.4956E-08
1.7613E-11
6.8030E-08
1.0003E-10
1.4283E-11
5.9873E-04
1.2147E-02
5.4484E-03
3.0523E-03
1.5509E-03
3.3672E-15
2.3193E-07
1.7833E-08
2.8187E-08
2.0624E-11
3.8460E-14

20 000
MWd/te1.2700E-05
2.3358E-08
1.7864E-05
1.7535E-07
1.0150E-10
2.2040E-07
5.8364E-10
6.4505E-11
5.9676E-04
1.1908E-02
5.4492E-03
3.0856E-03
1.5556E-03
1.0867E-14
4.1599E-07
3.2502E-08
5.0712E-08
4.8574E-11
1.7510E-13

30 000
MWd/te2.1061E-05
6.2987E-08
2.9674E-05
4.7606E-07
4.6140E-10
6.1147E-07
2.6991E-09
2.2003E-10
5.9380E-04
1.1552E-02
5.4486E-03
3.1340E-03
1.5632E-03
3.0006E-14
6.8959E-07
5.5160E-08
8.4451E-08
9.3792E-11
6.0459E-13

40 000
MWd/te2.7956E-05
1.0919E-07
3.9447E-05
8.3051E-07
1.0769E-09
1.0854E-06
6.3862E-09
4.2516E-10
5.9134E-04
1.1258E-02
5.4466E-03
3.1729E-03
1.5700E-03
5.3069E-14
9.1533E-07
7.4630E-08
1.1251E-07
1.3209E-10
1.1803E-12

Units are u1024 atoms cm–3

50 000
MWd/te3.4778E-05
1.6631E-07
4.9158E-05
1.2737E-06
2.0710E-09
1.6928E-06
1.2443E-08
6.9895E-10
5.8888E-04
1.0967E-02
5.4431E-03
3.2104E-03
1.5773E-03
8.2485E-14
1.1390E-06
9.4627E-08
1.4052E-07
1.7055E-10
1.9607E-12

60 000
MWd/te4.1524E-05
2.3335E-07
5.8808E-05
1.8008E-06
3.5236E-09
2.4327E-06
2.1442E-08
1.0410E-09
5.8644E-04
1.0677E-02
5.4382E-03
3.2464E-03
1.5849E-03
1.1815E-13
1.3607E-06
1.1514E-07
1.6847E-07
2.0910E-10
2.9513E-12

Table 3.20. Phase 2a – deterministic calculations – results from Participant A

70 000
MWd/te4.8185E-05
3.0938E-07
6.8400E-05
2.4071E-06
5.5099E-09
3.3042E-06
3.3945E-08
1.4509E-09
5.8402E-04
1.0391E-02
5.4318E-03
3.2808E-03
1.5931E-03
1.5994E-13
1.5803E-06
1.3617E-07
1.9636E-07
2.4774E-10
4.1578E-1280 000
MWd/te5.4757E-05
3.9347E-07
7.7935E-05
3.0886E-06
8.0998E-09
4.3063E-06
5.0502E-08
1.9282E-09
5.8162E-04
1.0106E-02
5.4241E-03
3.3137E-03
1.6017E-03
2.0775E-13
1.7979E-06
1.5771E-07
2.2417E-07
2.8645E-10
5.5862E-12

35

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0.00E+00

0
MWd/te
–
–
–
–
–
–
–
–
6.01E-04
1.24E-02
5.45E-03
3.01E-03
1.55E-03––––

10 000
MWd/te7.08E-06–
2.32E-07
5.62E-08
1.84E-11
7.44E-08
1.12E-10
3.05E-10
5.99E-04
1.21E-02
5.45E-03
3.05E-03
1.55E-03–
2.31E-07
1.79E-08
2.82E-08
2.11E-11
1.41E-10

20 000
MWd/te
1.41E-05–
4.62E-07
2.20E-07
1.45E-10
2.97E-07
8.95E-10
1.22E-09
5.96E-04
1.18E-02
5.45E-03
3.09E-03
1.56E-03–
4.59E-07
3.65E-08
5.64E-08
5.72E-11
2.82E-1030 000
MWd/te2.11E-05–
6.89E-07
4.86E-07
4.82E-10
6.68E-07
3.01E-09
2.73E-09
5.94E-04
1.15E-02
5.45E-03
3.14E-03
1.56E-03–
6.86E-07
5.57E-08
8.45E-08
9.59E-11
4.24E-1040 000
MWd/te2.79E-05–
9.15E-07
8.48E-07
1.12E-09
1.19E-06
7.13E-09
4.86E-09
5.91E-04
1.12E-02
5.45E-03
3.17E-03
1.57E-03–
9.10E-07
7.57E-08
1.13E-07
1.35E-10
5.66E-10

Units are u1024 atoms cm–3

50 000
MWd/te
3.48E-05–
1.14E-06
1.30E-06
2.16E-09
1.85E-06
1.39E-08
7.57E-09
5.89E-04
1.09E-02
5.45E-03
3.21E-03
1.58E-03–
1.13E-06
9.63E-08
1.41E-07
1.74E-10
7.09E-1060 000
MWd/te
4.15E-05–
1.36E-06
1.84E-06
3.67E-09
2.66E-06
2.39E-08
1.09E-08
5.86E-04
1.06E-02
5.44E-03
3.25E-03
1.58E-03–
1.35E-06
1.18E-07
1.69E-07
2.14E-10
8.53E-10

Table 3.21. Phase 2a – deterministic calculations – results from Participant B

70 000
MWd/te4.81E-05–
1.58E-06
2.46E-06
5.74E-09
3.61E-06
3.78E-08
1.47E-08
5.84E-04
1.03E-02
5.43E-03
3.28E-03
1.59E-03–
1.57E-06
1.39E-07
1.97E-07
2.53E-10
9.98E-1080 000
MWd/te5.47E-05–
1.79E-06
3.15E-06
8.44E-09
4.70E-06
5.63E-08
1.91E-08
5.81E-04
1.01E-02
5.43E-03
3.32E-03
1.60E-03–
1.78E-06
1.62E-07
2.24E-07
2.93E-10
1.14E-09

36

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
6.01E-04
1.24E-02
5.45E-03
3.01E-03
1.55E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00

10 000
MWd/te7.12E-06
9.16E-09
1.11E-05
4.57E-08
1.39E-11
7.15E-08
9.56E-11
3.99E-10
5.98E-04
1.21E-02
5.45E-03
3.05E-03
1.55E-03
3.58E-15
2.33E-07
1.79E-08
2.83E-08
1.04E-10
1.41E-10

20 000
MWd/te
1.42E-05
3.59E-08
2.17E-05
1.90E-07
1.20E-10
3.01E-07
8.00E-10
1.49E-09
5.96E-04
1.19E-02
5.45E-03
3.09E-03
1.56E-03
1.46E-14
4.63E-07
3.66E-08
5.65E-08
1.56E-10
2.84E-1030 000
MWd/te2.12E-05
7.88E-08
3.26E-05
4.27E-07
4.17E-10
6.86E-07
2.81E-09
3.22E-09
5.93E-04
1.16E-02
5.45E-03
3.12E-03
1.56E-03
3.43E-14
6.90E-07
5.60E-08
8.47E-08
1.98E-10
4.26E-1040 000
MWd/te2.81E-05
1.36E-07
4.28E-05
7.52E-07
9.83E-10
1.24E-06
6.68E-09
5.57E-09
5.90E-04
1.13E-02
5.44E-03
3.16E-03
1.57E-03
5.90E-14
9.16E-07
7.61E-08
1.13E-07
2.43E-10
5.69E-10

Units are u1024 atoms cm–3

50 000
MWd/te
3.50E-05
2.08E-07
5.31E-05
1.16E-06
1.91E-09
1.92E-06
1.33E-08
8.52E-09
5.87E-04
1.10E-02
5.44E-03
3.19E-03
1.58E-03
9.14E-14
1.14E-06
9.70E-08
1.41E-07
2.81E-10
7.13E-1060 000
MWd/te
4.17E-05
2.91E-07
6.34E-05
1.65E-06
3.23E-09
2.75E-06
2.32E-08
1.20E-08
5.85E-04
1.07E-02
5.44E-03
3.23E-03
1.58E-03
1.35E-13
1.36E-06
1.19E-07
1.69E-07
3.20E-10
8.57E-10

Table 3.22. Phase 2b – Monte Carlo calculations – results from Participant D

70 000
MWd/te
4.84E-05
3.85E-07
7.40E-05
2.20E-06
5.07E-09
3.73E-06
3.68E-08
1.62E-08
5.82E-04
1.04E-02
5.43E-03
3.26E-03
1.59E-03
1.84E-13
1.58E-06
1.41E-07
1.97E-07
3.75E-10
1.00E-09

80 000
MWd/te5.50E-05
4.90E-07
8.44E-05
2.83E-06
7.59E-09
4.87E-06
5.51E-08
2.09E-08
5.79E-04
1.01E-02
5.42E-03
3.29E-03
1.60E-03
2.42E-13
1.79E-06
1.64E-07
2.25E-07
4.14E-10
1.15E-09

37

242m

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
234
U
235
U
236
U
237
U
238
U

241

Nuclide0
MWd/te1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
6.0118E-04
1.2447E-02
5.4460E-03
3.0097E-03
1.5454E-03
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19
1.0000E-19

10 000
MWd/te7.0773E-06
7.3507E-09
9.9469E-06
5.4960E-08
1.7615E-11
6.8061E-08
1.0009E-10
1.4284E-11
5.9873E-04
1.2147E-02
5.4484E-03
3.0523E-03
1.5509E-03
3.3641E-15
2.3193E-07
1.7833E-08
2.8187E-08
2.0626E-11
3.8463E-14

20 000
MWd/te1.4100E-05
2.8698E-08
1.9845E-05
2.1567E-07
1.3884E-10
2.7219E-07
8.0104E-10
8.3620E-11
5.9626E-04
1.1848E-02
5.4492E-03
3.0939E-03
1.5569E-03
1.3391E-14
4.6179E-07
3.6224E-08
5.6339E-08
5.5982E-11
2.2744E-13

30 000
MWd/te2.1061E-05
6.2992E-08
2.9684E-05
4.7610E-07
4.6147E-10
6.1176E-07
2.7008E-09
2.2005E-10
5.9380E-04
1.1552E-02
5.4485E-03
3.1342E-03
1.5632E-03
2.9979E-14
6.8959E-07
5.5160E-08
8.4449E-08
9.3801E-11
6.0462E-13

40 000
MWd/te2.7956E-05
1.0920E-07
3.9460E-05
8.3058E-07
1.0771E-09
1.0859E-06
6.3901E-09
4.2520E-10
5.9134E-04
1.1258E-02
5.4464E-03
3.1731E-03
1.5700E-03
5.3021E-14
9.1533E-07
7.4630E-08
1.1251E-07
1.3210E-10
1.1804E-12

