X-Ray and Neutron Reflectivity: Principles and Applications

Lecture Notes in Physics: Monographs 58 ......Page 1
X-Ray and Neutron Reflectivity: Principles and Applications ......Page 3
Foreword ......Page 5
Contents ......Page 7
List of authors ......Page 15
Acknowledgements ......Page 17
Introduction ......Page 19
1.1 Introduction ......Page 25
1.2.1 Conventions ......Page 26
1.2.2 Wave Equation ......Page 27
1.2.3 Intensity, Current and Flux ......Page 29
1.2.4 Scattering Length and Cross-Sections ......Page 31
1.2.5 The Use of Green Functions ......Page 33
1.2.6 Green Functions: the Case of the Electromagnetic Field ......Page 35
1.3.1 Introduction ......Page 38
1.3.2 The Optical Theorem and its Extensions ......Page 39
1.3.3 The Extinction Length and the Born Approximation ......Page 45
1.3.4 When the Interaction Becomes Stronger ......Page 46
1.4.1 General Considerations ......Page 48
1.4.2 Classical Description: Thomson Scattering by a Free Electron ......Page 49
1.4.3 Classical Description: Thomson Scattering by the Electrons of an Atom, Rayleigh Scattering ......Page 50
1.4.4 Quantum Description: a General Expression for Scattering and Absorption ......Page 53
1.4.5 Quantum Description: Elastic and Compton Scattering ......Page 56
1.4.6 Resonances: Absorption, Photoelectric Effect ......Page 60
1.4.7 Resonances: Dispersion and Anomalous Scattering ......Page 63
1.4.8 Resonances: Dispersion Relations ......Page 65
1.5.1 Introduction ......Page 70
1.5.2 Non Resonant Magnetic Scattering ......Page 73
1.5.3 Resonant Magnetic Scattering ......Page 74
1.5.5 The Effect of an Anisotropy in the Index of Refraction ......Page 76
1.A Appendix: the Born Approximation ......Page 78
References ......Page 81
2.1 Introduction ......Page 82
2.2.2 Height Probability Distributions ......Page 83
2.2.3 Homogeneity and Ergodicity ......Page 85
2.2.4 The Gaussian Probability Distribution and Various Correlation Functions ......Page 87
2.2.5 More Complicated Geometries: Multilayers and Volume Inhomogeneities ......Page 89
2.3 Description of a Surface Scattering Experiment, Coherence Domains ......Page 90
2.3.1 Scattering Geometry ......Page 91
2.3.2 Scattering Cross-Section ......Page 92
2.3.3 Coherence Domains ......Page 94
2.4.1 To What Extent is a Statistical Formulation of the Diffraction Problem Relevant? ......Page 96
2.4.2 Notions on Coherent (Specular) and Incoherent (Diffuse) Intensity ......Page 101
2.5.1 The Differential Scattering Cross-Section ......Page 102
2.5.3 Self-Affine Rough Surfaces ......Page 105
References. ......Page 107
3.1.1 Basic Concepts ......Page 109
3.1.2 Fresnel Reflectivity ......Page 110
3.1.4 The Penetration Depth ......Page 118
3.2.1 The Matrix Method ......Page 121
3.2.2 The Refraction Matrix for X-Ray Radiation ......Page 125
3.2.3 Reflection from a Flat Homogeneous Material ......Page 126
3.2.4 A Single Layer on a Substrate ......Page 127
3.2.5 Two Layers on a Substrate ......Page 128
3.3 From Dynamical to Kinematical Theory ......Page 130
3.4 Influence of the Roughness on the Matrix Coefficients ......Page 135
3.A.1 Second-Order Rayleigh Calculation for a Sinusoidal Grating ......Page 138
3.A.3 Simple Derivation of the Debye-Waller and, Croce-Névot Factors ......Page 140
References ......Page 142
4 Diffuse Scattering ......Page 143
4.1.1 Propagation Equation ......Page 144
4.1.2 Integral Equation ......Page 145
4.1.3 Derivation of the Green Functions Using the Reciprocity Theorem ......Page 147
4.1.4 Green Function in a Vacuum ......Page 148
4.1.5 Green Function for a Stratified Medium ......Page 149
4.1.6 Differential Scattering Cross-Section ......Page 150
4.2.1 Expression of the Differential Scattering Cross-Section ......Page 152
4.2.2 Example: Scattering by a Single Rough Surface ......Page 153
4.3 Distorted-Wave Born Approximation ......Page 154
4.3.1 Case of a Single Rough Surface ......Page 155
4.3.2 General Case of a Stratified Medium ......Page 156
4.3.3 Particular Case of a Film ......Page 159
4.4 Polarisation Effects ......Page 162
4.5.1 Density Inhornogeneities in a Multilayer ......Page 163
4.5.2 Density Fluctuations at a Liquid Surface ......Page 164
4.6 Further Approximations ......Page 165
4.7.1 Expression of the Scattered Intensity ......Page 166
4.7.2 Wave-Vector Resolution Function ......Page 168
4.8 Reflectivity Revisited ......Page 170
4.A Appendix: the Reciprocity Theorem ......Page 172
4.B Appendix: Verification of the Integral Equation in the Case of the Reflection by a Thin Film on a Substrate ......Page 174
