Preface ......................................................... v
Part I Fundamental
1. Essential Properties of X-Rays ............................... 3
1.1. X-Rays as Electromagnetic Waves ............................ 3
1.2. Generation of X-Rays ....................................... 3
1.2.1. X-ray tube .......................................... 3
1.2.2. Synchrotron radiation ............................... 6
1.3. Properties and Effects of X-Rays ........................... 8
1.3.1. Absorption of X-rays ................................ 8
1.3.2. X-ray scattering ................................... 10
1.3.3. X-ray refraction ................................... 11
1.3.4. Effects used for the detection of X-rays ........... 11
1.3.5. Other effects ...................................... 12
References ..................................................... 13
2. X-Ray Scattering, Interference and Diffraction .............. 15
2.1. Scattering by a Single Electron ........................... 16
2.2. Interference and Diffraction of Scattered X-Rays .......... 18
2.2.1. The phenomena of interference and diffraction ...... 18
2.2.2. Basis for calculating the amplitudes of, and
phase differences between, diffracted waves ........ 21
2.2.3. The relationship between real and reciprocal
space .............................................. 23
2.3. Scattering of X-Rays by a Single Atom ..................... 25
2.3.1. Atomic scattering factor ........................... 25
2.3.2. Anomalous dispersion ............................... 27
2.3.3. Compton scattering intensity ....................... 27
2.4. Scattering of X-Rays by a Single Polyatomic Molecule ...... 28
2.5. X-Ray Scattering from a Dense, Disordered Assemblage
of Identical Atoms (a Monatomic Liquid) ................... 31
2.6. A Dense, Disordered Assemblage of Dissimilar Atoms ........ 33
2.7. A Dense, Disordered Assemblage of Polyatomic
Molecules (a Molecular Liquid) ............................ 34
2.8. Scattering of X-Rays by Amorphous Solids .................. 35
2.9. Scattering of X-Rays by Crystals .......................... 37
2.9.1. Amplitude and intensity of the scattered rays ...... 37
2.9.2. Form of the X-ray diffraction pattern .............. 41
2.10.Summary ................................................... 43
References ..................................................... 43
3. Crystal Structure ........................................... 45
3.1. Crystal Systems and the Unit Cell ......................... 45
3.2. Crystal Planes and Their Indices .......................... 47
3.2.1. Crystal planes ..................................... 47
3.2.2. Lattice plane indices .............................. 48
3.2.3. The spacing of lattice planes and the
relationship between plane indices and
Laue indices ....................................... 49
3.2.4. Coordinates of atoms, lattice points, and
reciprocal lattice points and indices of
crystal planes and zone axes ....................... 51
3.3. Crystal Symmetry .......................................... 52
3.3.1. Point groups and their symmetries .................. 52
3.3.2. Space groups ....................................... 55
3.3.3. Relationship between the atoms present in
the unit cell and the equivalent points of
the space group .................................... 59
References ..................................................... 60
4. Detailed Interpretation of the Diffraction of X-Rays
by Crystals ................................................. 61
4.1. The Bragg Diffraction Condition ........................... 61
4.2. Lattice Structure Factors ................................. 63
4.3. Reciprocal Space and Reciprocal Lattice ................... 63
4.4. Wider Applications of the Reciprocal Lattice .............. 68
4.4.1. Interpretation of rotating-crystal and
oscillating-crystal photographs .................... 68
4.4.2. Interpretation of Weissenberg photographs .......... 75
4.4.3. Interpretation of diffraction from crystalline
powders or polycrystalline specimens ............... 78
4.4.4. Fibrous polycrystalline specimens .................. 79
Reference ...................................................... 82
5. Diffraction of X-Rays by Imperfect Crystals and
Paracrystals ................................................ 83
5.1. Ideal Crystals and Imperfect Crystals ..................... 83
5.1.1. Lattice distortions of the first kind .............. 85
5.1.2. Lattice distortions of the second kind ............. 88
5.2. Fourier Transform Theory of X-Ray Diffraction ............. 89
5.2.1. Fourier transform theorem .......................... 89
5.2.2. Shape factor for the scattering body ............... 92
5.2.3. Scattering factor of atoms undergoing thermal
vibrations in a crystal ............................ 94
5.2.4. Optical experiments on Fourier transforms .......... 95
5.3. Diffraction of X-Rays by Paracrystals ..................... 96
5.3.1. Statistical representation of paracrystalline
lattice points and the derivation of their
function Q(r) ...................................... 97
5.3.2. Lattice factor and diffraction intensity for a
paracrystal ........................................ 98
5.4. Summary of the Relationship between Structure and
X-Ray Diffraction Intensity .............................. 104
References ............................................... 108
