Elements of X-Ray Diffraction

Elements of X-Ray Diffraction
B. D. Cullity
Associate Professor of Metallurgy
University of Notre Dame
Addison-wesley Publishing Company, Inc.
Reading, Massachusetts
Contents
Fundamentals
Chapter 1 Properties of X-Rays 1
1-1 Introduction 1
1-2 Electromagnetic Radiation 1
1-3 the Continuous Spectrum . 4
1-4 the Characteristic Spectrum 6
1-5 Absorption . 10
1-6 Filters 16
1-7 Production of
X-Rays 17
1 -8 Detection of
X-Rays 23
1 9 Safety Precautions . 25
Chapter 2 the Geometry of Crystals 29
^2-1 Introduction . 29
J2-2 Lattices . 29
2-3 Crystal Systems 30
^2-4 Symmetry 34
2-5 Primitive and Nonprimitive Cells 36
2-6 Lattice Directions and Planes * . 37
2-7 Crystal Structure J 42
2-8 Atom Sizes and Coordination 52
2-9 Crystal Shape 54
2-10 Twinned Crystals . 55
2-11 the
Stereographic Projection . . 60
Chapter 3 Diffraction I: the Directions of Diffracted Beams 78
3-1 Introduction . .78
3-2 Diffraction F . 79
3-3 the Bragg Law *
3-4 X-Ray Spectroscopy 85
3-5 Diffraction Directions – 88
3-6 Diffraction Methods . 89
3-7 Diffraction Under Nonideal Conditions . 96
Chapter 4 Diffraction Ii: the Intensities of Diffracted Beams . 104
4-1 Introduction 104
4-2
Scattering by an Electrons . . 105
4-3
Scattering by an Atom >, . / 108
4-4
Scattering by a Unit Cell */ . Ill
Centents
Fundamentals
Chapter 1 Pr.pertieh .f X-Rays
1-] Int1oduction
1-2 Eleetromagnctie Radiation
1-3 the Continuouh Spectrum
1-4 the Characterihtir Hpectrum
1-5 Absorption
1–6 Filters
1-7 Production of X-Rays
J-s Detection of X-Rays
1 9 Hafety Precautions
C’hapter 2 Th]o~ Geometuy of Crystals
Introduc,tion
Lattiees
Crystal Hystems
Symmetry
Primitive and Nonprimitive Cells
Lattiee Direetions and Planes ../
Crystal Strueture J
Atom Sizes and Roordination
Crystal Shape
Twinned Crystals
The Stereographic Projeetion
Chapteu 3 Difjo’ractlon I: the Dmections of Diffracted Beams
Intl’oduction
Diffraction
The Bragg Law’ –
X-Ray Speetroseopy
Diffmetion Dirertions •
Diffraction Methods
Diffraction Under Nonidcal Conditions
Chapter 4 Diffraction Ii: the Intensities of Diffra – Ted Beams
Introduetion
Seattering by an Electron ,;’
Scattering by an Atom. _/
Scattering by a Unit Cell
4-5 Some Useful Relations . 118
4-6 Structure-factor Calculations ^ 118
4-7 Application to Powder Method ‘ 123
4-8 Multiplicity Factor 124
4-9 Lorentz Factor 124
1-10 Absorption Factor 129
4-11 Temperature Factor 130
4-12 Intensities of Powder Pattern Lines 132
4-13 Examples of Intensity Calculations 132
4-14 Measurement of X-Ray Intensity 136
Experimental Methods
Lpter 5 Laue Photographs 138
5-1 Introduction 138
5-2 Cameras . 138
5-3 Specimen Holders 143
5-4 Collimators . .144
5-5 the Shapes of Laue Spots . 146
Kpter 6 Powder Photographs . .149
6-1 Introduction . 149
6-2 Debye-scherrer Method . 149
6-3 Specimen Preparation …. 153
6-4 Film Loading . . 154
6-5 Cameras for High and Low Temperatures . 156
6-6 Focusing Cameras … . 156
6-7 Seemann-bohlin Camera . 157
6-8 Back-reflection Focusing Cameras . . .160
6-9 Pinhole Photographs . 163
6-10 Choice of Radiation . .165
6-11 Background Radiation . 166
6-12 Crystal Monochromators . 168
6-13 Measurement of Line Position 173
6-14 Measurement of Line Intensity . 173
Vpter 7 Dlffractometer Measurements 177
7-1 Introduction . . . 177
7-2 General Features …. 177
7-3 X-Ray Optics . . . – 184
7-4 Intensity Calculations . … 188
7-5 Proportional Counters . . . . 190
7-6 Geiger Counters . . … .193
7-7 Scintillation Counters . – 201
7-8 Sealers . … …. .202
7-9 Ratemeters . – 206
7-10 Use of Monochromators 211
Contents
L-5 Some Useful Relations . 118
L-6 Structure-factor Calculations .,/’, 118
‘1:-7 Application to Powder Method J 123
