ISBN: 3-540-66195-6
TITLE: X-Ray and Neutron Reflectivity: Principles and Applications
AUTHOR: Daillant, Jean; Gibaud, Alain
TOC:

Part I. Principles
1 Interaction of X-rays (and Neutrons) with Matter 3
Franois de Bergevin
1.1 Introduction 3
1.2 Generalities and Definitions 4
1.2.1 Conventions 4
1.2.2 Wave Equation 5
1.2.3 Intensity, Current and Flux 7
1.2.4 Scattering Length and Cross-Sections 9
1.2.5 The Use of Green Functions 11
1.2.6 Green Functions: the Case of the Electromagnetic Field 13
1.3 From the Scattering by an Object to the Propagation in a Medium 16
1.3.1 Introduction 16
1.3.2 The Optical Theorem and Its Extensions 17
1.3.3 The Extinction Length and the Born Approximation 23
1.3.4 When the Interaction Becomes Stronger 24
1.4 X-Rays 26
1.4.1 General Considerations 26
1.4.2 Classical Description: Thomson Scattering by a Free Electron 27
1.4.3 Classical Description: Thomson Scattering by the Electrons of an Atom, Rayleigh Scattering 28
1.4.4 Quantum Description: a General Expression for Scattering and Absorption 31
1.4.5 Quantum Description: Elastic and Compton Scattering 34
1.4.6 Resonances: Absorption, Photoelectric Effect 38
1.4.7 Resonances: Dispersion and Anomalous Scattering 41
1.4.8 Resonances: Dispersion Relations 43
1.5 X-Rays: Anisotropic Scattering 48
1.5.1 Introduction 48
1.5.2 Non-resonant Magnetic Scattering 51
1.5.3 Resonant Magnetic Scattering 52
1.5.4 Templeton Anisotropic Scattering 54
1.5.5 The Effect of an Anisotropy in the Index of Refraction 54
1.A Appendix: the Born Approximation
Anne Sentenac, Franois de Bergevin, Jean Daillant, Alain Gibaud and Guillaume Vignaud 56
2 Statistical Aspects of Wave Scattering at Rough Surfaces 60
Anne Sentenac, Jean Daillant
2.1 Introduction 60
2.2 Description of Randomly Rough Surfaces 61
2.2.1 Introduction 61
2.2.2 Height Probability Distributions 61
2.2.3 Homogeneity and Ergodicity 63
2.2.4 The Gaussian Probability Distribution and Various Correlation Functions 65
2.2.5 More Complicated Geometries: Multilayers and Volume Inhomogeneities 67
2.3 Description of a Surface Scattering Experiment, Coherence Domains 68
2.3.1 Scattering Geometry 69
2.3.2 Scattering Cross-Section 70
2.3.3 Coherence Domains 72
2.4 Statistical Formulation of the Diffraction Problem 74
2.4.1 To What Extent Is a Statistical Formulation of the Diffraction Problem Relevant? 74
2.4.2 Notions on Coherent (Specular) and Incoherent (Diffuse) Intensity 79
2.5 Statistical Formulation of the Scattered Intensity Under the Born Approximation 80
2.5.1 The Differential Scattering Cross-Section 80
2.5.2 Ideally Flat Surfaces 83
2.5.3 Self-Affine Rough Surfaces 83
3 Specular Reflectivity from Smooth and Rough Surfaces 87
Alain Gibaud
3.1 The Reflected Intensity from an Ideally Flat Surface 87
3.1.1 Basic Concepts 87
3.1.2 Fresnel Reflectivity 88
3.1.3 The Transmission Coefficient 96
3.1.4 The Penetration Depth 96
3.2 X-Ray Reflectivity in Stratified Media 99
3.2.1 The Matrix Method 99
3.2.2 The Refraction Matrix for X-Ray Radiation 103
3.2.3 Reflection from a Flat Homogeneous Material 104
3.2.4 A Single Layer on a Substrate 105
3.2.5 Two Layers on a Substrate 106
3.3 From Dynamical to Kinematical Theory 108
3.4 Influence of the Roughness on the Matrix Coefficients 113
3.A Appendix: The Treatment of Roughness in Specular Reflectivity
