Inorganic Materials Series Preface ............................. xi
Preface ...................................................... xiii
List of Contributors ........................................... xv
1 Powder Diffraction ......................................... 1
Kenneth D.M. Harris and P. Andrew Williams
1.1 Introduction ............................................... 1
1.2 The Similarities and Differences between Single-Crystal
XRD and Powder XRD ......................................... 2
1.3 Qualitative Aspects of Powder XRD: 'Fingerprinting'
of Crystalline Phases ...................................... 6
1.4 Quantitative Aspects of Powder XRD: Some Preliminaries
Relevant to Crystal Structure Determination ................ 8
1.4.1 Relationship between a Crystal Structure and its
Diffraction Pattern ................................. 8
1.4.2 Comparison of Experimental and Calculated Powder
XRD Patterns ....................................... 10
1.5 Structure Determination from Powder XRD Data .............. 12
1.5.1 Overview ........................................... 12
1.5.2 Unit Cell Determination (Indexing) ................. 14
1.5.3 Preparing the Intensity Data for Structure
Solution: Profile Fitting .......................... 15
1.5.4 Structure Solution ................................. 16
1.5.5 Structure Refinement ............................... 21
1.6 Some Experimental Considerations in Powder XRD ............ 22
1.6.1 Synchrotron versus Laboratory Powder XRD Data ...... 22
1.6.2 Preferred Orientation .............................. 24
1.6.3 Phase Purity of the Powder Sample .................. 25
1.6.4 Analysis of Peak Widths in Powder XRD Data ......... 26
1.6.5 Applications of Powder XRD for In Situ Studies
of Structural Transformations and Chemical
Processes .......................................... 28
1.6.2 Powder Neutron Diffraction versus Powder XRD ....... 30
1.8 Validation of Procedures and Results in Structure
Determination from Powder XRD Data ........................ 33
1.8.1 Overview ........................................... 33
1.8.2 Validation before Direct-Space Structure Solution .. 34
1.8.3 Aspects of Validation following Structure
Refinement ......................................... 36
1.9 More Detailed Consideration of the Application of Powder
XRD as a 'Fingerprint' of Crystalline Phases .............. 40
1.10 Examples of the Application of Powder XRD in Chemical
Contexts .................................................. 44
1.10.1 Overview ........................................... 44
1.10.2 Structure Determination of Zeolites and Other
Framework Materials ................................ 45
1.10.3 In Situ Powder XRD Studies of Materials Synthesis .. 47
1.10.4 Structure Determination of New Materials Produced
by Solid-State Mechanochemistry .................... 49
1.10.5 In Situ Powder XRD Studies of Solid-State
Mechanochemical Processes .......................... 52
1.10.6 In Situ Powder XRD Studies of a Polymorphic
Transformation ..................................... 54
1.10.7 In Situ Powder XRD Studies of a Solid-State
Reaction ........................................... 57
1.10.8 Establishing Details of a Hydrogen-Bonding
Arrangement by Powder Neutron Diffraction .......... 57
1.10.9 Structure Determination of a Material Produced
by Rapid Precipitation from Solution ............... 59
1.10.10 Structure Determination of Intermediates in
a Solid-State Reaction ............................. 61
1.10.11 Structure Determination of a Novel Aluminium
Methylphosphonate .................................. 61
1.10.12 Structure Determination of Materials Prepared
by Solid-State Dehydration/Desolvation Processes ... 62
1.10.13 Structure Determination of the Product Material
from a Solid-State Photopolymerisation Reaction .... 65
1.10.14 Exploiting Anisotropic Thermal Expansion
in Structure Determination ......................... 67
1.10.15 Rationalisation of a Solid-State Reaction ......... 68
1.10.16 Structure Determination of Organometallic
Complexes ......................................... 70
1.10.17 Examples of Structure Determination of Some
Polymeric Materials ............................... 71
1.10.18 Structure Determination of Pigment Materials ...... 72
1.11 Concluding Remarks ........................................ 73
References ................................................ 74
2 X-Ray and Neutron Single-Crystal Diffraction .............. 83
William Clegg
2.1 Introduction .............................................. 83
2.2 Solid-State Fundamentals .................................. 86
2.2.1 Translation Symmetry ............................... 87
2.2.2 Other Symmetry ..................................... 91
2.2.3 An Introduction to Non-Ideal Behaviour ............. 98
2.3 Scattering and Diffraction ............................... 101
2.3.1 Fundamentals of Radiation and Scattering .......... 102
2.3.2 Diffraction of Monochromatic X-Rays ............... 103
2.3.3 Diffraction of Polychromatic X-Rays ............... 110
2.3.4 Diffraction of Neutrons ........................... 111
2.3.5 Some Competing and Complicating Effects ........... 114
2.4 Experimental Methods ..................................... 119
2.4.1 Radiation Sources ................................. 119
2.4.2 Single Crystals ................................... 124
2.4.3 Measuring the Diffraction Pattern ................. 126
2.4.4 Correcting for Systematic Errors .................. 127
2.5 Structure Solution ....................................... 128
2.5.1 Direct Methods .................................... 130
2.5.2 Patterson Synthesis ............................... 131
2.5.3 Symmetry Arguments ................................ 132
2.5.4 Charge Flipping ................................... 133
2.5.5 Completing a Partial Structure Model .............. 134
2.6 Structure Refinement ..................................... 138
2.6.1 Minimisation and Weights .......................... 139
2.6.2 Parameters, Constraints and Restraints ............ 139
2.6.3 Refinement Results ................................ 140
2.6.4 Computer Programs for Structure Solution and
Refinement ........................................ 141
2.7 Problem Structures, Special Topics, Validation
and Interpretation ....................................... 142
