Note: Intro -- The Modelling of Radiation Damage in Metals Using Ehrenfest Dynamics -- Supervisor's Foreword -- Acknowledgements -- Contents -- Part I Introductory Material -- 1 Introduction -- 1.1…Why Simulate Radiation Damage? -- 1.2…Semi-classical Simulation as a Link in the Multi-scale Chain -- 1.3…How to Read this Thesis -- References -- 2 A Radiation Damage Cascade -- 2.1…The Early Stages -- 2.1.1 Ion Channelling -- 2.1.2 Sub-cascade Branching -- 2.2…The Displacement Phase -- 2.3…The Thermal Spike -- References -- 3 The Treatment of Electronic Excitations in Atomistic Simulations of Radiation Damage---A Brief Review -- 3.1…The Theoretical Treatment of Radiation Damage -- 3.2…The Electronic Stopping Regime -- 3.2.1 General Concepts -- 3.2.2 Models of Fast, Light Particle Stopping -- 3.2.2.1 Early Models -- 3.2.2.2 The Bohr Formula -- 3.2.2.3 The Bethe Formula -- 3.2.3 Expanding the Realm of Stopping Power Theory -- 3.2.4 Models of Fast, Heavy Particle Stopping -- 3.2.4.1 The Effective Charge of the Projectile -- 3.2.4.2 Non-Perturbative Models of Heavy Ion Stopping -- 3.2.4.3 Empirical Models of Stopping Power -- 3.2.5 Models of Slow, Heavy Particle Stopping -- 3.2.5.1 Binary Models of Slow Particle Stopping -- 3.2.5.2 Electron Gas models of Slow Particle Stopping -- 3.2.5.3 Non-Linear Calculations of Electron Gas Stopping -- 3.2.6 The Gaps in Stopping Power Theory -- 3.3…The Electron--Phonon Coupling Regime -- 3.3.1 The Importance of Electron--Phonon Coupling in Radiation Damage -- 3.3.2 Two-Temperature Models -- 3.3.3 Representing the Electron--Phonon Coupling -- 3.3.4 Models of Electron--Phonon Coupling -- 3.4…Electronic Effects in Atomistic Models of Radiation Damage -- 3.4.1 The Binary Collision Approximation -- 3.4.2 Molecular Dynamics Models -- 3.4.2.1 Molecular Dynamics with Electronic Drag -- 3.4.2.2 Electrons as a Heat Bath. , 3.5…Improving the Models: Incorporating Electrons Explicitly -- References -- 4 Theoretical Background -- 4.1…Overview -- 4.2…The Semi-Classical Approximation -- 4.2.1 The Ehrenfest Approximation -- 4.2.2 The Approximations in Ehrenfest Dynamics -- 4.3…The Independent Electron Approximation -- 4.3.1 Density Functional Theory -- 4.3.2 Time-Dependent Density Functional Theory -- 4.4…Tight-Binding Models -- 4.4.1 Ab-Initio Tight-Binding -- 4.4.2 Semi-Empirical Tight-Binding -- 4.4.3 The Harris--Foulkes Functional -- 4.4.4 Towards Semi-Empirical Tight-Binding -- 4.4.5 Self-Consistent Tight-Binding -- 4.5…Time-Dependent Tight-Binding -- 4.5.1 The Description of the System -- 4.5.2 The Evolution of our System -- 4.5.2.1 The Evolution of the Density Matrix -- 4.5.2.2 The Evolution of the Ionic System -- 4.6…Ehrenfest Dynamics -- 4.6.1 Ehrenfest Dynamics versus Surface Hopping -- 4.6.2 Energy Transfer in Ehrenfest Dynamics -- 4.7…Conclusions -- References -- Part II Simulating Radiation Damage in Metals -- 5 A Framework for Simulating Radiation Damage in Metals -- 5.1…A Simple Model Metal -- 5.1.1 The Parameters of the Model -- 5.1.2 The Electronic Structure of the Model -- 5.1.3 A Note on the Truncation of the Hopping Integrals -- 5.2…Ehrenfest Dynamics -- 5.3…spICED: Our Simulation Software -- References -- 6 The Single Oscillating Ion -- 6.1…Simulations of a Single Oscillating Ion -- 6.2…Simulation Results -- 6.2.1 Frequency and Temperature Dependence of Energy Transfer -- 6.2.2 Position and Direction Dependence -- 6.3…Theoretical Analysis of the System -- 6.4…Explaining the Results -- 6.4.1 High Frequency Cut-off -- 6.4.2 Isotropic Damping About Equilibrium Lattice Site -- 6.4.3 Absence of Energy Transfer at Some Frequencies -- 6.4.4 Frequency Independence of beta at High Temperature -- 6.5…Conclusions -- References. , 7 Semi-classical Simulations of Collision Cascades -- 7.1…The Evolution of a Cascade -- 7.1.1 Thermalization of the Initial Distribution -- 7.1.2 The Evolution of the Ions -- 7.2…The Electronic Subsystem -- 7.2.1 The Evolving Electronic System -- 7.2.1.1 The Non-crossing Theorem -- 7.2.2 Adiabaticity, Non-Adiabaticity and Electronic Excitations -- 7.2.2.1 A Toy Model of an Avoided Crossing -- 7.2.3 Achieving Adiabatic Evolution by Altering the Electron--Ion Mass Ratio -- 7.2.3.1 Some Cascade Simulations at High Ion Mass -- 7.2.4 The Irreversible Energy Transfer -- 7.3…Conclusions -- References -- 8 The Nature of the Electronic Excitations -- 8.1…Patterns of Excitation in Collision Cascades -- 8.1.1 Fitting a Pseudo-temperature -- 8.1.2 