6. Scattering of X-Rays by Very Small Bodies .................. 109
6.1. Small-angle Diffuse Scattering ........................... 109
6.2. Small-angle Scattering Theory ............................ 110
6.2.1. X-ray scattering by a substance of any
structure ......................................... 110
6.2.2. Small-angle scattering from systems of
dilutely dispersed particles (orvoids) ............ 1ll
6.2.3. Correlation function and distance distribution
function .......................................... 121
6.2.4. Polydispersed system of particles with uniform
shape ............................................. 127
6.2.5. Small-angle scattering from systems of
densely packed particles .......................... 128
6.2.6. Small-angle scattering from a non-particulate
system ............................................ 130
References ............................................... 133
7. Structure of High Polymeric Substances ..................... 135
7.1. Structure of High Polymer Chains in the Liquid State
and in Solution .......................................... 136
7.1.1. Configuration and conformation .................... 136
7.1.2. Classification of chain molecules ................. 142
7.2. Molecular Aggregations in Solid High Polymers ............ 146
7.2.1. Globular proteins ................................. 146
7.2.2. Synthetic and some natural high polymers .......... 146
7.3. Structure of the Amorphous State and of Amorphous
Regions in Solid High Polymers ........................... 150
7.3.1. Random-coil model ................................. 150
7.3.2. Folded-chain-fringed-micellar-grain model ......... 151
7.4. Fine Texture in Solid High Polymers ...................... 151
References ............................................... 155
Part II Experimental
8. Experimental Methods ..................................... 159
8.1. Preliminary Considerations ............................... 159
8.2. X-Ray Equipment .......................................... 159
8.2.1. X-ray generators .................................. 159
8.2.2. X-ray detectors ................................... 163
8.2.3. X-ray cameras ..................................... 169
8.2.4. X-ray diffractometers ............................. 175
8.2.5. X-ray small-angle scattering cameras .............. 183
8.3. Selection of the X-Ray Parameters ........................ 192
8.3.1. X-ray wavelength .................................. 192
8.3.2. Production of monochromatic X-rays ................ 193
8.3.3. Elimination of unwanted scattered X-rays .......... 196
8.4. The Specimen ............................................. 197
8.4.1. Preparation of the specimen ....................... 197
8.4.2. Determination of the specimen density ............. 200
8.5. Diffraction Studies for Identification Purposes .......... 201
8.5.1. Qualitative identification using polycrystal
diffraction data (unoriented X-ray diagrams) ...... 201
8.5.2. Treatment of the results .......................... 201
8.6. Diffraction Studies for Crystal Structure Analysis ....... 202
8.6.1. General remarks ................................... 202
8.6.2. Weissenberg photographs ........................... 203
8.6.3. Precession photographs ............................ 214
8.7. Diffraction Studies for Analysis of Fine Textures ........ 219
8.7.1. Measurement of crystallinity ...................... 219
8.7.2. Analysis of crystallite orientation ............... 220
8.7.3. Measurement of the size and shape of and/or
lattice distortion in crystallites ................ 220
8.7.4. Measurement of diffuse halos due to amorphous
solids and liquids ................................ 222
8.7.5. Analysis of distorted crystalline diffraction ..... 222
8.7.6. Measurement of small-angle scattering
(or diffraction) .................................. 222
8.7.7. Special experimental methods ...................... 223
References .................................................... 223
Part III Analytical
9. Identification of Crystals by X-Ray Diffraction ............ 229
9.1. Principles of Identification ............................. 229
9.2. Identification by the Powder Method ...................... 229
9.2.1. The JCPDS system .................................. 229
9.2.2. Locating a JCPDS card ............................. 230
9.3. Identification by the Single Crystal Method .............. 231
9.3.1. Computer databases ................................ 231
9.3.2. Others ............................................ 231
9.4. Identification of High Polymers .......................... 231
9.4.1. Identification by unoriented X-ray patterns ....... 232
9.4.2. Identification by oriented X-ray patterns ......... 232
9.5. X-Ray Diffraction Patterns of Copolymers and Polymer
Blends ................................................... 232
9.5.1. X-ray diffraction patterns of copolymers .......... 232
9.5.2. X-ray diffraction patterns of polymer blends ...... 235
Notes and References ..................................... 236
10.Analysis of Crystallite Orientation ........................ 239
10.1.Crystallite Orientation and the X-Ray Diffraction
Diagram .................................................. 239
10.1.1.General survey .................................... 241
10.1.2.Types of orientation .............................. 244
10.1.3.Interpretation of inclined X-ray diagrams ......... 248