L-8 Multiplicity Factor 124
4-9 Lorentz Factor 124
‘1:-10 Absorption Factor 129
‘1:-11 Temperature Factor 130
4-12 Intensities of Powder Pattern Lines 132
4-13 Examples of Intensity Calculations 132
4-14 Measurement of X-Ray Intensity 136
Experimental Methods
,pter 5 Laue Photographs 138
5-1 Introduction 138
5-2 Cameras 138
5-3 Specimen Holdei’fl 143
5-4 Collimators 144
5-5 Theshapesoflauespo~ 146
~pter 6 Powder Photographs 149
6-1 Introduction 149
6-2 Debye-scherrer Method 149
6-3 Specimen Preparation 153
6-4 Film Loading 154
6-5 Cameras for High and Low Temperatures 156
6-6 Focusing Cameras 156
6-7 Seemann-bohlin Camera 157
6-8 Back-reflection Focusing Cameras 160
6-9 Pinhole Photographs 163
6-10 Choice of Radiation 165
6-11 Background Radiation 166
6-12 Crystal Monochromators 168
6-13 Measurement of Line Position 173
6-14 Measurement of Line Intensity 173
~pter 7 Diffractometer Measurements 177
7-1 Introduction 177
7-2 General Features 177
7-3 X-Ray Optics 184
7-4 Intensity Calculations 188
7-5 Proportional Counters 190
7-6 Gf’iger Counters 193
7-7 Scintillation Counters 201
7-8 Scalers 202
7-9 Ratemeters 206
7-10 Use of Monochromators . 211contents Xi
Applications
Chapter 8 Orientation of Single Crystals . . . 215
8-1 Introduction . . …. 215
8-2 Back-reflection Laue Method . . .215
8-3 Transmission Laue Method …. . 229
8-4 Diffractometer Method
8-5 Setting a Crystal in a Required Orientation . 240
8-6 Effect of Plastic Deformation . 242
8-7 Relative Orientation of Twinned Crystals 250
8-8 Relative Orientation of Precipitate and Matrix . . . 256
Chapter 9 the Structure of Polycrystalline Aggregates . 259
9-1 Introduction . 259
Crystal Size
9-2 Grain Size 259
9-3 Particle Size . 261
Crystal Perfection
9-4 Crystal Perfection . …. 263
9-5 Depth of X-Ray Penetration . . 269
Crystal Orientation
9-6 General . .272
9-7 Texture of Wire and Rod (Photographic Method) . . . 276
9-8 Texture of Sheet (Photographic Method) 280
9-9 Texture of Sheet (Diffractometer Method) . . 285
9-10 Summary . . 295
Chapter 10 the Determination of Crystal Structure . . . 297
10-1 Introduction . . 297
10-2 Preliminary Treatment of Data . . . 299
10-3 Indexing Patterns of Cubic Crystals 301
10-4 Indexing Patterns of Noncubic Crystals (Graphical Methods) 304
10-5 Indexing Patterns of Noncubic Crystals (Analytical Methods) . .311
10-6 the Effect of Cell Distortion on the Powder Pattern . . . 314
10-7 Determination of the Number of Atoms in a Unit Cell . .316
10-8 Determination of Atom Positions . 317
10-9 Example of Structure Determination …. . 320
Chapter 11 Precise Parameter Measurements . … 324
11-1 Introduction …. 324
11-2 Debye-scherrer Cameras …. …. 326
1 1-3 Back-reflection Focusing Cameras 333
11-4 Pinhole Cameras 333
11-5 Diffractometers 334
11-6 Method of Least
Squares .335
Contents
Applications
Chapter 8 Orientation of Single Crystals
8-1 Introduction
8-2 Back-reflection Laue Method
8-3 Transmission Laue Method
8-4 Diffractometer Method
8-5 Setting a Crystal in a Required Orientation
8-6 Effect of Plastic Deformation
8-7 Relative Orientation of Twinned Crystals
8-8 Relative Orientation of Precipitate and Matrix
Chapter 9 the Structure of Polycrystalline Aggregates
9-1 Introduction
Crystal Size
9-2 Grain Size
9-3 Particle Size
Crystal Perfection
9-4 Crystal Perfection
9-5 Depth of X-Ray Penetration
Crystal Orientation
9-6 General
9-7 Texture of Wire and Rod (Photographic Method)
9-8 Texture of Sheet (Photographic Method)
9-9 Texture of Sheet (Diffractometer Method)
9-10 Summary
Chapter 10 the Determination of Crystal Structure 297
Hh Introduction 297
10-2 Preliminary Treatment of Data 299