Units are u1024 atoms cm–3

50 000
MWd/te3.4780E-05
1.6632E-07
4.9174E-05
1.2738E-06
2.0713E-09
1.6935E-06
1.2451E-08
6.9901E-10
5.8888E-04
1.0967E-02
5.4428E-03
3.2107E-03
1.5772E-03
8.2411E-14
1.1390E-06
9.4628E-08
1.4052E-07
1.7056E-10
1.9608E-1260 000
MWd/te4.1526E-05
2.3338E-07
5.8827E-05
1.8009E-06
3.5242E-09
2.4338E-06
2.1455E-08
1.0411E-09
5.8644E-04
1.0677E-02
5.4379E-03
3.2467E-03
1.5849E-03
1.1804E-13
1.3607E-06
1.1515E-07
1.6847E-07
2.0912E-10
2.9514E-12

Table 3.23. Phase 2b – deterministic calculations – results from Participant A

70 000
MWd/te4.8188E-05
3.0941E-07
6.8422E-05
2.4074E-06
5.5107E-09
3.3056E-06
3.3965E-08
1.4510E-09
5.8402E-04
1.0391E-02
5.4315E-03
3.2812E-03
1.5931E-03
1.5980E-13
1.5803E-06
1.3618E-07
1.9635E-07
2.4775E-10
4.1580E-12

80 000
MWd/te5.4760E-05
3.9352E-07
7.7960E-05
3.0889E-06
8.1011E-09
4.3082E-06
5.0531E-08
1.9284E-09
5.8163E-04
1.0106E-02
5.4237E-03
3.3141E-03
1.6017E-03
2.0757E-13
1.7979E-06
1.5772E-07
2.2416E-07
2.8647E-10
5.5863E-12

3.3 Phase 3
3.3.1 Core reactivity
Table 3.24. Phase 3 – Monte Carlo calculations
Parameter
Core reactivity
at 293.6 K
Core reactivity
at 1 000 K
Core reactivity at an
irradiation of 1 500
MWd/teCore average 135Xe
number density at an
irradiation of
1 500 MWd/teCore average
135
Xe absorption
cross-section at
an irradiation of
1 500 MWd/Te

Units
–
–

A
1.3494 r
100 pcm
1.2998 r
100 pcm

B
1.35714 r
50 pcm1.31111 r
50 pcmC
1.35302D
1.347

1.30577

1.318

–

E

F
1.36127 r
50 pcm1.29879 r
42 pcmE
1.29827

F

1.294

u1024
atoms
cm–3
barns

Table 3.25. Phase 3 – deterministic calculations
Parameter
Core reactivity
at 293.6 K
Core reactivity
at 1 000 K
Core reactivity at an
irradiation of 1 500
MWd/teCore average 135Xe
number density at an
irradiation of
1 500 MWd/teCore average
135
Xe absorption
cross-section at
an irradiation of
1 500 MWd/Te

Units
–

A
1.34645B
1.35075

CD–

1.29598

1.29820

1.26230–

1.28540

1.278181.24099

u1024
atoms
cm–3

9.1062E-11

barns

1.4614E-05

The mean calculated Monte Carlo core reactivity of all participants at 293.6 K is 1.35356 with astandard deviation between results of r576 pcm. The outlying results are from Participant D (-657 pcm
from the mean) and Participant F (+770 pcm from the mean).
The difference between Participant A’s Monte Carlo and deterministic results for the core reactivity
at 293.6 K is 295 r 100 pcm, and that for Participant B is 639 r 50 pcm. This is relatively good
agreement considering the stochastic nature of the pebble arrangement.

38For the Monte Carlo calculations, the reactivity reduction in going to power temperatures is
4 960 pcm from Participant A, 4 603 pcm from Participant B, 2 900 pcm from Participant D and
6 248 pcm from Participant F. However, the deterministic calculations of Participants A and B givevalues of 5 047 pcm and 5 255 pcm, respectively. Participant A’s Monte Carlo and deterministic
values are in good agreement.
Participant A has included a deterministic evaluation of the whole core xenon reactivity effect,
1 058 pcm. Participant B’s deterministic calculation of the same quantity is 2 002 pcm. Participant D
gives a Monte Carlo determination of 2 400 pcm.
There is poor agreement between the deterministic calculations of Participant E and those from
Participants A and B, however, this may be in part due to ENDF/B-5 relative to JEF2.2 nuclear data.
Note, however, that for the plutonium cell calculations of Phase 2 Participant E’s results gave higher
rather than lower values.
3.3.2 Spectra
Figure 3.8 shows the Monte Carlo spectra provided by Participants A, D and F.
Figure 3.9 shows the Monte Carlo and deterministic spectra for Participant A’s calculation. Thereis a general good agreement but with a slight difference at the peak of the low-energy plutonium
resonance.
Figure 3.10 compares the deterministic spectra from Participant E and Participant A. There are
significant differences between the two spectra, the causes of which are unknown.
3.3.3 Cross-sections
Table 3.26. Phase 3 – Monte Carlo calculations – results from Participant A
Specified in units of barns

NuclideCapture
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
Fission
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu

10–5 eV to 0.625 eV

0.625 eV to 9.118 KeV

9.118 KeV to 20 MeV

119 r 15390 r 6148 r 6201 r 98r2

13 r 316 r 1
241 r 516 r 288 r 3

0.6 r 0.4
0.3 r 0.1
0.3 r 0.2
0.6 r 0.3
0.4 r 0.3

4r2
654 r 70.02 r 0.02
567 r 15
0r1

2r1
25 r 1
0.2 r 0.1
55 r 4
0.1 r 0.1

2r1
2.1 r 0.3
0.7 r 0.2
2.3 r 0.7
0.5 r 0.3

39

Table 3.27. Phase 3 – Monte Carlo calculations – results from Participant D
Specified in units of barns

NuclideCapture
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
Fission
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu

10–5 eV to 0.625 eV

0.625 eV to 9.118 KeV

9.118 KeV to 20 MeV

145.2
374.6
153.8
204.7
7.953

14.64
15.57
232.4
15.31
87.78

0.48540.25980.34580.26870.30074.410
643.2
0.0330586.05
3.9E-04

2.016
24.37
0.130453.10
0.0202

1.45641.84010.75232.20420.5915

Table 3.28. Phase 3 – Monte Carlo calculations – results from Participant F
Specified in units of barns

NuclideCapture
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
Fission
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu

10–5 eV to 0.625 eV

0.625 eV to 9.118 KeV

9.118 KeV to 20 MeV

144.8939
390.5491
155.5297
231.4431
7.9612

14.5820
16.9366
237.4493
17.8462108.7398

0.47500.25980.34300.30850.29294.3997659.8146
0.0299599.5869
3.918E-04

2.171424.5275
0.125354.15930.02101.37901.78670.68152.1294
0.5382

Table 3.29. Phase 3 – deterministic calculations – results from Participant A
Specified in units of barns

NuclideCapture
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
Fission
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu

10–5 eV to 0.625 eV

0.625 eV to 9.118 KeV

9.118 KeV to 20 MeV

127.39400.59151.25204.53
7.37

13.63
16.00
242.89
15.75
87.05

0.36
0.29
0.37
0.44
0.33

4.25
672.950.03
576.01
0.01

2.15
24.03
0.24
53.74
0.08

1.47
1.88
0.73
2.24
0.58

40

Table 3.30. Phase 3 – deterministic calculations – results from Participant B
Specified in units of barns

NuclideCapture
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
Fission
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu

10–5 eV to 0.625 eV

0.625 eV to 9.118 KeV

9.118 KeV to 20 MeV

127.00403.54151.68207.02
7.39

13.57
16.02
241.34
15.61
91.26

0.37
0.29
0.37
0.45
0.33

4.24
677.440.03
581.38
0.01

2.14
24.04
0.24
53.55
0.08

1.61
1.97
0.84
2.34
0.66

Table 3.31. Phase 3 – deterministic calculations – results from Participant E
Specified in units of barns

NuclideCapture
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
Fission
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu

Participant E
10–5 eV to 10 MeV

Participant A
10–5 eV to 10 MeV

26.6
75.0
158
40.5
33.3

27.58
73.87
138.45
41.35
41.94

2.03
122
0.349
118
0.184

2.25
123.270.39
121.21
0.25

The Monte Carlo cross-sections submitted by Participant A and Participant F show differences
greater than the scoring statistics for 238Pu, 241Pu and 242Pu capture and 241Pu fission reflecting theorigin of the nuclear data. Participant D has a lower value, more in agreement with Participant A for
241
Pu and 242Pu capture and 241Pu fission despite using the same nuclear data as Participant F.
There is relatively good agreement between the deterministic cross-sections at all energies for
Participants A and B reflecting the same origin of nuclear data (JEF2.2).
All deterministic cross-sections from Participant A agree with the Monte Carlo cross-sections
from Participant A within the one sigma uncertainty in the Monte Carlo values, except for the thermal
239
Pu fission and capture cross-sections. As these are the most dominant reactions in the problem, andtherefore have the best counting statistics, this appears anomalous. However, Figure 3.9 shows theMonte Carlo flux at these thermal energies slightly exceeds that in the deterministic calculation. The
reaction rates are therefore in very good agreement. This anomaly is attributed to the accuracy of
splitting the Monte Carlo reaction rates into flux and cross-section components.
41

Participant E was only able to provide one-group cross-sections. Table 3.31 shows thesecross-sections, as well as equivalent values from Participant A. It appears that most of the cross-sections
are in reasonable agreement for Participants E and A, with the notable exception of 240Pu and 242Pu
capture. The significantly higher 240Pu capture cross-section from Participant E could be a contributing
factor to the low core reactivities provided by this participant.
3.4 Phase 4
3.4.1 Core reactivity
Table 3.32. Phase 4 – Monte Carlo calculations
Parameter
UnitsCore reactivity (keff) at–
293.6 K
Core reactivity (keff) at–
1 000 K–
Core reactivity at an
irradiation of 1 500 MWd/te –with buckling–Core reactivity at an
irradiation of 1 500 MWd/te –without bucklingCore average 135Xe number
u1024
density at an irradiation of
atoms
1 500 MWd/te
cm–3
135
Core average Xe absorption
barns
cross-section at an irradiation of
1 500 MWd/TeCore migration lengthcm

AB
1.3697 r 1.38208 r
100 pcm 50 pcm
1.2863 r 1.30557 r
100 pcm 50 pcm

CDE
1.364541.30350F
1.36881 r
0.000481.27599 r
0.00046

1.24585

Table 3.33. Phase 4 – deterministic calculations
Parameter
Core reactivity (keff) at
293.6 K – no streaming
correction
Core reactivity (keff) at
293.6 K – with streaming
correction
Core reactivity (keff) at
1 000 K – no streaming
correction
Core reactivity (keff) at
1 000 K – with streaming
correction
Core reactivity at an
irradiation of 1 500 MWd/te– with buckling

Units
–

A
1.36818B
1.37663––DE
1.35018

1.34551

1.284431.29460––C

1.27078

1.26737

1.24109

1.2532742

1.21118

F

Table 3.33. Phase 4 – deterministic calculations (cont.)
Parameter
Core reactivity at an
irradiation of 1 500 MWd/te– without bucklingCore average 135Xe number
density at an irradiationof 1 500 MWd/teCore average 135Xe absorption
cross-section at an irradiationof 1 500 MWd/TeCore migration length

