i
Radiation Damage Assessment of Zircaloy and Stainless Steel Cladding materials
based on Neutron Flux and Energy Deposition using both Computational Tools and
Analytical Solution.
A Thesis Submitted to the Department of NUCLEAR ENGINEERING
COLLEGE OF BASIC AND APPLIED SCIENCES
UNIVERSITY OF GHANA
BY
(Samiru Alhassan, 10507887)
BSc (Kumasi), 2013
In Partial Fulfillment of the Requirements for the Degree of
MASTER OF PHILOSOPHY
In
NUCLEAR ENGINEERING
July 2016
University of Ghana http://ugspace.ug.edu.gh
ii
DECLARATION
I hereby declare that with the exception of references to other people’s work which have
duly been acknowledged, this Thesis is the result of my own research work and no part of
it has been presented for another degree in this University or elsewhere.
……………………………………… ………………………………
SAMIRU ALHASSAN Date
(Candidate)
We hereby declare that the preparation of this project was supervised in accordance with
the guidelines of the supervision of Thesis work laid down by the University of Ghana.
………………………………… …………………………………
Dr. K..A. Danso NANA (Prof.) A. AYENSU GYEABUO I
(PRINCIPAL SUPERVISOR) (CO-SUPERVISOR)
…..………………………….. …………………………………
Date Date
University of Ghana http://ugspace.ug.edu.gh
iii
ABSTRACT
The maintenance of the structural integrity of materials in the nuclear reactor is a crucial
issue both in-service and out of service. The cladding which forms an integral part of the
fuel assembly isolates and prevents the fuel from contaminating the coolant. Pure
Zirconium alloys and Steels have extensive use in the nuclear industry including its usage
as clad materials for both Pressurized Water Reactor and Boiling Water Reactor fuels.
These materials possess good combination of properties such as low neutron absorption,
creep behavior, stress-corrosion cracking resistance, reduced hydrogen uptake, reduced
corrosion. However, these structural components are susceptible to defects when exposed
to high heat, pressure and irradiation. In this regard, the research focused on the use of
computational tools to assess the radiation damage on zircaloy and stainless steel clad
materials. In this thesis, a modified Low Enriched Uranium core (LEU) input deck was
created with focus on the determination of neutron parameters with the MCNP5 code for a
typical MNSR operating at 34KW maximum power using Zircaloy-4 fuel clad. The neutron
energy deposition and neutron flux (neutron parameters) were employed in the SRIM-
TRIM code and analytical radiation model calculations respectively to ascertain the
radiation damage. The MCNP5 results as obtained from running the LEU input deck from
the Argonne National Laboratory (USA) registered an average energy of 9.871884MeV in
all ten lattice rings. The average fast neutron flux representative of all 344 fuel rods gave
5.29667E+11ncm-2s-1 while the average fast neutron flux in the ten lattice rings gave
7.46E+13n/cm2.s. In the SRIM code, the target width was determined as about 2.81μm for
neutron interaction and 40.9μm for the radiation interaction. The damage assessment in the
TRIM code established Zircaloy-4 as the best cladding material as it recorded the least
University of Ghana http://ugspace.ug.edu.gh
iv
number of vacancies sustained, least replacement collisions of 147 and recoiling energy of
0.09 whilst Eurofer-97 suffered the highest vacancies created with stainless steel type-308
experiencing the highest replacement collision. The analytical calculations of the radiation
damage on Zircaloy-4 using both the Kinchin-Pease and Norgett-Robinson Torrens models
was determined as recording displacement of 17 and 14 atoms from the lattice site after
10,000 collisions respectively for only 30 minutes of operation.
The calculation of the
radiation damage suggests that the zircaloy-4 clad material shows good resistance to defect
formation and propagation hence the displacement per atom after a much longer operation
time will still leave the clad intact.
University of Ghana http://ugspace.ug.edu.gh
v
DEDICATION
This research is dedicated to God Almighty the benevolent, the Compassionate, Most
Gracious the Most Merciful who saw me through the entire course of this work.
University of Ghana http://ugspace.ug.edu.gh
vi
ACKNOWLEDGEMENTS
I believe acknowledgments are in order, my first is to the Almighty Allah, Creator and
Controller of the Universe for His Guidance and Protection.
I wish to express my sincerest appreciation to all whose support, tolerance and guidance
made this study a stunning success. I am highly indebted to many especially:
My supervisors, Dr. K.A Danso and Nana (Prof.) Ayensu A. Gyeabour I for their
stupendous and tremendous hard work, tolerance, words of motivation and advice,
empathetic comments and guidance in the preparation of this thesis.
I also want to use this opportunity to thank Dr. Henry Cecil Odoi, the Reactor Manager at
the Ghana Atomic Energy Commission (GAEC) for the tremendous support he offered me
from the beginning to the end of this thesis work.
My sincere gratitude also goes to my Uncle Naa Yussif Mumuni and Mr. Mahama
Zachariah, Mr. Abdul Rahman Usman and Mr. Yussif Nupaya for the overwhelming
support, encouragement and guidance in this endeavor and my entire life. Allah reward
you all abundantly.
My sincere gratitude and special thanks go to my parents and sisters, whose never ending
prayers, encouragement and support helped me to accomplish my goals. I love you so
much.
Finally, to all who contributed to this work directly or indirectly, no amount of words or
money can explain my appreciation for the help you offered me. God richly bless you.
University of Ghana http://ugspace.ug.edu.gh
vii
TABLE OF CONTENTS
DECLARATION ................................................................................................................ ii
ABSTRACT ....................................................................................................................... iii
DEDICATION ................................................................................................................... iii
ACKNOWLEDGEMENTS ............................................................................................... vi
LIST OF FIGURES ........................................................................................................... xi
LIST OF TABLES ........................................................................................................... xiii
NOMENCLATURE ........................................................................................................ xvi
LIST OF SYMBOLS AND CONSTANTS .................................................................... xvii
CHAPTER ONE: INTRODUCTION ..................................................................................1
1.1 Overview ........................................................................................................................1
1.2 Study Background ..........................................................................................................1
1.3 Study Area .....................................................................................................................4
1.4 Problem Statement .........................................................................................................5
1.5 Relevance and Justification ............................................................................................7
1.6 Aims and Objectives ......................................................................................................8
1.7 General description of MNSR .......................................................................................9
1.8 Scope ............................................................................................................................12
1.9 Definitions....................................................................................................................13
CHAPTER TWO: LITERATURE REVIEW ....................................................................16
University of Ghana http://ugspace.ug.edu.gh
viii
2.1 Nuclear cladding materials ..........................................................................................16
2.1.1 Properties of Cladding ..................................................................................16
2.1.2 Zirconium Metals ................................................................................................19
2.1.2.1 Material properties of Zirconium ............................................................20
2.1.2.2 Zirconium alloy .......................................................................................22
2.1.3 Steel Metal...........................................................................................................24
2.1.3.1 Stainless Steels………. ...........................................................................24
2.1.4 Limitation of Zirconium and Stainless steel applications ...................................25
2.1.5 Irradiation of Clad ...............................................................................................25
2.2 Interactions of Particulate Matter.................................................................................27
2.2.1 Particle Solid interactions ...................................................................................28
2.2.1.1 Cross-sections ........................................................................................29
2.2.2 Interaction of γ-radiation with Structural components ........................................30
2.2.2.1 Photoelectric Effect .................................................................................31
2.2.2.2 Pair production ........................................................................................31
2.2.2.3 Compton scattering .................................................................................32
2.2.3 Radiation Damage ...............................................................................................33
2.2.3.1 Knock-on atoms and Displacement cascade ...........................................34
2.2.3.2 Radiation damage models .......................................................................35
2.2.3.3 Displacement caused by Neutron Irradiation ..........................................38
University of Ghana http://ugspace.ug.edu.gh
ix
2.2.3.4 Radiation induced defects .......................................................................40
2.3 Radiation effects on Structural components ................................................................41
2.3.1 Irradiation Cascades ............................................................................................42
2.3.2 Radiation Effects in Nuclear fuel .......................................................................45
2.3.3 Radiation effects on Cladding ............................................................................46
2.3.4 Ion Irradiation Effects ........................................................................................47
2.4 Radiation damage Simulation ......................................................................................48
2.4.1 Binary Collision Approximation (BCA) Method...............................................48
2.4.2 Molecular Dynamics (MD) Method ...................................................................50
2.4.3 Kinetic Monte Carlo (KMC) Methods ...............................................................51
2.4.4 SRIM-TRIM Code .............................................................................................52
2.4.4.1 TRIM Code ............................................................................................53
2.4.4.2 TRIM setup Windows ............................................................................53
2.4.4.3 Types of TRIM Calculations ..................................................................54
2.4.4.3.1 Ion Distribution & Quick Calculation of Damage ..................54
2.4.4.3.2 Detailed Calculation with full Damage Cascade.....................55
2.4.4.3.3 Surface Sputtering ...................................................................55
2.4.4.3.4 Monolayer Collisions ..............................................................55
2.4.4.3.5 Neutron /Electron/Photon Cascade .........................................55
2.4.4.3.6 Various Ion Energy/Angle/Position ........................................56
University of Ghana http://ugspace.ug.edu.gh
x
CHAPTER THREE: RESEARCH METHODOLOGY ....................................................57
3.1 MCNP5 Simulation ......................................................................................................58
3.2 MCNP Neutronic Calculations ....................................................................................58
3.2.1 MCNP (LEU) Input deck ....................................................................................61
3.2.2 Working principles of MCNP Code ....................................................................62
3.2.2.1 Nuclear Data & Reactions ......................................................................62
3.2.2.2 Source Specification ................................................................................63
3.2.2.3 Tallies & Output ......................................................................................64
3.2.2.4 Estimation of Monte Carlo Errors ...........................................................65
3.2.2.5 Kinchin – Pease Model Evaluation .........................................................67
3.2.2.6 Norgett – Robinson Torrens Model Evaluation ......................................67
3.3 Assessment of Nuclear Parameters ..............................................................................68
3.3.1 Neutron Fluence ..................................................................................................68
3.3.2 Normalization Factor...........................................................................................68
3.3.3 Energy Deposition ..............................................................................................69
3.4 Damage Assessment by SRIM-TRIM Code ................................................................70
3.4.1 Input Data for SRIM Code ..................................................................................70
3.4.2 Input Data parameter Windows for SRIM ..........................................................73
3.4.3 Transport of Ion in Matter (TRIM) Simulation .................................................75
3.5 Displacement Cascade Assessment .............................................................................80
University of Ghana http://ugspace.ug.edu.gh
xi
3.5.1 Kinchin-Pease Model ..........................................................................................81
3.5.2 Norgett-Robinson Torrens Model .......................................................................82
CHAPTER FOUR: RESULTS AND DISCUSSION ........................................................83
4.1 Neutron Flux Distribution ............................................................................................83
4.1.1 Neutron Flux Distribution for the Lattice Rings .................................................84
4.1.2 Neutron Energy Deposition ................................................................................88
4.2 Damage Assessment by SRIM-TRIM Code ................................................................89
4.2.1 SRIM Calculations ..............................................................................................89
4.2.2 TRIM Code Calculations ....................................................................................91
4.2.2.1 Full Damage Assessment of Zircaloy-4 ..................................................91
4.2.2.2 Full Damage Assessment of Zircaloy-2 ..................................................94
4.2.2.3 Full Damage Assessment of Stainless steel type-308. ............................98
4.2.2.4 Full Damage Assessment of Eurofer 97 ...............................................101
CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATIONS ............................105
5.1 Conclusions ................................................................................................................105
5.2 Recommendations ......................................................................................................106
REFERENCES ................................................................................................................107
APPENDICES .................................................................................................................113
University of Ghana http://ugspace.ug.edu.gh
xi
LIST OF FIGURES
Figure 1.1: A representation of a fuel elements for LEU Core reactor………......................4
Figure 1.2: A schematic view of the LEU fuel element assembly for the GHARR-1……....5
Figure 1.3: Pictorial View of GHARR - 1 Vessel…………………….…………………..10
Figure 1.4: Cross-sectional view of GHARR – 1 Vessel……………...…………………..10
Figure 2.1: Iron-iron Carbide phase diagram……………………………………..……....24
Figure 2.2: Particles interactions in crystal structure…………..……………………..…..28
Figure 2.3: A schematic representation of the cross-section concept………………..……30
Figure 2.4: Representation of Photoelectric Effect…………………………………….....31
Figure 2.5: Representation of Pair Production……………………………..……………..32
Figure 2.6: Representation of Compton Effect…………………..……………………….32
Figure 2.7: Defects susceptible to changing material properties of a lattice structure ……41
Figure 2.8: Beta-decay of Daughter Nuclei after Fission process………………………...41
Figure 2.9: Point defect history during avalanche of displacement cascade…. ……….....43
Figure 2.10: Schematic illustrations of Linear Collision and Thermal spikes…………….44
Figure 2.11: Light water Reactor fuel Performance………………………………………47
Figure 2.12: Principle of interaction between ions and solids………………………….....48
Figure 2.13: MD in the multiscale Modeling framework of Microstructural evolution…..51
University of Ghana http://ugspace.ug.edu.gh
xii
Figure 2.14: Display of TRIM setup windows…………………………..………………..54
Figure 3.1: Data Windows for Cobalt-Zircaloy- 2 & 4 in SRIM Code……………….......73
Figure 3.2: Data Windows for Cobalt-Stainless Steels & Eurofer 97 in SRIM Code …….73
Figure 3.3: Data Windows for Helium-Zircaloy- 2 & 4 in SRIM Code…………………..74
Figure 3.4: Data Windows for Helium-Stainless Steels & Eurofer 97 in SRIM Code…….75
Figure 3.5: Data Windows for Zircaloy- 2 & 4 in TRIM Code …………………..……….79
Figure 3.6: Data Windows for Stainless Steels & Eurofer 97 in TRIM Code ………..…..80
Figure 4.1: A graph of Normalized neutron flux against Lattice distance………………...86
Figure 4.2: A graph of Normalized average neutron flux against Lattice distance……......87
Figure 4.3: A graph of Energy deposition (MeV) against Lattice distance…………….....89
Figure 4.4: A graph of Projection Range of Clad Materials (Neutron-Interactions)….......90
Figure 4.5: A graph of Projection Range of Clad Materials (λ-Interactions)……………..90
Figure 4.6: TRIM output for λ-Interactions with Zircaloy-4………………………….......92
Figure 4.7: Atomic displacement by Collision Event and Sputtering (Zr-4)……...………93
Figure 4.8: Illustration of Ionization and Ion Range in Zircaloy-4………………………..93
Figure 4.9: TRIM output for Neutron interactions with Zircaloy-4………………………94
Figure 4.10: Atomic displacement by Collision Event and Ionization (Zr-4)………….....94
Figure 4.11: TRIM output for λ-Interactions with Zircaloy-2………………………….....95
University of Ghana http://ugspace.ug.edu.gh
xiii
Figure 4.12: Atomic displacement by Collision Event and Sputtering (Zr-2)………….....96
Figure 4.13: Illustration of Ionization and Ion Range in Zircaloy-2………………………96
Figure 4.14: Damage Assessment by TRIM on Zircaloy-2 (Neutron Interaction)………..97
Figure 4.15: TRIM output for λ-Interactions with Stainless Steel Type 308……………..98
Figure 4.16: Atomic Displacement by Collision Event and Sputtering (Fe - 308)………..99
Figure 4.17: Illustration of Ionization and Atom distribution (Fe - 308)………................99
Figure 4.18: Damage Assessment by TRIM on Fe - 308 (Neutron Interaction)…………100
Figure 4.19: TRIM output for λ-Interactions with Eurofer 97…………………………...101
Figure 4.20: Atomic Displacement by Collision Event and Sputtering (Eurofer 97)……102
Figure 4.21: Illustration of Atom Distribution and Ionization (Eurofer 97)……………..102
Figure 4.22: Damage Assessment by TRIM on Eurofer 97 (Neutron Interaction)………103
University of Ghana http://ugspace.ug.edu.gh
xiv
LIST OF TABLES
Table 1.1: Specifications of GHARR – 1…………………………………………………11
Table 2.1: Material behaviour of reactor at rising temperature…………………………...18
Table 2.2: Physical nature of Zirconium metal…………………………………………...21
Table 2.3: Structural properties of Zirconium metal………………………………….......22
Table 2.4: Mean Composition of Zirconium alloy by weight percentage………………...23
Table 3.1: Description of Mnemonics for Keff estimation………………………………...64
Table 3.2: Tally Mnemonics and Meanings………………………………………………65
Table 3.3: Guidelines for Interpreting Relative Error R…..………..…………………......66
Table 3.4: Ion (Cobalt) input data parameters for SRIM Code…………..……….……….70
Table 3.5: Target data parameters for Zircaloy-4 in SRIM Code………..…………….....71
Table 3.6: Target data parameters for Zircaloy-2 in SRIM Code…………..………...…..71
Table 3.7: Target data parameters for Stainless Steel type 308 in SRIM Code…...…......72
Table 3.8: Target data parameters for Eurofer 97 in SRIM Code…………………..…….72
Table 3.9: Ion (Helium) data parameters for SRIM Code………………..…………….....74
Table 3.10: Ion (Helium) data and input parameters for TRIM Code…..………………...76
Table 3.11: Ion (Cobalt) data and input parameters for TRIM Code…..………………….77
Table 3.12: Target data (Zr-4) data and input parameters for TRIM Code…..……………77
University of Ghana http://ugspace.ug.edu.gh
xv
Table 3.13: Target data (Zr-2) data and input parameters for TRIM Code…..……………78
Table 3.14: Target data (Fe-308) data and input parameters for TRIM Code…..…………78
Table 3.15: Target data (Eurofer-97) data and input parameters for TRIM Code…..….....79
Table 4.1: Average Normalized Neutron Flux in Clad from MCNP5 Simulations…….....83
Table 4.2: Normalized and Average Normalized Neutron Flux in Ten Lattice Rings……85
Table 4.3: Neutron Energy Deposition in Lattice Rings………………………………….88
Table 4.4: Percentage Energy Distribution in Zircaloy-4 Target Material……………….91
Table 4.5: Percentage Energy Distribution in Zircaloy-2 Target Material………………..95
Table 4.6: Percentage Energy Distribution in Fe - 308Target Material………………….98
Table 4.7: Percentage Energy Distribution in Eurofer 97 Target Material……………..101
Table 4.8: Vacancy Assessment on Clad Materials…………………………………….104
University of Ghana http://ugspace.ug.edu.gh
xvi
NOMENCLATURE
ACRONYM DEFINITIONS
ACTI Advanced Computational Technology Initiative
BCA Binary Collision Approximation
BCC Body Centered Cubic
CANDU Canadian Deuterium Uranium reactor
CIAE China Institute of Atomic Energy
DPA Displacement per Atom
ENDF Evaluated Nuclear Data File
FOM Figure of Merit
HCP Hexagonal Closed Packing
IAEA International Atomic Energy Agency
KMC Kinetic Monte Carlo method
LAMMPS Large Atomic/Molecular Massively Parallel Simulation
LEU Low Enriched Uranium
MCNP Monte Carlo N- Particle
MD Molecular Dynamics method
MNSR Miniature Neutron Source Reactor
NRT Norgett-Robinson Torrens
ORNL Oak Ridge National Laboratory
PKA Primary Knock-on Atom
SRIM Stopping Range of Ions in Matter
TRIM Transport of Ions in Matter
University of Ghana http://ugspace.ug.edu.gh
xvii
LIST OF SYMBOLS AND CONSTANTS
SYMBOLS DEFINITIONS
aB……………………………………………………………………..Bohr radius
…………………………………………………………………...Burgers vector
Å……………………………………………………………………..Angstrom
Ck…………………………………………………………………….Transformation atom
c………………………………………………………………………Speed of light
dP…………………………………………………………………..Differential probability
σd (En)…………………………………………………………Displacement cross-section
σe …………………………………………………………………....Elastic cross-section
Ed………………………………………………………………….....Displacement energy
∆E……………………………………………………………………Lost energy
nE ……………………………………………………………………Neutron flux energy
Keff…………………………………………………………………...Effective criticality
me…………………………………………………………………….Mass of electron
ɸ(E)…………………………………………………………………..Particle flux
ɸ(En)…………………………………………………………………Flux of neutrons
σpp……………………………………………………………Pair production cross-section
University of Ghana http://ugspace.ug.edu.gh
1
CHAPTER ONE: INTRODUCTION
1.1 Overview
This chapter contains the study background, the area under study, problem statement,
relevance and justification of the research, the Aims and objectives, scope, general
description of the fuel assembly, definitions of some terms that are relevant to the research
work.
1.2 Study Background
Since the demonstration in 1942 of the ability to sustain a nuclear chain reaction, nuclear
power has developed into a proven technology. This paved the way for the production of
electricity by means of nuclear power plants. To have a nominal appreciation in nuclear
power generation, it has to have low cost of production, suitable waste disposal
mechanisms and non-proliferation of fuel for other purposes and at the same time maintain
its long-standing safety and reliability records (IAEA, 2009). This greatly requires a
massive change in its fuel cycle. It is not surprising the increase in demand for nuclear
power dwells on various factors which includes the continuous improvement in living
conditions, concerns over ‘greenhouse’ emissions caused by burning fossil fuels, increased
population just to mention a few. These major achievements are challenged by irradiation
of reactor components as well as exposure of human and environment to radiation doses.
Structural materials sustain damage due to irradiation from fission or fusion processes.
High energy radiation such as α, β and γ rays as well as particles such as protons, neutrons
and electrons with crystallite structures give rise to defects and imperfections ranging from
University of Ghana http://ugspace.ug.edu.gh
2
vacancies to ionization (Mansur, et al., 1997). The major damaging process is as a result
of particle interaction with target nuclei of the material structure causing sputtering and
displacement cascades pinning dislocation motions and thus leading to reduction in the
materials resistance to indentation. Fast neutrons are regarded as the major source of
radiation effects which cause atomic displacement which leads to creep, swelling and
embrittlement. Additionally, precipitates (i.e. Cu and P) can be formed leading to
precipitation hardening of the material (Hännien, 1990). The resultant of these processes
are the irradiation embrittlement of the reactor components materials with the welds.
The maintenance of the integrity of structural materials in the nuclear power plant is a
crucial issue both in-service and out of service. The emission of gamma rays together with
other elements from the fission reaction processes causes interactions with structural
materials. These radiations are absorbed in the form of heat energy. This induced energy
causes the dislodgement of zirconium atoms from their lattice sites and then introducing
defects to the crystal structure of zirconium cladding material. The subjection of high heat,
pressure and irradiation to the cladding material leads to both elastic and plastic
deformation. In this research, our focus is on the assessment of radiation induced damage
on the zirconium and stainless steel cladding materials.
Zirconium metallurgy is very important to the nuclear industry. It is regarded as a proven
structural material as it possess a combination of properties such as good resistance
corrosion and capture of thermal (slow) neutrons (Douglass, 1971). Its application is found
in both the PWR and BWR as cladding tubes which encases the uranium dioxide fuel
pellets. The spacer grids are also made of either zircaloy or high nickel alloy Inconel. In
University of Ghana http://ugspace.ug.edu.gh
3
the CANDU reactor, the fuel-cladding tube, pressurized tubes, sometimes the calandria
tubes are all fabricated with zirconium alloys.
Steel is widely known to be composed mainly of iron and carbon in the presence of other
trace elements. These trace elements can be added deliberately to improve the property and
behaviour of the steel. Stainless steels are steels with about 10 - 12 wt % of chromium in
its structure. This gives the steel an enhanced property of high-resistance to corrosion.
There are generally three categories of stainless steels namely; ferritic, austenitic and
martensitic stainless steels. Stainless steels have a wide array of application in the nuclear
industry. Stainless steels of the type 308 and EUROFER 97 are used mostly as cladding
materials due to their superior properties in temperature and corrosion resistance (Robert,
et al., 1996).
