Structure from Diffraction Methods
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Duncan Bruce is Professor of Liquid Crystals and Materials Chemistry at the University of York. Prior to this he was Professor of Inorganic Chemistry at the University of Exeter, and co-director of the Sheffield Centre for Molecular Materials. His current research interests include liquid crystals and nanoparticle-doped, nanostructured, mesoporous silicates. He is immediate Past President of the Royal Society of Chemistry Materials Chemistry Division and Chair of the British Liquid Crystal Society. His work has been recognized by various awards including the RSC Tilden Prize for 2010, the RSC's Sir Edward Frankland Fellowship and Corday-Morgan Medal and Prize.
Dermot O'Hare is Professor in the Chemistry Research Laboratory at the University of Oxford. His interests are wide ranging, and include exploratory synthetic organometallic chemistry, intercalation chemistry, time-resolved, in situ diffraction studies and the synthesis of meso- and microporous solids. In 2010 he won the RSC Ludwig Mond award for outstanding research in Inorganic chemistry.
Dr Richard Walton is Associate Professor in Inorganic Chemistry at the University of Warwick. He was also formerly based in the Department of Chemistry at the University of Exeter. His research group works in the area of solid-state materials chemistry and has a number of projects focusing upon the synthesis, structural characterization and properties of inorganic materials.
Content
Preface xiii
List of Contributors xv
1 Powder Diffraction 1
Kenneth D. M. Harris and Andrew Williams
1.1 Introduction 1
1.2 The Similarities and Differences Between Single-Crystal nd Powder XRD 2
1.3 Qualitative Aspects of Powder XRD: 'Fingerprinting' of Crystalline Phases 6
1.4 Quantitative Aspects of Powder XRD: Some reliminaries 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
1.7 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 A 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 45
1.10.1 Overview 45
1.10.2 Structure Determination of Zeolites and Other Framework Materials 46
1.10.3 In Situ Powder XRD Studies of Materials Synthesis 48
1.10.4 Structure Determination of New Materials Produced by Solid-State Mechanochemistry 50
1.10.5 In Situ Powder XRD Studies of Solid-State Mechanochemical Processes 53
1.10.6 In Situ Powder XRD Studies of a Polymorphic Transformation 55
1.10.7 In Situ Powder XRD Studies of a Solid-State Reaction 58
1.10.8 Establishing Details of a Hydrogen-Bonding Arrangement by Powder Neutron Diffraction 58
1.10.9 Structure Determination of a Material Produced by Rapid Precipitation from Solution 60
1.10.10 Structure Determination of Intermediates in a Solid-State Reaction 62
1.10.11 Structure Determination of a Novel Aluminium Methylphosphonate 62
1.10.12 Structure Determination of Materials Prepared by Solid-State Dehydration/Desolvation Processes 63
1.10.13 Structure Determination of the Product Material from a Solid-State Photopolymerisation Reaction 66
1.10.14 Exploiting Anisotropic Thermal Expansion in Structure Determination 68
1.10.15 Rationalisation of a Solid-State Reaction 69
1.10.16 Structure Determination of Organometallic Complexes 71
1.10.17 Examples of Structure Determination of Some Polymeric Materials 72
1.10.18 Structure Determination of Pigment Materials 73
1.11 Conclusion 74
References 75
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 B. 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
4.2.3 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 TEM 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 TEM 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 255
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
5.4.5 Combination with Wide-Angle Scattering 290
5.4.6 Instrumental Smearing Effects 292
5.4.7 Sample Environments 293
5.5 Application of SAS Methods 294
5.5.1 Real-Time and In Situ Studies 295
5.5.2 Ultra Small-Angle Scattering 303
5.5.3 Contrast Variation in SAS 308
5.5.4 Grazing-Incidence SAS 314
5.6 Conclusion 318
Acknowledgements 318
References 319
Index 325
Chapter 1
Powder Diffraction
Kenneth D. M. Harris and P. Andrew Williams
School of Chemistry, Cardiff University, Cardiff, UK
1.1 INTRODUCTION
As discussed in Chapter 2, single-crystal X-ray diffraction[1–3] (XRD) is the most widely used and the most powerful technique for determining crystal structures, and this technique led to many monumental scientific discoveries in the 20th century. The wide-ranging scope and the routine application of single-crystal XRD in the modern day have arisen both through advances in instrumentation and through the development of powerful strategies for data analysis, such that crystal structures can now be determined rapidly and straightforwardly in all but the most challenging cases. The central importance of single-crystal XRD in the physical, biological and materials sciences will continue to be further developed and exploited in the years to come. Thus, provided a single crystal of sufficient size and quality is available for the material of interest, successful structure determination by analysis of single-crystal XRD data is nowadays very routine.
However, the requirement to prepare a suitable single crystal specimen for single-crystal XRD experiments represents a major limitation of this technique. As a consequence, the crystal structures of many important crystalline materials remain unknown simply because the material cannot be prepared as a crystal of appropriate size and quality for single-crystal XRD studies. In such cases, however, the material can usually be prepared as a microcrystalline powder, and therefore it is still feasible to record powder XRD data. The question that immediately arises is whether it is feasible to determine the crystal structure of a material from powder XRD data using techniques analogous to those employed with single-crystal XRD data. Furthermore, are there any aspects of structural science that might actually be more readily investigated by powder XRD than single-crystal XRD?
