The Modelling of Radiation Damage in Metals Using Ehrenfest Dynamics (eBook)

The Modelling of Radiation Damage in MetalsUsing Ehrenfest Dynamics 3
Supervisor’s Foreword 6
Acknowledgements 8
Contents 9
Part IIntroductory Material 15
1 Introduction 16
1.1…Why Simulate Radiation Damage? 16
1.2…Semi-classical Simulation as a Link in the Multi-scale Chain 19
1.3…How to Read this Thesis 20
References 21
2 A Radiation Damage Cascade 22
2.1…The Early Stages 22
2.1.1 Ion Channelling 22
2.1.2 Sub-cascade Branching 24
2.2…The Displacement Phase 25
2.3…The Thermal Spike 25
References 26
3 The Treatment of Electronic Excitations in Atomistic Simulations of Radiation Damage---A Brief Review 27
3.1…The Theoretical Treatment of Radiation Damage 29
3.2…The Electronic Stopping Regime 30
3.2.1 General Concepts 30
3.2.2 Models of Fast, Light Particle Stopping 33
3.2.2.1 Early Models 33
3.2.2.2 The Bohr Formula 34
3.2.2.3 The Bethe Formula 35
3.2.3 Expanding the Realm of Stopping Power Theory 35
3.2.4 Models of Fast, Heavy Particle Stopping 39
3.2.4.1 The Effective Charge of the Projectile 40
3.2.4.2 Non-Perturbative Models of Heavy Ion Stopping 41
3.2.4.3 Empirical Models of Stopping Power 45
3.2.5 Models of Slow, Heavy Particle Stopping 47
3.2.5.1 Binary Models of Slow Particle Stopping 47
3.2.5.2 Electron Gas models of Slow Particle Stopping 48
3.2.5.3 Non-Linear Calculations of Electron Gas Stopping 49
3.2.6 The Gaps in Stopping Power Theory 51
3.3…The Electron--Phonon Coupling Regime 54
3.3.1 The Importance of Electron--Phonon Coupling in Radiation Damage 54
3.3.2 Two-Temperature Models 56
3.3.3 Representing the Electron--Phonon Coupling 57
3.3.4 Models of Electron--Phonon Coupling 58
3.4…Electronic Effects in Atomistic Models of Radiation Damage 61
3.4.1 The Binary Collision Approximation 61
3.4.2 Molecular Dynamics Models 63
3.4.2.1 Molecular Dynamics with Electronic Drag 63
3.4.2.2 Electrons as a Heat Bath 65
3.5…Improving the Models: Incorporating Electrons Explicitly 69
References 73
4 Theoretical Background 79
4.1…Overview 79
4.2…The Semi-Classical Approximation 81
4.2.1 The Ehrenfest Approximation 82
4.2.2 The Approximations in Ehrenfest Dynamics 86
4.3…The Independent Electron Approximation 87
4.3.1 Density Functional Theory 88
4.3.2 Time-Dependent Density Functional Theory 92
4.4…Tight-Binding Models 93
4.4.1 Ab-Initio Tight-Binding 94
4.4.2 Semi-Empirical Tight-Binding 95
4.4.3 The Harris--Foulkes Functional 95
4.4.4 Towards Semi-Empirical Tight-Binding 97
4.4.5 Self-Consistent Tight-Binding 101
4.5…Time-Dependent Tight-Binding 103
4.5.1 The Description of the System 103
4.5.2 The Evolution of our System 104
4.5.2.1 The Evolution of the Density Matrix 106
4.5.2.2 The Evolution of the Ionic System 107
4.6…Ehrenfest Dynamics 107
4.6.1 Ehrenfest Dynamics versus Surface Hopping 107
4.6.2 Energy Transfer in Ehrenfest Dynamics 110
4.7…Conclusions 111
References 111
Part IISimulating Radiation Damage in Metals 113
5 A Framework for Simulating Radiation Damage in Metals 114
5.1…A Simple Model Metal 114
5.1.1 The Parameters of the Model 116
5.1.2 The Electronic Structure of the Model 117
5.1.3 A Note on the Truncation of the Hopping Integrals 119
5.2…Ehrenfest Dynamics 120
5.3…spICED: Our Simulation Software 121
References 122
6 The Single Oscillating Ion 123
6.1…Simulations of a Single Oscillating Ion 124
6.2…Simulation Results 126
6.2.1 Frequency and Temperature Dependence of Energy Transfer 127
6.2.2 Position and Direction Dependence 127
6.3…Theoretical Analysis of the System 129
6.4…Explaining the Results 133
6.4.1 High Frequency Cut-off 134
6.4.2 Isotropic Damping About Equilibrium Lattice Site 134
6.4.3 Absence of Energy Transfer at Some Frequencies 134
6.4.4 Frequency Independence of beta at High Temperature 138
6.5…Conclusions 141
References 141
7 Semi-classical Simulations of Collision Cascades 142
7.1…The Evolution of a Cascade 142
7.1.1 Thermalization of the Initial Distribution 142
7.1.2 The Evolution of the Ions 144
7.2…The Electronic Subsystem 145
7.2.1 The Evolving Electronic System 148
7.2.1.1 The Non-crossing Theorem 149
7.2.2 Adiabaticity, Non-Adiabaticity and Electronic Excitations 151
7.2.2.1 A Toy Model of an Avoided Crossing 151
7.2.3 Achieving Adiabatic Evolution by Altering the Electron--Ion Mass Ratio 154
7.2.3.1 Some Cascade Simulations at High Ion Mass 156
7.2.4 The Irreversible Energy Transfer 160
7.3…Conclusions 161
References 161
8 The Nature of the Electronic Excitations 162
8.1…Patterns of Excitation in Collision Cascades 162