Units
–

A
1.24162

B

u1024
atoms
cm–3
barns

1.0036E-10

4.1109E+05

cm

39.359CDEF

The difference between the Monte Carlo and deterministic (with no streaming correction) resultsof Participant A at 293.6K is 152 r 100 pcm. The equivalent difference at 1 000 K is 187 r 100 pcm.
This is good agreement, especially when consideration is made of the stochastic nature of the pebble
distribution. As the differences are comparable at both 293.6 K and 1 000 K, this indicates that
Participant A’s deterministic and Monte Carlo evaluations of the temperature coefficient are inagreement.
Participant B’s Monte Carlo estimate of k-effective at room temperature is 1 238 pcm higher thanthat of Participant A. The reactivity change to 1 000qC is calculated as 8 340 pcm, 7 651 pcm and9 282 pcm by Participants A, B and F respectively.
There is reasonable agreement between the k-effectives of the deterministic solutions, Participant B
being 845 pcm higher than Participant A and Participant E being 1 800 pcm lower. The reactivity to
1 000qC is calculated as 8 375 pcm, 8 203 pcm and 7 940 pcm by Participants A, B and E respectively.
3.4.2 Spectra
Figure 3.11 shows a comparison between the Monte Carlo spectra of Participants A and F. There
are significant differences between the spectra.
Figure 3.12 shows the Monte Carlo and deterministic 172-group spectra for Participant A’s
calculation. There is a general good agreement, but with a slight difference at the low-energy peak inthe spectrum.
Figure 3.13 shows the Monte Carlo and deterministic four-group spectra for Participant A’s
calculation. Again, there is a general good agreement between the spectra.
Figure 3.14 shows the deterministic spectrum from Participant E. It has not been possible to makea plot to directly compare the deterministic spectra from Participants E and A, since the VSOP code
applies an unknown normalisation factor to the flux data internally. The form of Participant E’s
deterministic spectrum, however, appears to be comparable to that of Participant A.

43

3.4.3 Cross-sections
Table 3.34. Monte Carlo calculations – results from Participant A
Specified in units of barns

NuclideCapture
235
U
238
U
Fission
235
U
238
U

10–5 eV to 20 MeV
19.956 r 0.630
3.696 r 0.065
98.138 r 1.300
0.027r 0.003

Table 3.35. Monte Carlo calculations – results from Participant D
Specified in units of barns

NuclideCapture
235
U
238
U
Fission
235
U
238
U

10–5 eV to 20 MeV
17.306
3.151885.8590.023774

Table 3.36. Monte Carlo calculations – results from Participant F
Specified in units of barns

NuclideCapture
235
U
238
U
Fission
235
U
238
U

10–5 eV to 20 MeV
24.46294.4870115.3191
0.0315

Table 3.37. Deterministic calculations – results from Participant A
Specified in units of barns

NuclideCapture
235
U
238
U
Fission
235
U
238
U

10–5 eV to 20 MeV
19.7883.722296.829
2.73E-02

44

Table 3.38. Deterministic calculations – results from Participant B
Specified in units of barns

10–5 eV to 20 MeV

NuclideCapture
235
U
238
U
Fission
235
U
238
U

19.94
3.67
97.63
0.03

Table 3.39. Deterministic calculations – results from Participant E
Specified in units of barns

10–5 eV to 20 MeV

NuclideCapture
235
U
238
U
Fission
235
U
238
U

20.7
3.53
96.5
0.026

The Monte Carlo cross-sections submitted by Participants A and F appear to be discrepant.
Participant A’s Monte Carlo and deterministic cross-sections are in good agreement. It is concluded
that Participant F’s Monte Carlo cross-sections are in error.
Cross-sections for Participants A, B and E are in reasonable agreement.
3.5 Phase 5
3.5.1 Phase 5a: Reactivities and reaction rates
Table 3.40. Phase 5a – Monte Carlo calculations
Parameter
kinf with zero buckling
and at T = 293.6 K
B2cr at T = 293.6 K
kinf(B2cr) at T = 293.6 K
M2 at T = 293.6 K
kinf with zero buckling
and at T = 1 000 K
keff(B2cr) at T = 1 000 K,
using B2cr at T = 239.6 K
B2cr at T = 1 000 K
kinf(B2cr) at
T = 1 000 K
M2 at T = 1 000 K

Units
–
cm–2
–
cm2
–

A
1.4442 r
50 pcmB

CD
1.46841
1.73E-4
1.45492607.81.45731.4452 r
50 pcm–0.89041

cm–2–

1.27E-4
1.4526cm2

3553.245

E

F

Table 3.40. Phase 5a – Monte Carlo calculations (cont.)
Parameter
Reactivity at apebble irradiationof 1 500 MWd/te
135
Xe number density
at a pebble irradiationof 1 500 MWd/te
135
Xe absorption
cross-section at apebble irradiationof 1 500 MWd/Te
U232
G233
G232
C

Units
–

A

B

C

D
0.9682

u1024
atoms
cm–3
barns

E

F

E

F
1.4645

1.95E-08(E < 0.625eV)
1.53935E+6(E > 0.625eV)
327.450.304520.049460.0001526
0.21156––––

U232 = 232Thcap(epithermal)/232Thcap(thermal): ratio of epithermal to thermal 232Th captures.
G233 = 233Ufis(epithermal)/233Ufis(thermal): ratio of epithermal to thermal 233U fissions.
G232 = 232Thfis/233Ufis: ratio of fissions in 232Th to fissions in 233U.
C = 232Thcap/233Ufis: ratio of captures in 232Th to fissions in 233U.

Table 3.41. Phase 5a – deterministic calculations
ParameterUnitskinf with zero buckling–
and at T = 293.6 K
B2cr at T = 293.6 K
cm–2
2
kinf(B cr) at T = 293.6 K–
M2 at T = 293.6 K
cm2
kinf with zero buckling–
and at T = 1 000 K
keff(B2cr) at T = 1 000 K,–
using B2cr at T = 239.6 K
B2cr at T = 1 000 K
cm–2
2
kinf(B cr) at T = 1 000 K–
M2 at T = 1 000 K
cm2–
Reactivity at a pebble
irradiation of 1 500
MWd/te135
Xe number density at
u1024
a pebble irradiation of
atoms
1 500 MWd/te
cm–3
135
Xe absorption crossbarns
section at a pebble
irradiation of
1 500 MWd/Te
–
U232
–
G233
232
–
G
C
–

A
1.46068
1.738E-04
1.45351
2609.1
1.46114B
1.46269

1.46276

CD

1.666E-04
1.457
2743.91.461630.86259

0.879171.189E-04
1.45134
3794.91.39580

1.195E-04
1.452
3781.2

1.9792E-08

1.0800E+06

0.33380
0.051605
0.0001366
0.2117646

As in the previous phases, where a spherical outer boundary is used, a significant difference isfound between Participant A’s deterministic and Monte Carlo reactivities due to the boundary
conditions. This is again the result of using an isotropic flux boundary condition as opposed to areflective boundary condition at the edge of the sphere.
There is close agreement (1.4645 – 1.46068 = 382 pcm) between the deterministic evaluations of
the cell reactivity at 193.6 K provided by Participants A and F. Their deterministic evaluation of cell
reactivity at 1 000 K provided are also in agreement (1.46163 – 1.46114 = 49 pcm). This indicates thatthe deterministic temperature coefficients calculated by these two participants are consistent.
The Monte Carlo calculations from Participant A indicate that changing the temperature from
293.6 K to 1 000 K leads to a small (100 r 71 pcm) positive change in cell reactivity, which is also
observed in the deterministic calculation from Participant A. The deterministic calculations from
Participant F, however, indicate a small (287 pcm) decrease in cell reactivity for the same change.
These opposite results may be a result of differences in the nuclear data used by these participants.
3.5.2 Phase 5b: Reactivities and reaction rates
Table 3.42. Phase 5b – Monte Carlo calculations
Parameter
kinf with zero buckling
and at T = 293.6 K
B2cr at T = 293.6 K
kinf(B2cr) at T = 293.6 K
M2 at T = 293.6 K
kinf with zero buckling
and at T = 1 000 K
keff(B2cr) at T = 1 000 K,
using B2cr at T = 239.6 K
B2cr at T = 1 000 K
kinf(B2cr) at T = 1 000 K
M2 at T = 1 000 K
Reactivity at apebble irradiationof 1 500 MWd/te
135
Xe number density
at a pebble irradiationof 1 500 MWd/te
135
Xe absorption
cross-section at apebble irradiationof 1 500 MWd/Te
U232
G233
G232
C

Units
–
cm–2
–
cm2
–
–
cm–2
–
cm2
–

A
1.4617 r
50 pcm
9.998E-05

BC

D
1.46778
9.83E-5
1.45994565.11.45271.4621 r
50 pcm
0.8659 r
100 pcm
6.929E-05

0.89624
7.35E-5
1.45436141.00.96210

x1024
atoms
cm–3
barns

1.95E-08(E < 0.625eV)
1.54044E+6(E > 0.625eV)
329.98––––0.301390.04906
0.00015247E

F
1.46470 r
34 pcm

1.46127 r
26 pcmTable 3.41. Phase 5b – deterministic calculations
ParameterUnitskinf with zero buckling–
and at T = 293.6 K
B2cr at T = 293.6 K
cm–2
2
kinf(B cr) at T = 293.6 K–
M2 at T = 293.6 K
cm2
kinf with zero buckling–
and at T = 1 000 K
keff(B2cr) at T = 1 000 K,–
using B2cr at T = 239.6 K
B2cr at T = 1 000 K
cm–2
2
kinf(B cr) at T = 1 000 K–
M2 at T = 1 000 K
cm2–
Reactivity at apebble irradiationof 1 500 MWd/te
135
x1024
Xe number density
at a pebble irradiation
atoms
of 1 500 MWd/te
cm–3
135
Xe absorption
barns
cross-section at apebble irradiationof 1 500 MWd/Te
–
U232
–
G233
–
G232
C
–

A
1.46066

B

C

D

E

F

1.278E-04
1.453133546.61.460100.862108.742E-05
1.450985159.01.39580

1.9792E-08

1.0802E+06

3.3342E-01
5.1553E-02
1.3674E-04
2.1171E-01

Participants A’s Phase 5a and Phase 5b deterministic calculations are in close agreement. There isalso good agreement (1.4617 – 1.46068 = 102 r 50 pcm) between the cell reactivity at 293.6 K
evaluated in Participant A’s deterministic and Monte Carlo calculations.
The Monte Carlo results from Participants A and F are in reasonable agreement (300 pcm at
293.6 K and 83 pcm at 1 000 K). As in Phase 5a, Participant A’s Monte Carlo calculations indicate asmall positive effect on reactivity due to increasing the temperature from 293.6 K to 1 000 K.
However, Participant A’s deterministic calculation for Phase 5b shows a small decrease in reactivity
with increased temperature.
3.5.3 Spectra
A comparison between the deterministic and Monte Carlo spectra of Participant A is given in
Figure 3.15. There is generally good agreement between the spectra, apart from a slight discrepancy at
the low-energy peak.
Figure 3.16 shows a comparison of the deterministic spectra for Phase 5a calculated byParticipants A and F.