SRIM-TRIM simulation code which stands for stopping and range of ion in matter (SRIM)
- transport of ion in matter (TRIM) are a compendium of softwares used in the calculation
of energy losses and range of ions distribution in matter. TRIM is a Monte Carlo code
embedded in the SRIM code responsible for the computation of the stopping range of ions
(10 – 2GeV/amu) into matter. SRIM - TRIM code is used in the simulation of collisions
between ions and target atoms. It is a widely used tool by both researchers and students for
analyzing a number of parameters relevant to understanding atomistic behaviour during
collision (Ziegler, et al., 2010).
University of Ghana http://ugspace.ug.edu.gh
4
1.3 Study Area
The fuel element is considered as the heart of a nuclear reactor. Energy released from
fission is converted to thermal energy. This heat is transferred to the coolant that flows
through the core. The fuel element is made of a rod containing nuclear materials, cladding,
end caps, and spacing parts. Cladding isolates and prevents the fuel from contaminating
the coolant. The end caps seal the fuel element and together with the spacing parts fix it in
the required position (Olander, 1976). For the fuel element to meet design objectives, they
must be able to withstand the power cycle. They therefore meet requirements like sufficient
heat transfer, complete retention of fission products, high reactivity, safety under accident
conditions and retention of material integrity (Stanislav, et al., 2004).
The Fuel component is constructed by inserting a stack of fissionable material such as 𝑈𝑂2
pellets into a length of cladding. Spring placed on pellet stack for mechanical stability in
the plenum region. Zircaloy (Zry) caps welded on the ends of the tube.
The fuel rods encompass a firm framework made from steel and zirconium which has fixed
grid supports that tightly grips the rods in their precise lattice positions.
Figure 1.1: A representation of the fuel elements for LEU Core.
University of Ghana http://ugspace.ug.edu.gh
5
Figure1.2: A schematic view of the LEU fuel element assembly for the GHARR-1.
1.4 Problem Statement
Radiation damage of structural materials is of major concern to the nuclear industry.
Structural components exposed to intense radiation from fission or fusion processes leads
to radiation damage. The heat generated from this process is dissipated from the fuel to the
clad and finally to the coolant by conduction, convection and radiation. Radiation
accompanying energy emitted from the fission process induces atomic displacement in the
cladding material there by introducing defects (Olander, 1976).
In-service, nuclear power reactor components are subjected to high heat and pressure, and
bombardment by radiation. The combination of these stresses causes most of these
components to deform elastically and permanently over time. For zirconium-based alloys,
University of Ghana http://ugspace.ug.edu.gh
6
thermal creep, irradiation creep and irradiation growth are the primary modes of
deformation (Stanislav, et al., 2004). Creep collapse results in cladding due to unequal
pressures from the coolant which is higher compared to the internal pressure between the
Helium filled gap and the clad. Also embrittlement due radiation damage or accumulation
of corrosion product thus hydrogen will result in a through -wall crack if a rapid power rise
ensues.
In this research, emphasis will be given to the processes of formation and growth of
radiation-induced defect in cladding material. The natural complexity of zircaloy cladding
material which has both metal sub lattice and an oxygen sub lattice makes it much more
challenging than in metals. This research will help us understand the radiation damage on
zircaloy using the SRIM-TRIM simulation codes since this will help in the fabrication
choice of a more suitable structural component as cladding material.
The threshold energy concept which demonstrates that particles of energy greater than a
specified value produces damage whilst those below a particular threshold do not produce
damage is a crude approximation. The use of Displacement per Atom (DPA) to represent
material damage has demonstrated to be a more conservative approach to solving threshold
value (McElroy, et al., 2004). Additionally, operational temperatures restores large number
of DPA with just small fraction causing residual damage. Its cross-section is accurate for
monoatomic crystals but not composites. Therefore, in-depth and precise data are required
on structural properties of both fuel and clad material to establish the relevant engineering
correlations.
University of Ghana http://ugspace.ug.edu.gh
7
1.5 Relevance and Justification
Structural components of the nuclear power plant need to keep their functional abilities so
as to maintain the material integrity of the fuel assembly. This is to prevent the released
radioactive substances of fission from getting to the coolant. Improved radiation resistant
materials needs to be fabricated for harsher irradiation environments, higher temperatures
and support higher burnups. This means the material should maintain integrity at higher
Displacement Per Atom (DPA) thus the frequency with which atom are dislodged from
their precise lattice site by atomic collision should not affect material property. Typically,
Light Water Reactor (LWR) fuel clad, at burn-up of 40 GWd/tU experiences about 20 DPA
(ANSI, 2007).
Advancement in technology which has evolved from the manufacture of civil nuclear
power such as the magnox to Generation IV reactors and fusion reactors will require
radiation resistant core structural components which can withstand imperfections or defects
at higher atomic displacements. Expectations on next generation fast reactor systems would
be to sustain atomic displacement reaching 150 – 200 DPA (Maisonnier, 2006).
Earlier studies on radiation damage have concentrated mostly on the energy causing this
radiation damage. This thesis work however concentrates more on the fluence of this
energy. This will in actual fact give a broader assessment of damage on the structural
material over specified area and time. Also conversion of the fuel of the MNSR GHARR-
1 from high to low enriched uranium with zircaloy cladding makes it an opportune time
to assess the effect of the radiation damage on zircaloy. This could be an additional boost
for regulatory approval.
University of Ghana http://ugspace.ug.edu.gh
8
Also advancement in reactor design and their core fabrication requires significant
experimental and theoretical progress. This challenging advancement demands continuous,
coordinated, synergetic efforts, with more intense research in the area of radiation damage
of materials. Emphasis must be placed on the transmogrification of structural components
at high temperatures and radiation exposures (ORNL, 2004).
There is therefore the need to study radiation damage effects on structural components as
this will help in the developments of new techniques relevant to the fabrication of materials
to meet the exigency of our time. It will also save cost as computational tool lessens the
time and cost of experimentation.
1.6 Aims and Objectives
Principally, the aim of this thesis is to assess radiation damage on the zircaloy, stainless
steel type 308 and EUROFER 97 cladding materials of the LEU fuel pins in a Miniature
Neutron Source Reactor at 34KW power with Zircaloy cladding by the use of computer
simulation program SRIM-TRIM code.
Specifically, this research is aimed at;
1. Understanding the irradiation-induced defects on structural components through
simulation efforts.
2. Use Monte Carlo Simulation MCNP5 to generate the neutron parameters.
3. To compare the defects generated on the target materials Zircaloy 2 and 4, stainless steel
308 and EUROFER 97 using SRIM-TRIM code.
4. Determine the Projection range, Lateral and Longitudinal Straggling of the various clad
materials using SRIM Code.
University of Ghana http://ugspace.ug.edu.gh
9
5. Compare the Collision Events, Sputtering, Ionization, vacancies generated and
Replacement collisions of all four clad materials.
6. Determining the displacement per atom (dpa) due to radiation damage by both the
Kinchin-Pease model and Norgett-Robinson Torrens model.
1.7 General Description of MNSR
The MNSR locally known as GHARR-1 is a typical research reactor with its maximum
power at 30 kW. It is generally a tank-in-pool type. It has a relatively small size with very
high levels of safety. It runs on highly enriched uranium (HEU) as fuel although there are
proposals for the conversion of the core to low enriched uranium. Water serves as coolant
and moderator, whilst beryllium shims serves as reflector (Nyarko and Debrah, 2012).
The reactor was fabricated and built by CIAE for the purposes of serving universities,
hospitals and research institutes. The reactor is designed with maximum flux of 1x1012
n/cm2.s and a neutron source. It can boost of 10 irradiation sides; 5 inside and 5 outside the
beryllium annulus reflector. This is why; it is precisely used for neutron activation analysis.
It also serves the purposes of producing short-lived radioisotopes. However, it’s used in
training nuclear engineers, nuclear physicists, health physicists and radio-chemists (Quaye,
2012). The MNSR has a vessel which is a cylindrical Aluminium (Al) alloy LT – 21 and
0.6 m in diameter, 5.6 m in length and 9.5x10-3 m in thickness and a volume of 1.5 m3. The
vessel was built in 2 sections and submerged in a water pool which is protected by a
reinforced concrete (Amuasi, et al., 2005). The lower and upper sections are held together
by 16 stainless steel tie rods of diameter 0.019 m (Darko, 2013). The rods are 2.48 m long
University of Ghana http://ugspace.ug.edu.gh
10
and extend from the lower section flange to the square container support flange near the
top of the upper section.
Figure 1.3: Pictorial View of GHARR - 1 Assembly.
Figure 1.4: Cross Sectional View of GHARR - 1 Core.
University of Ghana http://ugspace.ug.edu.gh
11
Table 1.1 Specifications of GHARR - 1 (Akaho, 2003).
NO. PARAMETER DESCRIPTION
1. Burn up / Cladding Material ~1% / Zr alloy
2. Continuous operating time at rated power 2.5 hours
3. Control Rod Absorber, Cladding Cadmium, Stainless Steel
4. Control Rod Length, Position 230 mm, Centre of Core
5. Coolant / Moderator Deionized H2O and 995.1 kg/m3
6. Coolant Flow Rate 400 L/hr.
7. Coolant Inlet Pressure and Temperature 101.3 KPa – 1 bar and 303 oK
8. Coolant Temperature (288 – 333) K
9. Core Diameter / Core Height 230 mm / 230 mm
10. Maximum Thermal Neutron Flux 1.0x1012 n/cm2s (rated power)
11. Number of Control Element 1
12. Number of Dummy Element 6
13. Outer Irradiation sites 5 (2 large, 3 small)
14. Prompt Neutron Lifetime (Λ) 58.12x10 s
15. Rated Thermal Power / Core shape 34 kW / Cylindrical
16. Effective delayed Neutron fraction (βeff)
38.08x10 k k
17. Excess Reactivity – Cold, Clean 4 mk
18. Fuel Composition U dispersed in Zr
19. Fuel Density 3.456 g/cm3
20. Fuel Element number in the Core 344
21. Fuel Element Shape / Fuel Lattice Pitch Thin rod / 10.95 mm
22. Fuel Rod Position / Fuel Type 350 / Rod
23. Inner Irradiation sites 5 (3 large, 2 small)
24. Reactor cooling mode Natural convection
25. Reactor type / Reflector Tank-in-pool / Beryllium
26. Temperature coefficient ~ 0.10 𝑚𝑘/ ℃
27. Total number of irradiation sites 10
University of Ghana http://ugspace.ug.edu.gh
12
1.8 Scope
The research work focused on modeling and simulation of radiation damage on cladding
materials, specifically zircaloy material and stainless steel of type 308 and EUROFER 97.
The background to the study, area of study, problem statement, the relevance and
justification of the study, the aims and objectives of the research are presented in Chapter
one.
In chapter two, a review of literature on cladding used as structural components in the
nuclear industry and also detailed information on radiation interaction with matter. This
chapter throws more light on the mechanism of radiation damage as well, with its growth
and propagation into cracks and eventually failure.
Chapter three focuses on the research methodology relevant to the application in radiation
damage. Since this research is geared towards the use of computer simulation codes, we
apply the codes MCNP5 purposely for neutron parameters (neutron flux and energy
deposition) and generating trim.dat file for the TRIM code.
In chapter four, the results obtained from the MCNP5 and SRIM-TRIM code are
simultaneously analyzed to ascertain the radiation damage caused by both neutrons and
gamma. The assessment of the radiation damage on the cladding material will lead to the
selection of the best suited clad material amongst the four cladding materials.
Chapter five will sum up all the finding from the results obtained in the research as provided
in the previous chapter. This chapter will include the conclusion and recommendation for
future works.
University of Ghana http://ugspace.ug.edu.gh
13
1.9 Definition
Creep: It occurs when a material plastically deforms to a subjected load /stress with respect
to time.
Defect: A crystal structure has a defect when it exhibits any deviation in the regularity in
the arrangement of atoms in their lattice structure. Depending on the dimension and degree
of the defect they are categorized as;
Point defects: They are zero or dimensionless defects. They include; vacancies, interstitials
Line (Linear) defects: These are one dimensional defect. They form from a cluster of point
defects close to each other. They include; screw and edge dislocations.
Planar (Surface) defects: They are two-dimensional defects. They are very common at the
interfaces of homogeneous materials. They include; grain boundaries, etc.
Volume defects: These are three dimensional defects. They include voids, pores, and
cracks.
Deformation: This refers to straining of a material. In elastic deformation, both shape and
dimension of the material is maintained. For plastic deformation, there is a permanent
change in shape or dimension.
Displacement: This refers to the removal of atom from its lattice site by an energetic
incident atom.
Displacement Energy: It refers to the least energy required to displace a particle of atom
about one atomic distance from its lattice position.
Displacement per Atom (dpa): This refers to the frequency at which a particle of atom is
dislodged from its lattice or interstitial site.
University of Ghana http://ugspace.ug.edu.gh
14
Final Energy (Efinal): This refers to the minimal amount of energy that brings an atom in
motion to a stop. As atoms move, they cause ionization of other particles near-by. They
lose energy in this process and eventually come to a stop when their energy is equal or
below the final energy.
Interstitial Site: Any area other than the normal lattice site of an atom possible for
accommodating an atom is an interstitial site. Any atom in this site is referred to as an
interstitial atom.
Irradiation Creep: This refers to plastic deformation caused by the evolution of the
varying defects induced by irradiation which is dependent upon their position relative to
an applied load.
Irradiation Growth: This refers to a phenomenon in which the dislocation present in a
lattice preferentially absorbs the defects causing the dislocation to climb in the absence of
applied stress.
Interface Mixing: It refers to the transport of particles from one layer of the interacting
ion to a target layer.
Ionization: This refers to the removable of an electron from an atom.
Lattice Binding Energy: This refers to the minimum energy needed to release a particle
of an atom from its site.
Primary Knock-on Atom (PKA): It refers to the first target particle ejected from the
lattice site by an incident particle.
Radiation Damage: This refers to the disruption of crystallographic nature of a solid
structure when particles (ions) with energy traverses through them. This produces
University of Ghana http://ugspace.ug.edu.gh
15
microscopic defects in the crystal structure due to irradiation, by the interactions between
incoming ion and target atom causing a Frenkel pairs.
Radiation Effect: Is the physical and mechanical property changes in materials induced
by radiation.
Replacement Collision: This is when the incident ion takes the lattice site of the target
atom after collision.
Recoil Implantation: This refers to when an incident ion is imbibed into a target atom for
the purposes of modification.
Sputtering: Is the opposite of recoil implantation.
Sputtering yield: This refers to the average number of sputted atom per incoming ion.
Surface Binding Energy: This refers to the amount of energy that is needed by an atom
in order to escape the surface of the target material. It is different from binding energy as
it considers surface roughness etc.
Vacancy: This refers to the vacant site when an atom is ejected from a crystal lattice.
University of Ghana http://ugspace.ug.edu.gh
16
CHAPTER TWO: LITERATURE REVIEW
2.1 Nuclear Cladding Material
Undoubtedly, one of the most extensive and substantial constituent of the fuel rod
composition is the all-important cladding tube. The cladding tube prevents the nuclear fuel
from making direct contact with the coolant material inside the reactor vessel responsible
for the transport of energy in the form of heat. Radioactive isotopes that accompany fission
products would have the potential to not only contaminate many different systems
throughout the primary loop, but also to be released into the neighboring environment.
The fuel clad acts as the layer of protection around the fuel pin from irradiation, corrosion.
It also prevent fuel material from getting throughout to the reactor coolant circuit. This is
achieved not only because it serves the purpose of trapping long-lived fission products, but
the sealed cladding tubes also contains dangerous gases produced during reactor
operations, such as Krypton and Xenon (Barrett, 2012). The existence of these gases in the
fuel element coalesces and results in fuel swelling. Cladding therefore provides a great
deal of protection in reducing the amount of radioactivity that is released into the primary
coolant cycle during operation.
On the contrary, if the design of the cladding tube fails to meet the adequate requirements,
there would be a drastic reduction in the amount of heat transferred from the pellets to the
coolant material.
2.1.1 Properties of Cladding Material
The material property for a cladding element includes;
Low Thermal Neutron Absorption Cross Section: For every fission process,
about 2.5 neutrons are produced to sustain the fission chain reaction. The prompt
neutrons are produced instantaneous whilst precursors undergo beta decay to
University of Ghana http://ugspace.ug.edu.gh
17
daughter nuclide which gives off delay neutrons. The cladding material should
exhibit low tendency to absorbing neutrons. Zirconium, the current cladding
material, has a neutron absorption cross section of 0.18 barns when compared to
nickel (4.5 barns) and iron (2.4 barns) (KAERI, 2013).
High Thermal Conductivity
The cladding material must demonstrate the ability to transfer high energy to the coolant
to prevent accumulation and subsequent burn-out or melt down. This should be done
effectively from the uranium pellet to the primary loop. Zirconium has a thermal
conductivity of approximately 22.6 W/m*K (Tritt, 2004).
Radiation Resilience
Nuclear fission products released from fission travel into the cladding (NRC, 2013). These
particles do not only displace lattice atoms from the crystal structure but they also activate
nearby atoms due to the released energy.
Corrosion Resistance
The material should be capable of creating an inert oxide layer, by a mechanism known as
passivation, which protects against corrosion. The zirconium and iron metals possess this
quality. The exposure of the cladding material to a large volume of highly volatile water
over an extended period of time makes it prone to corrosion. Cladding materials have to
have good corrosion resistance. At normal operating reactor temperature (300° C), Zircaloy
is an exceptional cladding and is considered to be the standard for fuel rods since the early
1950’s (Todreas and Kazimi, 2011). This is usually the operation temperatures of Light or
pressurized water reactors.
University of Ghana http://ugspace.ug.edu.gh
18
Table 2.1 Material behavior of reactor at rising temperatures (Ragheb, 2013).
Physical Phenomenon Temperature[oC]
Mean clad temperature during operations 350
Pressure buildup causes clad perforation or swell in volume
Release of fission gases thus I, Kr
Start of chemical reaction between stainless steel and Zircaloy
Coolant flow could be disrupted by clad ballooning
800 – 1,450
Steam reacts with Zircaloy clad producing hydrogen and
release of energy higher than the decay heat
Embrittlement of Zircaloy as a result of oxidation.
Melting of Steel alloy
1,450 – 1,500
Autocatalytic reaction of Zircaloy and steam occurs, this
reaction could only be quenched upon immersing zircaloy in
the coolant.
1,550 – 1,650
Melting of Zircaloy
The release of fission products becomes apparent above
temperatures of 2,150 K.
1,900
Melting of Uranium oxide and Zirconium 2,700
Temperature inevitably contributes significantly to the maintenance of the material
integrity of the fuel rods. Between 1,450 – 1500 oC, the steam begins to react with the
Zircaloy cladding. The oxidation process begins to produce zirconium hydride which forms
region of radiation embrittlement, crack growth and propagation.
University of Ghana http://ugspace.ug.edu.gh
19
2.1.2 Zirconium Metal
Zirconium metal is significantly key in the production of electricity through nuclear power.
This metal was first discovered in 1789 by renowned chemist Martin Heinrich Klaproth
while studying the makeup of the mineral jargon (ZrSiO4) within a jacinth stone from Sri
Lanka (Sepke, H. and Sepke, I., 1986). Early chemists were uninterested in studying the
jacinth stone very carefully as most believed that it was another form of aluminum oxide
alumina. Klaproth analyzed the mineral jargon by heating it in the presence of a highly
reactive compound, sodium hydroxide. It then formed an unknown oxide that turned out to
be zirconium oxide. It was named Zirconium from the Arabic word zargun meaning “gold
color” (Hoppe, et al., 1987).
For the next three and half decades, the struggle to produce the pure state of the metal
continued till a Swedish chemist Jöns Jacob Berzelius a founding father of modern
chemistry, succeeds in having a chemical reaction between potassium and potassium
zirconium fluoride by 1824 (Binnewies, et al., 2013). This only yielded a black powder of
about 93 percent zirconium which was not pure enough to be turned into a pure metal for
material characterization of its properties.
It was until 1925 when Anton Eduard van Arkel, Jan Hendrik de Boer, and Phillips Gloei-
Lampenfabriken invented the decomposition process for iodine, referred to as the Bar
Process. This process was used by researcher to test the material properties of zirconium.
The production of zirconium metal was on a small scale and in limited quantities in 1945.
This made it very expensive (Krebs, 1998). On-going research on the material
characterization then revealed the obvious desirable material properties like good
resistance to rust, high melting temperature, and high ultimate tensile strength. This then
University of Ghana http://ugspace.ug.edu.gh
20
boosted the interest to produce zirconium on a large scale. The break-through for
commercial production of zirconium metal was after the invention of magnesium reduction
method in 1937 (Kroll, 1940). The large scale production of zirconium combined with its
desired properties encouraged the wide use of the metal in different fields and industries.
In 1940 during the Manhattan Project, several groups of engineers and scientists exploring
different metals and alloys for their application in the nuclear industry particularly for
reactor fabrication and design, these scholars came to the realization that the desired
properties of the zirconium metal met all the requirements for fuel cladding applications.
The concern however, was that early research studies had shown a very strong thermal
neutron cross section. This was proven wrong by Dr. Sam Utermeyer and Dr. Albert R.
Kaufman of Massachusetts Institute of Technology. They discovered that early research
work and measurements taken on the cross section for zirconium with regards to thermal
neutron were wrong. This was because the hafnium content which occurred naturally with
zirconium in zircon gave it the high neutron absorption cross-section (Weinberg, 1994).
2.1.2.1 Material Properties of Zirconium
At STP, pure Zirconium solidifies out as 𝛼 phase for the hexagonal close packed (HCP)
crystal structure (SSMC, 1989). A study of the phase diagram of zirconium shows that at
temperatures of about 288oC, it stays in the 𝛼 phase. The crystallographic structure for the
𝛼 phase are reported as; a = 3.2312 Å and c = 5.1477 Å.
The β phase of zirconium is BCC, but starts to develop at the grain boundaries of the 𝛼
phase crystals as the temperature goes above 866°C (INSC, 1997). The β phase of the
crystallographic structure for the zirconium is a = 3.6090 Å (O'Donnell, 1994).
University of Ghana http://ugspace.ug.edu.gh
21
Zirconium exists in its pure form as a glossy, grayish white, ductile metal. It is soft with
desirable material properties such as low capture of neutron, high resistance to chemical
oxidation.
The need for improvement in the performance of zirconium alloys was initiated in order to
have high tensile and yield strength as well as good corrosion resistance.
Below is a representation of zirconium metal properties.
Table 2.2: Physical nature of Zirconium metal (Schweitzer, 2003).
Properties Zirconium
Atomic number 40
Atomic weight 91.22
Density at 20oC 6.510
Melting Point 1845oC
Boiling Point 3577 oC
Thermal conductivity 100 oC, 0,049 cal/s/cm/ oC
Specific heat 0.067 cal/g/ oC
Electrical resistivity 40𝜇Ω.cm
Zirconium belongs to a class of elements known as refractory metals. These are metals
with extraordinarily high resistance to heat and wear. Refractory metals include elements
such as titanium, niobium, molybdenum and tungsten. They exhibit properties including
high melting point above 1,850 °C. They are chemically inert with relatively high densities.
They are used in high temperature operations since they offer stability against creep
University of Ghana http://ugspace.ug.edu.gh
22
deformation at elevated temperatures and in corrosive environments. Below presents Table
2.3 demonstrating a table that demonstrates the mechanical properties of zirconium metal
which makes it preferable for alloying.
Table 2.3: Structural nature of Zirconium metal.