2.7.1 Disorder .......................................... 142
2.7.2 Twinning .......................................... 143
2.7.3 Pseudosymmetry, Superstructures and
Incommensurate Structures ......................... 145
2.7.4 Absolute Structure ................................ 147
2.7.5 Distinguishing Element Types, Oxidation States
and Spin States ................................... 148
2.7.6 Valence Effects ................................... 149
2.7.7 Diffraction Experiments under Non-Ambient
Conditions ........................................ 150
2.7.8 Issues of Interpretation and Validation ........... 151
Software Acknowledgements ................................ 153
References ............................................... 153
3 PDF Analysis of Nanoparticles ............................ 155
Reinhard В. Neder
3.1 Introduction ............................................. 155
3.2 Pair Distribution Function ............................... 160
3.3 Data Collection Strategies ............................... 168
3.4 Data Treatment ........................................... 170
3.4.1 Calculation of G(r) from a Structural Model ....... 175
3.4.2 Data Modelling .................................... 183
3.5 Examples ................................................. 184
3.5.1 Local Disorder versus Long-Range Average Order .... 185
3.5.2 ZnSe Nanoparticle ................................. 189
3.5.3 Decorated ZnO Nanoparticle ........................ 194
3.6 Complementary Techniques ................................. 197
References ............................................... 199
4 Electron Crystallography ................................. 201
Lu Han, Keiichi Miyasaka and Osamu Terasaki
4.1 Introduction ............................................. 201
4.2 Crystal Description ...................................... 203
4.2.1 Fourier Transformation and Related Functions ...... 203
4.2.2 Lattices .......................................... 204
4.2.2 Crystals and Crystal Structure Factors ............ 205
4.2.4 Simple Description of Babinet's Principle ......... 206
4.3 Electron Microscopy ...................................... 208
4.3.1 Interaction between Electrons and Matter .......... 208
4.3.2 Scanning Electron Microscopy ...................... 209
4.3.3 Transmission Electron Microscopy .................. 214
4.4 Electron Diffraction ..................................... 216
4.4.1 X-Rays (Photons) versus Electrons ................. 216
4.4.2 Scattering Power of an Atom ....................... 217
4.4.3 Crystal Structure and Electron Diffraction ........ 219
4.4.4 Relationship between Real and Reciprocal Space .... 221
4.4.5 Friedel's Law and Phase Restriction ............... 223
4.4.6 Information on the 0th, 1st and Higher-Order
Laue Zone ......................................... 224
4.4.7 Determining Unit Cell Dimensions and Crystal
Symmetry .......................................... 226
4.4.8 Convergent Beam Electron Diffraction .............. 227
4.5 Imaging .................................................. 229
4.5.1 Crystal Structure and ТЕМ Images .................. 229
4.5.2 Image Resolution .................................. 230
4.5.3 Limitation of Structural Resolution ............... 231
4.5.4 Electrostatic Potential and Structure Factors ..... 232
4.5.5 Image Simulation .................................. 235
4.6 The EC Method of Solving Crystal Structures .............. 235
4.6.1 1D Structures ..................................... 236
4.6.2 2D Structures ..................................... 239
4.6.3 3D Structures ..................................... 240
4.7 Other ТЕМ Techniques ..................................... 249
4.7.1 STEM and HAADF .................................... 249
4.7.2 Electron Tomography ............................... 249
4.7.3 3D Electron Diffraction ........................... 252
4.8 Conclusion ............................................... 254
Acknowledgment ........................................... 256
References ............................................... 256
5 Small-Angle Scattering ................................... 259
Theyencheri Narayanan
5.1 Introduction ............................................. 259
5.2 General Principles of SAS ................................ 261
5.2.1 Momentum Transfer ................................. 261
5.2.2 Differential Scattering Cross-Section ............. 262
5.2.3 Non-Interacting Systems ........................... 264
5.2.4 Influence of Polydispersity ....................... 266
5.2.5 Asymptotic Forms of I(q) .......................... 268
5.2.6 Multilevel Structures ............................. 269
5.2.7 Non-Particulate Systems ........................... 272
5.2.8 Structure Factor of Interactions .................. 273
5.2.9 Highly Ordered Structures ......................... 275
5.3 Instrumental Set-Up for SAXS ............................. 279
5.3.1 Synchrotron Source ................................ 280
5.3.2 X-Ray Optics ...................................... 281
5.3.3 X-Ray Detectors ................................... 283
5.3.4 SAXS Instrument Layout ............................ 284
5.4 Instrumental Set-Up for SANS ............................. 285
5.4.1 Neutron Sources ................................... 286
5.4.2 Neutron Optics .................................... 287
5.4.3 Neutron Detectors ................................. 288
5.4.4 SANS Instrument Layout ............................ 289
5.5 Additional Requirements for SAS .......................... 290
5.5.1 Combination with Wide-Angle Scattering ............ 290
5.5.2 Instrumental Smearing Effects ..................... 292
5.5.3 Sample Environments ............................... 293
5.6 Application of SAS Methods ............................... 294
5.6.1 Real-Time and In Situ Studies ..................... 295
5.6.2 Ultra Small-Angle Scattering ...................... 303
5.6.3 Contrast Variation in SAS ......................... 308
5.6.4 Grazing Incidence SAS ............................. 314
5.7 Conclusion ............................................... 318
Acknowledgements ......................................... 318
References ............................................... 319
Index ......................................................... 325
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