48

3.5.4 Burn-up
The results from Participant A’s deterministic Phase 5a and Phase 5b burn-up calculations are
essentially identical.
Table 3.44. Phase 5a – Monte Carlo calculations – results from Participant D
Units are u1024 atoms cm–3

Nuclide
241

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
239
Np
233
Pa
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
232
Th
233
U
234
U
235
U
236
U
237
U
238
U
242m

0
MWd/te0.000E+00
0.000E+00
0.000E+00
0.000E+00
0.000E+00
0.000E+00
0.000E+00
0.000E+00––
0.000E+00
0.000E+00
0.000E+00
0.000E+00
0.000E+00
2.190E-02
1.770E-03
0.000E+00
0.000E+00
0.000E+00
0.000E+00
0.000E+00

10 000
MWd/te1.490E-23
4.750E-26
1.370E-24
1.810E-25
7.800E-29
3.160E-27
1.840E-30
1.110E-12––
5.580E-15
1.120E-16
3.520E-18
6.950E-20
1.060E-21
2.190E-02
1.500E-03
2.510E-05
3.980E-07
3.660E-09
3.020E-12
7.940E-14

20 000
MWd/te2.410E-20
1.390E-22
6.650E-21
8.060E-22
7.890E-25
4.140E-23
5.200E-26
3.370E-11––
5.090E-13
1.970E-14
1.400E-15
5.930E-17
2.090E-18
2.180E-02
1.260E-03
4.920E-05
1.630E-06
3.390E-08
4.750E-11
2.850E-1249

30 000
MWd/te1.850E-18
1.450E-20
1.110E-18
1.130E-19
1.910E-22
1.230E-20
2.330E-23
2.520E-10––
7.110E-12
4.130E-13
5.010E-14
3.260E-15
2.010E-16
2.170E-02
1.030E-03
7.190E-05
3.710E-06
1.340E-07
2.480E-10
2.550E-11

40 000
MWd/te4.310E-17
4.070E-19
5.140E-17
4.180E-18
1.070E-20
8.890E-19
2.420E-21
1.130E-09––
4.880E-11
3.770E-12
6.960E-13
6.290E-14
6.100E-15
2.160E-02
8.210E-04
9.210E-05
6.710E-06
3.760E-07
9.090E-10
1.410E-10

Table 3.45. Phase 5a – Monte Carlo calculations – results from Participant A
Units are u1024 atoms cm–3

Nuclide
241

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
239
Np
233
Pa
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
232
Th
233
U
234
U
235
U
236
U
237
U
238
U
242m

0
MWd/te
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
2.1947E-02
1.7667E-03
–
–
–
–
–

10 000
MWd/te––1.5861E-19
1.2375E-19–
1.0697E-19–
9.9496E-13
2.0318E-17
4.4700E-05
1.3659E-14
2.1454E-16
8.9592E-18
2.6592E-19
1.2465E-19
2.1891E-02
1.5044E-03
2.5773E-05
3.8310E-07
3.5405E-09
2.6957E-12
7.0546E-14

20 000
MWd/te1.2208E-19–
2.3867E-19
1.5207E-19–
1.1785E-19–
2.9057E-11
1.0003E-15
7.8933E-05
9.0152E-13
2.8227E-14
2.6164E-15
1.0808E-16
4.5805E-18
2.1827E-02
1.2602E-03
5.1248E-05
1.5857E-06
3.3105E-08
4.0229E-11
2.4283E-1250

30 000
MWd/te2.9042E-18–
1.9677E-18
5.3034E-19–
1.5439E-19–
2.1229E-10
1.0589E-14
1.0861E-04
1.1152E-11
5.2146E-13
8.2270E-14
5.0540E-15
3.6357E-16
2.1752E-02
1.0311E-03
7.6114E-05
3.6722E-06
1.3213E-07
2.0526E-10
2.1371E-11

40 000
MWd/te5.7578E-17
5.2853E-19
6.3802E-17
1.1992E-17
1.1198E-19
1.3112E-18–
8.9948E-10
6.4397E-14
1.3858E-04
7.1504E-11
4.4200E-12
1.0675E-12
8.6047E-14
9.7929E-15
2.1663E-02
8.1643E-04
1.0006E-04
6.6604E-06
3.7556E-07
6.9980E-10
1.1246E-10

Table 3.46. Phase 5b – Monte Carlo calculations – results from Participant D
Units are u1024 atoms cm–3

Nuclide
241

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
239
Np
233
Pa
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
232
Th
233
U
234
U
235
U
236
U
237
U
238
U
242m

0
MWd/te0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00––
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.19E-02
1.77E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00

10 000
MWd/te
1.48E-23
4.73E-26
1.44E-24
1.81E-25
7.79E-29
3.34E-27
1.98E-30
1.08E-12––
5.49E-15
1.11E-16
3.47E-18
6.92E-20
1.05E-21
2.19E-02
1.50E-03
2.51E-05
3.95E-07
3.63E-09
2.96E-12
7.78E-14

20 000
MWd/te2.32E-20
1.34E-22
6.54E-21
7.77E-22
7.69E-25
4.06E-23
4.97E-26
3.29E-11––
4.97E-13
1.93E-14
1.36E-15
5.71E-17
2.02E-18
2.18E-02
1.26E-03
4.92E-05
1.63E-06
3.39E-08
4.64E-11
2.78E-1251

30 000
MWd/te
1.84E-18
1.44E-20
1.11E-18
1.12E-19
1.88E-22
1.22E-20
2.40E-23
2.61E-10––
7.18E-12
4.13E-13
4.95E-14
3.26E-15
2.00E-16
2.17E-02
1.03E-03
7.18E-05
3.74E-06
1.34E-07
2.61E-10
2.63E-1140 000
MWd/te
4.31E-17
4.05E-19
4.93E-17
4.15E-18
1.06E-20
8.50E-19
2.29E-21
1.12E-09––
4.93E-11
3.83E-12
7.07E-13
6.33E-14
6.10E-15
2.16E-02
8.21E-04
9.21E-05
6.68E-06
3.76E-07
8.80E-10
1.39E-10

Table 3.47. Phase 5b – deterministic calculations – results from Participant A
Units are u1024 atoms cm–3

Nuclide
241

Am
Am
243
Am
242
Cm
243
Cm
244
Cm
245
Cm
237
Np
239
Np
233
Pa
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
232
Th
233
U
234
U
235
U
236
U
237
U
238
U
242m

0
MWd/te
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
2.1947E-02
1.7667E-03
–
–
–
–
–

10 000
MWd/te––1.5861E-19
1.2375E-19–
1.0697E-19–
9.9493E-13
2.0317E-17
4.4700E-05
1.3658E-14
2.1453E-16
8.9589E-18
2.6591E-19
1.2465E-19
2.1891E-02
1.5044E-03
2.5772E-05
3.8309E-07
3.5405E-09
2.6956E-12
7.0544E-14

20 000
MWd/te1.2208E-19–
2.3867E-19
1.5207E-19–
1.1785E-19–
2.9057E-11
1.0002E-15
7.8933E-05
9.0149E-13
2.8226E-14
2.6163E-15
1.0807E-16
4.5803E-18
2.1827E-02
1.2602E-03
5.1248E-05
1.5857E-06
3.3105E-08
4.0228E-11
2.4283E-1252

30 000
MWd/te2.9041E-18–
1.9676E-18
5.3033E-19–
1.5439E-19–
2.1228E-10
1.0588E-14
1.0862E-04
1.1152E-11
5.2146E-13
8.2269E-14
5.0538E-15
3.6356E-16
2.1752E-02
1.0311E-03
7.6114E-05
3.6722E-06
1.3213E-07
2.0526E-10
2.1371E-11

40 000
MWd/te5.7577E-17
5.2852E-19
6.3800E-17
1.1992E-17
1.1198E-19
1.3111E-18
1.0000E-19
8.9948E-10
6.4397E-14
1.3858E-04
7.1503E-11
4.4200E-12
1.0675E-12
8.6045E-14
9.7927E-15
2.1663E-02
8.1643E-04
1.0006E-04
6.6604E-06
3.7556E-07
6.9980E-10
1.1246E-10

Chapter 4
CONCLUSIONS

A benchmark inter-comparison of results has been made for a series of cell and whole coreconfigurations relevant to uranium-, plutonium- and thorium-fuelled pebble bed modular reactors.
Contributions to the benchmark have been made by six teams: Serco Assurance (UK), CEA(France), the Nuclear Engineering Department of Hacettepe University (Turkey, two contributions),
Nexia Solutions (UK, formerly BNFL R&T, UK) and the Nuclear Science and Technology Divisionof ORNL (USA).
According to the participant, ENDF/B-V, ENDF/B-VI and JEF2.2 nuclear data have been appliedin both Monte Carlo and deterministic analyses of the benchmark.
Generally, participants using the same nuclear data report similar results, however, there are somedifferences, particularly in relation to the fuel temperature coefficients and the whole core xenon
fission product poisoning effect. There is also evidence of good agreement between Monte Carlo anddeterministic solutions for some of the participants despite the difficult nature of the problem
stochastic geometry.

53

REFERENCES[1]

Hosking, G., T.D. Newton, Proposed Benchmark Specification for an HTR Fuelled with
Reactor Grade Plutonium (or Reactor Grade Pu/Th and U/Th), NEA/NSC/DOC(2003)22(2003).

[2]

Coste, M., et al., “Self-shielding Calculations by APOLLO2 Code for Fuel Pins with aTemperature Distribution”, International Conference on Mathematical Methods andSupercomputing for Nuclear Applications, Saratoga Springs, New York, 5-9 October 1997.

[3]

Newton, T.D., J.L. Hutton, “The Next Generation WIMS Lattice Code: WIMS9”, PHYSOR 2002,
Seoul, Korea, October 2002.

[4]

Hutton, J.L., Advanced Monte Carlo Features, PHYSOR 1996, Japan, September 1996.

[5]

Herbert, A., et al., APOLLO2: Notice
SERMA/LENR/RT/98-2478/A (1998).

[6]

Andrieux, C., Notice d’identification des bibliothèques CEA93.V6 à 172 et 99 groupes, CEA
report, SERMA/LENR/RT/99-2724/A (1999).

[7]

Both, J.P., Y. Pénéliau, “The Monte Carlo Code TRIPOLI-4 and its First Benchmark
Interpretations”, International Conference on the Physics of Reactors PHYSOR 1996, Mito,
Ibaraki, Japan (1996).

