Nonrelativistic Quantum X-Ray Physics

Providing a solid theoretical background in photon-matter interaction, Nonrelativistic Quantum X-Ray Physics enables readers to understand experiments performed at XFEL-facilities and x-ray synchrotrons. As a result, after reading this book, scientists and students will be able to outline and perform calculations of some important x-ray-matter interaction processes. Key features of the contents are that the scope reaches beyond the dipole approximation when necessary and that it includes short-pulse interactions. To aid the reader in this transition, some relevant examples are discussed in detail, while non-relativistic quantum electrodynamics help readers to obtain an in-depth understanding of the formalisms and processes. The text presupposes a basic (undergraduate-level) understanding of mechanics, electrodynamics, and quantum mechanics. However, more specialized concepts in these fields are introduced and the reader is directed to appropriate references. While primarily benefiting users of x-ray light-sources, the material is equally of relevance to researchers in various disciplines, such as life sciences, biology, materials science, physics, and chemistry that plan on applying these new facilities in their respective fields.

Table of Contents:
Preface XIII Part I Introduction 1 1 Introduction 3 1.1 Motivation 3 1.2 Comparing X-Rays with Optical Radiation 3 1.3 Novel X-Ray Sources 5 1.4 Unit Systems 6 1.5 Overview of Lagrangian and Hamiltonian Mechanics 9 1.5.1 Lagrangian Mechanics 9 1.5.2 Hamiltonian Mechanics 10 1.6 Approximations 12 1.6.1 Semiclassical Approximation 12 1.6.2 Dipole Approximation 13 2 Review of Some Concepts in Quantum Mechanics 15 2.1 Introduction 15 2.2 Dirac’s Bra–Ket (Bracket) Notation 15 2.3 Eigenvalues and Eigenfunctions 16 2.4 Functions of Operators 18 2.5 Point Particle in a Radially Symmetric Potential 19 2.5.1 Radial Schrödinger Equation 19 2.5.2 Bound States in a Modified Attractive Coulomb Potential 21 2.5.3 Unbound States in a Coulomb Potential 21 2.5.4 Pure Coulomb Potential 22 2.6 Mixed States 23 2.6.1 Isolated Systems 23 2.6.2 Coupled Systems 25 2.7 Schrödinger and Heisenberg Pictures of Quantum Mechanics 26 2.7.1 Evolution Operator in the Schrödinger Picture 26 2.7.2 Equivalent Pictures of Quantum Mechanics 28 2.7.3 Schrödinger Picture 28 2.7.4 Heisenberg Picture 29 2.8 Representing Quantum Mechanics in Position and Momentum Space 29 2.9 Transition from Classical Mechanics to Quantum Mechanics 31 2.10 Molecular Orbital Approximation 31 2.10.1 Derivation of the Hartree–Fock Equations 32 2.10.2 Interpretation of Orbital Energies 38 2.10.3 Post-Hartree–Fock Methods 40 Part II Quantization of the Free Electromagnetic Field 41 3 Classical Electromagnetic Fields 43 3.1 Introduction 43 3.2 Maxwell’s Equations 43 3.3 Electromagnetic Potentials 44 3.3.1 Field Equations 44 3.3.2 Gauge Transformation 45 3.3.3 Coulomb Gauge 45 3.3.4 Lorenz Gauge 46 3.4 Transverse and Longitudinal Maxwell’s Equations 46 3.4.1 Helmholtz Decomposition of Maxwell’s Equations 47 3.4.2 Decomposition of the Field Equations in the Coulomb Gauge 47 3.5 The Free Electromagnetic Field as a Sum of Mode Oscillators 48 3.5.1 Density of States of the Radiation Field 53 3.5.2 