10-3 Indexing Patterns of Cubic Crystals 301
10-4 Indexing Patterns of Noncubic Crystals (Graphical Methods) 304
10-5 Indexing Patterns of Noncubic Crystals (Analytical Methods) 311
10-6 the Effect of Cell Distortion on the Powder Pattern . 314
10-7 Determination of the Number of Atoms in a Unit Cell 316
10-8 Determination of Atom Positions . 317
10-9 Example of Structure Determination . 320
Chapter 11 Precise Parameter Measurements. 324
11-1 Introduction 324
11-2 Debye-scherrer Cameras 326
11-3 Back-reflection Focusing Camems 333
11-4 Pinhole Cameras 333
11-5 Diffractometers 334
11-6 Method of Least Squares 335xll Contents
11-7 Cohen’s Method …. 338
11-8 Calibration Method . . 342
Chapter 12 Phase-diagram Determination . . . 345
12-1 Introduction . 345
12-2 General Principles . . 346
12-3 Solid Solutions . 351
12-4 Determination of Solvus Curves (Disappearing-phase Method) 354
12-5 Determination of Solvus Curves (Parametric Method) 356
12-6 Ternary Systems 359
Chapter 13 Order-disorder Transformations 363
13-1 Introduction . 363
13-2 Long-range Order in Aucus 363
13-3 Other Examples of Long-range Order 369
13-4 Detection of Superlattice Lines 372
13-5 Short-range Order and Clustering 375
Chapter 14 Chemical Analysis by Diffraction 378
14-1 Introduction 378
Qualitative Analysis
14-2 Basic
Principles 379
14-3 Hanawait Method 379
14-4 Examples of Qualitative Analysis 383
14-5 Practical Difficulties 386
14-6 Identification of Surface Deposits 387
Quantitative Analysis (Single Phase)
14-7 Chemical Analysis by Parameter Measurement 388
Quantitative Analysis (Multiphase)
14-8 Basic Principles . . . 388
14-9 Direct Comparison Method . . . 391
14-10 Internal Standard Method . . . 396
14-11 Practical Difficulties . . . 398
Chapter 15 Chemical Analysis by Fluorescence 402
15-1 Introduction . … 402
15-2 General
Principles . . 404
15-3 Spectrometers … . 407
15-4
Intensity and Resolution . . . 410
15-5 Counters …. . 414
15-6 Qualitative Analysis …. … 414
15-7 Quantitative Analysis … . . 415
15-8 Automatic
Spectrometers . . 417
15-9 Nondispersive Analysis ….. . 419
15-10 Measurement of Coating Thickness
Cohen’s Method
Calibration Method
Chapter 12 Phase-diagram Determination 345
12-1 Introduction 345
12-2 General Principles 346
12-3 Solid Solutions 351
12-4 Determination of Solvus Curves (Disappearing-phase Method) 354
12-5 Determination of Solvus Curves (Parametric Method) 356
12-6 Ternary Systems 359
Chapter 13 Order-disorder Transformations 363
13-1 Introduction 363
13-2 Long-range Order in Aucu3 363
13-:-3 Other Examples of Long-range Order 369
13-4 Detection of Superlattice Lines 372
13-5 Short-range Order and Clustering 375
Chapter 14 Chemical Analysis by Diffraction 378
14-1 Introduction 378
Qualitative Analysn;
14-2 Basic Principles 379
14-3 Hanawalt Method 379
14-4 Examples of Qualitative Analysis 383
14-5 Practical Difficulties 386
14-6 Identification of Surface Deposits 387
Quantitative Analysis (Single Phase)
14-7 Chemical Analysis by Parameter Measurement 388
Quantitative Analysis (Multiphase)
14-8 Basic Principles 388
14-9 Direct Comparison Method 391
]4-10 Internal Standard Method 396
14-11 Practical Difficulties 398
Chapter 15 Chemical Analysis by Fluorescence 402
15-1 Introduction 402
15-2 General Principles . 404
15-3 Spectrometers. 407
15-4 Intensity and Resolution 410
15-5 Counters. 414
15-6 Qualitative Analysis 414
15-7′ Quantitati~e Analysis 415
15-8 Automatic Spectrometers 417
15-9 Nondispersive Analysis 419
1iho Measurement of Coating Thickness 421contents Xiil
Chapter 16 Chemical Analysis by Absorption . . . 423
16-1 Introduction . . . … 423
16-2 Absorption-edge Method . . … 424