10.2.Analysis of the Type of Crystallite Orientation .......... 250
10.2.1.Establishing the presence or absence of
orientation ....................................... 250
10.2.2.Identification of the type of orientation ......... 251
10.3.Determination of the Degree of Orientation ............... 258
10.3.1.Criteria of the degree of orientation ............. 258
10.3.2.Determination of the mean of the crystallite
orientation distribution (orientation
coefficient) ...................................... 259
10.3.3.Analysis of the crystallite orientation
distribution (orientation distribution
functions) ........................................ 263
10.4.Preferred Orientation of Two-dimensional Lattices ........ 270
References .................................................... 271
11.Crystal Structure Analysis of High Polymers ................ 273
11.1.Use of Unoriented Diffraction Patterns ................... 273
11.1.1.Rietveld method ................................... 273
11.1.2.Pattern decomposition method ...................... 276
11.1.3.Extension to fibrous materials .................... 277
11.2.Structure Analyses Using Uniaxially Oriented
Diffraction Patterns ..................................... 277
11.2.1.Determination of fiber period ..................... 277
11.2.2.Indexing diffractions and determining unit
cell parameters ................................... 280
11.2.3.Determination of the space group .................. 281
11.2.4.Structure analysis ................................ 285
11.2.5.Fourier transforms and syntheses and Patterson
functions ......................................... 289
11.2.6.Determination of phases in Fourier syntheses ...... 293
11.2.7.Refinement of the structure ....................... 294
11.2.8.Crystal structure analysis of polyetylene ......... 295
11.3.Analyses Using Biaxially or Doubly Oriented
Diffraction Patterns ..................................... 302
11.4.Analyses Using Diffraction Patterns from Helical
Structures .......................................... 304
11.4.1.Diffraction of X-rays by a continuous helix ....... 304
11.4.2.Diffraction of X-rays by a discontinuous helix .... 305
11.4.3.Interpretation of the diffraction pattern and
structure analysis of helical polymers ............ 307
11.4.4.Determination of helical structures ............... 311
References .................................................... 318
12.Crystal Structure Determination of Macromolecules .......... 321
12.1.Characteristics of Protein Crystals ...................... 322
12.1.1.Solvent of crystallization ........................ 322
12.1.2.Special features of X-ray diffraction by a
protein crystal ................................... 322
12.2.Crystallization .......................................... 323
12.2.1.Solubility of protein ............................. 323
12.2.2.Techniques for crystallization .................... 324
12.2.3.Preparation of isomorphous heavy atom
derivative crystals ............................... 327
12.2.4.Crystal mounting .................................. 328
12.3.Data Collection .......................................... 329
12.3.1.Determination of preliminary crystallographic
data .............................................. 329
12.3.2.Collection of intensity data ...................... 330
12.4.Phase Determination ...................................... 331
12.4.1.Isomorphous replacement ........................... 331
12.4.2.Anomalous scattering .............................. 334
12.4.3. Determination of the position of heavy atoms ..... 337
12.5.Molecular Replacement Method ............................. 341
12.5.1. Structure solution of bacterial cytochrome
C2 from Rhodopseudomonas
viridis (Rps. viridis) ........................... 341
12.6.Interpretation of Electron Density Maps: Model
Building ................................................. 343
12.7.Refinement of the Structure .............................. 345
12.7.1.Restrained least-squares refinement ............... 345
12.7.2.Crystallographic refinement by simulated
annealing ......................................... 346
12.7.3.Further refinement ................................ 348
12.7.4.Expression of the result .......................... 348
12.8.Structure Analysis of Macromolecules by Image
Reconstruction from Electron Micrographs (Electron
Crystallography) ......................................... 350
12.8.1.Principle ......................................... 350
12.8.2.Procedures for the image reconstruction ........... 352
12.9.Structural Study of Macromolecules in Solution—NMR
Investigations ........................................... 354
References .................................................... 355
13.Analysis of the Breadth and Shape of Diffraction
Patterns ................................................... 359
13.1.Instrumental Broadening .................................. 360
13.1.1.Systematic errors in measured diffraction
breadths .......................................... 360
13.1.2.Methods of correcting the line profile ............ 361
13.2.Relationship between the Size and Shape of an Ideal
Crystal and the Broadening of Its Diffraction Pattern .... 363
13.2.1.Broadening due to the Laue function ............... 363
13.2.2.Variation in the shape of diffractions with
|F|2 · G .......................................... 364
