X-Ray and Neutron Diffraction in Nonideal Crystals

Mục Lục

 

1. Distribution of the Scattering Intensity.
General Aspects . . . . . . . . . . . . 1
1.1 Diffraction Techniques
for Analyzing Imperfections in Crystals 1
1.2 Kinematical Theory of Scattering … 8
1.2.1 Dynamical and Kinematical Theories . 8
1.2.2 X-Ray Scattering Intensity. . . . . . 9
1.2.3 Scattering Cross Section for Thermal Neutrons 15
1.2.4 Applicability Range for the Kinematic Theory 19
1.3 Scattering by Perfect Crystals of Finite Size ….. 22
1.3.1 Intensity Distribution in Reciprocal Lattice Space:
Form Function …… 22
1.3.2 Intensity Distribution in the
Debye Diffraction Pattern 27

1.4 Scattering in Undistorted Crystals Containing
Microscopic Cavities or Inclusions . . . . . 29
1.5 Scattering by Crystals Containing Defects
of Arbitrary Type. Classification of Defects. 33
1.5.1 Analysis of Scattering by Imperfect Crystals 34
1.5.2 Scattering by Crystals
with Randomly Distributed Defects. . . . 36
1.5.3 Classification of Defects . . . . . . . . . 42
1.5.4 Diffuse Scattering by Crystals Containing
First-Class Defects Under Weak Overlap
of the Displacement Fields of Individual Defects 48
1.5.5 Approximation of Smoothly Varying Distortions 49
1.5.6 Scattering Intensity with
Correlated Arrangement of Defects . . . . . . . 53
1.6 Harmonic Analysis of the X-Ray Line Shapes ….. 56
1.6.1 Fourier Coefficients for the Intensity Distributions
of X-Ray Lines . . . . . . . . 57
1.6.2 Limiting Cases of Nondistorted
and Large-Size Crystallites .. 61
1.6.3 Analysis of Crystallite Size and Distortions . 67

XVI Contents
2. Static Displacements in Crystals with Bounded Defects .
2.1 Fluctuation Waves of Defects Concentration
and Static Displacements …….. .
2.1.1 Symmetry of Defects . . . . . . .
2.1.2 The Defect Distribution in Terms of Static

75
75
77
Concentration Waves ………. 80
2.1.3 Static Displacement Waves. . . . . . . . 83
2.2 Macroscopic Theory for the Static Displacement Waves 85
2.2.1 Long-Wavelength Fluctuation Waves
and the Free Energy of the Anisotropic
Elastic Continuum . . . . . . . . . . 85
2.2.2 Amplitudes of the Fluctuation Waves
of Static Displacements . . . . . . . . . . . . 87
2.2.3 Fourier Components of the Static Displacements
in the Continuum Description . . . . . 92
2.2.4 Simplifications Introduced by Symmetry . . . 95
2.2.5 Fluctuation Waves in Thin Films . . . . . . . 99
2.3 Microscopic Theory for the Static Displacement Waves 103
2.3 .1 Free Energy of Distorted Crystal
with Bravais Lattice . . . . . . . . . . . . 103
2.3.2 Transition to the Long-Wave Approximation
and the Related Force Constants . . . . . . 106
2.3.3 Crystals of Arbitrary Structure . . . . . . . 111
2.4 Static Displacement Fields Around Bounded Defects 112
2.4.1 Atom Displacements Far from Defects . . . 112
2.4.2 Atomic Displacements Near Defects, Green Functions
and Mean Squares of Static Displacements 124

2.5 Static Distortions in Quasi-One-Dimensional
and Quasi-Two-Dimensional Crystals. . . . . . . 130
2.5.1 Discreteness of the Lattice and Spatial Dispersion. 130
2.5.2 Static Distortion Fields of Defects
in Strongly-Anisotropic Crystals . . . . . 136
3. Positions and Intensities of Regular Reflection Peaks . . . . . . . 147
3.1 Shift of X-Ray Lines in Imperfect Crystals
and the Determination of Defect Concentrations 147
3.1.1 Influence of Defects on X-Ray Line Positions
and Estimated Crystal Sizes . . . . . . . . . 147
3.1.2 Studies of Vacancies in Crystals . . . . . . . 152
3.1.3 Complexes in Solid Solutions and Their Effect
on the Lattice Parameters . . . . . . . . . . 155
3.1.4 Dilation Effects Caused by Dislocation Loops 164
3.2 Regular Reflection Intensities in Perfect Crystals 166
3.2.1 Intensity Attenuation Factors. . . . . . . . . 166