55

d’Utilisation

Version

2.4,

CEA

report,Figures57

Figure 3.1. Phase 1a – Monte Carlo calculation: Critical fuel spectrum normalised to unit integral
1.60E-01
Participant D

1.40E-01

59

Flux per unit Lethargy

1.20E-01

1.00E-01

8.00E-02

6.00E-02

4.00E-02

2.00E-02

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.2. Phase 1a – deterministic calculation: Critical fuel spectrum normalised to unit integral
1.80E-01

1.60E-01

Participant A, average spectrum in matrix and particles
Participant B

1.40E-01

60

Flux per unit lethargy

Participant F, average spectrum in matrix and particles
1.20E-01

1.00E-01

8.00E-02

6.00E-02

4.00E-02

2.00E-02

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.3. Phase 1b – Monte Carlo calculation: Critical fuel spectrum normalised to unit integral
1.60E-01

Participant D S(a,b)

1.40E-01

61

Flux per unit lethargy

1.20E-01

1.00E-01

8.00E-02

6.00E-02

4.00E-02

2.00E-02

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.4. Phase 1b – Monte Carlo calculation: Critical fuel spectrum normalised to unit integral
1.80E-01
Participant A, phase 1b, average spectrum in matrix and particles

1.60E-01

Participant A, phase 1a, average spectrum in matrix and particles

62

Flux per unit lethargy

1.40E-01
1.20E-01
1.00E-01
8.00E-02
6.00E-02
4.00E-02
2.00E-02
0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

Energy (eV)

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.5. Phase 2a – Monte Carlo calculation: Critical fuel spectrum normalised to unit integral
1.20E-01
Participant D
1.00E-01

63

Flux per unit lethergy

8.00E-02

6.00E-02

4.00E-02

2.00E-02

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.6. Phase 2a – deterministic calculation: Critical fuel spectrum normalised to unit integral
1.20E-01

Participant B
1.00E-01

64

Flux per unit lethergy

Participant A
Participant F, average spectrum in matrix and particles

8.00E-02

6.00E-02

4.00E-02

2.00E-02

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.7. Phase 2b – Monte Carlo calculation: Critical fuel spectrum normalised to unit integral
1.20E-01

Participant D -S(a,b)
1.00E-01

65

Flux per unit lethergy

8.00E-02

6.00E-02

4.00E-02

2.00E-02

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.8. Phase 3 – Monte Carlo calculation: Fuel region spectrum normalised to unit integral
1.20E-01
Participant A
Participant D - S(a,b)

1.00E-01

Participant F

66

Flux per unit lethargy

8.00E-02

6.00E-02

4.00E-02

2.00E-02

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)Figure 8

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.9. Phase 3 – Serco Assurance Monte Carlo and deterministic fuel region spectra normalised to unit integral
9.00E-02
Monte Carlo

8.00E-02

Deterministic

67

Flux per unit lethargy

7.00E-02

6.00E-02

5.00E-02

4.00E-02

3.00E-02

2.00E-02

1.00E-02

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.10. Phase 3 – deterministic fuel region spectra normalised to unit integral
9.00E-02

8.00E-02
Participant E

7.00E-02

Participant A

68

Flux Per Unit Lethargy

6.00E-02

5.00E-02

4.00E-02

3.00E-02

2.00E-02

1.00E-02

0.00E+00
1.00E-04

1.00E-03

1.00E-02

1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08
Energy (eV)

Figure 3.11. Phase 4 – comparison between Monte Carlo spectra of Participants A, D and F
2.50E-01

2.00E-01

69

Flux per unit lethargy

Participant A
Participant D - S(a,b)
Participant F

1.50E-01

1.00E-01

5.00E-02

0.00E+00
1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.12. Phase 4 – Serco Assurance Monte Carlo and deterministic fuel region spectra normalised to unit integral (172 energy groups)
0.25

Monte Carlo
Deterministic

70

Flux per unit lethargy

0.2

0.15

0.1

0.05

0
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.13. Phase 4 – Serco Assurance Monte Carlo and deterministic fuel region spectra normalised to unit integral (4 energy groups)
0.06

0.05

71

Flux per unit lethargy

0.04

0.03
Deterministic
Monte Carlo

0.02

0.01

0
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.14. Phase 4 – VSOP fuel region spectrum
1.40E+01

1.20E+01

1.00E+01

8.00E+00

72
6.00E+00

4.00E+00

2.00E+00

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

Figure 3.15. Phase 5 – Serco Assurance deterministic spectra: Comparison between Phase 5a and Phase 5b results
5.00E-01

4.50E-01

4.00E-01
Phase 5a73

Flux per unit lethargy

3.50E-01

Phase 5b

3.00E-01

2.50E-01

2.00E-01

1.50E-01

1.00E-01

5.00E-02

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08Figure 3.16. Phase 5 – Comparison between deterministic spectra for Participants A and F
5.00E-01

4.50E-01

4.00E-01

Participant A
Participant F - average spectrum in matrix and particles

74

Flux per unit lethargy

3.50E-01

3.00E-01

2.50E-01

2.00E-01

1.50E-01

1.00E-01

5.00E-02

0.00E+00
1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02
Energy (eV)

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

Appendix A
BENCHMARK SPECIFICATIONS

75

OECD Nuclear Energy Agency
Nuclear Science Committee
Working Party on the Physics of Plutonium Fuels and Innovative Fuel Cycles
Proposal version 2
Benchmark Specification for an HTR Fuelled with Reactor-grade Plutonium(or Reactor-grade Pu/Th and U/Th)
J.G. Hosking, T.D. Newton
Serco Assurance (sponsored by Nexia Solutions)P. Morris
Nexia Solutions

1.

Background and aims

This benchmark proposal builds upon that specified in Ref. [1]. In addition to the three phases
described in that reference, another two phases have now been defined. Additional items for calculation
have also been added to the existing phases. It is intended that further items may be added to thebenchmark after consultation with its participants.
Although the benchmark is specifically designed to provide inter-comparisons for plutonium- andthorium-containing fuels, it is proposed that phases considering simple calculations for a uranium fuelcell and uranium core be included. The purpose of these is to identify any increased uncertainties,
relative to uranium fuel, associated with the lesser-known fuels to be investigated in different phasesof this benchmark.
The first phase considers an infinite array of fuel pebbles fuelled with uranium fuel. Phase 2
considers a similar array of pebbles but for plutonium fuel. Phase 3 continues the plutonium fuelinter-comparisons within the context of whole core calculations. Calculations for Phase 4 are for auranium-fuelled core. Phase 5 considers an infinite array of pebbles containing thorium.
In setting the benchmark the requirements in the definition of the LEUPRO-12 PROTEUS
benchmark have been considered.
Participants are invited to submit both deterministic results as well as, where appropriate, resultsfrom Monte Carlo calculations.
Fundamental nuclear data have been taken from Refs. [2] and [3] and are given to five decimal
places. Avogadro’s number and natural abundance data have been taken from Ref. [2] and atomic
weights from Ref. [3].
2.

Phase 1: Uranium cell calculations – definition

2.1 Phase 1: General description
In this phase, an infinite array of fuel pebbles is considered by modelling a single fuel pebble andusing a white (isotropic flux) boundary condition. In the course of the definition of the benchmark,there has been some debate as to the form of the model to consider.
77

If coolant is modelled surrounding the fuel pebble, the flux on the outer boundary of the model
will be anisotropic. This isotropic flux boundary condition applied in many Monte Carlo codes will
not match the white boundary condition applied in deterministic collision probability calculations.
Results between Monte Carlo and deterministic methods should not agree in these circumstances. Thiscan be avoided by assuming a unit cell in a cubic array of spheres, with reflective boundary conditions.
However, the introduction of a buckling in such a system, to obtain a reactor-like spectrum, would
again introduce differences between Monte Carlo and deterministic solutions. To cater for the requestsof all parties two cases are considered, an infinite array of fuel pebbles with reflective boundary
conditions and an infinite array of pebbles arranged in a cubic lattice with reflective unit-cell boundary
conditions.
The fuel pebble has a diameter of 6 cm with a central 5 cm diameter sphere containing a matrix
material supporting uranium-fuelled coated particles. The pebbles have a packing fraction of 0.61,
giving the outer radius of the associated spherical coolant region as 3.53735 cm, 7.0747 cm diameter.
Preserving volumes would give the unit cell length for the cubic array of spheres as 5.70218 cm,
however this is smaller than the pebble diameter of 6 cm. In Phase 1b, a 6 cm cubic array is to be used
to match the pebble diameter. The pebble packing fraction is effectively reduced in the cubic arraycalculation. The coated particles have a fuel kernel of 8.2% enriched UO2 with a diameter of 0.05 cm.
This is surrounded by four coatings consisting of an inner carbon layer, a layer of dense pyro carbon, alayer of silicon carbide and an outer pyro carbon layer. There are 15 000 coated particles in eachpebble giving a UO2 mass per pebble of 10.210 g.
The absorption effect of impurities is specified in terms of equivalent-ppm natural boron content
by mass. The values used are typical of those for actual cores, although there is considerable variationin impurities between different suppliers and even different batches of material from a single supplier.
2.2 Phase 1: Pebble and coated particle specification
The detailed specifications of the fuel pebble and coated particle are given in Table A.1. Atomic
number densities for the different regions of the pebble and coated particle are given in Table A.2.
Additional data used to derive the atomic number densities are given in Table A.3.
For simplification, the He coolant composition is assumed as that at 293.6 K, even for
calculations at 1 000 K.
2.3 Phase 1: Definition of calculations
2.3.1 Phase 1a: Infinite array of fuel pebbles (spherical outer boundary)In Phase 1a, the benchmark considers a spherical system with an outer associated coolant regionof radius 3.53735 cm and reflective or white outer boundary conditions.
2.3.2 Phase 1b: Infinite array of fuel pebbles (cubic array)
In Phase 1b, the benchmark considers a cubic array of pebbles with an outer reflective boundary
condition. The unit cell pitch in the array is 6 cm.

78

2.3.3 Phases 1a and 1b: Items for calculation
Isothermal temperature of 293.6 K:a) Reactivity (kinf).
b) The energy-independent critical buckling B2cr and kinf(B2cr)c) The migration area M2.
Isothermal temperature of 1 000 K:d) Reactivity (kinf).
e) Reactivity keff(B2cr), using the buckling from step (b), at power temperatures.
f) The energy-independent critical buckling B2cr and kinf(B2cr).
g) The migration area M2.
h) The spectral indices, calculated with a critical buckling:x U238 = 238Ucap(epithermal)/238Ucap(thermal): ratio of epithermal to thermal 238U captures.
x G235 = 235Ufis(epithermal)/235Ufis(thermal): ratio of epithermal to thermal 235U fissions.
x G238 = 238Ufis/235Ufis: ratio of fissions in 238U to fissions in 235U.
x C = 238Ucap/235Ufis: ratio of captures in 238U to fissions in 235U.
Note: Epithermal is defined here as all energies above 0.625 eV and thermal as all energies
below 0.625 eV.
i) The neutron spectrum (calculated with a critical buckling) in the fuel kernel region (flux per
unit energy) with as much energy resolution as can be provided.
j) Compositions of the actinides at irradiations of 0, 10 000, 20 000, 30 000, 40 000, 50 000,
60 000, 70 000 and 80 000 MWd/tonne, to five significant figures. The burn-up should be
performed at an isothermal temperature of 1 000 K and a constant power of 556.25 watts perfuel ball. A time-dependent, energy-independent buckling should be included to give ak-effective of unity during the burn-up. Sufficient calculational steps should be used toremove the dependency of the results on the number of steps chosen.
k) Compositions of the actinides at irradiations of 0, 10 000, 20 000, 30 000, 40 000, 50 000,
60 000, 70 000 and 80 000 MWd/tonne, to five significant figures. The burn-up should be
performed at an isothermal temperature of 1 000 K and a constant power of 556.25 watts perfuel ball. No buckling should be applied during burn-up. Sufficient calculational steps should
be used to remove the dependency of the results on the number of steps chosen.
3.

