N. Kasai M. Kakudo X-Ray Diffraction by Macromolecules With 351 Figures and 56 Tables Kodansha ~Springer
... Contents Preface v Part I Fundamental 1. Essential Properties of X-Rays................. 3 1.1 X-Rays as Electromagnetic Waves.... 3 1.2 Generation ofx-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
VIII Contents 2.6 A Dense, Disordered Assemblage of Dissimilar Atoms...... 33 2.7 A Dense, Disordered Assemblage ofpolyatomic 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 ofweissenberg photographs... 75 4.4.3 Interpretation of diffraction from crystalline powders or polycrystalline specimens...... 78 4.4.4 Fibrous polycrystalline specimens... 79 Reference...... 82
Contents IX 5. Diffraction of X-Rays by Imperfect Crystals and Paracrystals.................... 83 5.1 Ideal Crystals and Imperfect Crystals..... 83 5.l.l 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 (or voids)........... Ill 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.I 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. l Globular proteins......................... 146 7.2.2 Synthetic and some natural high polymers...... 146
X Contents 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
Contents XI 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.I 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 I 0.3 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 I 0.4 Preferred Orientation of Two-dimensional Lattices............ 270 References......... 271
XII Contents 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 Detennination... 331 12.4.1 Isomorphous replacement..... 331 12.4.2 Anomalous scattering... 334
Contents XIII 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 Cz 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 IF 1 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
XIV Contents 13.6.1 Broadening of diffraction patterns from paracrystalline 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......................................................................................................... 41 0 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
Contents XV 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 metl"lod)... 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