13.3.Calculation of Crystallite Size from the Broadening
of the Diffraction Pattern ............................... 364
13.3.1.The Scherrer formula .............................. 364
13.3.2.Effect of crystallite size distribution ........... 366
13.3.3.Effect of crystallite shape ....................... 366
13.3.4.Application to very small crystallites ............ 367
13.4.Estimation of Lattice Distortion from Line Broadening .... 372
13.5.Separation of Line Broadenings Due to Crystallite Size
and Lattice Distortion ................................... 373
13.5.1.Method of integral breadths ....................... 373
13.5.2.Method of profile fitting ......................... 374
13.5.3.Method of Fourier transforms ...................... 375
13.6.Analyses Including Background Scattering Due to
Imperfect Crystals ....................................... 376
13.6.1.Broadening of diffraction patterns from
paracrystaliine structures ........................ 376
13.6.2.Analysis of the broadening of diffractions
from paracrystal structures ....................... 379
13.6.3.Shape of the diffraction pattern of
a three-dimensional paracrystal and
calculation of the degree of distortion ........... 388
References .................................................... 391
14.Analyses Using the Total Diffraction Intensity
Distribution Curves of High Polymers ....................... 393
14.1.Correction for Coherent Background Scattering ............ 393
14.1.1.Correction of the measured intensity for the
effect of polarization ............................ 393
14.1.2.Normalization of the scattering intensity ......... 393
14.2.Determination of Crystallinity ........................... 394
14.2.1.Principles of the measurement of crystallinity .... 395
14.2.2.Differentiation between crystalline and
amorphous scattering in coherent scattering ....... 396
14.2.3.Measurement of crystallinity ...................... 398
14.3.Analysis of the Radial Distribution Function P(r) ........ 402
14.3.1.Calculation of the radial distribution function ... 402
14.3.2.The radial distribution function of Nylon 6,6 ..... 403
14.3.3.Special cases where the shape of the molecular
chains can be deduced without determining the
radial distribution function ...................... 404
14.4.Recognition of Oriented Diffraction Mixed with
Unoriented Amorphous Scattering .......................... 405
14.4.1. Resolution of oriented diffraction masked
by unoriented amorphous scattering ............... 407
14.5.Analysis of the Orientation of Molecular Chains in
Amorphous Regions ........................................ 407
14.5.1.Orientation of molecular chains in amorphous
regions ........................................... 407
14.5.2.Degree of orientation of the molecular chains;
practical measure of parallelism of amorphous
chains ............................................ 408
14.5.3.Estimation of the degree of orientation of
molecular chains in amorphous regions by
methods other than X-ray methods .................. 409
14.6.Cylindrical Patterson Functions of Uniaxially
Oriented Fiber Diffraction Patterns ...................... 410
14.6.1.The cylindrical distribution function ............. 411
14.6.2.Representation of Q(r) in polar coordinates ....... 412
14.6.3.Where there is periodicity along the cylinder
axis .............................................. 416
References .................................................... 417
15.Analysis of X-ray Small-angle Scattering ................... 419
15.1.Preparative Procedure .................................... 419
15.1.1.Detection and recording of the small-angle
scattering ........................................ 419
15.1.2.Corrections to the scattering intensity
distribution ...................................... 420
15.2.Analysis of Particle Size and Shape ...................... 422
15.2.1.The Guinier plot .................................. 422
15.2.2.Comparison of the measured scattering intensity
curve with the theoretical curve (Curve
fitting method) ................................... 427
15.2.3.The distance distribution function ................ 431
15.2.4.Other analytical methods .......................... 440
15.3.Analysis of Small-angle Scattering for Solutions of
Chain Macromolecules ..................................... 441
15.3.1.Persistence of polymer chain ...................... 441
15.3.2.Scattering intensity from stiff chain molecules ... 445
15.4.Analysis of the "Long-period Pattern" .................... 450
15.4.1.Long-period small-angle scattering patterns ....... 450
15.4.2.Anisotropy in the small-angle scattering
pattern and in orientation and particle
distribution ...................................... 467
15.5.Analysis of Crystallinity from Small-angle Scattering .... 469
15.5.1.Analysis using the long-period pattern ............ 469
15.5.2.Analysis using the central diffuse scattering ..... 469
15.6.Analysis of Well-oriented Small- and Wide-angle
Diffractions ............................................. 469
15.6.1.X-ray diffraction patterns from contracting
muscle ............................................ 470
References .................................................... 478
Appendix ...................................................... 481
Index ......................................................... 497
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