Properties Zirconium
Tensile Stress 379 MPa
Yield Stress 207 MPa
Modulus of shear force 33 GPa
Young’s Modulus 68 GPa
Poisson’s Ratio 0.35
2.1.2.2 Zirconium Alloys
Zirconium alloy formation first begun with the identification of relevant elements which
could improve the corrosion resistance of the Kroll sponge zirconium to a level comparable
to a quality Iodine crystal bar. The elements considered included tin, tantalum and niobium
respectively. These metals showed decreasing order of effectiveness and avoided the
damaging effect of impurities present in the sponge zirconium. With a drop from 5wt% to
2.5wt% of tin, it was realized that after some time in service the alloy begins to yield to
rapid continuous corrosion. This led to the subsequent addition of other alloying elements.
Presently, the use of zirconium alloy has gained vast applications in various industries. The
two foremost series of zirconium alloys have become the major constituents for the
development of fuel claddings. They are namely: Zr-Nb and Zr-Sn. These alloying
elements play an important role in either strengthening the base solid or improving one or
more of the material properties. Tin is mainly used for solid solution hardening to increase
University of Ghana http://ugspace.ug.edu.gh
23
the creep resistance and strength of zirconium alloys (Liu, 2007). Niobium (Nb) increases
the strength, ductility, and corrosion resistance when added to zirconium (Crepin, 1995).
When oxygen is combined with the elements such as Iron (Fe) or Chromium (Cr) as
alloying metals, the resultant alloy tends to have improved plastic deformation resistance
since the oxygen tends to fill in empty voids in the lattice structure of the material.
A list of the zirconium cladding alloys used in thermal reactor applications and their
chemical compositions are shown in Table 2.4 below. Trace metals in zirconium alloys are
typically transition metals (Ni, Cr, Fe) and they are mostly insoluble within the alpha - Zr
phase which precipitate out as metallic flakes. These flakes, depending on the size
distribution, will have a substantial impact on the corrosion behavior of the alloys.
Table 2.4 Mean Composition of Zirconium alloy by weight percentage (%).
Mean Composition in Weight %
Zirconium Alloys Sn Fe Cr Ni O Nb
Zr - Sn Alloys
Zircaloy-1 2.5 - - - - -
Zircaloy-2 1.20–1.70 0.07-0.20 0.05-0.15 0.03 – 0.08 - -
Zricaloy-3 0.25 0.25 - - - -
Zircaloy-4 1.20–1.70 0.18-0.24 0.07-0.13 - - -
Zr-Sn-Nb Alloys
ZIRLOTM 1.00 0.10 - - - 1.00
Alloy 635 1.20 0.40 - - - 1.00
Zr - Nb Alloys
Zr – 1Nb - - - - - 1.00
Alloy M-5 - - - - 0.10 1.00
University of Ghana http://ugspace.ug.edu.gh
24
2.1.3 Steels
They are ferrous materials which are mostly alloys of mainly iron and carbon with other
trace elements. Steels are also classified according to the carbon content as mild, medium
and high carbon steels. The carbon content of the steels have a direct relationship with the
physical characteristics of the metal.
Figure 2.1: Iron-iron carbide phase diagram (Massalski, 1990).
2.1.3.1 Stainless Steels
They are Steels with about 11 wt% of chromium. They exhibit high corrosion resistance.
On the basis of microstructure, stainless steels are classified as ferritic, martensitic and
austenitic stainless steels. They exhibit high strength and toughness at the extremes of
temperature scales (Callister Jr., 2005).
University of Ghana http://ugspace.ug.edu.gh
25
2.1.4 Limitations of Zirconium and Stainless Steel alloys in Nuclear Application
Oxygen is central to the structure of Zirconium and Stainless Steel alloys. Its existence is
originally considered as an impurity. Oxygen influences the slip behavior of alloys in
proportion to its concentration within the matrix structure. Its interaction on the aqueous
environment forms an oxide layer. This layer serves as a protection site for further
corrosion. This oxide layer formation however reduces the amount of matrix atom in the
clad tube. This reduction in the amount of the matrix material puts more pressure on the
cladding tube thereby cracking the protective oxide layer. Eventually, failure occurs as a
result of increased localized corrosion. The oxygen atoms which hitherto were responsible
for filling the empty voids in the crystal lattice now compete for the oxide layer. They
therefore leave behind vacancy-base alloy which changes the microstructure and behaviour
of the component. This invariably changes the nature of the alloy. The increase in the
number of vacancies increases the paths for diffusion available to free atoms which lead to
high rate of failure. Creep occurs which leads to plastic deformation which is undesirable
in cladding materials.
2.1.5 Irradiation of cladding
The exposure of structural materials in the nuclear environment to fission products can lead
to unique phenomena. A crystal lattice describes atomic nuclei occupying lattice sites with
a sea of electrons filling the remaining volume. This crystallographic structure allows for
the entrance of alpha, beta and gamma particles as well as high-energy neutrons into the
lattice. The alpha particles tend to gather in regions around voids and other defects leading
to pockets of helium and alteration of creep behaviour within the crystal.
University of Ghana http://ugspace.ug.edu.gh
26
The high-energy (prompt/fast) neutrons, in the simple terms, act like a billiard ball striking
one nucleus and potentially rendering it loose from its lattice site. This now-free atom may
interact with other lattice sites or become a self-interstitial. In either case, this potentially
raises the energy of the crystal and leads to atomic displacement which contributes to easier
corrosion of the lattice.
The injected vacancies from the removal of a metal ion of the cladding element forming
the oxide layer have multiple potential paths. The first is in-situ vacancy annihilation at the
oxide/alloy interface. If this annihilation does not ensue, then the vacancy may precipitate
out to form a dislocation and later a void (Gibbs and Hales, 1977). The voids created,
coalesce and grow to affect the corrosion resistance, mechanical, and thermodynamic
properties of the alloy. The elevated levels of oxygen beneath the oxide layer leading to
reduced ductility and fracture toughness (Chung and Kassner, 1998).
Irradiation of zirconium alloys creates "black-spots", or clusters of irradiation-induced
defects, small dislocation loops, short line dislocations, and dislocation entanglements.
These three dislocations related defects occur on the prismatic plane with an - type
Burgers vector at lower fluence and associated fuel burnup. At higher burnups, there is an
increase in oxidation leading to mechanical disturbance, creep, decrease in ductility,
growth and increase in brittle-type fracture behavior (Chung and Kassner, 1998).
As the clad is exposed to a flux of high-energy particles, displacement of the lattice atoms
create dislocations and other crystalline defects and imperfections. This affects the
mechanical and crystalline properties of the alloy. With oxide layer formation, this creates
the potential for super saturation of vacancies in the base alloy. The coalescing of vacancies
into voids and increasing the oxygen content in the region below the oxide layer, the
properties of the alloy greatly changes with increased fuel burnup. The localized corrosion
University of Ghana http://ugspace.ug.edu.gh
27
aggravated by the incident high-energy neutrons, also enables diffusion of hydrogen into
the alloy creating embrittlement or diffusion of fission products into the overall system or
environment including the welds.
Several processes have been designed to alloy cladding material achieve good texture for
the best strength and material properties. These processes include annealing to reduce
existing defects and vacancies. Other mechanisms also used include the exploration of
different alloying compositions to improve upon corrosion and diffusion. This makes it
much easier to achieve the goal of higher burnups for better efficiencies.
2.2 Interactions of Particulate Matter
The interaction of particles such as photon or neutron with matter is dictated by short-range
forces. Maxwell's equations describe precisely interactions of long wave-length radiation,
which readily yields a wave equation for the electric and magnetic fields of radiant energy
(Olander, 2006). Neutrons, which are neutral particles, move in a straight line through a
medium, where they are intersperse by occasional "point" interactions, in which the
neutral particle is absorbed, scattered or cause some other forms of reaction. Interactions
are stochastic in nature, as particles travel between successive collisions within the
medium. These collisions are predicted only in some average or expected sense. The
interaction of a given type of radiation with matter may be classified according to the type
of interaction, the nature and manner with which the interaction takes place. The interaction
may take place with an electron, in which case the electron behaves as though it is free.
Likewise, interactions of particles may take place with an atomic nucleus, which in many
cases behaves as not bound in a molecule or crystal lattice. However, lattice energy of
crystalline materials also places an important role in the various forms of interactions. In
University of Ghana http://ugspace.ug.edu.gh
28
most instances, these interactions involve energy transfer from the radiation to the target
matter with which it interacts. Matter consists of atomic nuclei and extra nuclear electrons.
Radiation may interact with either the atomic nuclei, electrons in the shell or both
depending on the energy of the incident particle. The propensity of any particular category
of interaction, and consequently the penetrating power of the various radiations, depends
on the type and energy of the radiation as well as on the nature of the absorbing medium.
The interaction in most cases leads to excitation and ionization of the absorber atoms
resulting from their interaction. Energy in the form of heat is dissipated.
2.2.1 Particle - Solid Interactions
Nuclear interactions include the interaction of charged particles with atomic nuclei in a
solid medium (Olander, 2006).These nuclear interactions produce permanent atomic
displacements within solid. These displacements give rise to vacancies and interstitials
which develop into dislocation loops in picoseconds. These are detrimental to the crystal
structure as they lead to microstructural evolution. The effect of an external flux of
energetic particles in a crystal structure maybe categorized into two components: thus by
creation of primary knock-on atoms or the creation of transmuted atoms. This is shown
below:
i = neutron, gamma, ion
Figure 2.2: Particle interaction in crystal structure (Olander and Arthur, 2015).
i
i(Ei)
Transmutation atoms
Primary knock-on atom
(PKA)
Energy distribution NPKA(E)
University of Ghana http://ugspace.ug.edu.gh
29
i (Ei) represents all particle fluxes i incident on the solid (i=neutron, gamma), while
represents the probability of interaction of all relevant particle-solid reactions k,
transferring energy E to the atoms in the solid. The product of interaction of the flux of
energetic particles (represented by ) and the atoms in the solid (represented by ) is the
creation of NPKA(E) self-atom recoils called primary knock-on atoms (PKAs) and a
concentration of transmutation atoms Ck, where k represents the atomic species created.
2.2.1.1 Cross-Section
The probability of interaction of particles between the atoms in the solid and the incident
particle flux is commonly represented as the cross-section for the reaction. The notion of
the microscopic cross-section is illustrated in figure 2.3 below. Research shows that
reaction rates are influenced by the apparent size of the atoms in the solid. The finer the
size, the faster or higher the reaction rates and vice versa. (Olander and Arthur, 2015).
Given that, there exists particle flux in a solid of a given atomic density, the probability of
reaction increases with the apparent particle size. This is illustrated in figure 2.3 where two
types of atoms are present in the solid: atom A has a greater reaction rate for reaction 1
than atom B, and thus these atoms appear as large as when reaction 1 is considered. In
contrast, reaction 2 has atom B to have a much larger cross section and consequently these
atoms appear bigger for reaction 2.
The differential probability for solid particle interactions is defined as
( )idP N E dx =Σdx (2.1)
Where dP = differential Probability; N = target atoms per unit volume; Ei = Energy of
incident particle; σ = Microscopic cross-section; dx = Thickness of medium;
University of Ghana http://ugspace.ug.edu.gh
30
∑ = Macroscopic cross-section.
Reaction 1 Reaction 2
Figure 2.3: Schematic representation of the cross-section concept (Olander and Arthur,
2015).
2.2.2 Interaction of γ-radiation with structural components
Primary radiation is represented by γ-radiation entitled by the fission related activities in
the reactor core. The photons therefore interact with the lattice atoms of the crystal
structure. In these interactions, high energy electrons are produced which travels within the
structural components. Electrons are affected by electromagnetic collisions since they are
charged. Collision in the “electron cloud” slows down the electrons in their trajectory.
While interactions against nuclei are rarer and do not cause a big energy loss, since the
electron mass is much smaller than the nucleus mass (Tsegay, 2011) and (Festus, 2014).
It has been shown that, photons do not dissipate energy directly to the atomic nuclei, rather
electron are energized by one of the three known interactions of gamma rays with electrons
A
A
A
A
A
A
B
B
B
B B
B
B
B
B
B
B
B
B
B
B
B
B
B
University of Ghana http://ugspace.ug.edu.gh
31
in the outer orbits of the atom at the range of energies from (0.1MeV < E < 1.5MeV). They
are; Photoelectric effect, Pair production and Compton scattering (Knoll, 1989).
These processes lead to displacement cascades in solid as a result of the energetic electrons
that are excited. The processes are dominant at different energy ranges.
2.2.2.1 Photoelectric effect
The photoelectric effect is an absorption process which occurs for X-rays due to their low
energies. A photon of energy Ee < 0.1 MeV is transferred to an inner tightly-bound electron
in a K-shell orbital. This gives the electron sufficient energy to escape from the atom. In
photoelectric effect, the incident gamma ray interacts with an entire atom where photon is
absorbed. The resultant electron is a photoelectron.
Eγ Ee
Photoelectron
Figure 2.4: Representation of Photoelectric effect.
2.2.2.2 Pair production
This is an electromagnetic interaction which involves the disappearance of a photon for
two resultant electrons (positron and negatron). This process is possible only if the gamma
ray energy exceeds twice the rest-mass energy of the electron (i.e. 1.02MeV). The cross-
section for a pair production σpp, increases steadily with increasing energy of the photon
(Birikorang and Nyarko, 2014). Pair production takes place in the vicinity of the columbic
field which enables the motion of the electrons. These electrons lose energy through
ionization and excitation in collision. At very low energies the positron slows down and
University of Ghana http://ugspace.ug.edu.gh
32
combines with the electron. After the combination of the two particles they annihilate
radiation with energies of 0.511MeV each.
Positron – electron pair
Figure 2.5: Representation of Pair Production.
2.2.2.3 Compton scattering
Compton scattering is the most important of all the three interaction of gamma with
electrons. It is essentially an elastic collision process of a high-energy photon with the free
electron initially at rest. An incoming photon of high energy (>0.1 MeV) collides with an
electron in the valence band, ejecting the electron from the atom. The energy of the ejected
or Compton electron can be determined by knowledge of the energies of the incoming and
scattered photons.
'E
E
Ee
Figure 2.6: Representation of Compton Effect.
The expression that relates the energy transfer and the scattering angle for any given
interaction are given below.
E =
'E + Ee; (2.2)
University of Ghana http://ugspace.ug.edu.gh
33
Where;
)cos1(Ecm
cmE
'E 2
e
2
e
(2.3)
2.2.3 Radiation damage
Radiation damage refers to the localized microscopic defects which occurs in the crystal
lattice of a solid when high-energy radiation traverses through it. The end result is the
modification of the lattice structure which affects the physical, chemical and mechanical
properties of the structural component. Desired material properties such as strength,
toughness, ductility, dimensional stability are largely influenced by the nature of the defect
structure. Grain size, internal interface, size and density of second phase precipitation,
dislocation densities all affect the nature of the defect structure (Stoller, 2011). The
microscopic defects that are as a result of the interaction of high-energy subatomic particles
and radiation with crystal lattice atoms gives rise to vacancies and interstitials. These point
defects coalesce to form dislocation which degenerates into voids. Most radiation damages
are caused by neutrons and fission fragments. Some other types of radiation generally have
insufficient energy (or are not produced in large enough quantities) to cause major damage.
The main stages or development of radiation damage are composed of distinct processes.
These processes occur in the sequence below (Osetsky, 2008):
1. Formation and growth of defect cluster and dislocation loop. This causes matrix
hardening.
2. Radiation induced and enhanced diffusion. This second stage is characterized by phase
stability changes with segregation and precipitation.
University of Ghana http://ugspace.ug.edu.gh
34
3. Coagulation of voids and gas bubbles which induces swelling of matrix structural
component.
4. Anisotropic diffusion due to high degree of randomness of defects in different direction
of the crystallographic plane leads to radiation growth.
5. Finally, induced-stresses cause the component to creep.
2.2.3.1 Knock-on atoms and displacement cascades
Atomic displacement may occur in a ballistic manner through kinetic energy transfer or by
the transformation of radiation-induced excitation into atom motion by recoil cascade.
Charged particles pass through matter causing ionization. The particle energy is dissipated
by exciting orbital electrons and by elastic collisions with the material nuclei at relatively
low energies. An elastic collision can eject an atom from its normal lattice position. The
ejected atom is known as a primary knock-on which may cause a cascade of atomic
displacements of secondary knock-on atoms before eventually coming to rest. They create
a Frenkel pair of a vacancy and interstitial simultaneously (Olander, 1975).
The PKA interacts with its neighboring atoms causing many collisions and atomic
displacements in a localized region. The final damaged state evolves as a result of intra-
cascade clustering and recombination, and consists of a distribution of interstitial and
vacancy clusters of different sizes. The spatial distribution of the created microscopic
defects is non-directional and non-uniform. Because the interstitials require energetic
atoms, they form the replacement collision sequences where the incident particles replace
the primary knock-on atom.
University of Ghana http://ugspace.ug.edu.gh
35
For every displacement of a lattice atom there exists a displacement threshold energy Ed
within the crystal such that the incident particle energy T disperses. Primary knock-on is
possibly only feasible if; T > Ed. For an isotropic elastic collision, the maximum energy
dissipated is represented as
nET max Where
2)1(
4
A
A
(2.4)
A = atomic mass, En = neutron flux energy
The rate of atomic displacement will be represented as the product of the number of atoms
N and the displacement cross-section.
)()( nndd EENR (2.5)
0 )()()()(
.
nnndEd nnnd
d dEEEtdEEEtN
tRdpa
(2.6)
dEEVEE
E
E
E n
n
d
nnd )(),()(
(2.7)
2.2.3.2 Radiation damage model
There are basically three methods known for calculating the displacement per atoms (dpa).
They are namely;
a. Kinchin - Pease model method
b. Lindhard model method
c. Norgett - Robinson Torren model method
University of Ghana http://ugspace.ug.edu.gh
36
a. Kinchin - Pease model method
For a PKA with energy E, the total number of displaced PKA can be estimated by the
Kinchin-Pease method as follows (BNL, 2006): for a displacement cascade to take place,
at time t=0, E >2Ed; then displacement of PKA occurs. Assuming an elastic collision, then
the number of PKA created (N) PKA have an average energy T of
N=1,
2
ET
; for N=2,
4
ET
and N= n,
n
ET 2
(2.8)
Displacement cascade ends when dET 2 . If dET ; A single PKA is generated which
could lead to the formation of an interstitial atom.
d
dd
EEif
EEEif
EEEdif
Ed
E
EEif
Ed
E
Ev
..0
2.,1
.2.
2
.
2
)(
(2.9)
v(E) =Atomic displacement produced by PKA’s; Ed = Displacement Energy;
Kinchin-Pease model is based on the following assumption below (Olander, 1975):
i. Only two bodies engage in an elastic collision.
ii. Atomic displacement is triggered only dET
iii. Hard sphere models are preferred to realistic potentials.
iv. Atomic structure has imperfections.
v. No annihilation after collision.
University of Ghana http://ugspace.ug.edu.gh
37
b. Lindhard model
This model is only valid in the nuclear stopping regime. This regime is characterized by E
< E*. It’s mostly applicable in realistic potential than in hard sphere. Thomas-Fermi used
this model to predict the energy partition between electronic and nuclear stopping
(Greenwood, et al., 1985). Assuming nuclear stopping portion of the deposited energy is
converted to atomic displacement; the amount of displacement is evaluated as below:
d
L E
EZEV 2),(
(2.10)
Damage efficiency = ξ (E, Z) =
)4.04.3(88.01
1
4
3
6
1
6
1
Z
(2.11)
Where
aeZ
E
222
; Screening radius = a;
2132
2
32
1 )(
2
ZZ
aa B
with aB as Bohr radius.
c. Norgett - Robinson Torren model method
Norgett-Robinson Torrens (NRT) is a modified model of the Kinchin-Pease model which
is valid for all primary knock-on range particularly for Z > 20.The NRT model gives a
stable Frenkel pair produced through interaction (Parkin and Coulter, 1981). Norgett-
Robinson-Torrens model considers neutron irradiation recoil energy of less than 50keV
(Norgett, et al., 1975). The number of displacement is expressed in the form:
d
NRT E
EV 28.0
(2.12)
)(.).( EtEdpa (2.13)
dTTETE VtEdpa d NRT ),(..).( max
(2.14)
University of Ghana http://ugspace.ug.edu.gh
38
It is important to add that both Lindhard and Norgett-Robinson Torrens models have some
limitations. These are;
(i). Differences in atomic masses of colliding bodies are not taken into account.
(ii).The variation in displacement energies with regards to crystalline orientation is not
considered.
2.2.3.3 Displacements caused by Neutron Irradiation
In nuclear power reactors, neutrons and gamma rays bombard the nuclear fuel and reactor
components at a high velocity due to their high energies. Fast neutrons are therefore the
major causes of irradiation. To evaluate the displacement rate k in a neutron flux spectrum
(En), the displacement cross section d(En) is needed. k in units of dpa/s is given by:
dE
nnnd dEEEk )()(
(2.15)
The integral expression has as its lower limit the lowest energy for neutrons capable of
transferring at least Ed to the atoms in the solid. The displacement cross section is
n
d
E
E
nsnd dEEEEE )(),()(
(2.16)
For equal scattering in all directions, (PKA energies E are equally probable), then
Ed << En
n
nsns E
EEE
)(),(
(2.17)
University of Ghana http://ugspace.ug.edu.gh
39
n
d
E
En
ns
nd dEEE
EE )()()(
(2.18)
Substituting equation (2.18) into equation (2.15) gives
d
n
E
n
E
n
nS
n dEdEEE
EEk
0
)()()(
(2.19)
Using the Norgett-Robinson Torrens
d
NRT E
Ev 28.0
(2.20)
n
E
dn
ns
E nNRT dEdEE
E
E
EEk nd
)2
8.0(()()( 0
(2.21)
Solving
)()2
8.0(0 EdE
En
d
E
gives
d
n
E
E
5
)( 2 (2.22)
By substitution,
n
d
n
n
ns
E nNRT dEE
E
E
EEk d
)5
)(()(
)()(
2
(2.23)
Rewriting the equation above gives the generalized equation below;
dE
nnnSn
d
NRT dEEEEE
k )()(
5
(2.24)
The displacement rate from the known neutron flux can be obtained by integration.
Monochromatic neutron flux approximation is used to derive the magnitude of atomic
displacement that have ensued.
The overall average neutron flux is evaluated as given below:
tnnn EEE )()( (2.25)
University of Ghana http://ugspace.ug.edu.gh
40
Where )( nn EE is the Kronicka delta, t is the total damage producing neutron flux
given by
dE
nnt dEE )(
(2.26)
nE which is the average neutron energy is given as;
t
E
nnn
n
d
dEEE
E
)(
(2.27)
While, for a monochromatic neutron beam of energy
nE ,
n
d
nS
tNRT EE
Ek 5
)(
(2.28)
2.2.3.4 Radiation induced defects
Structural materials occur originally with defects. They are fabricated with imperfections
due to the fact that the orderliness in arrangement of lattice atoms is not perfect. The
imperfection makes them amenable to mechanical and chemical transformation (Callister
Jr., 2005). A point defect created can lead to a cluster of point defects which easily turns
into a dislocation in a matter of picoseconds. Figure 2.7 below shows defects susceptible
to changing material properties of a lattice structure.
University of Ghana http://ugspace.ug.edu.gh
41
Figure 2.7: Defects susceptible to changing material properties of a lattice structure (IAEA,
2014).
2.3 Radiation effects on structural components
Fission processes in the reactor core occurs as a result of splitting of heavy nuclei ( 𝑈92
235 )
by neutrons into two fragments releasing large amount of energy in the process. The new
neutrons can in turn cause new fissions. Each fission reaction releases an average of about
200 MeV. This energy is dissipated in the form of heat. Of the fragments that are formed
as fission products in the reactor, some are rare gases which are of special importance,
because they have very high absorption cross sections. Their presence has a large influence
on the reactivity (Dam, et al., 2005).