Franois de Bergevin, Jean Daillant, Alain Gibaud and Anne Sentenac 116
3.A.1 Second-Order Rayleigh Calculation for a Sinusoidal Grating 116
3.A.2 The Treatment of Roughness in Specular Reflectivity within the DWBA 118
3.A.3 Simple Derivation of the Debye-Waller and Croce-Nevot Factors 118
4 Diffuse Scattering 121
Jean Daillant, Anne Sentenac
4.1 Differential Scattering Cross-Section for X-Rays 122
4.1.1 Propagation Equation 122
4.1.2 Integral Equation 123
4.1.3 Derivation of the Green Functions Using the Reciprocity Theorem 125
4.1.4 Green Function in a Vacuum 126
4.1.5 Green Function for a Stratified Medium 127
4.1.6 Differential Scattering Cross-Section 128
4.2 First Born Approximation 130
4.2.1 Expression of the Differential Scattering Cross-Section 130
4.2.2 Example: Scattering by a Single Rough Surface 131
4.3 Distorted-Wave Born Approximation 132
4.3.1 Case of a Single Rough Surface 133
4.3.2 General Case of a Stratified Medium 134
4.3.3 Particular Case of a Film 137
4.4 Polarisation Effects 140
4.5 Scattering by Density Inhomogeneities 141
4.5.1 Density Inhomogeneities in a Multilayer 141
4.5.2 Density Fluctuations at a Liquid Surface 142
4.6 Further Approximations 143
4.7 The Scattered Intensity 144
4.7.1 Expression of the Scattered Intensity 144
4.7.2 Wave-Vector Resolution Function 146
4.8 Reflectivity Revisited 148
4.A Appendix: the Reciprocity Theorem 150
4.B Appendix: Verification of the Integral Equation in the Case of the Reflection by a Thin Film on a Substrate 152
4.C Appendix: Interface Roughness in a Multilayer Within the Born Approximation 153
4.D Appendix: Quantum Mechanical Approach of Born and Distorted-Wave Born Approximations
Tilo Baumbach and Petr Mikulk 155
4.D.1 Formal Theory 155
4.D.2 Formal Kinematical Treatment by First-order Born Approximation 157
4.D.3 Formal Treatment by a Distorted Wave Born Approximation 158
5 Neutron Reflectometry 163
Claude Fermon, Frdric Ott and Alain Menelle
5.1 Introduction 163
5.2 Schrdinger Equation and Neutron-Matter Interactions 165
5.2.1 Schrdinger Equation 165
5.2.2 Neutron-Matter Interaction 165
5.3 Reflectivity on Non-magnetic Systems 168
5.3.1 Neutron Optical Indices 168
5.3.2 Critical Angle for Total External Reflection 169
5.3.3 Determination of Scattering Lengths and Optical Indices 170
5.3.4 Reflection on a Homogeneous Medium 171
5.4 Neutron Reflectivity on Magnetic Systems 172
5.4.1 Interaction of the Neutron with an Infinite Homogeneous Layer 172
5.4.2 Solution of the Schrdinger Equation 174
5.4.3 General Solution 175
5.4.4 Continuity Conditions and Matrices 176
5.4.5 Reflection on a Magnetic Dioptre 179
5.5 Non-perfect Layers, Practical Problems and Experimental Limits 183
5.5.1 Interface Roughness 183
5.5.2 Angular Resolution 185
5.5.3 Analysis of Experimental Data 186
5.6 The Spectrometers 186
5.6.1 Introduction 186
5.6.2 Time of Flight Reflectometers 187
5.6.3 Monochromatic Reflectometers 187
5.7 Polymer Examples 188
5.8 Examples on Magnetic Systems 189
5.8.1 Absolute Measurement of a Magnetic Moment 189
5.8.2 Bragg Peaks of Multilayers 190
5.8.3 Measurement of the In-Plane and Out-of-Plane Rotation of Moments. Measurement of the Moment Variation in a Single Layer 191
5.8.4 Selective Hysteresis Loops 193
5.9 Conclusion on Neutron Reflectometry 194
Part II. Applications
6 Statistical Physics at Crystal Surfaces 199