Phase 2: Plutonium cell calculations – definition

3.1 Phase 2: General description
As for Phase 1 of the benchmark, an infinite array of fuel pebbles is again considered, as aspherical system for Phase 2a and as a cubic array for Phase 2b.
The fuel pebble has a diameter of 6 cm with the central 5 cm diameter sphere containing a matrix
material supporting plutonium-fuelled coated particles. The coated particles have a fuel kernel of PuO2
with diameter of 0.024 cm. This is surrounded by four coatings consisting of an inner carbon layer,79a layer of dense pyro carbon, a layer of silicon carbide and an outer pyro carbon layer. There are
15 000 coated particles in each pebble giving a plutonium mass per pebble of 1.129 g. In Phase 2a ofthe benchmark, the outer diameter of the coolant associated with each pebble is 5.70218 cm as inbenchmark Phase 1a. In Phase 2b a 6 cm cubic array is considered as in benchmark Phase 1b.
The isotopic plutonium vector defined for this study was suggested by B. Bonin (Cogema).
It represents that arising from the irradiation of 3.7% enriched UOX fuel in a PWR with a cycle lengthof 274 full power days and a dwell time of four cycles. This corresponds to a burn-up of
41.2 GWd/tonne, typical of current practice. A cooling time of three years following irradiation has
been allowed.
The absorption effects of impurities are specified in terms of equivalent-ppm natural boron
content by mass.
3.2 Phase 2: Pebble and coated particle specification
The detailed specifications of the fuel pebble and coated particle are given in Table A.4. Atomic
number densities for the different regions of the pebble and coated particle are given in Table A.2.
Additional data used to derive the atomic number densities are given in Table A.3 and the plutonium
isotopic vector is given in Table A.6.
3.3 Phase 2a and 2b: Items for calculation
Infinite array of fuel pebbles:
Isothermal temperature of 293.6 K:a) Reactivity (kinf).
b) The energy-independent critical buckling B2cr and kinf(B2cr).
c) The migration area M2.
Isothermal temperature of 1 000 K:d) Reactivity (kinf).
e) Reactivity keff(B2cr), using the buckling from step (b), at power temperatures.
f) The energy-independent critical buckling B2cr and kinf(B2cr).
g) The migration area M2.
h) The spectral indices, calculated with a critical buckling:x U239 = 239Pucap(epithermal)/239Pucap(thermal): ratio of epithermal to thermal 239Pu captures.
x
x
x
x
x
x

G239 = 239Pufis(epithermal)/239Pufis(thermal): ratio of epithermal to thermal 239Pu fissions.
G241 = 241Pufis(epithermal)/241Pufis(thermal): ratio of epithermal to thermal 241Pu fissions.
F241 = 241Pufis/239Pufis: ratio of fissions in 241Pu to fissions in 239Pu.
C8 = 238Pucap/239Pufis: ratio of captures in 238Pu to fissions in 239Pu.
C9 = 239Pucap/239Pufis: ratio of captures in 239Pu to fissions in 239Pu.
C0 = 240Pucap/239Pufis: ratio of captures in 240Pu to fissions in 239Pu.
80

x
x

C1 = 241Pucap/239Pufis: ratio of captures in 241Pu to fissions in 239Pu.
C2 = 242Pucap/239Pufis: ratio of captures in 242Pu to fissions in 239Pu.

Note: Epithermal is defined here as all energies above 0.625 eV and thermal as all energies
below 0.625 eV.
i) The neutron spectrum (calculated with a critical buckling) in the fuel kernel region (flux per
unit energy) with as much energy resolution as can be provided.
j) Xenon reactivity effect: Reactivity (keff) at a pebble burn-up of 1 500 MWd/tonne. The burn-up
to be performed at a constant power of 556.25 watts per fuel ball with the addition of anenergy independent buckling to give a k-effective of unity during the burn-up. In addition,
135
Xe number density and absorption cross-section at 1 500 MWd/tonne irradiation.
k) Compositions of the actinides at irradiations of 0, 10 000, 20 000, 30 000, 40 000, 50 000,
60 000, 70 000 and 80 000 MWd/tonne, to five significant figures. The burn-up should be
performed at an isothermal temperature of 1 000 K and a constant power of 556.25 watts perfuel ball. A time-dependent, energy-independent buckling should be included to give ak-effective of unity during the burn-up. Sufficient calculational steps should be used toremove the dependency of the results on the number of steps chosen.
4.

Phase 3: Plutonium core calculations – definition

4.1 Phase 3: General description
This part of the benchmark considers a whole core calculation using plutonium fuel. The schematicof the core is shown in Figure A.1 and has been taken from the first draft specification of thisbenchmark by H.J. Rutten of Jülich and M. Delpech and F. Dolci of the CEA. For the plutonium fuelproposed, this core design will probably be too reactive. However, as during the course of thisbenchmark different fuels may be investigated in different phases, there is no specific reason to
optimise the design for this particular fuel type.
4.2 Phase 3: Core specification
The core layout is shown in Figure A.1 where the composition numbers are indicated for eachregion. Core parameters are defined in Table A.4 and compositions for each region are given in
Table A.5. The plutonium isotopic vector is given in Table A.6. Table A.7 gives a naming convention
for the different material regions and Tables A.3 and A.8 give the additional data used in deriving thecomposition data given in Table A.5.
The pebble and coated particle specification for Phase 3 of the benchmark is the same as for
Phase 2a of the benchmark. For simplification, the He coolant temperature and pressure have been
defined for zero power conditions for both the low power and power operation parts of the benchmark.
The reflector regions have been assumed to have an impurity level equivalent to 2 ppm by mass ofnatural boron.

814.3 Phase 3: Definition of calculations
Calculations for Phase 3 of the benchmark should provide:a) Core reactivity (k-effective) at an isothermal temperature of 293.6 K.
b) In full-power conditions, all materials are assumed to be at an average temperature of
1 000 K. In principle, the reactor structure will be at an average temperature about 50 degrees
lower than this. However, as it is not intended to model the distribution of temperature in thecore, using an approximation of an isothermal temperature is adequate for the benchmark.
x The core reactivity (k-effective).
x The neutron spectrum averaged over the core pebble-containing region (flux per unit
energy) with as much energy resolution as can be provided.
x The absorption and fission cross-sections for the plutonium isotopes 238Pu, 239Pu, 240Pu,
241
Pu and 242Pu, averaged over the core pebble containing region, in the three energy
ranges 10–5 eV to 0.625 eV, 0.625 eV to 9.118 KeV and 9.118 KeV to 20 MeV.
c) Xenon reactivity effect: Reactivity (k-effective) at an isothermal temperature of 1 000 K andaverage pebble burn-up of 1 500 MWd/tonne. The burn-up is to be performed at full power,
200.0 MW with the addition of an energy-independent buckling to give a k-effective of unity
during the burn-up. The buckling is of course set to zero following the burn-up to evaluate thexenon reactivity effect. In addition, the core average 135Xe number density and absorption
cross-section at 1 500 MWd/tonne irradiation.
5.

Phase 4: Uranium core calculations – definition

This part of the benchmark considers a whole core calculation using the uranium fuel definedpreviously in Phase 1. The schematic of the core is shown in Figure A.1 and is identical in terms ofcore geometry and structural material composition to the plutonium-fuelled core of Phase 3. The only
difference here is the use of the uranium fuel pebble.
5.1 Phase 4: General description
The core layout is shown in Figure A.1 where the composition numbers are indicated for eachregion. Core parameters are defined in Table A.1 and structural material compositions for each region
are given in Table A.5. Fuel pebble compositions are as defined in Table A.2. Table A.7 gives anaming convention for the different material regions and Tables A.3 and A.8 give the additional data
used in deriving the composition data given in Table A.5.
The pebble and coated particle specification for Phase 5 of the benchmark is the same as for
Phase 1a of the benchmark. For simplification, the He coolant temperature and pressure have been
defined for zero power conditions for both the low power and power operation parts of the benchmark.
The reflector regions have been assumed to have an impurity level equivalent to 2 ppm by mass ofnatural boron.

82

5.2 Phase 4: Definition of calculations
Calculations for Phase 4 of the benchmark should provide:a) Core reactivity (k-effective) at an isothermal temperature of 293.6 K. Results with andwithout a neutron streaming correction if available.
In full-power conditions, all materials are assumed to be at an average temperature of 1 000 K.
In principle, the reactor structure will be at an average temperature about 50 degrees lower than this.
However, as it is not intended to model the distribution of temperature in the core, using anapproximation of an isothermal temperature is adequate for the benchmark.
b) The core reactivity (k-effective) at an isothermal temperature of 1 000 K. Results to beprovided both with and without a neutron streaming correction, if available.
c) The tabulated, normalised (sum of flux per unit lethargy equal to unity) neutron spectrum
averaged over the core pebble-containing region in as much energy resolution as can beprovided. In addition, the tabulated neutron spectrum, normalised as above, for the same coreregion in four energy groups (lowest energy 10–5 eV, upper group boundaries at 1.86 eV,
29 eV, 100 KeV and 20 MeV or as close as possible to these). This spectrum should be
calculated at an isothermal temperature of 1 000 K.
d) The absorption and fission cross-sections for the uranium isotopes 238U and 235U averaged
over the core pebble-containing region, in the single energy range 10–5 eV to 20 MeV. Thesecross-sections should be calculated at an isothermal temperature of 1 000 K.
e) Reactivity (keff) at an average pebble burn-up of 1 500 MWd/tonne and isothermal temperatureof 1 000 K. The burn-up is to be performed at full power, 200.0 MW:
x with the addition of an energy independent buckling to give a k-effective of unity during
burn-up but with zero buckling applied following burn-up to evaluate the reactivity effect;x with no buckling applied;x the core average 135Xe number density and absorption cross-section at 1 500 MWd/tonne
irradiation for Case 1 with the addition of an energy-independent buckling.
f) The averaged fuelled core migration length, if available.
Participants should provide details of the calculational route they have chosen to derive values foritems e) and f).
6.