Xenon – 135 reactivity effects are dependent on its concentration during operation of the
reactor and they buildup some hours after shut down as their half-life is about 9.2h.
Fission
3.3 % 3.1 0.25 %
𝛽− 𝛽− 𝛽− 𝛽−
𝑇𝑒 52
135 𝐼53
135 𝑋𝑒54
135 𝐶𝑠 55
135
𝐵𝑎 56
135 (Stable)
19.2s 6.6h 9.2h 2.106 a
Figure 2.8: Beta decay of Daughter Nuclei born from Fission process.
University of Ghana http://ugspace.ug.edu.gh
42
During irradiation in a nuclear reactor, some of the burnable poisons such as Krypton and
Xenon cause the fuel to swell due to their precipitation into bubbles (Matzke and Turos,
1992). Iodine which is a volatile element has the possibility of causing fuel failure by stress-
corrosion (Hocking, et al., 2001). The other elements however form solid precipitates in
the fuel and with the cladding material (Matzke, et al., 1994). Other gases such as helium
are of major concern in fuel technology.
2.3.1 Irradiation cascade
The essential, microscopic events that precede the appearance of visible changes in the
solid are termed irradiation damage. When energy transferred to the lattice atom
(displacement energy) exceeds the lattice binding energy of the atom in its lattice, the
lattice atom is displaced from its original position. The displaced atom (recoil atom) might
carry high enough kinetic energy to create series of lattice displacements before it finally
comes to rest. The displaced atom ultimately appears in the lattice as an interstitial atom.
The empty lattice sites left behind by the displaced atoms become vacancies or are replaced
by the incident atom. The collection of point defects as a result of vacancies and interstitials
created by a single primary knock-on atom is known as a displacement cascade.
During atomic collision cascades, vacancies and interstitials can be produced so close to
each other that clustering of the point defects occurs spontaneously within the short time
forming dislocations and voids. Due to the proximity of the clustering point defects, many
of the vacancies and interstitials produced by the high-energy collisions obliterate. Just a
little as low as 1% of the initially generated defects survives and are capable of producing
visible radiation damage (Olander, 1976).
University of Ghana http://ugspace.ug.edu.gh
43
The bombarding particle accelerated at particular projected range transfers energy on the
order of ten to hundreds of kilo electron volts for stationary lattice atoms. The energy
dissipated by incident ion in the solid is separated into two parts:
(i). Discrete elastic atom - atom collision that is useful reducing the energy of the incident
atom and also responsible for lattice displacements.
(ii). A continuous process of electronic excitation contributes to energy loss.
Interaction between the moving atoms or ions with the outer electrons (valency) of the solid
constitutes the major energy loss process at high energies. Energy Transfer from a moving
atom to an electron does not lead to displacement but excitation. Energy is transferred to
the electrons in small increments so closely spaced that the process can be regarded as a
continuous loss of energy. Furthermore, displacements are only caused during elastic atom-
atom collisions, where a significant portion of the initial kinetic energy possessed by
incoming atom is transferred.
Figure 2.9: Point defect history during avalanche of displacement cascade (Was, 2007).
Mutual recombination
outside of cascade
Loss to sinks in
matrix
Loss at grain
boundaries
Total dpa
Loss to
displacement
cascades
Freely
migrating
defects
University of Ghana http://ugspace.ug.edu.gh
44
Recent researches on MD simulations creates the impression that displacement cascades
are in two main forms (Diaz de la Rubia, 1989) and (Heinish, 1993). These include the first
ballistic stage where several particles are extricated from their lattice sites and then
secondly, thermal spike phase where the cascade region attains thermal equilibrium with
the immediate environment. The recalculation of the energy of atoms in the dense collision
region into temperature (k) with the aid of the equation E = 3/2.N.kBT, gives initial
temperatures of the order of 10,000 K. This high temperature region encourages the
creation delocalized defects with radius less than 50 Angstrom (Å) inside the solid. This
phenomenon is called a “thermal spike”.
Thermal spike maybe seen as localized melting of the irradiation affected regions which is
followed by fast quenching leading to a change in phase which forms a damaged
amorphous structure. Thermal spikes normally cool down to the ambient temperature in 1-
100 ps. Some experiments have shown that thermal spike are able to induce a phase
transition which requires a very high temperature (Bacon and Diaz de la Rubia, 1994).
a. linear collision cascade b. thermal spikes
Figure 2.10: Schematic illustrations of linear collision and thermal spikes (Meldrum,
1998).
University of Ghana http://ugspace.ug.edu.gh
45
The cooling of displacement cascade occurs by way of conduction through the lattice and
heat conductivity by electronic means. The final stage of a cascade is the relaxation phase,
when the defects migrate and possibly recombine; they last for some picoseconds to an
infinitely small times, depending on its structure, its defect migration and recombination
properties, at ambient temperature.
2.3.2 Radiation effects in nuclear fuels
The main source of energy in nuclear fuel is slowing down of high-energy fission products.
It is this slowing down of high-energy that leads to damage by radiation. Fission products
passing through reactor materials lose part of their energy to the atoms of these structural
components and eject some lattice atoms from their normal positions in the materials,
replace or stay as interstitial atoms. The collective result is manifested in drastic
modification of the material and structural properties of irradiated materials. These
properties include changes in dimensions, strength and hardness, conductivity of heat and
electricity, magnetism, resistance to corrosion, resistance to irradiation, toughness and
creep just to mention a few. These properties have a direct relationship with service life of
most structural components in nuclear application.
It is generally known atomic displacements contribute significantly to the source of
radiation effects in nuclear materials (Matzke, 1992). These displacements are
predominantly caused by the fission process which occurs in the fuels. Displacements often
lead to the localized and delocalized changes in microstructure, composition and
stoichiometry. These effects result in the modification of physical properties of core
materials. For example, the lattice parameter in UO2 increases as a function of irradiation
University of Ghana http://ugspace.ug.edu.gh
46
dose. This is due to the variation in the population of defects (interstitials and vacancies)
and their clusters.
2.3.3 Radiation Effects in Cladding material
Radiation-induce defects are sustained on the cladding material. These point defects are
formed instantaneously within picoseconds. These defects generally results into changes
in the mechanical and physical properties of the cladding material. At low homologous
temperatures and low radiation dose (0.001- 0.1 dpa), defects lead to matrix hardening
when they pin dislocation motion. The pinning of dislocation movement promotes strain-
hardening and reduces the ductility of the material especially in body centered cubic (BCC)
crystals. At intermediate homologous temperatures 0.4≤ T ≥0.6 and higher radiation doses
(1-10 dpa), radiation-induced segregation and radiation-induced precipitation can lead to
localized corrosion or mechanical property degradation, such as grain boundary
embrittlement (Mansur, et al., 1997).
Radiation-induced segregation and diffusion leads to phase change of the structural
material which affects the distribution of alloy elements. Voids created as a result of this
phase change leads to swelling when voids get occupied with displaced atoms thereby
changing the crystallographic dimensions.
University of Ghana http://ugspace.ug.edu.gh
47
Figure 2.11: Light water Reactor Fuel Performance. (Bouffioux, 2001).
2.3.4 Ion irradiation effects
Energetic beam of ions are bombarded on target surfaces. Ionized particles are emitted
from the surface as a result of this interaction. These ions include backscattered ions,
secondary ions, and sputtered ions (Nastasi, 2004). Some of the energetic ions penetrate
the solid surface experiencing numerous collisions with the target surface before coming
to a stop with their final energies. The lost in energy of the incoming distance is expressed
as:
ldldEE 11
(2.29)
The path length is also expressed as:
0
/
)(
0
0
E dldE
dEERt
(2.30)
∆E1 = Energy loss along ion path ∆l; dE1/dl = Energy loss per distance; ∆l = Path length;
Rt(Eo) = Projected range Eo = Initial energy of ion;
E1 = Energy of incident particle;
University of Ghana http://ugspace.ug.edu.gh
48
Figure 2.12: Principle of interaction between ions and solid.
Ion irradiation effects are encouraged by sputtering, cascade mixing, and implantation of
recoils. These effects develop into 3 – dimensional defects through enhanced diffusion and
induced segregation of ions.
2.4 RADIATION DAMAGE SIMULATION
Most commonly used techniques employed in the simulation study of the behaviour of
lattice atoms during collision cascades include (Was, 2007):
Binary Collision Approximation (BCA) method
Molecular Dynamics (MD) method
Kinetic Monte Carlo (KMC) method
2.4.1 Binary Collision Approximation (BCA) method
The BCA method is useful in ion irradiation physics. This makes it proficient for computer
simulation of defect production and penetration depth of energetic ions by sputtering or ion
University of Ghana http://ugspace.ug.edu.gh
49
implantation in target materials (Robinson, et al., 1974). This method is employed at the
collisional stages of recoil cascades in substantial numbers. The BCA method discriminates
on other forms of interactions considering only interaction occurring between just two
colliding atoms at a specific time. This makes it suited with the Kinchin-Pease model.
The BCA method agrees with the threshold energy theory of discriminating interactions at
very high energies, however it considers low energies, ballistic features which includes
replacement collision sequence and focused collision. Damage regions are proportionally
related to the energies.
The Binary collision approximation methods are categorized into two types according to
the degree of orderliness in the atom arrangements. They are namely:
Binary crystal model
Monte Carlo model
Binary crystal model (BCM) is of a much ordered crystal structure and employs a
deterministic approach in calculation. The later has an amorphous (thus unordered) crystal
structure and employs a stochastic method in locating and determining collision
parameters. This is achieved by using analytical approach to track particle population in a
media.
TRIM code in the SRIM-TRIM code is a typical example of the binary collision
approximation code which utilizes Monte Carlo techniques in simulating particle
behaviour and associated damage sustained in target solids. It employs maximum impact
parameters executed at constant mean free path and specified density for each binary
collision (Was, 2007).
University of Ghana http://ugspace.ug.edu.gh
50
2.4.2 Molecular Dynamics (MD) method
The MD simulation method provides detail atomistic processes in microstructural
evolution. MD uses algorithms to solve classical motion equations as represented below
for as atomic system;
ii frim .. urf ii
(2.31)
mi = mass of particle i; ri = position vector of i; fi = force acting on particle i; u
= interatomic potential
This equation above requires that the force be evaluated using the interatomic potential of
the atom.
Molecular dynamics is categorized into two forms, namely:
Classical molecular dynamics
Ab initio molecular dynamics
The first category is useful in treating classical entities for their positions and momentum
whilst ab initio molecular dynamics is used separately in resolving electronic and ionic
degrees of freedom using wave function distribution for electrons.
Operationally, MD method generates under suitable boundary conditions a projectile path
of an atomic system through the integration of the classical motion equations with precise
interatomic potential (Cai, et al., 2012).
University of Ghana http://ugspace.ug.edu.gh
51
Figure 2.13: MD in the multiscale modeling framework of microstructural evolution
(Kopetskii, 1974).
Molecular dynamics is significant in the fact that it gives a unified study of all physical
properties. This is achieved by providing structural, mechanical and crystallographic
properties of the atomic system. Furthermore, it gives direct link between potential models
and physical properties (Cai, et al., 2012).
2.4.3 Kinetic Monte Carlo (KMC) method
The KMC method presents very efficient approach to predicting stochastic behaviour of
particles at very relatively infinitesimal scales such as the mesoscale level. The random
walk model presents a probabilistic event of tracking particle motion between successive
interactions.
The KMC method attempts to subdue the time step of the simulation by exploiting the long
processing time of the dynamic system to a state - state diffusive jump. Therefore KCM
method neglects the tracing of trajectories from their vibrational periods (Was, 2007).
University of Ghana http://ugspace.ug.edu.gh
52
2.4.4 SRIM – TRIM Code
The SRIM-TRIM code calculates the interactions of energetic ions within the structure of
the target media. This code develops a general interatomic potential that is dependent on
ionic energy of the interacting atoms. The potential of these ions are of the form
demonstrated below:
a
r
r
eZZrV
2
21)(
(2.32)
Where; Z1 - Atomic number; Z2 – Energetic ion; e – Electric charge; r – Interatomic
distance; a – Empirical screening length
23.0
2
23.0
1
8854.0
ZZ
aa Bohr
(2.33)
aBohr is the Bohr radius (the radius of the hydrogen atom, 0.53 Å). Φ is the “universal”
screening function determined by precisely fitting of the evaluated interatomic potentials
of 521 randomly selected element permutations given by:
4
1
exp
i
ii a
rBAa
r (2.34)
The “free flight” distance is calculated at the start of the code using the interatomic distance
of target media at low energies. At higher energies, a more complex function of the ionic
energy and scattering density is applied. With good interaction, the ions undergoes nuclear
collision with a particle possessing an impact parameter which randomly selected. For
University of Ghana http://ugspace.ug.edu.gh
53
transferred energy E lower than the displacement energy Ed, the calculation is terminated.
Otherwise the code recalculates “free flight” distance original ion and struck atom. This
continues until a final energy is reached.
The SRIM-TRIM code keeps record of rate of displacements, percentage of energy lost,
vacancies created, replacement collisions, and various other parameters. Nuclear reactions
for amorphous structures are not considered (Olander, et al., 2015).
2.4.4.1 TRIM Code
The TRIM is a compendium of software which calculates the stopping and range of ions
(between 10 eV- 2GeV/amu) in matter using ion-atom interaction. This calculation is made
very easy and efficient by the use of algorithms that allows the ions to make jumps between
calculated collisions. The collisions are averaged to get results over the intervening gap.
During the collisions, the ion and atom have a screened coulomb collision, this including
exchange and correlation interactions between the overlapping electron shells.
2.4.4.2 TRIM Setup Windows
The TRIM setup Windows is a Monte Carlo Transport calculation which is used to run a
type of TRIM calculation with the data on the ion and target material. The TRIM Demo is
used to familiarize one with the use of the program.
University of Ghana http://ugspace.ug.edu.gh
54
Figure 2.14: Display of TRIM setup Widows.
2.4.4.3 Types of TRIM calculations
2.4.4.3.1 Ion Distribution and Quick Calculation of Damage
This execution of this calculation is employed when the estimation of damage is required
in a short time. It is therefore reliable for the evaluation of final distribution of ions in the
University of Ghana http://ugspace.ug.edu.gh
55
target, the Ionization energy loss, Energy transferred to recoil atoms, Backscattering Ions
and Transmitted Ions. It is however not suitable for analysis on detailed damage or
sputtering.
(Ziegler, et al., 2010).
2.4.4.3.2 Detailed Calculation with full Damage Cascades
This calculation takes into account all collision cascades until the final energy is reached.
The program however has a limitation of 8000 atoms collision in a single cascade. TRIM
will therefore run out of memory when this limit is reached.
2.4.4.3.3 Surface Sputtering
Surface sputtering is referred to as the removal of atoms from the surface of a target
material as a result of collision. It is the opposite of recoil implantation. For sputtering,
every near-surface interaction should be considered, and specifying this type of calculation
requires TRIM to evaluate every atom collision in detail. The atomic no of the ion should
be proportional to the depth of penetration. Therefore heavier ions interaction requires
larger depth compared to lighter ions.
2.4.4.3.4 Monolayer Collisions
The execution of this calculation in the TRIM takes into consideration collision in each
monolayer of the target material. The result of this calculation includes mean range,
ionization and damage cascade.
2.4.4.3.5 - Neutron / Electron / Photon Cascades
The particle cascade option has input file, the TRIM.DAT which has the necessary
University of Ghana http://ugspace.ug.edu.gh
56
description of target surface for the initiation of recoil cascades. TRIM code takes these
description and calculates the defects sustained on the target from the avalanche of
collision. These types of calculations can be run in 3 modes: Full damage cascades; Full
damage cascades and monolayer steps and the approximate calculations using the Kinchin-
Pease (K-P) model for approximations.
2.4.4.3.6 Various Ion Energy / Angle / Positions
Ion energy /Angle /Positions makes evaluation of various energies of the ions with diverse
trajectory angles of the target surface possible using the various depths in the target. Its
application is vast and considers varying ion parameters such as Ion energy, angle and
position of ion.
University of Ghana http://ugspace.ug.edu.gh
57
CHAPTER THREE: RESEARCH METHODOLOGY
The methodology employed in the determination of radiation damage effects by both
neutron and gamma radiation on zircaloy-2, zircaloy-4, stainless steel type 308 and Eurofer
97 cladding materials which involved the use of computational codes MCNP5 and SRIM-
TRIM codes. These codes are expounded upon in this chapter.
The MCNP5 simulation code was employed in the determination of the necessary neutron
parameters such as the neutron flux on each cladding material. This was followed by the
simulation of the energy deposited on the cladding materials by the neutron flux. The above
milestones were achieved by the modification of the input deck of the GHARR-1 Miniature
Neutron Source Reactor which has embedded in its clad structure zircaloy-4.
The average energy deposited on the clad materials from the MCNP5 code was used for
the simulation of gamma radiation damage on each cladding material in the SRIM-TRIM
code. SRIM was for the estimation of projected range or depth of neutron and gamma
penetration in each material. Transport of Ions in Matter (TRIM) was employed in the
simulation of gamma radiation damage. This involved the estimation of vacancies created,
ion distribution, displacement per atom (dpa) and the estimation of whether radiations lead
to changing the material integrity to an amorphous (disordered) crystal structure. Finally,
the analytical solution to the radiation damage on Zircaloy-4 clad material using both K-P
and NRT models are carried-out in this chapter.
University of Ghana http://ugspace.ug.edu.gh
58
3.1 MCNP5 Simulation
Monte Carlo methods are a set of rules used in the calculation process for analyzing 3-
dimensional, time dependent computational challenges using pseudo-random numbers. It
is particularly suited for interaction of nuclear particles with materials. This method
accommodates algorithms which are large enough to necessitate the use of digital
computers. It gives results only specified by the user in the tally cards.
MCNP is a general 3-dimensional combinatorial geometry transport code which is used to
evaluate neutron, electron or photon flux, current, charge or energy deposited, reaction rate,
heating and so on. For this thesis which deals with neutrons, the energy regime lies between
10-11 MeV to 20 MeV (X-5 MCNP, 2003).
The input data created for the MCNP simulation contained the following:
Definition of geometry
Material description for choosing cross-sections.
Specified location and nature of the neutron source.
Tally specification.
3.2 MCNP Neutronic Calculations
The secret to Monte Carlo codes are largely built upon on the principle of pseudo sampling.
There are basically two methods employed in Monte Carlo methods for resolving transport
equations.
They are namely; Computer simulation of physical processes and Mathematical methods
for numerical integration.
Computer simulation method is most appropriately used for collision physics tracking,
tallying for specific outputs, just to mention a few whilst mathematical approach is
University of Ghana http://ugspace.ug.edu.gh
59
appropriate in random sampling techniques, variance reduction, convergence, calculating
Eigen value (Booth, 1992). Monte Carlo method is chiefly used in solving integral
problems such as the Boltzmann equation. In resolving the integral Boltzmann transport
equation, the equation simulates the resultant events such as particle cascade and fission
processes. Eigen value is a prerequisite for reactor physics and criticality calculations. The
nature of the source models the arrangement of materials (Carter, et al., 1975). The time-
dependent linear Boltzmann transport equation is represented below:
dvvrrTvrQdvrvvCvrvr ),(.),(),(),(),( (3.1)
Where
),( vr Collision density
),( vrQ Source term
),( rvvC Collision kernel,
),( vrrT Transport Kernel,
Angular Flux =
),(
),(
vr
vr
(3.2)
Scalar Flux =
vvdvr vrvr ,),( ),(),(
(3.3)
Source term for the Boltzmann equation
dvrvvFvr
k
dvrvvFvrvrS
vrS
vrQ
),(),(
1
),(),(),(
),(
),(
(3.4)
Where
),( vrS Fixed source
University of Ghana http://ugspace.ug.edu.gh
60
),( rvvF Creation operator (due to fission particle at ),( vr creates particles at ),( vr
k Eigen value
The Monte Carlo method makes assumptions in the Boltzmann equation.
These assumptions are;
A homogeneously static medium
Time-dependent system
Markovian (thus the next event depends only on the current ),,( Evr event).
Intra-particle interactions do not exist.
Relativistic effects are not considered.
No long range forces (Particles interact in a straight path between events).
Particle reactions must be independent of material properties.
The particle histories after repeated substitution for ψk will give
dpppRpp kk )().()( 1 (3.5)
1...01211000 )()...().().()( kkk dpdpppRppRppRpp (3.6)
Particles normally start from a source by creating a track. The track may then split into two
for that one track at collision or the splitting surface where a second track is generated.
These two tracks possess half the track weight of the first particle track. When the track
attains (n, 2n) reaction, an additional track is created making three tracks. Particle tracks
consider the component of the first particle during its history. The length of a track in a cell
is useful in determining fluxes, energy fluence and energy deposition. Fluence, flux and
pulse-height energy deposition is analyzed using the crossing surfaces. Criticality collision
estimators and multiplication are calculated using collisions from the tracks (Foster, 1991).
University of Ghana http://ugspace.ug.edu.gh
61
For a cell with specified composition, the theory below is used in collecting collision along
its track (X-5 MCNP, 2003). The occurrence of the first collision between l and dll
along its flight is given as;
tl dledllp t)( (3.7)
Where t the macroscopic total cross section of the medium.
Setting ξ the random number on [0 1] to be
t ls tt edse 110 (3.8)
It shows that
)1(1 Inl t (3.9)
But, because the collision particle is distributed in the same manner as 1-ξ. Then, the
expression may be expanded to;
)(1 Inl
t
(3.10)
Having gone through the physics of how the code works below is the outline of the
methodologies that are employed in the MCNP code.
3.2.1 MCNP (LEU) Input file
The LEU input data file of the Miniature Neutron Source Reactor designed for the Ghana
Research Reactor -1 was modified to produce the flux of neutrons generation after 830
cycles at a criticality of 1.004 mk simulating 500000 neutron histories. The other control
runs at similar neutron history for 500 and 700 cycles at the same criticality dumping the
University of Ghana http://ugspace.ug.edu.gh
62
first 30 cycles were added. The flux of neutrons on each cladding of the 344 fuel pins where
ascertained as well as the flux of the clad in each of the ten concentric rings using the f4
tally at seven energies of 5.8e-8, 2.5e-7, 6.25e-7, 4.0e-6, 5.53e-3, 8.21e-1 and 20MeV
respectively. Each fuel pin was sectioned into ten subdivisions: 4 7710 7720 7730 7740
7750 7760 7780 7790 -3 for more accurate results to be achieved.
The energy deposition of the flux of neutrons on the cladding of all fuel pins was
determined. The energy deposition on each of the ten concentric rings was also determined.
This was possible by using the f7 tally cards at similar energies of flux. A copy of the flux
tallies will be found in the Appendix I. The output files of the MCNP5 simulation run in
the Femi-lab of the Argonne National Laboratory (ANL, 2016).
3.2.2 Working principle of MCNP Simulation tool
The MCNP program like other computational codes has a specific and particular trait and
attribute that serves as requirement for successfully running a problem. These features are
necessary in understanding the principle and manner in which the program is executed.
Below are elucidations on a number of features used in the MCNP code.
3.2.2.1 Nuclear Data and Reactions
The essential source of data employed in running the MCNP5 program are expressions
obtained in the Evaluated Nuclear Data File (ENDF) system, Advanced Computational
Technology Initiative (ACTI), data from the Nuclear Physics (T-16) group at Los Alamos
(Rose, 1991). Nuclear data libraries are written in a language that is easily understood by
University of Ghana http://ugspace.ug.edu.gh
63
MCNP. This increases the chance of retaining most details from the original evaluations in
order to reflect the intent of the user (MacFarlane, 1982).