Alberto Pimpinelli
6.1 Surface Thermodynamics 199
6.1.1 Surface Free Energy 199
6.1.2 Step Free Energy 200
6.1.3 Singularities of the Surface Tension 201
6.1.4 Surface Stiffness 201
6.1.5 Surface Chemical Potential 202
6.2 Morphology of a Crystal Surface 204
6.2.1 Adatoms, Steps and Thermal Roughness of a Surface 204
6.2.2 The Roughening Transition 206
6.2.3 Smooth and Rough Surfaces 206
6.2.4 Diffraction from a Rough Surface 208
6.2.5 Capillary Waves 210
6.3 Surface Growth and Kinetic Roughening 210
6.3.1 Equilibrium with the Saturated Vapour 210
6.3.2 Supersaturation and Vapour Deposition 210
6.3.3 Nucleation on a High Symmetry Substrate 211
6.3.4 Kink-Limited Growth Kinetics 212
6.3.5 Scaling 213
6.3.6 Surface-Diffusion-Limited Growth Kinetics 215
7 Experiments on Solid Surfaces 217
Jean-Marc Gay and Laurent Lapena
7.1 Experimental Techniques 217
7.1.1 Reflectivity Experiments 217
7.1.2 Roughness Investigations with Other Experimental Tools 223
7.2 Examples of Investigations of Solid Surfaces/Interfaces 223
7.2.1 Co/Glass - Self-Affine Gaussian Roughness 223
7.2.2 Si Homoepitaxy on Misoriented Si Substrate. Structured Roughness 226
7.3 Conclusion 229
8 X-ray Reflectivity by Rough Multilayers 232
Tilo Baumbach and Petr Mikulk
8.1 Introduction 232
8.2 Description of Rough Multilayers 234
8.2.1 Ideal Planar Multilayers 235
8.2.2 Multilayers with Rough Interfaces 235
8.2.3 Correlation Properties of Different Interfaces 235
8.3 Setup of X-Ray Reflectivity Experiments 237
8.3.1 Experimental Setup 238
8.3.2 Experimental Scans 239
8.4 Specular X-Ray Reflection 241
8.4.1 Roughness with a Gaussian Interface Distribution Function 241
8.4.2 Stepped Surfaces 248
8.4.3 Reflection by "Virtual Interfaces" Between Porous Layers 249
8.5 Non-specular X-Ray Reflection 250
8.5.1 Interfaces with a Gaussian Roughness Profile 251
8.5.2 The Main Scattering Features of Non-specular Reflection by Rough Multilayers 254
8.5.3 Stepped Surfaces and Interfaces 259
8.5.4 Non-coplanar NSXR 262
8.6 Interface Roughness in Surface Sensitive Diffraction Methods 264
8.7 X-Ray Reflection from Multilayer Gratings 266
8.7.1 Theoretical Treatments 267
8.7.2 Discussion 272
8.7.3 Reflectivity from Rough Multilayer Gratings 273
8.A Appendix: Reciprocal Space Constructions for Reflectivity 275
8.A.1 Reflection from Planar Surfaces and Interfaces 275
8.A.2 Periodic Multilayer 277
8.A.3 Reciprocal Space Representation of DWBA 278
9 Reflectivity of Liquid Surfaces and Interfaces 281
Jean Daillant
9.1 Statistical Description of Liquid Surfaces 281
9.1.1 Capillary Waves 282
9.1.2 Relation to Self-Affine Surfaces 285
9.1.3 Bending Rigidity 286
9.2 Experimental Measurement of the Reflectivity of Liquid Surfaces 287
9.2.1 Specific Experimental Difficulties 287
9.2.2 Reflectivity 288
9.2.3 Diffuse Scattering 290
9.3 Some Examples 294
9.3.1 Simple Liquids Free Surface 295
9.3.2 Liquid Metals 296
9.3.3 Surfactant Monolayers 297
9.4 Liquid-Liquid Interfaces 300
10 Polymer Studies 305
Gunter Reiter
10.1 Introduction 305
10.2 Thin Polymer Films 306
10.3 Polymer Bilayer Systems 310
10.4 Adsorbed Polymer Layers 314
10.5 Polymer Brushes 315
10.6 Polymer-Metal Interfaces 319
10.7 Spreading of Polymers 320
10.8 Dewetting of Polymers 321
Main Notation Used in This Book 325
Index 327
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