Contents XVII
3.2.2 Debye-Waller Factor in Perfect Harmonic Crystals 168
3.2.3 Chain-Like and Layered Crystals …….. 177
3.2.4 Effect of Anharmonicity on the Debye-Waller Factor 182
3.3 Effect of Static Displacements on Intensities
of Regular Reflections . . . . . . . . . . . . . . . . . . 194
3.3.1 Debye-Waller Factor Due to Static Displacements 194
3.3.2 Effects in Crystals Containing Particles
of a New Phase or Dislocation Loops 201
3.3.3 Layered and Chain-Like Crystals … 206
3.3.4 Concentrated Solutions …….. 211
3.3.5 Experimental Results on Regular Reflection
Intensities in Imperfect Crystals . . . . . 218
3.4 Effect of Thermal Vibrations in Imperfect Crystals 223
3.4.1 Crystals with Low Defect Concentrations. 223
3.4.2 Concentrated Solutions ……… 228
3.5 Debye-Waller Factors in Dynamical Diffraction Effects 234
3.5.1 Anomalous Transmission . 234
3.5.2 X-Ray Fluorescence . . . . 237
3.5.3 Spatial Intensity Oscillations 239
3.5.4 Critical Potentials . . . . . 240
4. Diffuse Scattering of X-Rays and Neutrons
by Crystal Defects . . . . . . . . . . . 241
4.1 Weakly Distorted Crystals . . . . . . . 241
4.1.1 Scattering by Single Defects . . 241
4.1.2 Scattering Intensity Near Reciprocal Lattice Points:
Symmetry of Defects and Force Dipole Tensors. . 244
4.1.3 Scattering Intensity Distribution at Large Distances
from Reciprocal Lattice Points and Determination
of the Defect Configuration and the Force Field. 251
4.1.4 Diffuse Scattering and the Correlation
in Defect Positions ………… 254
4.1.5 Experiments on Scattering by Point Defects
in Irradiated Crystals and Dilute Solutions . 257
4.1.6 Scattering by Self-localized Electrons . . . 268
4.1.7 Diffuse Scattering Representation in Various
Experimental Techniques ……… 271
4.2 Effects of Groups of Point Defects, New-Phase Particles,
or Small-Radius Dislocation Loops ….. 275
4.2.1 Scattering by Large Bounded Defects
in Weakly Distorted Crystals. . . . . 275
4.2.2 Diffuse Scattering by Weakly Distorted Crystals
with Particles of a Second Phase
and Ageing of Solutions . . . . . . . . . . . . . . . 283

XVIII Contents
4.2.3 Diffuse Scattering by Small-Radius Dislocation Loops
in Strained and Irradiated Materials .. 294

4.3 Intensity Distribution for Scattering by Strongly
Distorted Crystals with Finite Defects ….. 303
4.3.1 Change in Scattering Intensity Distribution
with Increasing Defect Strength . 303
4.3.2 Integrated Intensity from Strongly
Distorted Crystals . . . . . . . . 305
4.3.3 Intensity Distribution in the Reciprocal Space. 308
4.3.4 The Debye Diffraction Pattern . . . . . . . . 314
4.3.5 Experiments on Strongly Distorted Ageing Alloys
and Irradiated Materials . . . . . . . . . 323
4.3.6 Nonrandom Arrangement of Finite Defects 327
4.4 Strongly Anisotropic Crystals . . . . . . 331
4.4.1 Quasi-Two-Dimensional Crystals 332
4.4.2 Quasi-One-Dimensional Crystals 342
4.5 Effect of Finite Defects in Thin Films
and Surface Layers on X-Ray Scattering . 349
4.5.1 Scattering Intensity for Imperfect Finite Crystals 350
4.5.2 Diffuse Scattering by Defects in Thin Films 352
4,5.3 Broadening of Regular Reflection Peaks
in Free Films with a Large Surface Area 354
4.5.4 Diffuse Scattering by Defects
in a Thin Surface Layer . . . . . 355

S. Scattering of X-Ray and Neutrons in Crystals
with Dislocations . . . . . . . . . . . . . . .
5.1 Broadening of Peaks by Randomly Distributed Defects

357
of the Second Class . . . . . . . . . . 358
5.1.1 Linear Dislocations . . . . . . 358
5.1.2 Large-Radius Dislocation Loops 369
5.1.3 Dislocation Dipoles . . . . . . 372
5.1.4 Stacking Faults and Split Dislocations 376
5.2 Effect of Nonrandom Dislocation Arrangement
on Scattering Intensity Distribution. . . . . . 381
5.2.1 Scattering by Crystals with Dislocation Walls
and a Dislocation Description for the Effects
Caused by Blocks and Cells . . . . . . .. …. 381
5.2.2 Correlation in the Uniform Dislocation
Ensemble and in Crystals with Nonuniform
Dislocation Arrangement. . 392

5.3 Diffraction Methods of Investigation
of Dislocation Ensembles . . . . . 406
5.3.1 Determination of Dislocation Density . 407

Contents XIX

5.3.2 Correlation and Inhomogeneity
in Dislocation Arrangement . . . . . . . . . . . . . 411
5.3.3 Dislocations in Narrow Small-Angle Walls (Boundaries)
and Excess Dislocations of a Given Sign 414
5.3.4 Diffraction Techniques for Analyzing
the Grain Boundaries 418

Appendices . .
A. Cumu1ant Expansion. . . . .
B. Equations for Amplitudes of Static Displacement Waves
for Various Crystal and Defect Symmetries . . . . . .
C. Microscopic Theory of Ak in Cubic Crystals . . . . .
D. Mean Squares of Static Displacements in fcc Crystals.
E. Calculation of the Function Til (p)
and the First Moment of the Intensity Distribution
for Strongly Deformed Crystals Containing
Limited-Size Defects ……. .
F. Calculation of T(p) for Homogeneous
Dislocation Ensemble
References

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