The Nuclear data libraries are tabulated for interactions of neutron, neutron - photons,
particle interactions and thermal scattering (𝛼, β). These data tables accessible to MCNP
are cataloged on directory file XSDIR. ZAID is the distinct identification used for finding
data. In this thesis work, the neutron data table was specifically used for the cladding
materials under study.
Reaction rates can be determined when cross-sections are used as energy-dependent
response functions. The Nuclear data also considers reactions involving molecular binding
and crystalline effects. This becomes relevant when the neutron’s energy get sufficiently
low. Nuclear data takes into account information on the type of moderator, reflector
materials such as beryllium metal, graphite, and other metals such as beryllium oxide,
benzene, polyethylene, zirconium and hydrogen in zirconium hydride at varying
temperatures (MacFarlane, 1982).
3.2.2.2 Source Specification
The user is obliged to specify the source of neutron, electron or photon for MCNP program.
One has a vast array of choice of source to use. A source variable of energy, time, position,
direction may be appropriate for an independent stochastic distribution (X-5 MCNP, 2003).
This is also specified for other parameters like starting cells or surfaces. Some built-in
functions for fission and fusion energy spectra have been made available.
Source specification allows decoupling of calculations into several regions simultaneously
without running the problem for each region separately.
University of Ghana http://ugspace.ug.edu.gh
64
For a fission source, criticality source is used to estimate keff. The Table 3.1 below shows
the description of a fission source.
Table 3.1: Description of Mnemonics used in the Data Card to estimate keff.
Mnemonics Descriptions
Kcode Card name for criticality calculation.
nsrck Neutrons per cycle.
rkk Initial keff.
ikz Cycles dumped before accumulating data.
kct Cycles to run.
Ksrc Card name for initial fission source location
3.2.2.3 Tallies and Output
The MCNP output on a large scale depends on the tallies used. These tallies relate to
particle current, flux and energy deposition. But for criticality sources, these are tallies
normalized for each particle from inception. Current is tallied as dependent on direction
across surfaces, segments or sum of surfaces.
Fluxes are taken across surfaces, segments, in the cell, cell segments and sum of surfaces.
Fluxes are also taken at assigned detector locations as standard tallies with the F5 cards.
Special tally such as the super imposed mesh tally is independent of the geometry to which
the user tallies a particle on a mesh. Energy deposition is also specified for particular cells
University of Ghana http://ugspace.ug.edu.gh
65
for the heating and fission. The energy distribution created in the detector by pulses of
radiations is provided by the pulse height tally.
The tally results can be presented graphically except for the mesh tallies. These graphical
representations could be generated while the code still runs or separately in a post
processing mode. Below is Table 3.2 of tally mnemonics with their various descriptions.
Table 3.2: Tally Mnemonics and meanings.
Tally Mnemonic Description
F1:N/ F1:P/ F1:E Current of surface
F2:N/ F2:P/ F2:E Flux of surface
F4:N/ F4:P/ F4:E Track length
F5a:N/ F5a:P Flux at a point
F6:N/ F6:P/ F6:N,P Track length on deposited energy
F7:N Track length on deposited energy
F8:P/ F8:E/ F8:P,E Pulse-energy distributed in a detector
3.2.2.4 Estimation of Monte Carlo Errors
The tally and its second moment are the quantities used for error estimation R. This error
estimation is evaluated for every complete Monte Carlo history.
The error estimate is expressed as
N1
and N represents the histories. The true results
of the simulation could be related to the calculated relative error which is evaluated to give
University of Ghana http://ugspace.ug.edu.gh
66
the confidence intervals of the estimated mean. The confidence statement made based on
the results relies entirely on the precision of the Monte Carlo calculations. Table 3.3 shows
the error estimation (R) values and their interpretations.
Table 3.3: Guidelines for Interpreting the Relative Error R.
Range of R Quality of Tally
0.5 - 1.0 Unreliable
0.2 - 0.5 Factor of a few
0.1 - 0.2 Doubtful
< 0.10 Reliable
< 0.05 Reliable confidence interval
x
SR x
(3.11)
Where
XS
- Standard deviation of the mean, x - estimated mean
The Figure of Merit serves as means for quality assurance by the code in ensuring that the
output has a good confidence factor. It is printed out in the tally chart found in the output
file. FOM can be expressed in the form:
)(1 2TRFOM
(3.12)
Where T - Computer time in minutes; R - Relative Error;
The efficacy of the MCNP calculation is directly proportional to the FOM, which implies
the larger the FOM the better the results.
University of Ghana http://ugspace.ug.edu.gh
67
3.2.2.5 Kinchin-Pease Model Evaluation
d
n
E
EN 4
(3.13)
N = Number of displacement per atom for each collision
Ed = Displacement energy
nE Interacting energy
d s
d
N E . t σ Edpa = E . t . EN AE
(3.14)
3.2.2.6 Norgett-Robinson Torrens model Evaluation
d
NRT E
EV 28.0
(2.12)
n
d
nS
tNRT EE
Ek 5
)(
(2.28)
NRTV Displacement per atom
NRTk Rate of displacement per atom
University of Ghana http://ugspace.ug.edu.gh
68
3.3 Assessment of Nuclear Parameters
3.3.1. Neutron Fluence
The neutron fluence was evaluated in the MCNP Simulation code using the equation
Neutron Fluence = dEdttEr ),,( (3.15)
),,( tEr represents the neutrons flux integrated over the entire energy and time. The
analysis was carried out considering the fact that neutron fluence represents particle-track
lengths per unit volume. Furthermore,
V
WT was tallied with a cell for all particle-tracks
within the stipulated energy and time range. W is the particle weight, T is the track length
taken as the product of the time taken and velocity and V is the volume of the cell (Ziegler,
et al., 2008).
3.3.2 Normalization factor
The conversion factor used was
Ws
fissionEMeV
fission
E
MeV
W
sJ 10450908.388.180160205.1
1/1
(3.16)
The source strength of the reactor is evaluated using the factor 3.450908E+10P (W) where,
P represents reactor power in Watts. At steady state, the number of neutrons/fission v is
estimated as 2.4. The flux tallies were normalized with the expression below (Odoi, et al.,
2011).
volume
tallyvWPE *))((*10450908.3 (3.17)
University of Ghana http://ugspace.ug.edu.gh
69
3.3.3. Energy deposition
The neutron heating value represents the total energy deposition within the volume of a
cell in MeV/g. F7 gives an estimation of the heat of fissionable materials, that is only
achievable in a physical experiment with no photons while those generated from fission
are immediately captured. Actual heating could be estimated as a sum of the neutron and
photons in the F6 tallies. F6 estimates the energy of materials considered as light in an
experiment when photons are captured from the structure in a neutron-problem.
The relation below was used for the estimation of fission energy deposition in each cell in
the MCNP5 code with the help of F7 tally:
dEdtdVtErEEQgVaF ),,()()(7
(3.18)
Where V = volume,
σ (E) = microscopic cross-section,
Q (E) = fission Q-value and
ɸ (r, E, t) = neutron flux
All parameters undergo triple integration under the limits of energy and time. F7 scores
fission energy deposition was expressed as
gV
a(E) T1QW f
and was therefore available
for neutrons.
In the MCNP5 neutron simulation, F7 energy deposition is delivered locally with photons
from fission captured instantaneously. For F6 gamma heating was delivered somewhere
and the photons were traced. Afterwards the actual heating is determined by merging
neutron and photon tallies in a coupled neutron/photon calculation, the F6: N and Photon
tally (Ziegler, et al., 2008).
University of Ghana http://ugspace.ug.edu.gh
70
3.4 Damage assessment by SRIM – TRIM CODE
3.4.1 Input data for stopping range of ion in matter (SRIM)
Gamma radiations are emitted at high energies during fission. These energies propel them
from the fuel meat to interactions with the cladding materials. For each cladding material,
SRIM was used to evaluate the energy required for atomic displacement by the gamma
radiation using Cobalt-60 which is a gamma source. Table 3.4 represents the ion data
parameters for the gamma source.
Table 3.4: Ion data input parameters used in the SRIM Code.
Ion Data Details
Incident Ion name Cobalt
Symbol for the incident ion Co
Atomic number 27
Atomic mass 58.93 amu
Density 8.9 g/cm3
Ion lower energy 10 keV
Ion higher energy 10,000 keV
The Ion data table above describes the details of the incident particle: Cobalt-60. The
following tables below represent the target data parameters for each cladding material
under study. Cobalt-60 serves as Ion data parameter of gamma radiation for all cladding
materials used in the SRIM code.
University of Ghana http://ugspace.ug.edu.gh
71
Table 3.5: Target data parameters for Zircaloy-4 used in the SRIM Code.
Target data Details
Target description Co in Zr-Sn-Fe-O- Cr
Target density 6.498 g/cm3
Zirconium 91.22 amu
Tin 118.71 amu
Iron 55.847 amu
Oxygen 15.999 amu
Chromium 51.996 amu
Table 3.6: Target data parameters for Zircaloy-2 used in the SRIM Code.
_____________________________________________________________________________
Target data Details
Target description Co in Zr-Sn-Fe-O-Cr-Ni
Target density 6.49951 g/cm3
Zirconium 91.22 amu
Tin 118.71 amu
Iron 55.847 amu
Oxygen 15.999 amu
Chromium 51.996 amu
Nickel 58.69 amu
University of Ghana http://ugspace.ug.edu.gh
72
Table 3.7: Target data parameters for Stainless steel type 308 used in the SRIM Code.
Target data Details
Target description Co in Fe-Cr-W-Mn-V-Ta-C
Target density 7.933 g/cm3
Iron 55.847 amu
Chromium 51.996 amu
Tungsten 183.85 amu
Manganese 54.938 amu
Vanadium 50.942 amu
Tantalum 180.95 amu
Carbon 12.011 amu
Table 3.8: Target data parameters for Eurofer 97 used in the SRIM Code.
_____________________________________________________________________________
Target data Details
Target description Co in Fe-Cr-Ni-Mn-Si-C-P-S
Target density 7.8296 g/cm3
Iron 55.847 amu
Chromium 51.996 amu
Nickel 58.69 amu
Manganese 54.938 amu
Silicon 28.086 amu
Carbon 12.011 amu
Phosphorus 30.974 amu
Sulfur 32.066 amu
University of Ghana http://ugspace.ug.edu.gh
73
3.4.2 Input Data Parameter Window for SRIM
Figure 3.1: Data window for Cobalt- Zircaloy- 2 & 4 in SRIM Code.
Figure 3.2: Data window for Cobalt-Stainless Steel materials in the SRIM code.
University of Ghana http://ugspace.ug.edu.gh
74
Replicating the same analysis this time with the point of interest being neutrons
represented as Helium atom. Below is Table 3.9 of the input parameter of He used in the
SRIM Code.
Table 3.9: Ion data input parameters used in the SRIM Code.
Ion Data Details
Incident Ion name Helium
Symbol for the incident ion He
Atomic number 4
Atomic mass 4.003 amu
Density 0.1259 g/cm3
Ion lower energy 10 keV
Ion higher energy 10,000 keV
Figure 3.3: Data window for Helium-Zircaloy materials in the SRIM code.
University of Ghana http://ugspace.ug.edu.gh
75
Figure 3.4: Data window for Helium-Steel materials in the SRIM code.
The energy range of interest for this thesis work is between 0.9 - 10 MeV for an appreciable
projection range to revile much details. The results of the projection range determined from
the SRIM code used are found in chapter four.
3.4.3 Transport of Ions in Matter (TRIM) Simulation Code
The TRIM code is used specifically to analyze the damage cascade into details on the
cladding materials using the energy derived from the MCNP5 code. The energy is
possessed by both neutrons and gamma radiation for the purposes of the study of radiation
damage caused by either of the two. Therefore energy possessed by either neutron
independently or gamma radiation independently is directed to the crystal structure of
zircaloy-4 which is the material clad used in the MNSR (GHARR-1) reactor of the country.
This is aimed at accessing the radiation damage for the purposes of boosting the approval
University of Ghana http://ugspace.ug.edu.gh
76
of the core convention from HEU with Aluminium cladding to LEU with zircaloy-4
cladding. Also, similar independent neutron and gamma radiation interaction with zircaloy-
2, stainless steel type 308 and Eurofer 97 are estimated to ascertain their performance in
relation or comparison to zircaloy-4 cladding material.
A full detailed calculation with the Full Damage Cascades option was selected as the
damage type, to compute the defects created within the target atoms thus cladding materials
in a total of 100 ions at energy of 9871.9KeV.
Every primary knock-on atom created as a result of the collision cascade ignited in the
cladding materials were detected by the SRIM code and followed till it losses all its energy.
Input data parameters used for the TRIM simulation exercise for ion data and target data
for all cladding materials are indicated in the Tables 3.10 – 3.15 below.
Table 3.10: Ion (He) data input parameters used in the TRIM Code.
Ion Data Details
Incident Ion name Helium
Symbol for the incident ion He
Atomic number 4
Atomic mass 4.003 amu
Density 0.1259 g/cm3
Energy 9871.9KeV
Damage Type Full Damage Cascades
Angle of incident 0
University of Ghana http://ugspace.ug.edu.gh
77
Table 3.11: Ion (Co) data and input parameters used in the TRIM Code.
Ion Data Details
Incident Ion name Cobalt
Symbol for the incident ion Co
Atomic number 27
Atomic mass 58.93 amu
Density 8.9 g/cm3
Energy 9871.9KeV
Damage Type Full Damage Cascades
Angle of incident 0
Table 3.12: Target data (Zr-4) and input parameters used in the TRIM Code.
Ion Data Name/Value
Layer name Zircaloy-4
Symbol for the Target atom Zr-Sn-Fe-Cr-O
Width 2.81μm /40.9 μm
Atomic Density 4.2824E+22 atom/cm3
Displacement energy 25 eV
Lattice energy 3 eV
Surface binding energy 4.12 eV
Calculated Ions 100
University of Ghana http://ugspace.ug.edu.gh
78
Table 3.13: Target data (Zr-2) and input parameters used in the TRIM Code.
Ion Data Name/Value
Layer name Zircaloy-2
Symbol for the Target atom Zr-Sn-Fe-Cr-O-Ni
Width 2.81μm /40.88 μm
Atomic Density 4.2854E+22 atom/cm3
Displacement energy 25 eV
Lattice energy 3 eV
Surface binding energy 4.12 eV
Calculated Ions 100
Table 3.14: Target data (Fe-308) and input parameters used in the TRIM Code.
Ion Data Name/Value
Layer name Stainless Steel type 308
Symbol for the Target atom Fe-Cr-W-Mn-V-Ta-C
Width 2.11μm /26.99 μm
Atomic Density 8.3775E+22 atom/cm3
Displacement energy 25 eV
Lattice energy 3 eV
Surface binding energy 4.46 eV
Calculated Ions 100
University of Ghana http://ugspace.ug.edu.gh
79
Table 3.15: Target data (Eurofer-97) and input parameters used in the TRIM Code.
Ion Data Name/Value
Layer name Eurofer-97
Symbol for the Target atom Fe-Cr-Ni-Mn-Si-C-P-S
Width 2.07μm /26.89 μm
Atomic Density 8.52775E+22 atom/cm3
Displacement energy 25 eV
Lattice energy 3 eV
Surface binding energy 4.34 eV
Calculated Ions 100
Figure 3.5: Data Windows for Zircaloy 2 & 4 in TRIM Code.
University of Ghana http://ugspace.ug.edu.gh
80
Figure 3.6: Data Windows for Stainless Steels & Eurofer 97 in TRIM Code.
After the simulations, using SRIM – TRIM Code, a number of output files on radiation
damage as described in were produced. The result has been presented in Chapter Four.
3.5 Displacement Cascade Assessment
The SRIM-TRIM software evaluates the displacement cascade using the Kinchin – Pease
(K - P) model. Where only binary collisions are considered but in actual sense a multitude
of collision occur haphazardly with no regards to binary collision. The analytical approach
used in this thesis considers the evaluation of the displacement cascade with the Kinchin –
Pease (K - P) and Norgett – Robinson Torrens models. Therefore the K-P model is
calculated again using the analytical approach. Additionally, the Norgett-Robinson
Torrens model was also used to analytically calculate the displacement cascade.
University of Ghana http://ugspace.ug.edu.gh
81
The analytical solution for zircaloy-4 using Kinchin Pease and Norgett-Robinson Torrens
models is carried-out concurrently. The solution considers collisions on the zircaloy-4 clad
material which lasted averagely about 30 minutes from the TRIM analysis.
3.5.1 Kinchin-Pease Model
Zr [A= 90], En= 9.87188MeV, σel = 3b, φ (En) = 7.46 x 1013n/cm2.s, t = 1800s
Ed = 25eV
Where;
A = Mass number, En= Neutron Energy, σel = Elastic cross section, φ (En) = Neutron
flux
Ed = Displacement Energy, Rd = Rate of atomic displacement, dpa = Displacement per
atom.
04347.0)190(
904
)1(
4
22
x
A
A
Rd =
d
nnel
xE
EEN
4
)().( , dpa =
N
tRd. =
xNxE
tEEN
d
nnel
)4(
).().( =
)4(
).().(
d
nnel
xE
tEE
dpa =
eVx
sscmnxeVxcmx
254
)1800)(.1046.7)(1087188.9)(103)(04347.0( 2136224
dpa = 0.00172870980411024
306236.4291254
)1087188.904347.0(
.4
6
eVX
eVxx
E
E
d
n dpa per neutron collision
University of Ghana http://ugspace.ug.edu.gh
82
3.5.2 Norgett-Robinson Torrens Model
157950252
)1087188.9(8.0
2
8.0 6 eVx
eVxx
E
ENRT
d
n
The displacement per atom using the NRT Model is computed as follows;
255
)1800)(.1046.7)(1087188.9)(103)(04347.0(
5
).().( 2136224
x
sscmnxeVxcmx
xE
tEEEV
d
nneln
NRT
VNRT= 0.001382967843288192 displacement per atom
University of Ghana http://ugspace.ug.edu.gh
83
CHAPTER FOUR: RESULTS AND DISCUSSIONS
The results obtained from the Argonne National Laboratory (USA) on the MCNP5
simulation of the (LEU) input deck of GHARR-1 research reactor is expounded upon in
Section 4.1. Similarly, the outcome of the energy deposition (f7) is employed in the SRIM-
TRIM code for the radiation damage assessment in all four clad materials. This elaboration
is conspicuous in Section 4.2.
The output data from the MCNP5 simulation regarding the flux distribution on all 344 fuel
pins are presented in the Appendix I. Thus, flux distribution for the three levels of energy
on each clad. Also the output for the SRIM simulations are presented for energies between
(10KeV- 10MeV) showing their respective projected range, longitudinal and lateral
straggling.
4.1 Neutron Flux distributions
The neutron flux distribution over the Zircaloy-4 cladding material of all 344 fuel pins
sectioned into 4 7710 7720 7730 7740 7750 7760 7780 7790 -3 for all three energy groups
gave the following results.
Table 4.1: Average Normalized Neutron Flux in Zircaloy-4 at 34KW Power.
Energy group Average Neutron Flux (ncm-2s-1)
Thermal 4.0896E+11
Epithermal 9.02759E+11
Fast 5.29667E+11
Total 1.84139E+12
University of Ghana http://ugspace.ug.edu.gh
84
The outcome of the MCNP5 simulation of the axial thermal, epithermal, fast and total
normalized neutron fluxes for the 12.6% Low Enriched Uranium (LEU) UO2 is shown in
Table 4.1 above. The epithermal neutron flux showed a much higher flux compared to the
fast and thermal neutron fluxes respectively. The total neutron flux gave the highest flux
values. This behaviour of flux distribution is generally the trend in most simulations where
the epithermal fluxes are higher than the fast and thermal respectively. Comparing the
normalized average thermal flux achieved to that estimated by other works using the High
Enriched Uranium at a data description of 500000 1.004 100 500 gave a flux of 3.179E+11
(Boafo, 2012). The 22% increment in neutron flux is attributable to the fact that the Low
Enriched Uranium (LEU) core generates more thermal neutrons that undergo decay to
produce more daughter nuclei for interactions. It is attributed to the fact that the zircaloy
clad material has better conditions to sustaining more neutrons. It is also due to the fact
that the flux of neutrons generated in this thesis work is after 830 cycles which is higher
than that which was used in the HEU simulation.
4.1.1 Neutron flux distribution for the Lattice Rings
The Zircaloy-4 cladding material engulfs all the 344 fuel pins at the reactor core. The power
house of the reactor (control rod) is conspicuously located at the center of the concentric
rings with its stainless steel clad. The rest of the ten lattice positions left, serve other
purposes which are not under study in this work.
The core has ten (10) concentric rings of fuel pins arranged in it. These concentric rings
have in the center a control rod which controls neutron population upon its introduction or
withdrawal. These ten lattice rings have a varying number of fuel pins in the order of 6 12
19 26 32 39 45 48 58 59.
University of Ghana http://ugspace.ug.edu.gh
85
Table 4.2 below shows the output of the MCNP5 simulation representing the normalized
average flux of neutrons in all the ten lattice rings. Normalized fluxes refers to flux values
that have been normalized with the generally known normalization factor
Ws
fissionEMeV
fission
E
MeV
W
sJ 10450908.388.180160205.1
11
Whereas the Average normalized flux refers to the average of the normalized flux for each
lattice which has a specific number of fuel element.
Table 4.2: Normalized and Average Normalized Neutron Flux in Ten Lattice Ring.
Thermal flux Average Epithermal Flux Average Fast flux Average
9.24E+11 1.54E+11 2.45E+12 4.08E+11 1.32E+12 2.20E+11
1.79E+12 1.49E+11 5.57E+12 4.64E+11 3.13E+12 2.61E+11
2.94E+12 1.55E+11 9.51E+12 5.01E+11 5.41E+12 2.85E+11
4.23E+12 1.63E+11 1.40E+13 5.38E+11 7.93E+12 3.05E+11
4.81E+12 1.50E+11 1.58E+13 4.94E+11 8.97E+12 2.80E+11
5.75E+12 1.47E+11 1.88E+13 4.82E+11 1.06E+13 2.72E+11
6.10E+12 1.36E+11 1.95E+13 4.33E+11 1.09E+13 2.42E+11
5.72E+12 1.19E+11 1.78E+13 3.71E+11 9.76E+12 2.03E+11
6.11E+12 1.05E+11 1.71E+13 2.95E+11 9.32E+12 1.61E+11
7.86E+12 1.27E+11 1.36E+13 2.19E+11 7.24E+12 1.17E+11
From Table 4.2, it is obvious that the normalized flux in all three energy group recorded
higher neutron flux values than the average normalized neutron flux estimated for each fuel
pin. This observation is attributable to the fact that Normalized flux values are
University of Ghana http://ugspace.ug.edu.gh
86
representative of the whole ring or lattice whilst the Average normalized flux values are
representative of each fuel element in the lattice or ring.
There is a general increase in the flux values for the three energy groups from the first
lattice ring to the last lattice ring. However, there appears to be a decline in the average
normalized neutron flux for all the three energy groups in varying ways. The normalized
average fast neutron flux shows a more continuous decline in flux. This is followed by the
normalized average epithermal neutron flux and lastly the normalized average thermal
neutron flux registering a more distorted decline.
The graph below demonstrates the variation in the normalized flux distribution for the three
energy groups (levels) in the ten concentric lattice rings.
figure 4.1: A graph of Normalized neutron flux against Lattice distance.
0.00E+00
5.00E+12
1.00E+13
1.50E+13
2.00E+13
2.50E+13
0 5 10 15 20 25 30 35 40
N
o
rm
al
iz
e
d
F
lu
x
(n
/c
m
2
. s
)
Lattice Distance (mm)
Thermal Flux
Epithermal Flux
Fast Flux
University of Ghana http://ugspace.ug.edu.gh
87
For a true reflection of neutron flux in the individual lattice rings, the graph in figure 4.2
exhibits the normalized average neutron flux distribution for all the three energy levels
(groups). The specific number of fuel pins in each ring were taken into consideration.
figure 4.2: A graph of Normalized average neutron flux against Lattice distance.
