X-Ray Diffraction by Polycrystalline Materials

René Guinebretière is a senior lecturer at the Ecole Nationale Supérieure de Céramiques Industrielles in Limoges, France, and teaches X-ray diffraction. His research activities are the study of materials using X-ray diffraction at the SPCTS International Laboratory in France.

• Preface xiAcknowledgements xvAn Historical Introduction: The Discovery of X-rays and the First Studies in X-ray Diffraction xviiPart 1. Basic Theoretical Elements, Instrumentation and Classical Interpretations of the Results 1Chapter 1. Kinematic and Geometric Theories of X-ray Diffraction 31.1. Scattering by an atom 31.1.1. Scattering by a free electron 31.1.1.1. Coherent scattering: the Thomson formula 31.1.1.2. Incoherent scattering: Compton scattering [COM 23] 61.1.2. Scattering by a bound electron 81.1.3. Scattering by a multi-electron atom 111.2. Diffraction by an ideal crystal 141.2.1. A few elements of crystallography 141.2.1.1. Direct lattice 141.2.1.2. Reciprocal lattice 161.2.2. Kinematic theory of diffraction 171.2.2.1. Diffracted amplitude: structure factor and form factor 171.2.2.2. Diffracted intensity 181.2.2.3. Laue conditions [FRI 12] 221.2.3. Geometric theory of diffraction 231.2.3.1. Laue conditions 231.2.3.2. Bragg's law [BRA 13b, BRA 15] 241.2.3.3. The Ewald sphere 261.3. Diffraction by an ideally imperfect crystal 281.4. Diffraction by a polycrystalline sample 33Chapter 2. Instrumentation used for X-ray Diffraction 392.1. The different elements of a diffractometer 392.1.1. X-ray sources 392.1.1.1. Crookes tubes 412.1.1.2. Coolidge tubes 422.1.1.3. High intensity tubes 472.1.1.4. Synchrotron radiation 492.1.2. Filters and monochromator crystals 522.1.2.1. Filters 522.1.2.2. Monochromator crystals 552.1.2.3. Multi-layered monochromators or mirrors 592.1.3. Detectors 622.1.3.1. Photographic film 622.1.3.2. Gas detectors 632.1.3.3. Solid detectors 682.2. Diffractometers designed for the study of powdered or bulk polycrystalline samples 722.2.1. The Debye-Scherrer and Hull diffractometer 732.2.1.1. The traditional Debye-Scherrer and Hull diffractometer 742.2.1.2. The modern Debye-Scherrer and Hill diffractometer: use of position sensitive detectors 762.2.2. Focusing diffractometers: Seeman and Bohlin diffractometers 872.2.2.1. Principle 872.2.2.2. The different configurations 882.2.3. Bragg-Brentano diffractometers 942.2.3.1. Principle 942.2.3.2. Description of the diffractometer; path of the X-ray beams 972.2.3.3. Depth and irradiated volume 1032.2.4. Parallel geometry diffractometers 1042.2.5. Diffractometers equipped with plane detectors 1092.3. Diffractometers designed for the study of thin films 1102.3.1. Fundamental problem 1102.3.1.1. Introduction 1102.3.1.2. Penetration depth and diffracted intensity 1112.3.2. Conventional diffractometers designed for the study of polycrystalline films 1162.3.3. Systems designed for the study of textured layers 1182.3.4. High resolution diffractometers designed for the study of epitaxial films 1202.3.5. Sample holder 1232.4. An introduction to surface diffractometry 125Chapter 3. Data Processing, Extracting Information 1273.1. Peak profile: instrumental aberrations 1293.1.1. X-ray source: g1(epsilon) 1303.1.2. Slit: g2(epsilon) 1303.1.3. Spectral width: g3(epsilon) 1313.1.4. Axial divergence: g4(epsilon) 1313.1.5. Transparency of the sample: g5(epsilon) 1333.2. Instrumental resolution function 1353.3. Fitting diffraction patterns 1383.3.1. Fitting functions 1383.3.1.1. Functions chosen a priori 1383.3.1.2. Functions calculated from the physical characteristics of the diffractometer 1433.3.2. Quality standards 1443.3.3. Peak by peak fitting 1453.3.4. Whole pattern fitting 1473.3.4.1. Fitting with cell constraints 1473.3.4.2. Structural simulation: the Rietveld method 1473.4. The resulting characteristic values 1503.4.1. Position 1513.4.2. Integrated intensity 1523.4.3. Intensity distribution: peak profiles 153Chapter 4. Interpreting the Results 1554.1. Phase