X-RAY DIFFRACTION PROCEDURES
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1 X-RAY DIFFRACTION PROCEDURES For Polycrystalline and Amorphous Materials HAROLD P. KLUG HEAD OF THE DEPARTMENT OF RESEARCH IN CHEMICAL PHYSICS LEROY E. ALEXANDER SENIOR FELLOW IN X-RAY DIFFRACTION Mellon Institute NEW YORK JOHN WILEY & SONS, INC. LONDON CHAPMAN & HALL, LIMITED
2 CONTENTS. 1. ELEMENTARY CRYSTALLOGRAPHY. I The Crystalline State 1 Crystalline and amorphous solids, I. Definition of a crystal, 2. Characteristics of the crystalline and vitreous states, 3. Crystal Geometry 4. Exte-rnal form and habit variation of crystals, 4. Constancy of interfacial angles, 6. Symmetry elements of crystals, 6. Pseudosymmetry, 12. Crystallographic axes, 13. Axial ratios, 13. The six crystal symmetry systems, 14. Miller indices, 15. The law of rational indices, 18. Crystal forms, 19. Composite crystals and twinning, 21. Equation for the plane (hkl), 23. Zones and zone relationships, 24. Space Lattices Historical introduction, 26.' Definition, 29. The unit cell, 30. The fourteen Bravais lattices, 30. Some crystallographic implications of space lattices, 33. Distance between neighboring lattice planes in the series (hkl), 34. The reciprocal lattice, 37. Point Groups and Space Groups. * 40 The point group or crystal symmetry class, 41: The space group, 48. : Literature THE PRODUCTION AND PROPERTIES OF X-RAYS, 56 X-Ray Safety and Protection The Production of X-Rays.. 58 The origin of x-rays, 58. X-ray tubes, 58. Power equipment for the production of x-rays, 71. Commercial x-ray generators for diffraction, 78. Properties of X-Rays and Their Measurement The x-ray spectrum of an element, 78. The precise determination of x-ray wavelengths, 88. Absorption of x-rays, 92. Secondary fluorescent and scattered x-rays, 95. Refraction of x-rays, 98.,, Monochromatization of x-radiation, 100. The photographic effects of x-rays, 108., Literature, FUNDAMENTAL PRINCIPLES OF X-RAY DIFFRACTION 111 The Geometry of Diffraction.,. Ill Scattering of x-rays by electrons and atoms, 111. Scattering
3 < CONTENTS by a regularly spaced row of atoms, 112. Conditions for diffraction by a linear lattice of atoms, 113. Diffraction by a simple cubic lattice, 115. Proof that the "diffracting plane" is a lattice plane, 119. The Bragg equation, 120. Derivation of the Bragg equation from the "reflection" analogy, 120. The geometrical picture of diffraction in reciprocal space, ,. The Intensity of Diffraction 125 Perfect and imperfect crystals, 125. Primary and secondary extinction, 129. Relative and absolute intensities, 130. Factors affecting the diffraction' intensities-, Expressions for the relative intensity of diffraction by the various techniques, 156. Lattice-centering and space-group extinctions, 157. Literature PHOTOGRAPHIC POWDER TECHNIQUES 162 The Debye-Scherrer Method 163 Introduction, 163. Camera design, 165. Preparation of the powder, 193. Mounting the powder, 197. Making the exposure, 206. Processing the film, 211. Parafocusing Methods. 211 Monochromatic-Pinhole Techniques 218 Forward-reflection method, 218. Back-reflection method, 221. High- and Low-Temperature Diffraction Techniques. 226 Miscellaneous Techniques 232 Literature SPECTROMETRIC POWDER TECHNIQUE " 235 Description of the Geiger-Counter Spectrometer. 236 Spectrometer geometry, 236. Effect of instrumental factors upon profiles and positions of diffraction maxima, 243. Geiger-Miiller tubes for the detection of x-rays, 261. Features of spectrometer circuit design, 265. Other important features, 270. Operation of the Geiger-Counter Spectrometer General principles, 270. Adjustment and calibration, 274. Specimen preparation, 290. Counting technique, 305. Recorder technique, 305. Literature THE INTERPRETATION OF POWDER DIFFRACTION DATA. '319 The Viewing and Precision Measurement of Powder Photographs 320