For the purposes of this thesis work, the fast neutron energy group is known from literature
to be responsible for causing majority of radiation damage in materials. For this reason, the
normalized fast neuton flux of 7.46E+13n/cm2.s in all ten concentric rings is employed for
the radiation assessment. Comparing the normalized thermal neuton flux of 1.41E+12
n/cm2.s from this thesis to (9.8±0.0017)E+11 n/cm2.s of thermal neutron flux from the
inner irradition channel using the same code shows good trend (Odoi, et al., 2011).
0.00E+00
1.00E+11
2.00E+11
3.00E+11
4.00E+11
5.00E+11
6.00E+11
0 5 10 15 20 25 30 35 40
A
ve
ra
ge
N
o
rm
al
iz
e
d
f
lu
x(
n
/c
m
2 )
Lattice Distances (mm)
Ave. Thermal
Flux
Ave. Epithermal
Flux
Ave. Fast Flux
University of Ghana http://ugspace.ug.edu.gh
88
4.1.2 Neutron Energy deposition
The neutron heating value from the MCNP5 output represented with the F7 tally card in
the input deck is given in Table 4.3. The energy deposition is estimated for each lattice
ring.
Table 4.3: Neutron energy deposition in Lattice rings.
Lattice Ring Lattice distance (mm) Energy(MeV)
A 3.5 1.83932
B 7 3.67863
C 11.5 5.8245
D 15.5 8.27692
E 19 9.50313
F 23.5 7.31204
G 27 13.7949
H 28.5 14.7145
I 34.5 13.0047
J 35.5 20.7702
9.871884
It is observed that, a general increase in the energy deposition is recorded as the position
of the ring is moved further from the control rod. This increase is as a result of the increase
University of Ghana http://ugspace.ug.edu.gh
89
in neutron flux as was stated above. Neutron flux is directly proportional to the energy
deposition.
The figure 4.3 below is a graph of energy deposition on the zircaloy clad in each concentric
ring.
Figure 4.3: A graph of Energy deposition (MeV) Against Lattice distance (mm).
4.2 Damage Assessment by SRIM – TRIM Code
4.2.1 SRIM Calculations
The results of the projection range for all four clad materials from SRIM have been
presented in this section. At the maximum energy of 10MeV closest to the neutron energy
deposited in the clad 9.871884MeV, the projection range was taken. This is carried-out for
both neutron and gamma interactions. The detailed output text files for SRIM analysis are
shown in the Appendices. The results presented below are for both neutron and gamma
interactions respectively.
0
5
10
15
20
25
0 5 10 15 20 25 30 35 40
D
ep
o
si
te
d
e
n
er
g
y
(
M
eV
)
Lattice distance (mm)
University of Ghana http://ugspace.ug.edu.gh
90
Figure 4.4: A graph of Projection Range of Clad Materials (Neutron Interaction).
For neutron interaction, both Zircaloy-4 and Zircaloy-2 have a targeted width of 2.81μm.
At this width all much of the collision cascade details are reviled for both metals. Stainless
steel type 308 registered 2.11 μm. Eurofer-97 had a targeted width of 2.07 μm though it
has a higher density than the two zircaloy materials.
Figure 4.5: A graph of Projection Range of Clad Materials (λ-Interaction).
For Gamma interaction, similar trend of behaviour is seen this time at higher target width.
Zircaloy-4 recorded the highest width at 40.9 μm and Eurofer-97 recording the least at
0
0.5
1
1.5
2
2.5
3
Zr-4 Zr-2 Steel type 308 Eurofer-97
T
a
rg
et
W
id
th
(μ
m
)
Clad Materials
0
5
10
15
20
25
30
35
40
45
Zr-4 Zr-2 Steel type 308 Eurofer-97
Ta
rg
et
W
id
th
(μ
m
)
Clad Materials
University of Ghana http://ugspace.ug.edu.gh
91
26.89 μm. The result is reasonable as it is known from literature gamma radiations are
electromagnetic and travel long distance for interactions unlike neutrons.
4.2.2 TRIM Code Assessment
This section solely deals with the elucidations of the results from the full damage cascades
calculation of the Transport of Ions in Matter (TRIM). The target width estimated for both
neutron and gamma radiations are used as guide to achieving the results where collision is
at its pinnacle. The simulation uses the energy deposition as determined from the MCNP5
code for the full damage cascade calculation for neutron and gamma interactions with all
four clad materials. Our interest however is mainly with Zircaloy-4 clad material since it
is the clad for the LEU core which has been proposed.
4.2.2.1 Full damage assessment of Zircaloy-4
The output of the damage cascade after 100 collisions of a gamma- beta particle (Co-60)
with the Zircaloy-4 clad is shown below in fig. 4.6. The interaction generated 30356.2
vacancies.
Table 4.4: Percentage Energy distribution in Zircaloy-4 Target Material.
% Energy Loss Ions (%) Recoils (%)
Ionization 85.12 3.58
Vacancies 0.02 0.09
Phonons 0.08 10.29
The percentage of energy lost by the incoming ion to the target material is represented
above in Table 4.4. This table shows the loss of energy from the ion as it interacts with the
University of Ghana http://ugspace.ug.edu.gh
92
target zircaloy-4 material. Not all vacancies created leads to primary radiation. The ions
contribute 0.02% of its energy to vacancies whilst the recoil contributes 0.09%. This shows
that the recoil contributes to the creation of vacancies more than the ions though only 3.58%
of the energy was gained from the incoming ion. The recoil ions are therefore more
susceptible to causing primary radiation.
Figure 4.6: TRIM output for λ-Interaction with Zircaloy-4.
The Collision events in the figure below shows the vacancies created by virtue of the
constituent atoms in the target alloy. It is clear most of the vacancies created where those
of the zirconium metal which makes up about 98% of the entire material. Tin records some
few vacancies with chromium and the other trace elements recording almost nil in terms of
vacancies. The same trend is exhibited in sputtering.
University of Ghana http://ugspace.ug.edu.gh
93
Figure 4.7: Atomic Displacement by Collision event and sputtering (Zr-4).
Figure 4.8: Illustration of Ionization and Ion Range in Zircaloy-4.
For the neutron interaction with zircaloy-4, a whole different behaviour ensues. The total
number of vacancies recorded was 327 even after 1000 collisions. Only 0.01% of this was
as a result of recoils generated.
University of Ghana http://ugspace.ug.edu.gh
94
Figure 4.9: TRIM output for Neutron Interaction with Zircaloy-4.
Figure 4.10: Atomic Displacement by Collision event and Ionization (Zr-4).
4.2.2.2 Full damage assessment of Zircaloy-2.
After 100 collisions of the Co-60 particles with Zircaloy-2, the results are provided below.
This table shows the contributions of ions and recoil ions in the interaction.
University of Ghana http://ugspace.ug.edu.gh
95
Table 4.5: Percentage Energy distribution in Zircaloy-2 Target Material.
% Energy Loss Ions (%) Recoils (%)
Ionization 84.72 3.55
Vacancies 0.02 0.94
Phonons 0.08 10.69
Zorcaloy-2 interaction with gamma generated 31569 vacancies. 0.94% of these vacancies
created where as a result of recoil ions. This is greater than 0.09% recorded for zircaloy-4.
This implies zircaloy-2 will generate recoils with more energy to cause vacancies that will
lead to primary radiation.
Figure 4.11: TRIM output for λ-Interaction with Zircaloy-2.
From the sputtering graph, zirconium experiences the most lost at energies between 4 -
20eV. Tin has a sputtering yield lower than 0.5. The collision event on the other hand,
records 31533 vacancies and 1662 replacement collision. A 5.27% replacement collision.
University of Ghana http://ugspace.ug.edu.gh
96
Figure 4.12: Atomic Displacement by Collision event and sputtering (Zr-2).
Figure 4.13: Illustration of Ionization and Ion Range in Zircaloy-2.
For neutron interaction, Zicaloy-2 experienced 348 displacements. 17 of these collisions
resulted into replacement and the rest as target displacement. Below is a representation of
the assessment made using the simulation for neutron interactions.
University of Ghana http://ugspace.ug.edu.gh
97
Figure 4.14: Damage Assessment by TRIM on Zircaloy-2 (Neutron Interaction).
University of Ghana http://ugspace.ug.edu.gh
98
4.2.2.3 Full damage assessment of Stainless Steel type-308
The full damage cascade results for 100 collision of the Co-60 particle with Stainless steel
type 308 are given below. Table 4.6 shows the contributions of ions and recoil ions in the
interaction.
Table 4.6: Percentage Energy distribution in Stainless steel type-308 Target Material.
% Energy Loss Ions (%) Recoils (%)
Ionization 83.71 5.38
Vacancies 0.02 0.97
Phonons 0.11 9.8
Stainless steel type-308 interaction with gamma generated 32860 vacancies. 0.97% of these
vacancies created where as a result of recoil ions. This is comparatively greater than that
of zircaloy-4 and zircaloy-2. Since recoil atoms are susceptible to causing primary
radiation, a great deal of vacancies, interstitials and other defects are created in Stainless
steel type 308 than will be created in zircaloy 2 and 4. They also recorded higher number
of vacancies than zircaloy 2 and 4.
Figure 4.15: TRIM output for λ-Interaction with Stainless steel type-308.
University of Ghana http://ugspace.ug.edu.gh
99
From the collision event, 34637 displacements were recorded in the steel. Almost the same
for vacancies created with about 1890 replacement collisions ensuing. As much as 5.77%
of the vacancies recorded resulted into replacement which means more defect generation.
The sputtering of Iron and Chromium was relatively very high. This means reduction in
material thickness.
Figure 4.16: Atomic Displacement by Collision event and Sputtering (Steel type-308).
Figure 4.17: Illustration of Ionization and Atom distribution (Steel type-308).
University of Ghana http://ugspace.ug.edu.gh
100
For neutron interaction, a total of 415 vacancies were created. This is much lower than that
recorded for gamma interaction. In comparison to that recorded for neutron interaction with
zircaloy 2 and 4, this number of vacancies far exceeds those of zircaloy 2 and 4. After 1000
collisions, only 327 vacancies were recorded in zircaloy-4 which is still lower than 415
vacancies created in Stainless steel type 308 after only 100 collisions. Below are the
exhibitions of the results obtained.
Figure 4.18: Damage Assessment by TRIM on Steel type-308 (Neutron Interaction).
University of Ghana http://ugspace.ug.edu.gh
101
4.2.2.4 Full damage assessment of Eurofer 97
After 100 collisions the results of the Co-60 gamma particle interaction with Eurofer 97 is
almost similar to that of Stainless steel type-308 with gamma. Below is an illustration of
the percentage energy lost during the interaction.
Table 4.7: Percentage Energy distribution in Eurofer 97 Target Material.
% Energy Loss Ions (%) Recoils (%)
Ionization 83.55 5.49
Vacancies 0.02 0.96
Phonons 0.11 9.87
Eurofer 97 interaction with gamma generated 33647 vacancies. 5.49% of the ionization
taking place was as a result of the recoil. This is 0.11% more than that recorded for Stainless
steel type 308. Both Iron carbide alloy recorded about the same recoil ions causing the
primary radiation. The behaviour of Eurofer in this simulation showed similar traits as
Stainless steel type 308 with a slightly higher outcome. This is eminent in fig. 4.28 below.
Figure 4.19: TRIM output for λ-Interaction with Eurofer 97.
University of Ghana http://ugspace.ug.edu.gh
102
A record high of 34739 displacements occurred in the collision event. Out of which 1137
lead to replacement of ions in the target lattice positions. This represents 3.27% as against
5.77% recorded for Stainless steel type 308. The trend remain the same for sputtering with
Iron, Chromium and Nickel as the highest victims.
Figure 4.20: Atomic Displacement by Collision event and Sputtering (Eurofer 97).
Figure 4.21: Illustration of Atom Distribution and Ionization (Eurofer 97).
University of Ghana http://ugspace.ug.edu.gh
103
For neutron interaction, a total of 366 vacancies were created. This is much lower than that
recorded for neutron interaction in Eurofer 97. For 379 target displacement only 13 resulted
into replacement collision. Below are the exhibitions of the results obtained.
Figure 4.22: Damage Assessment by TRIM on Eurofer 97 (Neutron Interaction).
University of Ghana http://ugspace.ug.edu.gh
104
From the TRIM simulation, it has become crystal clear that zircaloy-4 sustained the least
number of vacancies resulting from the collision. Zircaloy-4 also records the least
percentage of energy for recoil ions which essentially contribute to the creation of primary
radiation. The trend shown from the simulation agrees with the fact that the Iron chromium
carbide (Fe-Cr-C) alloys have more vacancies ensuing from the collisions with Eurofer 97
having the highest vacancies of 33647. However, Stainless steel type 308 recorded the
highest percentage of recoil contributing to the creation of vacancies as well as the highest
replacement collision of 1890. The table below summarizes the outcome.
Table 4.8: Vacancy Assessment on clad materials.
Clad Materials Vacancies % Recoil Energy Replacement Collision
Zircaloy-4 30356 0.09 147
Zircaloy-2 31569 0.94 1162
Stainless Steel type 308 32860 0.97 1890
Eurofer 97 33647 0.96 1137
Based on the findings from the SRIM-TRIM simulation, the analytical determination of
the radiation damage using the radiation models will be carried-out on zircaloy-4. This is
because it has shown to be a much better clad material than the other three cladding
materials. Also, it suits this research work to ascertain the radiation damage of zircaloy-4
since it is used as the LEU clad material.
University of Ghana http://ugspace.ug.edu.gh
105
CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusion
The MCNP5 simulation results for neutron (particle) flux and energy deposition on the
Zircaloy-4 cladding material of the proposed LEU core at a Kcode specification of 500000
1.004 30 830 showed good trend of values for the three energy groups under study. TRIM
data file was generated for the actual radiation damage assessment on the clad materials.
The neutron flux distribution over the zircaloy-4 cladding material of all 344 fuel pins
sectioned into 4 7710 7720 7730 7740 7750 7760 7780 7790 -3 recorded an average fast
neutron flux which was evaluated from the simulation as 5.29667E+11n/cm2.s. The
average thermal and epithermal flux gave 4.0896E+11 and 9.02759E+11 respectively. The
average fast neutron flux in each of the ten concentric rings was 7.46E+13n/cm2.s.
The calculated average neutron energy deposition was 9.871884 MeV as was evaluated
from the F7 tally card using the MCNP5 simulation code.
The TRIM simulation established zircaloy-4 cladding material as the best since it sustained
the least amount of vacancies of 30356 as against 31569, 32860 and 33647 from zircaloy-
2, stainless steel type-308 and Eurofer 97 respectively. It registered the least energy for the
recoil which was directly responsible for creating defects. Zircaloy-4 also showed little
susceptibility to replacement collision.
The analytical calculation of radiation damage on zircaloy-4 cladding material using both
the Kinchin-Pease model and the Norgett –Robinson Torrens model gave 0.00173 and
0.00138 dpa respectively for only 30 minutes of clad exposure. The displacement per atom
(dpa) for each neutron collision gave 4291.
University of Ghana http://ugspace.ug.edu.gh
106
5.2 Recommendation
First of all, it is highly recommended to have a more in-depth analysis on the radiation
damage assessment at the atomistic level using molecular dynamics code LAMMPS.
LAMMPS will afford us the opportunity to ascertain the DPA of the clad material and the
overall outlook or behaviour before and after collision from the VMD.
Also, it is recommended that another approach is used for the determination of energy
deposition by both neutron and gamma particles using MCNPX to compare the findings.
University of Ghana http://ugspace.ug.edu.gh
107
REFERENCES
Akaho E.H., Maakuu B.T., Anim - Sampong, Emi - Reynolds, Boadu H.O., Osae E.K.,
Akoto - Bamford and Dodoo - Amoo D.N.A. (2003). GHARR-1 Safety Analysis
Report. Ghana: GAEC.
American National Standards Institute (ANSI). (2007). Standard practice for
characterizing neutron exposures in Iron and low alloy Steels in terms of (DPA).
Retrieved from
http://webstore.ansi.org/ansidocstore/product.asp?sku=ASTM+E693-01
Amuasi J.H., Schandorf C. and Yeboah J.,. (2005). Safety of Ghana Research Reactor
(GHARR-1). Ghana: GAEC.
Argonne National Laboratory. (2016, March). MCNP5 Simulation of LEU for GHARR-1.
USA.
Arthur E.D., Young P.G., Smith A.B. and Philis C.A. (1981). New tungsten Isotope
evaluation for neutron energies between 0.1 and 20 MeV. Trans. Am. Nucl., Soc.
39, pp. 793.
Bacon D.J., Diaz de la Rubia T. (1994). Molecular Dynamics computer simulations of
Displacement cascade in metal. Journal of Nuclear Materials, volume 216, pp. 275
-290.
Barett K., Bragg - Sitton, Galicki S.D. (2012). Advanced LWR Nuclear fuel cladding
systems development trade - off study. Idaho National Laboratory.
Binnewies M., Glaum, Schmidt R. and Schmidt M.P. (2013). Chemical vapor transport
reaction: A historical review. Zeitschrift fur anorganische und allgemeine chemie,
vol. 693, pp 219 - 229.
Birikorang S.A. and Nyarko B.J.B. (2014). Radiation detection. University of Ghana,
Legon., Department of Nuclear Engineering . Accra: Department of Nuclear
Engineering.
Booth E. Thomas. (1992). Monte carlo variance reduction approaches for non-Boltzman
tallies. Los Alamos National laboratory report, LA - 12433.
Bouffioux P. (EDF R&D) and Cheng B. (2001). Review of degradation phenomenon
affecting fuel rod cladding. Light water reactor fuel perfomance.
Brian Linn. (2013). Radiation damage effects in materials. Damage formation models.
MANE 6960.
Brookhaven National Laboratory. (2006). Evaluated Nuclear data file VI. ENDF - VI.
Cai W., Li J. and Yip S. (2012). Molecular dynamics in: Konings R.J.M., (ed)
comprehensive nuclear materials. Elsievier, Volume 1, pp. 249 - 265.
University of Ghana http://ugspace.ug.edu.gh
108
Carter L.L. and Cashwell E.D. (1975). Particle transport simulation with the Monte carlo
method. ERDH critical review series TID - 26607.
Chung H.M. and Kassner T.F. (1998). Cladding metallurgy and fracture behaviour during
reactivity - initiated accidents at high burnup. 186(3).
Crepin J., Bretheau T., Caldemaison D. (1995). Plastic deformation mechanisms of Beta -
treated Zirconium. Acta Metall Mater, Volume 43(10), pp. 3709 - 3719.
Darko J.,. (2013). Computation of thermal stresses in GHARR-1 vessel due to unsteady
flow of reactor coolant using the finite element method. MPhil Thesis, University
of Ghana,Legon, Department of Nuclear Engineering, Accra.
Diaz dela Rubia T., Averback R.S., Hsieh Horngming. and Benedek R. (1989). Molecular
dynamics simulation of displacement cascade in Cu and Ni: Thermal spike
behavior. Journal of Materials Research, volume 4, pp. 579 - 586.
Donald Olander and Arthur Motta. (2006). Light water reactor materials. volume 1, pp. 3 -
4.
Donald R. Olander. (1975). Fundamentals aspect of nuclear reactor element.
Donald R. Olander. (1976). Fundament aspects of nuclear reactor fuel elements. Energy
Research and Development Administration.
Donald R. Olander. (1976). Radiation damage: Fundamental aspects of nuclear reactor
elements. University of California, Department of Nuclear Engineering, Berkeley.
Douglass, D. (1971). The metallurrgy of Zirconium. Atomic energy review supplement.
Festus f. Appiah - Ofori. (2014). Assessment of Gamma irradiation heating and damage of
Miniature Neutron Source Reactor vessel using computaional methods and SRIM
- TRIM Code. MPhil Thesis, University of Ghana, legon, Department of Nuclear
Engineering, Accra. Retrieved March 2014
Foster R.A. (1991). A new method of assessing the statistical convergence of Monte carlo
solutions. Trans.Am.Nucl.soc, volume 64, pp. 305.
Gibbs G.B. and Hales R. (1977). The influence of metal lattice vacancies on the oxidation
of high temperature materials (Vol. 17). Corrsion Science.
Greenwood L.R. and Smither R.K. (1985). SPECTER: Neutron damage calculations for
materials irradiations. ANL/FPP/TM - 197.
Hannien H. (1990). Phenomena of material degradation with relevance to reactor pressure
vessel. NEA/CSNI Workshop on Safety assessment of reactor pressure vessels,
volume 1, p. pp. 13. Espoo, Finland.
Heinish H.J., and Singh B.N. (1993). On the structure of irradiation induced collision
cascades in metals as a function of recoil energy and crystal structure. Philosophy
magazine, volume 67, pp. 407 - 424.
University of Ghana http://ugspace.ug.edu.gh
109
Hocking W.H., Verrall R.A. and Muir I.J. (2001). Migration behaviour of Iodine in nuclear
fuel. Journal of Nuclear Materials, volume 294, pp. 45 - 52.
Hoppe G., Damaschun F. and Wappler G. (1987). An appreciation of Martin Heinrich
Klaproth as a mineral chemist. Pharmazie, 44(4), pp. 266 - 7.
International Atomic Energy Agency. (2009). Intergrity of reactor pressure vessels in
nuclear power plant. Assessment of irradiation embrittlement effects in reactor
pressure vessels steels. Vienna: IAEA Nuclear Energy Series No.NP-T-3.11.
International Atomic Energy Agency. (n.d.). Development of radiation resistance reactor
core structural materials. Retrieved from google:
http://www.iaea.org/About/Policy/GC/GC51/GC51InfDocuments/English/gc51inf
-3-att7_en.pdf
Knoll G.F. (1989). Radiation detection and measurements. New York: John Wiley & Sons.
Korean Atomic Energy Research Institute. (2013). Table of nuclides. Retrieved February
1st, 2014, from http://atom.kaeri.re.kr/
Krebs R. (1998). The history and use of our earth's chemical elements. (Westport, Ed.)
Connecticut: Greenwood Press.
Kroll W. (1940). The production of ductile Titanium. Transactions of the electrochemical
society, volume 78, pp. 35 - 47.
Lee J.G. (2005). Computational materials science: An introduction. New York, USA: 1st
Edition.
Liu H., Chen Y., Tang Y., Wei S. and Niu G,. (2007). The microstructure, tensile properties
and creep behaviour of As - cast Mg - (1 - 10)% Sn alloys. Journal of alloys and
compounds, volume 448, pp. 123 - 4.
MacFarlane R.E., Muir D.W. and Boicourt R.M. (1982). The NJOY Nuclear data
processing system. Los Alamos National Laboratory, volume I(ENDF-324).
Maisonnier D. (2006). DEMO and fusion power plant conceptual studies in Europe. Fusion
Engineering and Design, volume 81, pp. 1123.
Mansur L.K. and Farell K. (1997). Mechanisms of radiation - induced degradation of
reactor vessel materials. Journal of Nuclear Materials, volume 244, pp. 213.
Matzke H.J. (1992). Radiation damage in nuclear materials. Nuclear Instruments and
methods in Physics Research B, volume 65, pp. 30 - 39.
Matzke H.J. and Turos A. (1992). Ion implantation studies of UO2 and UN. Journal of
Nuclear Materials, volume 188, pp. 285 - 292.
Matzke H.J., Turos A. and Linker G. (1994). Polygonization of single crystals of the
flourite - type oxide UO2 due to dose ion implantation. Nuclear Instruments and
Methods in Physics Research B, volume 91, pp. 294 - 300.
University of Ghana http://ugspace.ug.edu.gh
110
McElroy R.J. (2004). Low temperature embrittlement of LWR PRV support structures.