Radiation Cavity in Thermodynamic Equilibrium 54 3.6 Charged Particle in an Electromagnetic Field and the Minimal-Coupling Hamiltonian 56 4 Harmonic Oscillator 59 4.1 Introduction 59 4.2 Classical Harmonic Oscillator with One Degree of Freedom 59 4.3 Quantum Mechanical Harmonic Oscillator 60 4.4 N-Dimensional Quantum Mechanical Harmonic Oscillator 64 5 Quantization of the Electromagnetic Field 67 5.1 Introduction 67 5.2 Transition to a Quantum Mechanical Description 67 5.3 Photon Number States (Fock States) 71 5.4 Photons 73 5.4.1 Photon Momentum and Poynting Vector 73 6 Continuous Fock Space 77 6.1 Introduction 77 6.2 Three-Dimensional Continuum Field 77 6.2.1 Number States in the Continuum Field 80 6.3 One-Dimensional Treatment 84 6.3.1 Intensity 85 6.3.2 Description in the Time Domain 86 7 Coherence 89 7.1 Introduction 89 7.2 Review of Classical Coherence Theory 89 7.2.1 First-Order Coherence 90 7.2.2 Second-Order Coherence 92 7.2.3 Chaotic Light 93 7.3 Quantum Coherence Theory 96 7.3.1 Coincidence Detection Using an Ideal Photon Detector 96 7.3.2 Field Correlations 98 7.3.3 Coherence 101 8 Examples for Electromagnetic States 103 8.1 Introduction 103 8.2 Quantum Phase of Radiation Fields 103 8.2.1 Dirac’s Phase Operator 104 8.2.2 Quantum Sine and Cosine Operators 105 8.2.3 Phase State Projectors 108 8.3 Single-Mode States 109 8.3.1 Pure Single-Mode States 110 8.3.2 Statistical Mixtures of Single-Mode States 112 8.3.3 Coherent States 113 8.4 Multimode States 117 8.4.1 Multimode Fock States 117 8.4.2 Multimode Coherent States 119 8.4.3 Localized Radiation (Wave Packets Describing Localized Photons) 120 8.4.4 Chaotic Light 123 8.5 One-Dimensional Continuum Mode States 124 Part III Interaction of X-Rays with Matter 125 9 Interaction of the Electromagnetic Field with Matter 127 9.1 Introduction 127 9.2 Tensor Product of Matter and Radiation Hilbert Spaces 127 9.3 Interaction Hamiltonian for the Electromagnetic Field and Matter 128 10 Time-Dependent Perturbation Theory 133 10.1 Introduction 133 10.2 Interaction Picture 134 10.2.1 Pure States 134 10.2.2 Mixed States 136 10.3 Transition Probabilities 137 10.3.1 Time Dependence of Perturbations 137 10.3.2 Transition Probabilities 139 10.4 Perturbative Expansion of Transition Amplitudes 141 10.4.1 Transition Amplitude in First Order 144 10.4.2 Transition Amplitude in Second Order 145 10.4.3 Transition Between Discrete States 148 10.4.4 Transition from Discrete to Continuous States 149 10.4.5 Transition Between Continuous States 152 10.4.6 Scattering (̂S) and Transition ( ̂ T) Matrices 153 10.5 Time-Dependent Perturbation Theory for Mixed States 154 10.5.1 Isolated System 154 10.5.2 Coupled Systems 155 11 Application of Perturbation Theory to the Interaction of Electromagnetic Fields with Matter 159 11.1 Introduction 159 11.2 Feynman Diagrams 160 11.3 Mixed States 161 11.3.1 Transition Probabilities 162 Part IV Applications of X-Ray–Matter-Interaction Theory 165 12 X-Ray Scattering by Free Electrons 167 12.1 Introduction 167 12.2 Energy and Momentum Conservation 167 12.2.1 Scattering of Photons by Free Electrons 167 12.2.2 A Free Electron Cannot Absorb a Photon 170 12.3 Scattering Cross Section 171 12.4 Scattering From an Electron at Rest 176 12.4.1 Kinematics 176 12.4.2 