With the aim of addressing these types of question, the present chapter provides an overview of the current state of the art in the application of powder XRD within chemical and materials sciences, focusing in particular on contemporary opportunities for determining crystal structures directly from powder XRD data. Fundamental aspects of the techniques used to carry out crystal structure determination from powder XRD data are described, and several illustrative examples of the application of these techniques in determining the structural properties of materials across a wide range of areas of chemistry are highlighted. In addition, we discuss the wide-ranging utility of powder XRD in other aspects of the characterisation of solid materials, from routine applications in the identification (‘fingerprinting’) of crystalline phases to more advanced applications in which in situ powder XRD studies are exploited to investigate structural transformations associated with phase transitions, solid-state chemical reactions, crystallisation processes and materials synthesis. While the chapter is focused primarily on powder XRD, the complementary opportunities offered by powder neutron diffraction are also discussed.
With the exception of some brief mention of certain specific aspects of experimental techniques for the measurement of powder XRD data, details of the instrumentation used to record powder XRD data lie outside the scope of this chapter, which is focused primarily on the application of powder XRD to determine structural information in chemical contexts rather than on the technical details of experimental techniques. Descriptions of the variety of experimental set-ups that may be used to record powder XRD data and comparisons of their relative merits (e.g. transmission mode versus reflection mode, Debye–Scherrer versus Bragg–Brentano, angle dispersive versus energy dispersive, point detectors versus position-sensitive detectors) may be found in more detailed monographs on instrumentation.[4–7]
1.2 THE SIMILARITIES AND DIFFERENCES BETWEEN SINGLE-CRYSTAL XRD AND POWDER XRD
As discussed above, the form of the sample studied in single-crystal and powder XRD is intrinsically different: a large, individual crystal in the former case and a powder comprising a huge number of small, randomly oriented crystallites in the latter case. However, while the nature of the sample is different, the physical phenomenon underlying both techniques is the same. In each case, X-ray radiation (usually monochromatic radiation) is incident on the sample. As the wavelength of the X-rays is comparable to the periodic repeat distances within the crystalline material (i.e. within the single crystal in the single-crystal XRD experiment and within each crystallite present in the powder sample in the powder XRD experiment), coherent/elastic scattering of the X-rays by the sample gives rise to an ‘XRD pattern’ (Figure 1.1) in which the radiation is scattered with significant intensity only in certain specific directions, while in all other directions the intensity of scattered radiation is zero.
Figure 1.1 Comparison of single-crystal and powder XRD measurements. In powder XRD, the diffraction phenomenon for each individual crystallite in the powder is the same as the diffraction phenomenon in single-crystal XRD. However, the powder comprises a large collection of crystallites with (in principle) a random distribution of crystallite orientations. As a consequence, the three-dimensional diffraction data are effectively compressed into one dimension (intensity versus diffraction angle ) in the powder XRD measurement.
Because the underlying physical phenomenon is the same, single-crystal and powder XRD patterns contain essentially the same information. However, as a result of the different nature of the sample used in each case, the form of the XRD pattern and the way in which it can be measured are different (Figure 1.1). Thus, in the single-crystal XRD pattern, the intense diffraction ‘peaks’ are well separated from each other in three-dimensional (3D) space (‘reciprocal space’) and both the scattering direction and the intensity of each individual intensity maximum can be measured very accurately. In the case of powder XRD, on the other hand, although each individual crystallite in the powder behaves in a similar manner to the single crystal sample in single-crystal XRD, the fact that the powder sample comprises a huge number of randomly oriented crystallites means that only the collective X-ray scattering from the whole sample can be measured. Because of the randomly oriented nature of the crystallites within the powder, the collective X-ray scattering from the whole powder sample comprises a set of coaxial cones of scattered radiation (in contrast to the sharp beams of scattered radiation that arise in single-crystal XRD). The semi-angle of each cone is the diffraction angle, As shown in Figure 1.1, the powder XRD pattern comprises the measured diffracted intensity as a function of a single spatial variable, the diffraction angle Effectively, in making the powder XRD measurement, the 3D information contained in the diffraction data is ‘compressed’ into one dimension, as the diffraction intensity is measured as a function of only one spatial variable.
The peaks in the powder XRD pattern arise at specific values of that satisfy Bragg's law:
1.1where is the wavelength of the X-rays, the indices and are three integers (the Miller indices) that uniquely label each intensity maximum (‘peak’) in the diffraction pattern and is the interplanar spacing of a specific set of lattice planes (also uniquely labelled with the same Miller indices, and ) in the crystal structure (the set of interplanar spacings, for a crystal structure depends on the dimensions of the unit cell – see below).
As a consequence of the fact that the diffraction data are ‘compressed’ into one dimension in the powder XRD measurement, there is usually considerable overlap of peaks in the powder XRD pattern. Such peak overlap serves to obscure information on the position (i.e. the value) and the intensity of each peak in the powder XRD pattern, and the difficulty of obtaining reliable and accurate information on the peak positions and intensities can impede (or, in severe cases, prohibit) the process of carrying out crystal structure determination from powder XRD data. For materials with large unit cells and low symmetry (such as most molecular solids), there is a very high density of peaks in the powder XRD pattern (especially at high values of ) and the problem of peak overlap can be particularly severe (see Figure 1.2). The ‘problem’ of peak overlap presents specific challenges in several aspects of the analysis of powder XRD data, particularly in the context of structure determination. Clearly, the extent of peak overlap may be reduced by recording the data under experimental conditions that give rise to narrow peaks in the powder XRD pattern (i.e. ‘high-resolution’ powder XRD data). As discussed in Section 1.6.4, the widths of peaks in a powder XRD pattern depend both on features of the instrumentation used to record the data and on features of the powder sample, and higher resolution can be achieved (at least in principle) by optimisation of these features. However, while peak overlap may be alleviated by appropriate...