4 CONTENTS Determination of Interplanar (d) Spacings Debye-Scherrer patterns, 324. Monochromatic-pinhole (flatfilm) patterns, 333. Indexing Cubic Powder Patterns Determination of Lattice Type 346 Indexing Noncubic Powder Patterns 348 The Measurement of Intensities from Photographic Blackening 364 Introduction, 364. Preparation of a graded intensity scale, 365. Visual estimation of intensities, 366. Photometer techniques, 368. The Measurement of Intensities with the X-Ray Spectrometer 376 Putting Intensities on an Absolute Scale 378 Special Scattering and Diffraction Effects Background effects, 379. Reflections of unusual character, 384. Spurious lines, 387. Literature QUALITATIVE AND QUANTITATIVE ANALYSIS OF CRYSTALLINE POW- DERS 390 Routine Qualitative Identification of Crystalline Powders The ASTM diffraction data file, 391. Experimental technique of the ASTM method, 394. Complications and limitations of the ASTM method, 402. Special identification techniques, 405. Quantitative Analysis of Powder Mixtures. ~ Basic aspects of absorption in quantitative analysis, 411. Photographic-microphotometric techniques, 416. Geigercounter spectrometric technique, 422. Selected examples and applications, 430. Literature THE PRECISION DETERMINATION OF LATTICE CONSTANTS FROM POWDER PHOTOGRAPHS 440 General Considerations... ; Sources of Systematic Errors in the Debye-Scherrer Method 442 Radius errors and film shrinkage, 442. Specimen eccentricity, 444. Sample absorption and divergence of the beam, 446. Vertical divergence, of the beam, 451. Methods of Correcting for Errors in the Debye-Scherrer "" Method ' Use of calibrating substances, 452. The Straumanis method of refined experimental technique, 454. Use of mathematical methods, 461. Precision Lattice-Constant Measurements by the Symmetrical- Focusing Method. 474 xi
5 xii CONTENTS Sources of systematic errors, 474. The elimination of errors, 476. The Precision Determination of Lattice Constants of Noncubic Materials Special simple cases, 477. Application of the Straumanis technique, 478. Graphical method of successive approximations, 481. Extrapolation procedure of Taylor and Floyd, 483. Cohen's least-squares method, 485. Summary Literature CRYSTALLITE-SIZE DETERMINATION FROM LINE BROADENING. 491 Determination of the Pure Diffraction Breadth, Fourier analysis of line profiles, 495. Warren's correction for instrumental broadening, 500. Jones' correction curves for instrumental broadening, 501. The correction of x-ray spectrometer profiles for instrumental broadening, 507. Determination of. the Shape Factor, K 511 Proper choice of K in the absence of shape or size-distribution information, 512. Choice of K for random-layer-lattice reflections, 516. Evaluation of K for more precise investigations, 517., Practical Considerations. 525 Illustrative Crystallite-Size Determinations Literature FURTHER APPLICATIONS OF POLYCRYSTALLINE DIFFRACTION. 539 Stress Measurement in Metals Advantages and disadvantages of diffraction methods, 540. Elastic stress-strain relationships, 541. Sum of the principal stresses in a surface, 544. Component of stress in any desired direction in a surface, 547. Illustrative examples, 550. Preferred-Orientation Determination 555 Orientation in fibers, wires, and sheet materials, 555. Preparation "of fiber patterns, 558. Analysis of fiber patterns, 561. The stereographic projection, 568. Pole figures and their preparation, 573. Microtechniques for fibers, 583. Literature. : ' DIFFRACTION STUDIES OF NONCRYSTALLINE MATERIALS Radial-Distribution Analysis Theory, 588. Experimental requirements, 592. Illustrative experimental procedure, 596. Sources of error, 607. Practical examples, 612. Further remarks on experimental technique, 620..
6 CONTENTS xiii Determination of Crystalline and Amorphous Fractions in Polymers 621 Crystallinity in rubber, 621. Crystalline-amorphous ratio in textile fibers and other polymers, 626. Identification of Noncrystalline Patterns 631 Literature SMALL-ANGLE X-RAY SCATTERING 634 Measurement of Very Long Periodicities Equipment and experimental conditions, 636. Applications of long-periodicity measurements, 645. Particle Size from Small-Angle Scattering Elementary theory of small-angle scattering by particles, 647. Camera geometry, 650. Camera design and experimental details, 653. Peak analysis of latex, 656. Slope analysis of colloidal gold, 658. Spectrometric measurement of continuous small-angle scattering, 659. Particle-size distribution, 661. Literature 664 APPENDIX 665 I. Layout for a Diffraction Laboratory 665 II. The Handling and Processing of X-Ray Film. 671 III. Miscellaneous Constants and Numerical Data 675 IV. International Atomic Weights 676 V. Mass Absorption Coefficients of the Elements. 677 VI. Quadratic Forms for the Cubic System 679 VII. Atomic and Ionic Scattering Factors. 680 VIII. Lorentz and Polarization Factors 682 IX. Temperature Factor Tables AUTHOR INDEX 687 SUBJECT INDEX 695
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