NEA/CSNI Workshop on safety assessment of reactor pressure vessels, volume 2,
p. 311. Espoo, Finland.
Meldrum A., Zinkle S.J., Boatner L.A and Ewing R.C. (1998). A transient liquid - like
phase in the displacement cascades of Zircon. Hafnon and Thorite, volume 395, pp.
56 - 58.
Nastasi M., Mayer J. and Hirvonen J.K. (2004). Ion - Solid interactions: Fundamentals and
Applications. Cambridge University Press.
Norgett M.J., Robinson M.T. and Torren I.M. (1975). A proposed method for calculating
displacement dose rates. Nuclear Engineering and Design, volume 33, pp. 50 - 54.
Nuclear regulatory commission. (2013). Pressurized Water Reactors. Retrieved January
22, 2014, from http://www.nrc.gov/reactors/pwers.html.
Nyarko B.J.B. and Deborah S. (2012). Types of reactors. University of Ghana,Legon,
Department of Nuclear Engineering. Accra: Department of Nuclear Engineering.
Oak Ridge National Laboratory (ORNL). (n.d.). Application to fusion and generation IV
fusion reactors. Workshop on Advanced computational materials
science(ORNL/TM - 2004/132). Retrieved June 2004
Odoi H.C., Akaho E.H.K. and Anim - Sampong. (2011). Investigative studies on effect of
reflector thickness on the performance of low enriched uranium-fueled Miniature
Neutron Source Reactors. (N. E. Design, Ed.) Elsevier, volume 241, pp. 2909 -
2915.
Olander, D. R. (2006). Radiation damage and neutron irradiation (Vol. section 9).
Fundamental aspects of nuclear reactor fuel elements.
Parkin D.M., Coulter C.A. (1981). Total and net displacement functions for polyatomic
materials. Journal of Nuclear Materials, volume 101, pp. 261 - 276.
Quaye C.R. (2012). Finite element modelling of transient heat conduction of fuel element
of Ghana research reactor - 1. School of Nuclear and Allied Sciences, Department
of Nuclear Engineering. Accra: University of Ghana.
Ragheb M. (2013). Loss of coolant accidents (LOCA). Retrieved January 9th, 2014, from
http://mragheb.com
Robert W.C. and Haasen P. (1996). Physical metallurgy. PhD Thesis, Technical University
Darmstadt, Department of materials science.
Robinson, Mark, Torrens and Ian. (1974). Computer simulation of atomic - displacement
cascade in solids in the binary - collision approximation. Physical review, B9(12),
5008.
University of Ghana http://ugspace.ug.edu.gh
111
Roger E. Stoller. (2011). Radiation damage fundamentals: Primary damage production.
Materials science and technology division, Oak Ridge National Laboratory, Joint
EFRC Summer school.
Rose P.F. (1991). ENDF - 201, ENDF/B - VI, Compiler and Editor, Summary
documentation. Brookhaven National Laboratory.
Sandvik special metal corporation (1989). (1989). Zirconium alloy fuel clad tubing.
Engineering guide, 1st Edition.
Schaeublin R., Leguey T., Spatig P., Baluc N. and Victoria M. (2002). Microstructure and
mechanical properties of two ODS ferritic/martensitic steels. Journal of Nuclear
Materials , pp. 307 - 311.
Schweitzer P. (2003). Zirconium and Zirconium alloy metallic materials. Physical,
mechanical and corrosion properties, pp. 647 - 666.
Sekpe H. and Sekpe I. (1986). The history of Physiologic chemistry in the first years of its
existence at the Berlin University: Contributions of the chemist M.H. Klaproth and
others. Zeitschrift fur die gesamte Hygiene und ihre Grenzgebiete, volume 32(8),
pp. 504 - 6.
Sika Boafo. (2012). Determination of neutron parameters in the GHARR-1 reactor using
MCNP and TRIM codes. School of Nuclear and Allied Sciences, Department of
Nuclear Engineering. Accra: University of Ghana.
Stanislav Golubov and Roger Stoller. (2004). Materials and technical division. Oak Ridge
National Laboratory .
Todreas N. and Kazimi M. (2011). Nuclear systems: 2nd Edition. Taylor & Francis,
volume I: Thermal hydraulic fundamentals, pp. 20 - 22, 33 - 34.
Tritt T. (2004). Thermal conductivity: Theory, properties and Applications. Physics of
solids and liquid, Springer - Verlag, pp. 41 - 47.
Tsegay H. (2011). Measurement of energy of Gamma radiation. M.Sc Thesis, Addis Ababa
University, Department of Physics, Addis Ababa.
Van Dam H., Van der Hagen T.H.J.J. and Hoogenboom J.E. (2005). Nuclear reactor
physics. Delft University of Technology, Department of Nuclear Engineering.
Verlet L. (1968). Computer experiments on classical fluid II: Equilibrium correlation
functions. Phys.Rev., volume 165, pp. 201 - 214.
Was G.S. (2007). Fundamentals of radiation materials science metal and alloys. Journal of
Materials Science.
Weinberg A.M. (1994). The first nuclear era: The life and times of technological fixers.
New York: AIP Press.
University of Ghana http://ugspace.ug.edu.gh
112
William D. Callister Jr. (2005). Materials science and Engineering. Book, University of
Utah, Department of Metallurgical Engineering.
X - 5 Monte carlo team. (2003). MCNP - A General Monte carlo N - Particle transport code
version 5. Los Alamos National Laboratory, volume I: Overview & Theory, pp. 1 -
11.
Yuri Osetsky. (2008). Introduction to radiation damage. Oak Ridge National Laboratory.
Zenngliang Y. and Liangdeng Y. (2006). Introduction to Ion beam biotechnology.
Springer.
Ziegler J.F., Ziegler M.D and Biersack J.P. (2010). SRIM - The stopping and range of Ions
in matter. Pergamum Press.
Zircaloy heat capacity. (1997). Retrieved from International Nuclear Safety centre:
http://www.insc.anl.gov/matprop/zircalov/zirccp/fmt html.pdf
University of Ghana http://ugspace.ug.edu.gh
113
APPENDIX I: Data from MCNP5 Simulation of Neutron Parameters for all 344 Fuel
Pins and 4 Tie rods Clads (Argonne National Laboratory).
Clad Thermal
Epithermal
Fast flux
Total flux
7 4.449E+11
1.09E+12
6.51E+11
2.18E+12
9 4.474E+11
1.09E+12
6.50E+11
2.18E+12
11 4.444E+11
1.09E+12
6.49E+11
2.18E+12
13 4.437E+11
1.08E+12
6.50E+11
2.18E+12
15 4.414E+11
1.09E+12
6.47E+11
2.18E+12
17 4.445E+11
1.09E+12
6.48E+11
2.19E+12
19 4.532E+11
1.08E+12
6.50E+11
2.18E+12
21 4.553E+11
1.08E+12
6.5E+11
2.18E+12
23 4.538E+11
1.08E+12
6.50E+11
2.18E+12
25 4.554E+11
1.08E+12
6.50E+11
2.19E+12
27 4.505E+11
1.08E+12
6.48E+11
2.18E+12
29 4.459E+11
1.08E+12
6.49E+11
2.18E+12
31 4.447E+11
1.09E+12
6.48E+11
2.18E+12
33 4.441E+11
1.08E+12
6.47E+11
2.18E+12
35 4.377E+11
1.08E+12
6.48E+11
2.17E+12
37 4.433E+11
1.08E+12
6.49E+11
2.17E+12
39 4.402E+11
1.08E+12
6.49E+11
2.17E+12
41 4.487E+11
1.08E+12
6.49E+11
2.18E+12
43 4.506E+11
1.06E+12
6.42E+11
2.16E+12
45 4.507E+11
1.06E+12
6.40E+11
2.15E+12
47 4.494E+11
1.07E+12
6.40E+11
2.16E+12
49 4.534E+11
1.07E+12
6.40E+11
2.16E+12
51 4.544E+11
1.06E+12
6.44E+11
2.16E+12
53 4.495E+11
1.07E+12
6.38E+11
2.15E+12
55 4.490E+11
1.06E+12
6.37E+11
2.15E+12
University of Ghana http://ugspace.ug.edu.gh
114
57 4.517E+11
1.06E+12
6.40E+11
2.16E+12
59 4.376E+11
1.06E+12
6.43E+11
2.14E+12
61 4.287E+11
1.07E+12
6.49E+11
2.14E+12
63 4.229E+11
1.07E+12
6.45E+11
2.13E+12
65 4.248E+11
1.07E+12
6.43E+11
2.13E+12
67 4.239E+11
1.07E+12
6.42E+11
2.13E+12
69 4.227E+11
1.07E+12
6.42E+11
2.13E+12
71 4.243E+11
1.07E+12
6.44E+11
2.13E+12
73 4.238E+11
1.07E+12
6.43E+11
2.14E+12
75 4.269E+11
1.07E+12
6.46E+11
2.14E+12
77 4.314E+11
1.06E+12
6.42E+11
2.14E+12
79 4.387E+11
1.06E+12
6.45E+11
2.15E+12
81 4.394E+11
1.04E+12
6.25E+11
2.10E+12
83 4.424E+11
1.04E+12
6.21E+11
2.1E+12
85 4.389E+11
1.03E+12
6.27E+11
2.09E+12
87 4.409E+11
1.04E+12
6.23E+11
2.09E+12
89 4.43E+11
1.04E+12
6.28E+11
2.11E+12
91 4.437E+11
1.04E+12
6.27E+11
2.11E+12
93 4.419E+11
1.04E+12
6.27E+11
2.11E+12
95 4.400E+11
1.04E+12
6.28E+11
2.11E+12
97 4.447E+11
1.03E+12
6.25E+11
2.11E+12
99 4.424E+11
1.04E+12
6.26E+11
2.11E+12
101 4.396E+11
1.04E+12
6.24E+11
2.1E+12
103 4.27E+11
1.04E+12
6.27E+11
2.09E+12
105 4.16E+11
1.04E+12
6.3E+11
2.09E+12
107 4.126E+11
1.03E+12
6.29E+11
2.08E+12
109 4.108E+11
1.04E+12
6.31E+11
2.08E+12
111 4.118E+11
1.04E+12
6.29E+11
2.08E+12
University of Ghana http://ugspace.ug.edu.gh
115
113 4.124E+11
1.04E+12
6.28E+11
2.08E+12
115 4.114E+11
1.04E+12
6.27E+11
2.08E+12
117 4.108E+11
1.04E+12
6.28E+11
2.08E+12
119 4.131E+11
1.04E+12
6.28E+11
2.08E+12
121 4.106E+11
1.04E+12
6.27E+11
2.08E+12
123 4.131E+11
1.04E+12
6.30E+11
2.08E+12
125 4.134E+11
1.04E+12
6.3E+11
2.08E+12
127 4.131E+11
1.04E+12
6.28E+11
2.08E+12
129 4.188E+11
1.04E+12
6.31E+11
2.09E+12
131 4.27E+11
1.04E+12
6.29E+11
2.09E+12
133 4.235E+11
1.00E+12
6.06E+11
2.03E+12
135 4.229E+11
9.99E+11
6.09E+11
2.03E+12
137 4.239E+11
1.00E+12
6.04E+11
2.03E+12
139 4.214E+11
9.98E+11
6.07E+11
2.03E+12
141 4.227E+11
9.99E+11
6.09E+11
2.03E+12
143 4.248E+11
1.00E+12
6.07E+11
2.03E+12
145 4.259E+11
1.00E+12
6.06E+11
2.04E+12
147 4.255E+11
1.00E+12
6.07E+11
2.04E+12
149 4.232E+11
9.99E+11
6.09E+11
2.03E+12
151 4.244E+11
1.00E+12
6.05E+11
2.03E+12
153 4.226E+11
1.00E+12
6.08E+11
2.04E+12
155 4.205E+11
1.00E+12
6.07E+11
2.03E+12
157 4.213E+11
1.01E+12
6.06E+11
2.03E+12
159 4.202E+11
1.00E+12
6.05E+11
2.03E+12
161 4.203E+11
1.00E+12
6.08E+11
2.03E+12
163 4.158E+11
1.01E+12
6.06E+11
2.03E+12
165 4.148E+11
1.01E+12
6.06E+11
2.03E+12
167 4.075E+11
1.01E+12
6.07E+11
2.02E+12
University of Ghana http://ugspace.ug.edu.gh
116
169 4.073E+11
1.01E+12
6.05E+11
2.02E+12
171 4.069E+11
1.01E+12
6.06E+11
2.02E+12
173 4.06E+11
1.01E+12
6.05E+11
2.02E+12
175 4.07E+11
1.01E+12
6.04E+11
2.02E+12
177 4.085E+11
1.01E+12
6.05E+11
2.03E+12
179 4.063E+11
1.01E+12
6.04E+11
2.02E+12
181 4.082E+11
1.01E+12
6.08E+11
2.02E+12
183 4.077E+11
1.01E+12
6.04E+11
2.02E+12
185 4.082E+11
1.01E+12
6.07E+11
2.02E+12
187 4.081E+11
1.01E+12
6.07E+11
2.02E+12
189 4.076E+11
1.01E+12
6.07E+11
2.02E+12
191 4.103E+11
1.00E+12
6.08E+11
2.02E+12
193 4.114E+11
1.01E+12
6.09E+11
2.03E+12
195 4.153E+11
1.00E+12
6.08E+11
2.03E+12
197 4.108E+11
9.62E+11
5.82E+11
1.95E+12
199 4.088E+11
9.63E+11
5.80E+11
1.95E+12
201 4.070E+11
9.59E+11
5.79E+11
1.95E+12
203 4.06E+11
9.63E+11
5.79E+11
1.95E+12
205 4.061E+11
9.57E+11
5.79E+11
1.94E+12
207 4.071E+11
9.59E+11
5.80E+11
1.95E+12
209 4.092E+11
9.66E+11
5.82E+11
1.96E+12
211 4.1E+11
9.64E+11
5.83E+11
1.96E+12
213 4.111E+11
9.59E+11
5.82E+11
1.95E+12
215 4.111E+11
9.59E+11
5.80E+11
1.95E+12
217 4.109E+11
9.59E+11
5.80E+11
1.95E+12
219 4.051E+11
9.55E+11
5.80E+11
1.94E+12
221 4.052E+11
9.64E+11
5.81E+11
1.95E+12
223 4.076E+11
9.63E+11
5.8E+11
1.95E+12
University of Ghana http://ugspace.ug.edu.gh
117
225 4.074E+11
9.66E+11
5.79E+11
1.95E+12
227 4.047E+11
9.63E+11
5.79E+11
1.95E+12
229 4.067E+11
9.60E+11
5.79E+11
1.95E+12
231 4.072E+11
9.65E+11
5.79E+11
1.95E+12
233 4.06E+11
9.65E+11
5.81E+11
1.95E+12
235 3.991E+11
9.63E+11
5.81E+11
1.94E+12
237 3.945E+11
9.64E+11
5.82E+11
1.94E+12
239 3.969E+11
9.63E+11
5.78E+11
1.94E+12
241 3.958E+11
9.65E+11
5.80E+11
1.94E+12
243 3.936E+11
9.66E+11
5.79E+11
1.94E+12
245 3.962E+11
9.63E+11
5.79E+11
1.94E+12
247 3.946E+11
9.65E+11
5.82E+11
1.94E+12
249 3.959E+11
9.63E+11
5.79E+11
1.94E+12
251 3.961E+11
9.63E+11
5.81E+11
1.94E+12
253 3.981E+11
9.66E+11
5.81E+11
1.95E+12
255 3.968E+11
9.65E+11
5.81E+11
1.94E+12
257 3.945E+11
9.65E+11
5.79E+11
1.94E+12
259 3.967E+11
9.64E+11
5.81E+11
1.94E+12
261 3.946E+11
9.66E+11
5.79E+11
1.94E+12
263 3.934E+11
9.60E+11
5.81E+11
1.94E+12
265 3.966E+11
9.66E+11
5.79E+11
1.94E+12
267 3.949E+11
9.69E+11
5.82E+11
1.95E+12
269 3.952E+11
9.63E+11
5.79E+11
1.94E+12
271 3.983E+11
9.62E+11
5.81E+11
1.94E+12
273 4.068E+11
9.61E+11
5.81E+11
1.95E+12
275 4.006E+11
9.13E+11
5.48E+11
1.86E+12
277 3.918E+11
9.12E+11
5.46E+11
1.85E+12
279 3.921E+11
9.15E+11
5.50E+11
1.86E+12
University of Ghana http://ugspace.ug.edu.gh
118
281 3.907E+11
9.16E+11
5.49E+11
1.86E+12
283 3.902E+11
9.13E+11
5.49E+11
1.85E+12
285 3.927E+11
9.14E+11
5.49E+11
1.86E+12
287 3.944E+11
9.14E+11
5.49E+11
1.86E+12
289 3.939E+11
9.15E+11
5.51E+11
1.86E+12
291 3.967E+11
9.14E+11
5.48E+11
1.86E+12
293 4.023E+11
9.15E+11
5.47E+11
1.86E+12
295 4.129E+11
9.14E+11
5.46E+11
1.87E+12
297 4.015E+11
9.17E+11
5.48E+11
1.87E+12
299 3.945E+11
9.13E+11
5.48E+11
1.86E+12
301 3.927E+11
9.15E+11
5.48E+11
1.86E+12
303 3.951E+11
9.16E+11
5.50E+11
1.86E+12
305 3.923E+11
9.15E+11
5.48E+11
1.86E+12
307 3.931E+11
9.16E+11
5.51E+11
1.86E+12
309 3.939E+11
9.14E+11
5.51E+11
1.86E+12
311 3.917E+11
9.14E+11
5.49E+11
1.86E+12
313 3.948E+11
9.15E+11
5.51E+11
1.86E+12
315 4.008E+11
9.16E+11
5.45E+11
1.86E+12
317 4.077E+11
9.14E+11
5.45E+11
1.87E+12
319 3.929E+11
9.14E+11
5.46E+11
1.85E+12
321 3.85E+11
9.16E+11
5.51E+11
1.85E+12
323 3.854E+11
9.16E+11
5.48E+11
1.85E+12
325 3.83E+11
9.18E+11
5.47E+11
1.85E+12
327 3.825E+11
9.21E+11
5.48E+11
1.85E+12
329 3.828E+11
9.19E+11
5.49E+11
1.85E+12
331 3.830E+11
9.18E+11
5.46E+11
1.85E+12
333 3.831E+11
9.19E+11
5.49E+11
1.85E+12
335 3.852E+11
9.19E+11
5.48E+11
1.85E+12
University of Ghana http://ugspace.ug.edu.gh
119
337 3.895E+11
9.19E+11
5.51E+11
1.86E+12
339 3.985E+11
9.16E+11
5.49E+11
1.86E+12
341 4.015E+11
9.11E+11
5.47E+11
1.86E+12
343 3.871E+11
9.2E+11
5.49E+11
1.86E+12
345 3.84E+11
9.19E+11
5.47E+11
1.85E+12
347 3.811E+11
9.22E+11
5.47E+11
1.85E+12
349 3.828E+11
9.21E+11
5.50E+11
1.85E+12
351 3.832E+11
9.19E+11
5.52E+11
1.85E+12
353 3.818E+11
9.17E+11
5.50E+11
1.85E+12
355 3.819E+11
9.19E+11
5.48E+11
1.85E+12
357 3.840E+11
9.16E+11
5.49E+11
1.85E+12
359 3.856E+11
9.17E+11
5.49E+11
1.85E+12
361 3.946E+11
9.18E+11
5.49E+11
1.86E+12
363 4.094E+11
9.12E+11
5.44E+11
1.87E+12
365 4.035E+11
8.61E+11
5.08E+11
1.77E+12
367 3.833E+11
8.66E+11
5.14E+11
1.76E+12
369 3.770E+11
8.65E+11
5.13E+11
1.76E+12
371 3.763E+11
8.64E+11
5.14E+11
1.75E+12
373 3.791E+11
8.62E+11
5.14E+11
1.76E+12
375 3.799E+11
8.62E+11
5.14E+11
1.76E+12
377 3.762E+11
8.67E+11
5.13E+11
1.76E+12
379 3.770E+11
8.63E+11
5.13E+11
1.75E+12
381 3.782E+11
8.66E+11
5.16E+11
1.76E+12
383 3.835E+11
8.67E+11
5.14E+11
1.77E+12
385 3.863E+11
8.64E+11
5.14E+11
1.76E+12
387 4.034E+11
8.61E+11
5.11E+11
1.78E+12
390 3.981E+11
8.63E+11
5.09E+11
1.77E+12
392 3.857E+11
8.63E+11
5.12E+11
1.76E+12
University of Ghana http://ugspace.ug.edu.gh
120
394 3.771E+11
8.6E+11
5.11E+11
1.75E+12
396 3.787E+11
8.66E+11
5.13E+11
1.76E+12
398 3.787E+11
8.66E+11
5.13E+11
1.76E+12
400 3.773E+11
8.65E+11
5.12E+11
1.75E+12
402 3.823E+11
8.69E+11
5.13E+11
1.77E+12
404 3.824E+11
8.64E+11
5.15E+11
1.76E+12
406 3.792E+11
8.64E+11
5.12E+11
1.76E+12
408 3.766E+11
8.64E+11
5.11E+11
1.75E+12
410 3.844E+11
8.61E+11
5.13E+11
1.76E+12
412 4.012E+11
8.63E+11
5.09E+11
1.77E+12
415 3.967E+11
8.63E+11
5.06E+11
1.77E+12
417 3.776E+11
8.67E+11
5.12E+11
1.76E+12
419 3.769E+11
8.64E+11
5.09E+11
1.75E+12
421 3.767E+11
8.66E+11
5.10E+11
1.75E+12
423 3.743E+11
8.69E+11
5.12E+11
1.76E+12
425 3.739E+11
8.72E+11
5.11E+11
1.76E+12
427 3.743E+11
8.68E+11
5.12E+11
1.75E+12
429 3.75E+11
8.69E+11
5.12E+11
1.76E+12
431 3.752E+11
8.69E+11
5.12E+11
1.76E+12
433 3.781E+11
8.69E+11
5.13E+11
1.76E+12
435 3.801E+11
8.68E+11
5.15E+11
1.76E+12
437 3.971E+11
8.68E+11
5.07E+11
1.77E+12
440 3.966E+11
8.64E+11
5.08E+11
1.79E+12
442 3.788E+11
8.69E+11
5.15E+11
1.76E+12