Why do We Obtain Hot Electrons? -- 8.1.3 The Importance of the Result -- 8.1.4 Thermalization or Thermal Excitation? -- 8.2…Electronic Entropy in Ehrenfest Simulations -- 8.2.1 Two Definitions of Electronic Entropy -- 8.2.2 Reconciling the Two Entropies -- 8.2.3 A Thought Experiment -- 8.3…Conclusions -- References -- 9 The Electronic Forces -- 9.1…Understanding the Electronic Force -- 9.2…The Effect of Electronic Excitations on the 'Conservative' Force -- 9.2.1 The Importance of the Reduction in the Attractive Electronic Force -- 9.2.1.1 The Effective Strain Due to Electronic Heating -- 9.2.2 Replacement Collision Sequences -- 9.2.2.1 Does the Non-adiabatic Force Have an Effect on RCS Dynamics? -- 9.3…Conclusions -- References -- 10 Channelling Ions -- 10.1…Semi-Classical Simulations of Ion Channelling -- 10.1.1 The Simulation Set-Up -- 10.1.2 The Evolution of a Channelling Simulation -- 10.1.3 Challenges in Simulating Ion Channelling -- 10.2…Steady State Charge -- 10.2.1 Results for a Non-Self-Consistent Model -- 10.2.2 A Perturbation Theory Analysis. , 10.2.2.1 A More Detailed Look at the Perturbation Theory Expression -- 10.2.2.2 A Toy Model -- 10.2.3 The Effect of Channelling Direction -- 10.2.4 The Effect of Charge Self-Consistency Parameters U and V -- 10.3…Electronic Stopping Power for a Channelling Ion -- 10.3.1 Results -- 10.3.2 The Origin of the Stopping Power: A Tight-Binding Perspective -- 10.3.2.1 Bond-Orders in Channelling Simulations -- 10.3.3 The 'Knee' in the Stopping Power for U = V = 0 -- 10.3.4 Effect of Onsite Charge Self-Consistency -- 10.3.4.1 A U-Dependent Mechanism for Suppressing the Bond-Orders -- 10.4…Conclusions -- References -- 11 The Electronic Drag Force -- 11.1…Is a Simple Drag Model Good Enough? -- 11.1.1 An Investigation of Damping Models for Total Energy Loss in Collision Cascades -- 11.2…The Microscopic Behaviour of the Non-Adiabatic Force -- 11.2.1 The Non-Adiabatic Force in Ehrenfest Dynamics -- 11.2.2 The Character of the Non-Adiabatic Force -- 11.3…An Improved Model of the Non-Adiabatic Force -- 11.3.1 A ''Non-Adiabatic Bond Model'' -- 11.3.2 The Performance of Our Proposed Model -- 11.3.2.1 The Irreversible Energy Transfer -- 11.3.2.2 The Non-Adiabatic Force -- 11.3.2.3 Model Performance at the Cascade Level -- 11.4…Conclusions -- References -- 12 Concluding Remarks -- 12.1…Our Aims -- 12.2…Our Results -- 12.2.1 The Nature of the Electronic Excitations -- 12.2.2 The Effect of Electronic Excitations on the Conservative Forces -- 12.2.3 Non-Adiabatic Effects on Channelling Ions -- 12.2.4 The Non-Adiabatic Force in Collision Cascades -- 12.3…Possible Directions for Further Research -- References -- 13 Appendices -- 13.1…Appendix A: Selected proofs -- 13.1.1 Proof of Equation (4.10)-(i) -- 13.1.2 Proof of Equation (4.10)-(ii) -- 13.1.3 Proof of Equation (4.12) -- 13.1.4 Proof of Equation (4.28) -- 13.1.5 Proof of Equation (4.85) -- 13.1.6 Proof of Equation (4.86). , 13.1.7 Proof of Equation (4.102) -- 13.1.8 Proof of Equation (4.131) -- 13.1.9 Proof of Equation (4.135) -- 13.1.10 Proof of Equation (4.137) -- 13.1.11 Proof of Equation (4.141): The Conservation of Total Energy -- 13.1.12 Proof of Increase of Pseudo-Entropy (8.16) -- 13.1.13 Proof of Equation (9.4) -- 13.1.14 Proof that Im{f4} = 0 -- 13.1.15 Proof of Equation (11.9) -- 13.1.16 Proof of Equation (11.12) -- 13.1.17 Proof of Equation (11.18) -- 13.2…Appendix B: Perturbation Theory -- 13.2.1 A Periodic Perturbation -- 13.2.2 The Effect of a Sinusoidal Perturbation on an Electronic System -- 13.2.2.1 The Irreversible Energy Transfer -- 13.2.2.2 Charge Transfer -- 13.2.2.3 First-Order Perturbation Theory Approximations -- The Time Dependence of the Energy and Charge Transfer -- Interpreting the Perturbation Theory Expressions -- Can We Neglect the Off-Diagonal Elements of \hat{\rho} in our Expression for {\Updelta }q_{\alpha}? -- Some Numerical Results -- 13.2.3 A Quantum Mechanical Oscillator -- 13.3…Appendix C: The Coupling Matrix for a Single Oscillating Ion -- 13.4…Appendix D: Some Features of the Electronic Excitation Spectrum in Collision Cascades -- 13.4.1 Anomalous Excitations Early in the Cascade -- 13.4.2 The Width of the Temperature Fitting Window -- 13.4.3 The Sommerfeld Expression for the Heat Capacity of Our Model -- 13.4.4 Behaviour of the Fitted Temperature Early in the Cascade -- 13.5…Appendix E: The Strain on an Inclusion due to Electronic Heating -- References -- Index.