16-3 Direct-absorption Method (Monochromatic Beam) . 427
16-4 Direct-absorption Method (Polychromatic Beam) 429
16-5 Applications . . 429
Chapter 17 Stress Measurement . … 431
17-1 Introduction . 431
17-2 Applied Stress and Residual Stress . . 431
17-3 Uniaxial Stress
. . 434
17-4 Biaxial Stress . 436
17-5 Experimental Technique (Pinhole Camera) 441
17-6 Experimental Technique (Diffractometer) 444
17-7 Superimposed Macrostress and Microstress 447
17-8 Calibration 449
1 7-9 Applications 451
Chapter 18 Suggestions for Further Study . 454
18-1 Introduction 454
18-2 Textbooks . 454
18-3 Reference Books . 457
18-4 Periodicals 458
Appendixes
Appendix 1 Lattice Geometry . 459
Al-1 Plane Spacings 459
Al-2 Cell Volumes . . 460
Al-3
Interplanar Angles . . . 460
Appendix 2 the Rhombohedral-hexagonal Transformation 462
Appendix 3 Wavelengths (in Angstroms) of Some Characteristic
Emission Lines and Absorption Edges . . . 464
Appendix 4 Mass Absorption Coefficients and Densities . 466
Appendix 5 Values of Sin2 8 . 469
Appendix 6 Quadratic Forms of Miller Indices . . . 471
Appendix 7 Values of (Sin 0)/x . . . 472
Appendix 8 Atomic Scattering Factors . 474
Appendix 9 Multiplicity Factors for Powder Photographs . * . 477
Appendix 10 Lorentz-polarization Factor 478
Appendix 11 Physical Constants . 480
Contents
Chapter 16 Chemical Analysis by Absorption
Hh Introduction
16-2 Absorption-edge Method
16-3 Direct-absorption Method (Monochromatic Beam)
16-4 Direct-absorption Method (Polychromatic Beam)
16-5 Applications
Chapter 17 Stress Measurement
17-1 Introduction
17-2 Applied Stress and Residual Stress
17-3 Uniaxial Stress
17-4 Biaxial Stress
17-5 Experimental Technique (Pinhole Camera)
17-6 Experimental Technique (Diffractometer)
17-7 Superimposed Macrostress and Microstress
17- X Calibration
17-9 Applications
Chapter 18 Suggestions for Fultther Study
18-1 Introduction
18-2 Textbooks
18-3 Reference Books
18-4 Periodir.als
Appendixes
Appendix I Lattice Geometry
Ai-l Plane Spacings
Ai-2 Cell Volumes
Ai-3 Interplanar Angles
Appendix 2 the Rhombohedral-hexagonal Transformation 462
Appendix 3 Wavelengths (in Angstroms) of Some Characteristic
Emission Lines and Absorption Edges 464
Appendix 4 Mass Absorption Coefficients and Densities 466
Appendix 5 Values of Sin! (J • 469
Appendix 6 Quadratic Forms of Miller Indices 471
Appendix 7 Values of (Sin (J)/,a 472
Appendix 8 Atomic Scattering Factors 474
Appendix 9 Multiplicity Factors for Powder Photographs 477
Appbndix 10 Lorentz-polarization Factor
Appendix 11 Physical Constants .
478
480xiv Contents
Appendix 12 International Atomic Weights, 1953 481
Appendix 13 Crystal Structure Data 482
Appendix 14 Electron and Neutron Diffraction 486
A14-1 Introduction . … . . 486
A14r-2 Electron Diffraction … . 486
A14-3 Neutron Diffraction …. . 487
Appendix 15 the Reciprocal Lattice . . 490
A15-1 Introduction . …. .490
A15-2 Vector
Multiplication . … 490
A15-3 the
Reciprocal Lattice . . … 491
A15-4 Diffraction and the
Reciprocal Lattice . 496
A15-5 the
Rotating-crystal Method . 499
A15-6 the Powder Method . 500
A15-7 the Laue Method . . 502
Answers to Selected Problems . 506
Index… 509
Xiv Contents
Appendix 12 International Atomic Weights, 1953
Appendix 13 Crystal Structure Data .
Appendix 14 Electron and Neutron Diffraction
Al4-1 Introduction .
Ai4-2 Electron Diffraction
Ai4-3 Neutron Diffraction
Appendix 15 the Reciprocal Lattice
Al5-1 Introduction
Ai5-2 Vector Multiplication
Ai5-3 the Reciprocal Lattice
Ai5-4 Diffraction and the Reciprocal Lattice
Ai5-5 the Rotating-crystal Method
Ai5-6 the Powder Method
Ai5-7 the Laue Method .
Answers to Selected Problems
Index.
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