8.1.1 Fitting a Pseudo-temperature 164
8.1.2 Why do We Obtain Hot Electrons? 167
8.1.3 The Importance of the Result 170
8.1.4 Thermalization or Thermal Excitation? 172
8.2…Electronic Entropy in Ehrenfest Simulations 175
8.2.1 Two Definitions of Electronic Entropy 176
8.2.2 Reconciling the Two Entropies 177
8.2.3 A Thought Experiment 177
8.3…Conclusions 178
References 179
9 The Electronic Forces 180
9.1…Understanding the Electronic Force 181
9.2…The Effect of Electronic Excitations on the ‘Conservative’ Force 184
9.2.1 The Importance of the Reduction in the Attractive Electronic Force 188
9.2.1.1 The Effective Strain Due to Electronic Heating 189
9.2.2 Replacement Collision Sequences 190
9.2.2.1 Does the Non-adiabatic Force Have an Effect on RCS Dynamics? 194
9.3…Conclusions 195
References 196
10 Channelling Ions 197
10.1…Semi-Classical Simulations of Ion Channelling 198
10.1.1 The Simulation Set-Up 198
10.1.2 The Evolution of a Channelling Simulation 199
10.1.3 Challenges in Simulating Ion Channelling 200
10.2…Steady State Charge 201
10.2.1 Results for a Non-Self-Consistent Model 201
10.2.2 A Perturbation Theory Analysis 206
10.2.2.1 A More Detailed Look at the Perturbation Theory Expression 208
10.2.2.2 A Toy Model 211
10.2.3 The Effect of Channelling Direction 214
10.2.4 The Effect of Charge Self-Consistency Parameters U and V 215
10.3…Electronic Stopping Power for a Channelling Ion 218
10.3.1 Results 218
10.3.2 The Origin of the Stopping Power: A Tight-Binding Perspective 220
10.3.2.1 Bond-Orders in Channelling Simulations 221
10.3.3 The ‘Knee’ in the Stopping Power for U = V = 0 222
10.3.4 Effect of Onsite Charge Self-Consistency 225
10.3.4.1 A U-Dependent Mechanism for Suppressing the Bond-Orders 226
10.4…Conclusions 229
References 230
11 The Electronic Drag Force 231
11.1…Is a Simple Drag Model Good Enough? 232
11.1.1 An Investigation of Damping Models for Total Energy Loss in Collision Cascades 232
11.2…The Microscopic Behaviour of the Non-Adiabatic Force 235
11.2.1 The Non-Adiabatic Force in Ehrenfest Dynamics 235
11.2.2 The Character of the Non-Adiabatic Force 237
11.3…An Improved Model of the Non-Adiabatic Force 238
11.3.1 A ‘‘Non-Adiabatic Bond Model’’ 239
11.3.2 The Performance of Our Proposed Model 243
11.3.2.1 The Irreversible Energy Transfer 243
11.3.2.2 The Non-Adiabatic Force 244
11.3.2.3 Model Performance at the Cascade Level 247
11.4…Conclusions 249
References 252
12 Concluding Remarks 254
12.1…Our Aims 254
12.2…Our Results 255
12.2.1 The Nature of the Electronic Excitations 255
12.2.2 The Effect of Electronic Excitations on the Conservative Forces 256
12.2.3 Non-Adiabatic Effects on Channelling Ions 257
12.2.4 The Non-Adiabatic Force in Collision Cascades 258
12.3…Possible Directions for Further Research 258
References 260
13 Appendices 261
13.1…Appendix A: Selected proofs 261
13.1.1 Proof of Equation (4.10)-(i) 261
13.1.2 Proof of Equation (4.10)-(ii) 262
13.1.3 Proof of Equation (4.12) 262
13.1.4 Proof of Equation (4.28) 263
13.1.5 Proof of Equation (4.85) 264
13.1.6 Proof of Equation (4.86) 265
13.1.7 Proof of Equation (4.102) 266
13.1.8 Proof of Equation (4.131) 267
13.1.9 Proof of Equation (4.135) 268
13.1.10 Proof of Equation (4.137) 269
13.1.11 Proof of Equation (4.141): The Conservation of Total Energy 270
13.1.12 Proof of Increase of Pseudo-Entropy (8.16) 272
13.1.13 Proof of Equation (9.4) 273
13.1.14 Proof that Im{f4} = 0 274
13.1.15 Proof of Equation (11.9) 275
13.1.16 Proof of Equation (11.12) 276
13.1.17 Proof of Equation (11.18) 276
13.2…Appendix B: Perturbation Theory 278
13.2.1 A Periodic Perturbation 280
13.2.2 The Effect of a Sinusoidal Perturbation on an Electronic System 282
13.2.2.1 The Irreversible Energy Transfer 282
13.2.2.2 Charge Transfer 284
13.2.2.3 First-Order Perturbation Theory Approximations 285
The Time Dependence of the Energy and Charge Transfer 285
Interpreting the Perturbation Theory Expressions 286
Can We Neglect the Off-Diagonal Elements of hat{ 
287 
Some Numerical Results 289
13.2.3 A Quantum Mechanical Oscillator 289
13.3…Appendix C: The Coupling Matrix for a Single Oscillating Ion 292
13.4…Appendix D: Some Features of the Electronic Excitation Spectrum in Collision Cascades 295
13.4.1 Anomalous Excitations Early in the Cascade 295
13.4.2 The Width of the Temperature Fitting Window 296
13.4.3 The Sommerfeld Expression for the Heat Capacity of Our Model 297
13.4.4 Behaviour of the Fitted Temperature Early in the Cascade 300
13.5…Appendix E: The Strain on an Inclusion due to Electronic Heating 300
References 303
Index 304