Nonrelativistic Scattering Cross Section 177 12.4.3 Polarization 178 12.4.4 Relativistic Klein–Nishima Cross Section 179 12.5 Doppler Effect 179 13 Radiative Atomic Bound–Bound Transitions 183 13.1 Introduction 183 13.2 Emission of Photons 183 13.3 Lifetime and Natural Line Width 187 13.3.1 Weisskopf–Wigner Theory 187 13.3.2 Frequency Spectrum 191 13.3.3 Breit–Wigner Procedure 191 13.4 Absorption of Photons 192 13.5 Einstein’s A and B Coefficients 194 13.6 Radiative Atomic Bound–Bound Transitions in Mixed States 197 14 One-Photon Photoionization 201 14.1 Introduction 201 14.2 Photoionization in a Pure-State Radiation Field 201 14.3 Photoionization in a Mixed-State Radiation Field 204 14.4 Single-Electron Approximation for Photoionization 207 14.5 Photoionization of Hydrogen-Like Atoms 210 14.5.1 Large Photon Energies 211 14.5.2 Small Photon Energies 214 14.5.3 Comparing Small and Large Photon Energies 217 15 Bremsstrahlung 221 15.1 Introduction 221 15.2 Electron–Nucleus Bremsstrahlung 221 15.3 Electron–Positron Bremsstrahlung 225 15.4 Electron–Electron Bremsstrahlung 229 15.4.1 Quadrupole Nature of Bremsstrahlung 229 15.4.2 Indistinguishable Particles 230 15.5 Inverse Bremsstrahlung Absorption 231 16 X-Ray Scattering 235 16.1 Introduction 235 16.2 Steady-State Scattering Formalism 236 16.2.1 Dipole Approximation 241 16.3 Elastic Scattering (Rayleigh Scattering) 241 16.3.1 Elastic Scattering for Large X-Ray Energies 242 16.3.2 Elastic Scattering for Intermediate X-Ray Energies 243 16.4 Raman Scattering 244 16.5 Compton Scattering 246 16.5.1 Nonresonant Compton Scattering 247 16.5.2 Resonant Raman–Compton Scattering 252 16.5.3 Infrared Divergence for Soft Scattered Photon Energies 252 16.6 Single-Electron Approximation for X-Ray Scattering 253 16.7 Short-Pulse Scattering 255 16.7.1 General Formalism 256 16.7.2 Plane-Parallel Light Pulse 260 16.7.3 Coherent Pulses 261 17 Relaxation Processes 265 17.1 Introduction 265 17.2 Auger Decay 266 17.2.1 Eigenstates Due to Coupling of a Discrete Level to a Continuum 266 17.2.2 Autoionization in First-Order Perturbation Theory 269 17.3 X-Ray Fluorescence following Photoionization 271 17.4 Branching Ratio 274 18 Multiphoton Photoionization 277 18.1 Introduction 277 18.2 Above-Threshold Ionization 278 18.3 Sequential Two-Photon Absorption 279 19 Threshold Phenomena 285 19.1 Introduction 285 19.2 One-Step Treatment of Threshold Excitations 286 19.3 Nonradiative Threshold Processes 288 19.3.1 Shake-Modified Resonant Autoionization 289 19.3.2 Post-Collision Interaction 289 References 293 Index 299

About the Author :
Stefan Hau-Riege is the X-ray Science and Technology Group Leader at the Lawrence Livermore National Laboratory (LLNL), where he works on x-ray free-electron-laser interactions with materials, x-ray instrumentation, and ultrafast imaging, drawing on computational and experimental physics. Previously, he worked on extreme-ultraviolet lithography and laser-assisted recrystallization. Dr. Hau-Riege received his Ph.D. in materials science from the MIT in 2000, and a M.S. in solid-state physics and applied mathematics from the University of Hamburg, Germany. He has authored and co-authored more than 100 scientific journal publications, and is co-inventor of more than 20 patents.