identification 1554.2. Quantitative phase analysis 1584.2.1. Experimental problems 1584.2.1.1. Number of diffracting grains and preferential orientation 1584.2.1.2. Differential absorption 1614.2.2. Methods for extracting the integrated intensity 1624.2.2.1. Measurements based on peak by peak fitting 1624.2.2.2. Measurements based on the whole fitting of the diagram 1634.2.3. Quantitative analysis procedures 1654.2.3.1. The direct method 1654.2.3.2. External control samples 1664.2.3.3. Internal control samples 1664.3. Identification of the crystal system and refinement of the cell parameters 1674.3.1. Identification of the crystal system: indexing 1674.3.2. Refinement of the cell parameters 1714.4. Introduction to structural analysis 1724.4.1. General ideas and fundamental concepts 1734.4.1.1. Relation between the integrated intensity and the electron density 1734.4.1.2. Structural analysis 1754.4.1.3. The Patterson function 1774.4.1.4. Two-dimensional representations of the electron density distribution 1804.4.2. Determining and refining structures based on diagrams produced with polycrystalline samples 1834.4.2.1. Introduction 1834.4.2.2. Measuring the integrated intensities and establishing a structural model 1844.4.2.3. Structure refinement: the Rietveld method 185Part 2. Microstructural Analysis 195Chapter 5. Scattering and Diffraction on Imperfect Crystals 1975.1. Punctual defects 1975.1.1. Case of a crystal containing randomly placed vacancies causing no relaxation 1985.1.2. Case of a crystal containing associated vacancies 2015.1.3. Effects of atom position relaxations 2035.2. Linear defects, dislocations 2055.2.1. Comments on the displacement term 2075.2.2. Comments on the contrast factor 2105.2.3. Comments on the factor f(M) 2125.3. Planar defects. 2125.4. Volume defects 2185.4.1. Size of the crystals 2185.4.2. Microstrains 2265.4.3. Effects of the grain size and of the microstrains on the peak profiles: Fourier analysis of the diffracted intensity distribution 231Chapter 6. Microstructural Study of Randomly Oriented Polycrystalline Samples 2356.1. Extracting the pure profile 2366.1.1. Methods based on deconvolution 2376.1.1.1. Constraint free deconvolution method: Stokes' method 2386.1.1.2. Deconvolution by iteration 2426.1.1.3. Stabilization methods 2446.1.1.4. The maximum entropy or likelihood method, and the Bayesian method 2446.1.1.5. Methods based on a priori assumptions on the profile 2456.1.2. Convolutive methods 2466.2. Microstructural study using the integral breadth method 2476.2.1. The Williamson-Hall method 2486.2.2. The modified Williamson-Hall method and Voigt function fitting 2506.2.3. Study of size anisotropy 2526.2.4. Measurement of stacking faults 2556.2.5. Measurements of integral breadths by whole pattern fitting 2576.3. Microstructural study by Fourier series analysis of the peak profiles 2626.3.1. Direct analysis: the Bertaut-Warren-Averbach method 2626.3.2. Indirect Fourier analysis 2686.4. Microstructural study based on the modeling of the diffraction peak profiles 270Chapter 7. Microstructural Study of Thin Films 2757.1. Positioning and orienting the sample 2767.2. Study of disoriented or textured polycrystalline films 2797.2.1. Films comprised of randomly oriented crystals 2797.2.2. Studying textured films 2857.2.2.1. Determining the texture 2857.2.2.2. Quantification of the crystallographic orientation: studying texture 2897.3. Studying epitaxial films 2927.3.1. Studying the crystallographic orientation and determining epitaxy relations 2927.3.1.1. Measuring the normal orientation: rocking curves 2937.3.1.2. Measuring the in-plane orientation: phi-scan 2957.3.2. Microstructural studies of epitaxial films 3007.3.2.1. Reciprocal space mapping and methodology 3047.3.2.2. Quantitative microstructural study by fitting the intensity distributions with Voigt functions 3077.3.2.3. Quantitative microstructural study by modeling of one-dimensional intensity distributions 312Bibliography 319Index 349