4.C Appendix: Interface Roughness in a Multilayer Within the Born Approximation ......Page 175
4.D.1 Formal Theory ......Page 177
4.D.2 Formal Kinematical Treatment by First Order Born Approximation ......Page 179
4.D.3 Formal Treatment by a Distorted Wave Born Approximation ......Page 180
References ......Page 183
5.1 Introduction ......Page 185
5.2.2 Neutron-Matter Interaction ......Page 187
5.3.1 Neutron Optical Indices ......Page 190
5.3.2 Critical Angle for Total External Reflection ......Page 191
5.3.4 Reflection on a Homogeneous Medium , ......Page 192
5.4.1 Interaction of the Neutron with an Infinite Homogeneous Layer ......Page 194
5.4.2 Solution of the Schr8dinger Equation ......Page 196
5.4.3 General Solution ......Page 197
5.4.4 Continuity Conditions and Matrices ......Page 198
5.4.5 Reflection on a Magnetic Dioptre ......Page 201
5.5.1 Interface Roughness ......Page 205
5.5.2 Angular Resolution ......Page 207
5.6.1 Introduction ......Page 208
5.6.2 Time of Flight Reflectometers ......Page 209
5.7 Polymer Examples ......Page 210
5.8.1 Absolute Measurement of a Magnetic Moment ......Page 211
5.8.2 Bragg Peaks of Multilayers ......Page 212
5.8.3 Measurement of the In-Plane and Out-Of-Plane Rotation of Moments. Measurement of the Moment Variation in a Single Layer ......Page 213
5.8.4 Selective Hysteresis Loops ......Page 215
References ......Page 216
6.1.1 Surface Free Energy ......Page 221
6.1.2 Step Free Energy ......Page 222
6.1.4 Surface Stiffness ......Page 223
6.1.5 Surface Chemical Potential ......Page 224
6.2.1 Adatoms, Steps and Thermal Roughness of a Surface ......Page 226
6.2.3 Smooth and Rough Surfaces ......Page 228
6.2.4 Diffraction from a Rough Surface ......Page 230
6.3.2 Supersaturation and Vapour Deposition ......Page 232
6.3.3 Nucleation on a High Symmetry Substrate ......Page 233
6.3.4 Kink-Limited Growth Kinetics ......Page 234
6.3.5 Scaling ......Page 235
6.3.6 Surface-Diffusion-Limited Growth Kinetics ......Page 237
References ......Page 238
7.1.1 Reflectivity Experiments ......Page 239
7.2.1 Co/Glass - Self-Affine Gaussian Roughness ......Page 245
7.2.2 Si Hornoepitaxy on Misoriented Si Substrate. Structured Roughness ......Page 248
7.3 Conclusion ......Page 251
References ......Page 253
8.1 Introduction ......Page 254
8.2 Description of Rough Multilayers ......Page 256
8.2.3 Correlation Properties of Different Interfaces ......Page 257
8.3 Setup of X-Ray Reflectivity Experiments ......Page 259
8.3.1 Experimental Setup ......Page 260
8.3.2 Experimental Scans ......Page 261
8.4.1 Roughness with a Gaussian Interface Distribution Function ......Page 263
8.4.2 Stepped Surfaces ......Page 270
8.4.3 Reflection by \"Virtual Interfaces\" Between Porous Layers ......Page 271
8.5 Non-Specular X-Ray Reflection ......Page 272
8.5.1 Interfaces with a Gaussian Roughness Profile ......Page 273
8.5.2 The Main Scattering Features of Non-Specular Reflection by Rough, Multilayers ......Page 276
8.5.3 Stepped Surfaces and Interfaces ......Page 281
8.5.4 Non-Coplanar NSXR ......Page 284
8.6 Interface Roughness in Surface Sensitive Diffraction Methods ......Page 286
8.7 X-Ray Reflection from Multilayer Gratings ......Page 288
8.7.1 Theoretical Treatments ......Page 289
8.7.2 Discussion ......Page 294
8.7.3 Reflectivity from Rough Multilayer Gratings ......Page 295
Acknowledgments ......Page 296
8.A.1 Reflection from Planar Surfaces and Interfaces ......Page 297
8.A.2 Periodic Multilayer ......Page 299
8.A.3 Reciprocal Space Representation of DWBA ......Page 300
References ......Page 301
9.1 Statistical Description of Liquid Surfaces ......Page 303
9.1.1 Capillary Waves ......Page 304
9.1.2 Relation to Self-Affine Surfaces ......Page 307
9.1.3 Bending Rigidity ......Page 308
9.2.1 Specific Experimental Difficulties ......Page 309
9.2.2 Reflectivity ......Page 310
9.2.3 Diffuse Scattering ......Page 312
9.3 Some Examples ......Page 316
9.3.1 Simple Liquids Free Surface ......Page 317
9.3.2 Liquid Metals ......Page 318
9.3.3 Surfactant Monolayers ......Page 319
9.4 Liquid-liquid Interfaces ......Page 322
References ......Page 325
10.1 Introduction ......Page 327
10.2 Thin Polimer Films ......Page 328
10.3 Polymer Bilayer Systems ......Page 332
10.4 Adsorbed Polymer Layers ......Page 336
10.5 Polymer Brushes ......Page 337
10.6 Polymer-Metal Interfaces ......Page 341
10.7 Sperading of Polymers ......Page 342
10.8 Dewetting of Polymers ......Page 343
References ......Page 345
Main Notation Used in This Book ......Page 347
Index ......Page 349