444 3.739E+11
8.68E+11
5.16E+11
1.76E+12
446 3.748E+11
8.69E+11
5.14E+11
1.76E+12
448 3.756E+11
8.64E+11
5.11E+11
1.75E+12
450 3.728E+11
8.68E+11
5.12E+11
1.75E+12
University of Ghana http://ugspace.ug.edu.gh
121
452 3.727E+11
8.67E+11
5.14E+11
1.75E+12
454 3.73E+11
8.67E+11
5.16E+11
1.76E+12
456 3.755E+11
8.68E+11
5.12E+11
1.76E+12
458 3.776E+11
8.66E+11
5.12E+11
1.76E+12
460 3.798E+11
8.64E+11
5.13E+11
1.76E+12
462 3.968E+11
8.63E+11
5.09E+11
1.77E+12
465 4.056E+11
8.04E+11
4.67E+11
1.68E+12
467 3.845E+11
8.10E+11
4.71E+11
1.67E+12
469 3.765E+11
8.13E+11
4.73E+11
1.66E+12
471 3.723E+11
8.09E+11
4.74E+11
1.66E+12
473 3.785E+11
8.13E+11
4.71E+11
1.66E+12
475 3.869E+11
8.06E+11
4.67E+11
1.66E+12
477 3.870E+11
8.07E+11
4.69E+11
1.66E+12
479 3.756E+11
8.09E+11
4.71E+11
1.66E+12
481 3.732E+11
8.09E+11
4.75E+11
1.66E+12
483 3.764E+11
8.09E+11
4.75E+11
1.66E+12
485 3.879E+11
8.10E+11
4.68E+11
1.67E+12
487 3.956E+11
8.12E+11
4.68E+11
1.68E+12
489 3.877E+11
8.09E+11
4.74E+11
1.67E+12
491 3.948E+11
8.08E+11
4.69E+11
1.67E+12
493 3.916E+11
8.07E+11
4.69E+11
1.67E+12
495 3.873E+11
8.09E+11
4.71E+11
1.67E+12
497 3.955E+11
8.05E+11
4.66E+11
1.67E+12
499 3.844E+11
8.08E+11
4.68E+11
1.66E+12
501 3.752E+11
8.13E+11
4.73E+11
1.66E+12
503 3.735E+11
8.11E+11
4.71E+11
1.66E+12
505 3.770E+11
8.11E+11
4.73E+11
1.66E+12
507 3.884E+11
8.12E+11
4.73E+11
1.67E+12
University of Ghana http://ugspace.ug.edu.gh
122
509 3.918E+11
8.08E+11
4.66E+11
1.67E+12
511 3.799E+11
8.11E+11
4.71E+11
1.66E+12
513 3.743E+11
8.09E+11
4.72E+11
1.66E+12
515 3.746E+11
8.12E+11
4.71E+11
1.66E+12
517 3.834E+11
8.10E+11
4.69E+11
1.66E+12
519 4.026E+11
8.04E+11
4.66E+11
1.67E+12
521 4.021E+11
8.06E+11
4.68E+11
1.68E+12
523 3.822E+11
8.08E+11
4.71E+11
1.66E+12
525 3.769E+11
8.14E+11
4.70E+11
1.66E+12
527 3.768E+11
8.11E+11
4.72E+11
1.66E+12
529 3.877E+11
8.1E+11
4.70E+11
1.67E+12
531 3.88E+11
8.15E+11
4.71E+11
1.67E+12
533 3.722E+11
8.14E+11
4.72E+11
1.66E+12
535 3.711E+11
8.17E+11
4.72E+11
1.66E+12
537 3.732E+11
8.11E+11
4.70E+11
1.65E+12
539 3.828E+11
8.13E+11
4.69E+11
1.67E+12
541 3.896E+11
8.12E+11
4.67E+11
1.67E+12
543 3.73E+11
8.14E+11
4.69E+11
1.66E+12
545 3.697E+11
8.14E+11
4.71E+11
1.66E+12
547 3.791E+11
8.13E+11
4.68E+11
1.66E+12
549 3.973E+11
8.08E+11
4.68E+11
1.67E+12
551 4.103E+11
8.10E+11
4.69E+11
1.69E+12
553 3.864E+11
8.11E+11
4.73E+11
1.67E+12
555 3.733E+11
8.14E+11
4.73E+11
1.66E+12
557 3.734E+11
8.17E+11
4.73E+11
1.66E+12
559 3.753E+11
8.12E+11
4.72E+11
1.66E+12
561 3.902E+11
8.11E+11
4.69E+11
1.67E+12
563 3.817E+11
8.09E+11
4.72E+11
1.66E+12
University of Ghana http://ugspace.ug.edu.gh
123
565 3.725E+11
8.13E+11
4.74E+11
1.66E+12
567 3.727E+11
8.08E+11
4.68E+11
1.65E+12
569 3.738E+11
8.12E+11
4.71E+11
1.66E+12
571 3.859E+11
8.11E+11
4.66E+11
1.66E+12
573 3.875E+11
8.11E+11
4.71E+11
1.67E+12
575 3.813E+11
8.12E+11
4.74E+11
1.67E+12
577 3.88E+11
8.11E+11
4.70E+11
1.67E+12
579 4.045E+11
8.11E+11
4.68E+11
1.68E+12
581 5.505E+11
7.19E+11
3.05E+11
1.57E+12
583 4.256E+11
7.52E+11
4.13E+11
1.59E+12
585 4.052E+11
7.58E+11
4.21E+11
1.58E+12
587 4.007E+11
7.55E+11
4.19E+11
1.57E+12
589 4.021E+11
7.57E+11
4.19E+11
1.58E+12
591 4.2E+11
7.52E+11
4.14E+11
1.59E+12
593 5.416E+11
7.21E+11
3.06E+11
1.57E+12
595 4.190E+11
7.56E+11
4.16E+11
1.59E+12
597 4.05E+11
7.54E+11
4.19E+11
1.58E+12
3598 4.041E+11
7.55E+11
4.23E+11
1.58E+12
600 4.054E+11
7.56E+11
4.19E+11
1.58E+12
602 4.239E+11
7.57E+11
4.16E+11
1.59E+12
604 5.488E+11
7.26E+11
3.04E+11
1.58E+12
606 4.273E+11
7.54E+11
4.16E+11
1.59E+12
608 4.099E+11
7.56E+11
4.19E+11
1.59E+12
610 4.071E+11
7.54E+11
4.23E+11
1.59E+12
612 4.076E+11
7.49E+11
4.19E+11
1.58E+12
614 4.243E+11
7.49E+11
4.16E+11
1.59E+12
616 5.435E+11
7.19E+11
3.04E+11
1.57E+12
3617 4.213E+11
7.51E+11
4.14E+11
1.59E+12
University of Ghana http://ugspace.ug.edu.gh
124
619 4.061E+11
7.57E+11
4.2E+11
1.58E+12
621 4.027E+11
7.56E+11
4.22E+11
1.58E+12
623 4.057E+11
7.56E+11
4.21E+11
1.58E+12
625 4.242E+11
7.57E+11
4.15E+11
1.59E+12
627 5.509E+11
7.25E+11
3.04E+11
1.58E+12
629 4.261E+11
7.55E+11
4.18E+11
1.59E+12
631 4.064E+11
7.54E+11
4.22E+11
1.58E+12
633 4.011E+11
7.55E+11
4.22E+11
1.58E+12
635 4.074E+11
7.56E+11
4.19E+11
1.58E+12
3636 4.217E+11
7.50E+11
4.15E+11
1.59E+12
638 5.457E+11
7.19E+11
3.06E+11
1.57E+12
640 4.205E+11
7.46E+11
4.13E+11
1.58E+12
642 4.02E+11
7.54E+11
4.18E+11
1.57E+12
644 3.988E+11
7.58E+11
4.20E+11
1.58E+12
646 4.004E+11
7.55E+11
4.20E+11
1.58E+12
648 4.191E+11
7.57E+11
4.16E+11
1.59E+12
650 5.407E+11
7.26E+11
3.06E+11
1.57E+12
652 4.212E+11
7.57E+11
4.18E+11
1.59E+12
654 4.026E+11
7.57E+11
4.22E+11
1.58E+12
3655 4.017E+11
7.59E+11
4.21E+11
1.58E+12
657 4.03E+11
7.58E+11
4.22E+11
1.58E+12
659 4.197E+11
7.57E+11
4.15E+11
1.59E+12
661 5.361E+11
7.26E+11
3.05E+11
1.57E+12
663 4.161E+11
7.54E+11
4.17E+11
1.59E+12
665 3.996E+11
7.59E+11
4.19E+11
1.58E+12
667 3.969E+11
7.57E+11
4.18E+11
1.57E+12
669 4.035E+11
7.56E+11
4.21E+11
1.58E+12
671 4.235E+11
7.57E+11
4.17E+11
1.59E+12
University of Ghana http://ugspace.ug.edu.gh
125
673 5.448E+11
7.28E+11
3.07E+11
1.58E+12
3674 4.215E+11
7.55E+11
4.16E+11
1.59E+12
676 4.037E+11
7.59E+11
4.23E+11
1.59E+12
678 3.992E+11
7.60E+11
4.23E+11
1.58E+12
680 4.007E+11
7.57E+11
4.22E+11
1.58E+12
682 4.180E+11
7.54E+11
4.17E+11
1.59E+12
684 5.412E+11
7.27E+11
3.08E+11
1.58E+12
686 4.176E+11
7.55E+11
4.15E+11
1.59E+12
688 4.0E+11
7.58E+11
4.22E+11
1.58E+12
690 3.971E+11
7.58E+11
4.19E+11
1.57E+12
692 4.005E+11
7.56E+11
4.21E+11
1.58E+12
3693 4.177E+11
7.53E+11
4.14E+11
1.58E+12
695 5.421E+11
7.24E+11
3.05E+11
1.57E+12
697 4.220E+11
7.54E+11
4.15E+11
1.59E+12
699 4.071E+11
7.54E+11
4.21E+11
1.58E+12
701 4.074E+11 7.57E+11 4.19E+11 1.58E+12
University of Ghana http://ugspace.ug.edu.gh
126
APPENDIX II: Data from Stopping Range of Ion in Matter (SRIM) on Target
Width
APPENDIX II (a): SRIM Outputs\Cobalt in Cr-Fe- C-Mn- W- V-Ta.
Ion = Cobalt [27], Mass = 58.93 amu
Target Density = 7.9331E+00 g/cm3 = 8.3775E+22 atoms/cm3
Target Composition
Atom Atom Atomic Mass
Name Number Percent Percent
Cr 24 008.90 008.12
Fe 26 089.08 087.24
C 6 000.11 000.02
Mn 25 000.47 000.45
W 74 001.10 003.55
V 23 000.20 000.18
Ta 73 000.14 000.44
Ion dE/dx dE/dx Projected Longitudinal Lateral
Energy Elec. Nuclear Range Straggling Straggling
-------------- ---------- ---------- ---------- ------- ----------
10.00 keV 1.169E-01 2.108E+00 51 A 30 A 22 A
11.00 keV 1.226E-01 2.156E+00 55 A 32 A 23 A
12.00 keV 1.281E-01 2.198E+00 58 A 34 A 24 A
13.00 keV 1.333E-01 2.236E+00 61 A 35 A 26 A
14.00 keV 1.383E-01 2.271E+00 65 A 37 A 27 A
15.00 keV 1.432E-01 2.302E+00 68 A 39 A 28 A
16.00 keV 1.479E-01 2.331E+00 71 A 40 A 29 A
17.00 keV 1.524E-01 2.357E+00 74 A 42 A 30 A
18.00 keV 1.568E-01 2.381E+00 78 A 44 A 32 A
20.00 keV 1.653E-01 2.423E+00 84 A 47 A 34 A
22.50 keV 1.753E-01 2.467E+00 92 A 50 A 36 A
25.00 keV 1.848E-01 2.504E+00 99 A 54 A 39 A
27.50 keV 1.938E-01 2.535E+00 107 A 58 A 42 A
30.00 keV 2.025E-01 2.560E+00 115 A 61 A 44 A
32.50 keV 2.107E-01 2.581E+00 122 A 65 A 47 A
35.00 keV 2.187E-01 2.599E+00 130 A 68 A 49 A
37.50 keV 2.264E-01 2.614E+00 137 A 72 A 52 A
40.00 keV 2.338E-01 2.626E+00 144 A 75 A 54 A
45.00 keV 2.480E-01 2.645E+00 159 A 82 A 59 A
50.00 keV 2.614E-01 2.657E+00 174 A 89 A 64 A
55.00 keV 2.741E-01 2.664E+00 189 A 96 A 68 A
University of Ghana http://ugspace.ug.edu.gh
127
60.00 keV 2.863E-01 2.667E+00 204 A 102 A 73 A
65.00 keV 2.980E-01 2.666E+00 218 A 109 A 77 A
70.00 keV 3.093E-01 2.663E+00 233 A 116 A 82 A
80.00 keV 3.306E-01 2.652E+00 263 A 129 A 91 A
90.00 keV 3.507E-01 2.635E+00 293 A 142 A 99 A
100.00 keV 3.696E-01 2.614E+00 323 A 154 A 108 A
110.00 keV 3.877E-01 2.591E+00 354 A 167 A 117 A
120.00 keV 4.098E-01 2.566E+00 384 A 180 A 125 A
130.00 keV 4.394E-01 2.540E+00 415 A 193 A 134 A
140.00 keV 4.575E-01 2.514E+00 446 A 205 A 142 A
150.00 keV 4.687E-01 2.487E+00 478 A 218 A 151 A
160.00 keV 4.762E-01 2.460E+00 509 A 231 A 160 A
170.00 keV 4.828E-01 2.433E+00 541 A 244 A 168 A
180.00 keV 4.905E-01 2.406E+00 574 A 256 A 177 A
200.00 keV 5.115E-01 2.353E+00 640 A 282 A 194 A
225.00 keV 5.456E-01 2.290E+00 723 A 314 A 215 A
250.00 keV 5.838E-01 2.229E+00 808 A 346 A 237 A
275.00 keV 6.243E-01 2.171E+00 895 A 377 A 260 A
300.00 keV 6.665E-01 2.116E+00 982 A 408 A 282 A
325.00 keV 7.102E-01 2.065E+00 1070 A 439 A 305 A
350.00 keV 7.549E-01 2.015E+00 1158 A 469 A 327 A
375.00 keV 8.003E-01 1.969E+00 1247 A 499 A 350 A
400.00 keV 8.460E-01 1.924E+00 1337 A 528 A 373 A
450.00 keV 9.370E-01 1.842E+00 1516 A 585 A 419 A
500.00 keV 1.026E+00 1.768E+00 1696 A 640 A 465 A
550.00 keV 1.111E+00 1.700E+00 1877 A 693 A 510 A
600.00 keV 1.193E+00 1.639E+00 2057 A 745 A 555 A
650.00 keV 1.271E+00 1.582E+00 2238 A 794 A 600 A
700.00 keV 1.346E+00 1.530E+00 2417 A 842 A 644 A
800.00 keV 1.484E+00 1.436E+00 2776 A 934 A 729 A
900.00 keV 1.611E+00 1.356E+00 3133 A 1021 A 813 A
1.00 MeV 1.730E+00 1.285E+00 3488 A 1102 A 893 A
1.10 MeV 1.842E+00 1.222E+00 3839 A 1179 A 971 A
1.20 MeV 1.950E+00 1.166E+00 4188 A 1251 A 1046 A
1.30 MeV 2.055E+00 1.116E+00 4532 A 1320 A 1119 A
1.40 MeV 2.157E+00 1.071E+00 4873 A 1385 A 1190 A
1.50 MeV 2.259E+00 1.030E+00 5210 A 1447 A 1258 A
1.60 MeV 2.360E+00 9.919E-01 5542 A 1505 A 1324 A
1.70 MeV 2.462E+00 9.572E-01 5869 A 1561 A 1388 A
1.80 MeV 2.563E+00 9.253E-01 6192 A 1613 A 1449 A
2.00 MeV 2.768E+00 8.683E-01 6821 A 1711 A 1566 A
2.25 MeV 3.028E+00 8.076E-01 7577 A 1819 A 1702 A
2.50 MeV 3.292E+00 7.558E-01 8300 A 1915 A 1825 A
2.75 MeV 3.561E+00 7.111E-01 8989 A 1998 A 1939 A
3.00 MeV 3.834E+00 6.721E-01 9647 A 2072 A 2042 A
3.25 MeV 4.109E+00 6.377E-01 1.03 um 2138 A 2137 A
3.50 MeV 4.385E+00 6.071E-01 1.09 um 2196 A 2224 A
University of Ghana http://ugspace.ug.edu.gh
128
3.75 MeV 4.661E+00 5.796E-01 1.14 um 2248 A 2304 A
4.00 MeV 4.937E+00 5.549E-01 1.20 um 2295 A 2378 A
4.50 MeV 5.483E+00 5.119E-01 1.30 um 2377 A 2509 A
5.00 MeV 6.019E+00 4.758E-01 1.40 um 2444 A 2622 A
5.50 MeV 6.540E+00 4.450E-01 1.49 um 2501 A 2721 A
6.00 MeV 7.045E+00 4.183E-01 1.57 um 2549 A 2807 A
6.50 MeV 7.533E+00 3.951E-01 1.65 um 2590 A 2884 A
7.00 MeV 8.003E+00 3.745E-01 1.72 um 2626 A 2952 A
8.00 MeV 8.888E+00 3.398E-01 1.86 um 2689 A 3069 A
9.00 MeV 9.703E+00 3.116E-01 1.99 um 2739 A 3167 A
10.00 MeV 1.045E+01 2.881E-01 2.11 um 2781 A 3249 A
APPENDIX II (b): SRIM Outputs\He in Zr-Sn-Fe-Cr-O.
Ion = Helium [2], Mass = 4.003 amu
Target Density = 6.4980E+00 g/cm3 = 4.2824E+22 atoms/cm3
Atom Atom Atomic Mass
Name Number Percent Percent
Zr 40 097.97 097.80
Sn 50 001.50 001.95
Fe 26 000.24 000.15
Cr 24 000.13 000.07
O 8 000.16 000.03
Ion dE/dx dE/dx Projected Longitudinal Lateral
Energy Elec. Nuclear Range Straggling Straggling
--------- ------- ---------- ---------- ------------- -----------
10.00 keV 1.103E-01 1.643E-02 506 A 505 A 389 A
11.00 keV 1.164E-01 1.606E-02 554 A 536 A 415 A
12.00 keV 1.224E-01 1.570E-02 602 A 566 A 440 A
13.00 keV 1.281E-01 1.536E-02 650 A 595 A 465 A
14.00 keV 1.337E-01 1.503E-02 697 A 622 A 488 A
15.00 keV 1.391E-01 1.472E-02 744 A 649 A 511 A
16.00 keV 1.444E-01 1.442E-02 791 A 674 A 533 A
17.00 keV 1.495E-01 1.413E-02 837 A 698 A 554 A
18.00 keV 1.545E-01 1.386E-02 884 A 722 A 575 A
20.00 keV 1.641E-01 1.334E-02 976 A 766 A 614 A
22.50 keV 1.756E-01 1.276E-02 1089 A 817 A 661 A
25.00 keV 1.866E-01 1.223E-02 1201 A 864 A 705 A
University of Ghana http://ugspace.ug.edu.gh
129
27.50 keV 1.971E-01 1.175E-02 1310 A 908 A 747 A
30.00 keV 2.073E-01 1.131E-02 1419 A 949 A 786 A
32.50 keV 2.170E-01 1.091E-02 1525 A 987 A 824 A
35.00 keV 2.264E-01 1.055E-02 1630 A 1022 A 859 A
37.50 keV 2.356E-01 1.021E-02 1733 A 1056 A 893 A
40.00 keV 2.445E-01 9.892E-03 1835 A 1088 A 926 A
45.00 keV 2.615E-01 9.326E-03 2034 A 1145 A 987 A
50.00 keV 2.777E-01 8.831E-03 2228 A 1198 A 1043A
55.00 keV 2.932E-01 8.394E-03 2417 A 1245 A 1096A
60.00 keV 3.080E-01 8.005E-03 2601 A 1288A 1144A
65.00 keV 3.222E-01 7.656E-03 2780 A 1328 A 1190A
70.00 keV 3.359E-01 7.341E-03 2955 A 1364A 1233A
80.00 keV 3.620E-01 6.793E-03 3294 A 1429A 1313A
90.00 keV 3.864E-01 6.332E-03 3619 A 1486A 1384A
100.00 keV 4.093E-01 5.938E-03 3931 A 1536A 1448A
110.00 keV 4.310E-01 5.597E-03 4233 A 1580A 1508A
120.00 keV 4.516E-01 5.298E-03 4525 A 1620A 1562A
130.00 keV 4.711E-01 5.033E-03 4808 A 1656A 1613A
140.00 keV 4.897E-01 4.797E-03 5083 A 1689A 1660A
150.00 keV 5.074E-01 4.585E-03 5350 A 1719A 1704A
160.00 keV 5.243E-01 4.394E-03 5611 A 1746A 1745A
170.00 keV 5.404E-01 4.219E-03 5867 A 1772A 1784A
180.00 keV 5.558E-01 4.060E-03 6116 A 1796A 1821A
200.00 keV 5.846E-01 3.779E-03 6600 A 1839A 1889A
225.00 keV 6.172E-01 3.484E-03 7181 A 1887A 1966A
250.00 keV 6.463E-01 3.237E-03 7740 A 1928A 2035A
275.00 keV 6.725E-01 3.026E-03 8279 A 1965A 2097A
300.00 keV 6.959E-01 2.844E-03 8803 A 1997A 2155A
325.00 keV 7.168E-01 2.685E-03 9313 A 2027A 2209A
350.00 keV 7.355E-01 2.545E-03 9812 A 2054A 2259A
375.00 keV 7.520E-01 2.420E-03 1.03 um 2079A 2306A
400.00 keV 7.667E-01 2.308E-03 1.08 um 2103A 2350A
450.00 keV 7.909E-01 2.115E-03 1.17 um 2146A 2432A
500.00 keV 8.092E-01 1.955E-03 1.26 um 2186A 2508A
550.00 keV 8.224E-01 1.820E-03 1.36 um 2221A 2578A
600.00 keV 8.315E-01 1.703E-03 1.45 um 2255A 2645A
650.00 keV 8.370E-01 1.603E-03 1.53 um 2285A 2707A
700.00 keV 8.395E-01 1.514E-03 1.62 um 2315A 2768A
800.00 keV 8.375E-01 1.366E-03 1.80 um 2375A 2883A
900.00 keV 8.285E-01 1.246E-03 1.98 um 2433A 2992A
1.00 MeV 8.149E-01 1.147E-03 2.17 um 2488A 3098A
1.10 MeV 7.981E-01 1.064E-03 2.35 um 2541A 3202A
1.20 MeV 7.795E-01 9.935E-04 2.55 um 2594A 3305A
1.30 MeV 7.599E-01 9.322E-04 2.74 um 2647A 3408A
1.40 MeV 7.399E-01 8.786E-04 2.94 um 2700A 3512A
1.50 MeV 7.198E-01 8.314E-04 3.15 um 2754A 3616A
1.60 MeV 7.001E-01 7.893E-04 3.37 um 2808A 3723A
University of Ghana http://ugspace.ug.edu.gh
130
1.70 MeV 6.808E-01 7.517E-04 3.59 um 2864A 3831A
1.80 MeV 6.622E-01 7.177E-04 3.81 um 2920A 3941A
2.00 MeV 6.270E-01 6.589E-04 4.29 um 3073A 4168A
2.25 MeV 5.873E-01 5.986E-04 4.91 um 3296A 4467A
2.50 MeV 5.522E-01 5.491E-04 5.58 um 3531A 4784A
2.75 MeV 5.213E-01 5.077E-04 6.29 um 3776A 5120A
3.00 MeV 4.940E-01 4.725E-04 7.04 um 4033A 5476A
3.25 MeV 4.700E-01 4.422E-04 7.83 um 4299A 5851A
3.50 MeV 4.488E-01 4.158E-04 8.67 um 4576A 6245A
3.75 MeV 4.299E-01 3.926E-04 9.53 um 4861A 6658A
4.00 MeV 4.130E-01 3.720E-04 10.44 um 5155A 7089A
4.50 MeV 3.843E-01 3.371E-04 12.36 um 6070A 8004A
5.00 MeV 3.608E-01 3.086E-04 14.41 um 6981A 8983A
5.50 MeV 3.412E-01 2.848E-04 16.59 um 7890A 1.00um
6.00 MeV 3.246E-01 2.646E-04 18.88 um 8801A 1.11 um
6.50 MeV 3.104E-01 2.473E-04 21.29 um 9714A 1.23 um
7.00 MeV 2.981E-01 2.322E-04 23.81 um 1.06 um 1.34 um
8.00 MeV 2.777E-01 2.073E-04 29.12 um 1.35 um 1.59 um
9.00 MeV 2.601E-01 1.875E-04 34.82 um 1.63 um 1.86 um
10.00 MeV 2.436E-01 1.713E-04 40.90 um 1.89 um 2.13 um
University of Ghana http://ugspace.ug.edu.gh