Phase 5: U/Th cell calculations – definition

6.1 Phase 5: General description
As with Phase 1 and Phase 2 of the benchmark, Phase 5 considers an infinite array of fuelpebbles. This array takes the form of a spherical system in Phase 5a and a cubic array in Phase 5b.
The fuel pebble has a diameter of 6 cm, with the central 5 cm-diameter sphere containing a matrix
material supporting U/Th-fuelled coated particles. The particle design is identical to that for theplutonium system in Phase 2. Hence, each particle consists of a central fuel kernel with diameter
0.024 cm. This kernel is surrounded by four coatings, consisting of an inner carbon layer, a layer ofdense pyro carbon, a layer of silicon carbide and an outer pyro carbon layer. There are 15 000 coatedparticles in each pebble. In Phase 5a of the benchmark, the outer diameter of the coolant associatedwith each pebble is 5.70218 cm. In Phase 5b, a 6 cm cubic array is considered.
83

The fuel is a 233U/232Th mixed oxide, with the relative enrichment by mass of the 233U being
7.48%. This enrichment was chosen such that, when the particle/pebble design described above wasused, the cell k-infinity at room temperature and start-of-life was comparable to that of the equivalent
plutonium cell in Phase 2 of this benchmark.
As in the other phases, the absorption effects of impurities are again specified in terms ofequivalent ppm natural-boron content by mass, Table 2.
Although the reactivity of this pebble configuration decreases rapidly during depletion (andtherefore would not be practical for lone use in a core), the items to be calculated for this phaseinclude some relating to burn-up. The purpose of these items is to help provide an inter-comparison ofthorium burn-up in participants’ codes.
6.2 Phase 5: Pebble and coated particle specification
The detailed specifications of the fuel pebble and coated particle are given in Table A.9. Atomic
number densities for the different regions of the pebble and coated particle are given in Table A.2.
Additional data used to derive the atomic number densities are given in Table A.3.
6.3 Phase 5a and 5b: Items for calculation
Infinite array of fuel pebbles:
Isothermal temperature of 293.6 K:a) Reactivity (kinf).
b) The energy-independent critical buckling B2cr and kinf(B2cr).
c) The migration area M2.
Isothermal temperature of 1 000 K:d) Reactivity (kinf).
e) Reactivity keff(B2cr), using the buckling from step (b), at power temperatures.
f) The energy-independent critical buckling B2cr and kinf(B2cr).
g) The migration area M2.
h) The spectral indices, calculated with a critical buckling:x U232 = 232Thcap(epithermal)/232Thcap(thermal): ratio of epithermal to thermal 232Th captures.
x G233 = 233Ufis(epithermal)/233Ufis(thermal): ratio of epithermal to thermal 233U fissions.
x G232 = 232Thfis/233Ufis: ratio of fissions in 232Th to fissions in 233U.
x C = 232Thcap/233Ufis: ratio of captures in 232Th to fissions in 233U.
Note: Epithermal, here, is defined as all energies above 0.625 eV and thermal as all energies
below 0.625 eV.
i) The neutron spectrum (calculated with a critical buckling) in the fuel kernel region (flux per
unit energy) with as much energy resolution as can be provided.

84j) Xenon reactivity effect: Reactivity (keff) at a pebble burn-up of 1 500 MWd/tonne. The burn-up
to be performed at a constant power of 556.25 watts per fuel ball with the addition of anenergy independent buckling to give a k-effective of unity during the burn-up. In addition, the135
Xe number density and absorption cross-section at 1 500 MWd/tonne irradiation.
k) Compositions of the actinides at irradiations of 0, 10 000, 20 000, 30 000 and
40 000 MWd/tonne, to five significant figures. The burn-up to be performed at an isothermal
temperature of 1 000 K and a constant power of 556.25 watts per fuel ball. A time-dependent,
energy-independent buckling to give a k-effective of unity during the burn-up should be
included. Sufficient calculational steps should be used to remove the dependency of the resultson the number of steps chosen.
7.

References[1]

Hosking, G., T.D. Newton, Benchmark Specification for an HTR Fuelled with Reactor
Grade Plutonium (or Reactor Grade Pu/Th and U/Th), Proposal, NEA/NSC/DOC(2003)22,
November 2003.

[2]

Tuli, J.K., Nuclear Wallet Cards, National Nuclear Data Centre, Brookhaven National
Laboratory, P.O. Box 5000 Upton, New York 11973-5000 USA, January 2000.

[3]

Atomic Data and Nuclear Data Tables, Volume 39, Number 2, Academic Press, July 1988.

85

Table A.1. U benchmark: Core, pebble and coated particle parameter listing
Item
Reactor parameters:
PowerCore volume
He core inlet temperature (at full power)
He core outlet temperature (at full power)
He pressure (at full power)
Coolant mass flow
Core specification:
Core heightCore radius
Number of pebbles per m3
Number of pebbles in core
Packing fraction of pebbles in core
Graphite structure natural boron impurity by mass
(materials 1, 3, 4, 5, 6, 7, 8, see Table A.7)
Fuel pebble specification:
Unit cell square pebble array pitch (cubical outer boundary)
Unit cell coolant outer radius (spherical outer boundary)
Pebble diameter
Radius of fuel zone
Outer carbon coating thickness
Outer carbon natural boron impurity by mass
Number of coated particles per pebble
Packing fraction of coated particles
Graphite matrix density
Graphite matrix natural boron impurity by mass
Outer carbon coating density
UO2 fuel mass per pebble
Coated particle specification:
Fuel in kernel UO2
UO2 fuel density
Uranium enrichment (by mass 235U/(235U + 238U)
Fuel natural boron impurity by mass
Outer coated particle radius
Fuel kernel radius
Coating materials
Coating thickness
Coating densities

86

Units

Value

MW
m3
°C
°C
MPa
Kg/s

200
66.657400950
470.024

m
m
–
–
%
ppm

9.43
1.5
5 394
359 548
61
2

cm
cm
cm
cm
cm
ppm
–
%
g/cm3
ppm
g/cm3
g

6.0
3.53735
6.0
2.5
0.5
0.5
15 000
9.043
1.75
0.5
1.75
10.210

g/cm3
%
ppm
mm
mm
–
mm
g/cm3

10.4
8.2
1
0.455
0.25
C/C/SiC/C
0.09/0.04/0.035/0.04
1.05/1.9/3.18/1.9

Table A.2. Pebble calculations: Nuclide number density data
Material

Nuclide
238
U
235
U
O
10
B
11
B
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
O
10
B
11
B
232
Th
233
U
O
10
B
11
B
C
C
C
Si
C
10
B
11
B
C
10
B
11
B
3
He
4
He

UO2 fuel

PuO2 fuel233

U/232Th mixed oxide fuelInner low-density carbon kernel coating
Pyro carbon kernel coatings (inner and outer)
Silicon carbide kernel coating
Carbon matrix

Carbon pebble outer coating
Helium coolant (at 273.16 K,105 Pa)

87

Atoms per cm3 (u 10–24)
2.12877e-02
1.92585e-03
4.64272e-02
1.14694e-07
4.64570e-07
6.01178E-04
1.24470e-02
5.44599e-03
3.00965e-03
1.54539e-03
4.60983e-02
1.14694e-07
4.64570e-07
2.19473E-02
1.76668E-03
4.74279E-02
1.14694E-07
4.64570E-07
5.26449e-02
9.52621e-02
4.77240e-02
4.77240e-02
8.77414e-02
9.64977e-09
3.90864e-08
8.77414e-02
9.64977e-09
3.90864e-08
3.71220e-11
2.65156e-05

Table A.3. Additional composition data
Quantity
Physical constants:
Avagadro’s number
Universal gas constant, R
Atomic weights:
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
233
U
235
U
238
U
232
Th
28
Si
29
Si
30
Si
Si (natural)
12
C
13
C
C (natural)
10
B
11
B
B (natural)
16
O
17
O
18
O
O (natural)
3
He
4
He
Fractional abundances (by atom):3
He in natural He (coolant)
4
He in natural He (coolant)
10
B in natural B (impurity)
11
B in natural B (impurity)28
Si in natural Si
29
Si in natural Si
30
Si in natural Si
12
C in natural C
13
C in Natural C
16
O in natural O
17
O in natural O
18
O in natural O

88

Units

Value

Mol–1
J mol–1 K–1

6.02214E+23
8.31

gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1
gm Mol–1

238.04955
239.05216
240.05381
241.05685
242.05874
233.04500
235.04392
238.05078
232.03300
27.97693
28.97649
30.97536
28.11630
12.00000
13.00335
12.01114
10.01294
11.00931
10.81203
15.99491
16.99913
17.99916
15.99930
3.016034.00260

–
–
–
–
–
–
–
–
–
–
–
–

0.0000014
0.9999986
0.19800
0.80200
0.92230
0.046830.03087
0.988900.011100.997620.00038
0.00200

Table A.4. Pu benchmark: Core, pebble and coated particle parameter listing
Item
Reactor parameters:
PowerCore volume
He core inlet temperature (at full power)
He core outlet temperature (at full power)
He pressure (at full power)
Coolant mass flow
Core specification:
Core heightCore radius
Number of pebbles per m3
Number of pebbles in core
Packing fraction of pebbles in core
Graphite structure natural boron impurity by mass
(materials 1, 3, 4, 5, 6, 7, 8, see Table A.7)
Fuel pebble specification:
Unit cell square pebble array pitch (cubical outer boundary)
Unit cell coolant outer radius (spherical outer boundary)
Pebble diameter
Radius of fuel zone
Outer carbon coating thickness
Outer carbon natural boron impurity by mass
Number of coated particles per pebble
Packing fraction of coated particles
Graphite matrix density
Graphite matrix natural boron impurity by mass
Outer carbon coating density
PuO2 fuel mass per pebble
Average power per pebble
Coated particle specification:
Fuel in kernel PuO2
PuO2 fuel density
Fuel natural boron impurity by mass
Outer coated particle radius
Fuel kernel radius
Coating materials
Coating thickness
Coating densities

89

Units

Value

MW
m3
°C
°C
MPa
Kg/s

200
66.657400950
470.024

m
m
–
–
%
ppm

9.43
1.5
5 394
359 548
61
2

cm
cm
cm
cm
cm
ppm
–
%
g/cm3
ppm
g/cm3
g
watts

6.0
3.53735
6.0
2.5
0.5
0.5
15 000
3.45
1.75
0.5
1.75
1.129
556.25

g/cm3
ppm
mm
mm
–
mm
g/cm3

10.4
1
0.33
0.12
C/C/SiC/C
0.095/0.04/0.035/0.04
1.05/1.9/3.18/1.9

Table A.5. Core calculations: Nuclide number density data
Material

Nuclide
238
U
235
U
O
10
B
11
B
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu
O
10
B
11
B
C
C
C
Si
C
10
B
11
B
C
10
B
11
B
3
He
4
He
C
10
B
11
B
4
He
C
10
B
11
B
C
10
B
11
B
C
10
B
11
B
C
10
B
11
B
C
10
B
11
B
C
10
B
11
B

UO2 fuel

PuO2 fuelInner low-density carbon kernel coating
Pyro carbon kernel coatings (inner and outer)
Silicon carbide kernel coating
Carbon matrix

Carbon pebble outer coating
Helium coolant (at 273.16 K,105 Pa)
Reflector 1

Void
Void + graphite

Reflector 2

Carbon layer surrounding system

Reflector + coolant channel

Reflector + control rods

Reflector 3

90

Atoms per cm3 (u10–24)
2.12877e-02
1.92585e-03
4.64272e-02
1.14694e-07
4.64570e-07
6.01178E-04
1.24470e-02
5.44599e-03
3.00965e-03
1.54539e-03
4.60983e-02
1.14694e-07
4.64570e-07
5.26449e-02
9.52621e-02
4.77240e-02
4.77240e-02
8.77414e-02
9.64977e-09
3.90864e-08
8.77414e-02
9.64977e-09
3.90864e-08
3.71220e-11
2.65156e-05
7.72000e-02
3.39617e-08
1.37562e-07
1.00000e-11
2.00000e-03
8.79837e-10
3.56378e-09
9.03000e-02
3.97246e-08
1.60905e-07
8.53000e-03
3.75250e-08
1.51995e-07
7.02000e-02
3.08823e-08
1.25089e-07
3.51000e-02
1.54411e-08
6.25444e-08
3.51000e-02
1.54411e-08
6.25444e-08

Table A.6. Plutonium isotopic vector
Isotope
238
Pu
239
Pu
240
Pu
241
Pu
242
Pu

Units
wt.%
wt.%
wt.%
wt.%
wt.%

% by mass
2.59
53.85
23.66
13.13
6.77

Table A.7. Key to core regions shown in Figure A.1
RegionReflector 1
Void
Void + graphite
Reflector 2
Carbon layer surrounding system
Reflector + coolant channels
Reflector + control rods
Reflector 3

Number on core diagram
1
2
3
4
5
6
7
8

Table A.8. Core calculations: Material mass densities
Material
UO2 fuel
PuO2 fuelLow-density carbon kernel coating
Pyro carbon kernel coating
Silicon carbide kernel coating
Carbon matrix
Carbon pebble coating
Helium coolant (at 273.16 K, 105 Pa)
Reflector 1
Void
Void + graphite
Reflector 2
Carbon layer surrounding system
Reflector + coolant channels
Reflector + control rods
Reflector 3

91

Mass density (g/cm3)
10.4
10.4
1.05
1.9
3.18
1.75
1.75
1.78E-04
1.53901–
3.98707e-2
1.800161.70049
1.39946
6.99731E-01
6.99731E-01

Table A.9. U/Th benchmark: Pebble and coated particle parameter listing
Item
Fuel pebble specification:
Unit cell square pebble array pitch (cubical outer boundary)
Unit cell coolant outer radius (spherical outer boundary)
Pebble diameter
Radius of fuel zone
Outer carbon coating thickness
Outer carbon natural boron impurity by mass
Number of coated particles per pebble
Packing fraction of coated particles
Graphite matrix density
Graphite matrix natural boron impurity by mass
Outer carbon coating density
Fuel mass per pebble
Coated particle specification:
Fuel in kernel: 233U/232Th mixed oxide
Fuel density
233
U enrichment (by mass 233U/(233U + 232Th)
Fuel natural boron impurity by mass
Outer coated particle radius
Fuel kernel radius
Coating materials
Coating thickness
Coating densities

92

Units

Value

cm
cm
cm
cm
cm
ppm
–
%
g/cm3
ppm
g/cm3
g

6.0
3.53735
6.0
2.5
0.5
0.5
15 000
3.45
1.75
0.5
1.75
1.129

g/cm3
%
ppm
mm
mm
–
mm
g/cm3

10.4
7.48
1
0.455
0.12
C/C/SiC/C
0.095/0.04/0.035/0.04
1.05/1.9/3.18/1.9

Figure A.1. Schematic diagram of core R-Z calculation model. Dimensions shown are in cm.
1693
5

5

4

4

3

3

1633
1573
1538

1
1438
1388

2

7

1140

4

4

5

CORE
6

8

445
100
4
1

275

4
175
3

5
0

150

0

156 162

93

181

216.5

227.5

250

Appendix B
ENERGY GROUP SCHEMES USEDIN THE DETERMINISTIC CALCULATIONS

95

Table B.1. 172-group energy scheme used byParticipants A and B in their deterministic calculations
Group
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58

Upper energy boundary(eV)
1.964E+07
1.733E+07
1.492E+07
1.384E+07
1.162E+07
1.000E+07
8.187E+06
6.703E+06
6.065E+06
5.488E+06
4.493E+06
3.679E+06
3.012E+06
2.466E+06
2.231E+06
2.019E+06
1.653E+06
1.353E+06
1.225E+06
1.108E+06
1.003E+06
9.072E+05
8.209E+05
6.081E+05
5.502E+05
4.979E+05
4.505E+05
4.076E+05
3.020E+05
2.732E+05
2.472E+05
1.832E+05
1.228E+05
1.111E+05
8.230E+04
6.738E+04
5.517E+04
4.087E+04
3.698E+04
2.928E+04
2.739E+04
2.479E+04
1.662E+04
1.503E+04
1.114E+04
9.119E+03
7.466E+03
5.531E+03
5.005E+03
3.527E+03
3.355E+03
2.249E+03
2.035E+03
1.507E+03
1.434E+03
1.234E+03
1.010E+03
9.142E+02

Group
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116

Upper energy boundary(eV)
7.485E+02
6.773E+02
4.540E+02
3.717E+02
3.043E+02
2.040E+02
1.486E+02
1.367E+02
9.166E+01
7.567E+01
6.790E+01
5.560E+01
5.158E+01
4.825E+01
4.552E+01
4.017E+01
3.727E+01
3.372E+01
3.051E+01
2.761E+01
2.498E+01
2.260E+01
1.945E+01
1.593E+01
1.371E+01
1.122E+01
9.906E+00
9.190E+00
8.315E+00
7.524E+00
6.160E+00
5.346E+00
5.043E+00
4.129E+00
4.000E+00
3.381E+00
3.300E+00
2.768E+00
2.720E+00
2.600E+00
2.550E+00
2.360E+00
2.130E+00
2.100E+00
2.020E+00
1.930E+00
1.840E+00
1.755E+00
1.670E+00
1.590E+00
1.500E+00
1.475E+00
1.440E+00
1.370E+00
1.337E+00
1.300E+00
1.235E+00
1.170E+00

97

Group
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172

Upper energy boundary(eV)
1.150E+00
1.123E+00
1.110E+00
1.097E+00
1.071E+00
1.045E+00
1.035E+00
1.020E+00
9.960E-01
9.860E-01
9.720E-01
9.500E-01
9.300E-01
9.100E-01
8.600E-01
8.500E-01
7.900E-01
7.800E-01
7.050E-01
6.250E-01
5.400E-01
5.000E-01
4.850E-01
4.330E-01
4.000E-01
3.910E-01
3.500E-01
3.200E-01
3.145E-01
3.000E-01
2.800E-01
2.480E-01
2.200E-01
1.890E-01
1.800E-01
1.600E-01
1.400E-01
1.340E-01
1.150E-01
1.000E-01
9.500E-02
8.000E-02
7.700E-02
6.700E-02
5.800E-02
5.000E-02
4.200E-02
3.500E-02
3.000E-02
2.500E-02
2.000E-02
1.500E-02
1.000E-02
6.900E-03
5.000E-03
3.000E-03

Table B.2. 10-group energy scheme used by Participant A in their deterministic calculations
Group
1
2
3
4
5
6
7
8
9
10

Upper energy boundary(eV)
1.964E+07
1.162E+07
3.679E+06
2.231E+06
1.228E+05
3.043E+02
1.170E+00
9.720E-01
4.330E-01
2.480E-01

98

Table B.3. 238-group energy scheme used by participant F
Group
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60

Upper energy
boundary (eV)
2.00E+07
1.73E+07
1.57E+07
1.46E+07
1.38E+07
1.28E+07
1.00E+07
8.19E+06
6.43E+06
4.80E+06
4.30E+06
3.00E+06
2.48E+06
2.35E+06
1.85E+06
1.50E+06
1.40E+06
1.36E+06
1.32E+06
1.25E+06
1.20E+06
1.10E+06
1.01E+06
9.20E+05
9.00E+05
8.75E+05
8.61E+05
8.20E+05
7.50E+05
6.79E+05
6.70E+05
6.00E+05
5.73E+05
5.50E+05
5.00E+05
4.70E+05
4.40E+05
4.20E+05
4.00E+05
3.30E+05
2.70E+05
2.00E+05
1.50E+05
1.28E+05
1.00E+05
8.50E+04
8.20E+04
7.50E+04
7.30E+04
6.00E+04
5.20E+04
5.00E+04
4.50E+04
3.00E+04
2.50E+04
1.70E+04
1.30E+04
9.50E+03
8.03E+03
6.00E+03

Group
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120

Upper energy
boundary (eV)3.90E+03
3.74E+03
3.00E+03
2.58E+03
2.29E+03
2.20E+03
1.80E+03
1.55E+03
1.50E+03
1.15E+03
9.50E+02
6.83E+02
6.70E+02
5.50E+02
3.05E+02
2.85E+02
2.40E+02
2.10E+02
2.08E+02
1.93E+02
1.86E+02
1.22E+02
1.19E+02
1.15E+02
1.08E+02
1.00E+02
9.00E+01
8.20E+01
8.00E+01
7.60E+01
7.20E+01
6.75E+01
6.50E+01
6.10E+01
5.90E+01
5.34E+01
5.20E+01
5.06E+01
4.92E+01
4.83E+01
4.70E+01
4.52E+01
4.40E+01
4.24E+01
4.10E+01
3.96E+01
3.91E+01
3.80E+01
3.70E+01
3.55E+01
3.46E+01
3.38E+01
3.33E+01
3.18E+01
3.13E+01
3.00E+01
2.75E+01
2.50E+01
2.25E+01
2.10E+01

Group
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180

99

Upper energy
boundary (eV)
2.00E+01
1.90E+01
1.85E+01
1.70E+01
1.60E+01
1.51E+01
1.44E+01
1.38E+01
1.29E+01
1.19E+01
1.15E+01
1.00E+01
9.10E+00
8.10E+00
7.15E+00
7.00E+00
6.75E+00
6.50E+00
6.25E+00
6.00E+00
5.40E+00
5.00E+00
4.75E+00
4.00E+00
3.73E+00
3.50E+00
3.15E+00
3.05E+00
3.00E+00
2.97E+00
2.87E+00
2.77E+00
2.67E+00
2.57E+00
2.47E+00
2.38E+00
2.30E+00
2.21E+00
2.12E+00
2.00E+00
1.94E+00
1.86E+00
1.77E+00
1.68E+00
1.59E+00
1.50E+00
1.45E+00
1.40E+00
1.35E+00
1.30E+00
1.25E+00
1.22E+00
1.20E+00
1.17E+00
1.15E+00
1.14E+00
1.13E+00
1.12E+00
1.11E+00
1.10E+00

Group
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238

Upper energy
boundary (eV)
1.09E+00
1.08E+00
1.07E+00
1.06E+00
1.05E+00
1.04E+00
1.03E+00
1.02E+00
1.01E+00
1.00E+00
9.75E-01
9.50E-01
9.25E-01
9.00E-01
8.50E-01
8.00E-01
7.50E-01
7.00E-01
6.50E-01
6.25E-01
6.00E-01
5.50E-01
5.00E-01
4.50E-01
4.00E-01
3.75E-01
3.50E-01
3.25E-01
3.00E-01
2.75E-01
2.50E-01
2.25E-01
2.00E-01
1.75E-01
1.50E-01
1.25E-01
1.00E-01
9.00E-02
8.00E-02
7.00E-02
6.00E-02
5.00E-02
4.00E-02
3.00E-02
2.53E-02
1.00E-02
7.50E-03
5.00E-03
4.00E-03
3.00E-03
2.50E-03
2.00E-03
1.50E-03
1.20E-03
1.00E-03
7.50E-04
5.00E-04
1.00E-04

OECD PUBLICATIONS, 2 rue André-Pascal, 75775 PARIS CEDEX 16
Printed in France.