Theory of crystal dislocations

Contents......Page nabarro1_page0007.djvu
Short Titles......Page nabarro1_page0015.djvu
1.1. History of dislocation theory: reviews......Page 1
1.2. The geometry of single dislocations......Page 8
1.2.1. Edge and screw dislocations......Page 9
1.2.2. A dislocation of Burgers’s type cannot end within the medium......Page 13
1.2.3. Dislocations of the Weingarten-Volterra type......Page 16
1.2.4. Dislocations of the Somigliana type......Page 20
1.3.1. Conservative and non-conservative motions......Page 21
1.3.2. Dislocation sources......Page 24
1.4.1. Dislocation models of grain boundaries......Page 30
1.4.2. The motion of dislocation walls......Page 37
1.5. Continuous distributions of dislocations......Page 39
1.5.1. Dislocation movement, shear strain, and dislocation density......Page 42
1.5.2. Lattice curvature and torsion......Page 44
1.5.3. Lattice strain and incompatibility......Page 46
2.1.1.1. The edge dislocation......Page 53
2.1.1.2. The screw dislocation......Page 57
2.1.1.4. The inserted disk......Page 58
2.1.1.5. Tilt and twist boundaries......Page 59
2.1.2.1. Burgers’s solution for a dislocation loop......Page 62
2.1.2.2. Stresses in a bent or twisted crystal......Page 66
2.1.2.3. Kronei-’s formulae for the stresses......Page 69
2.1.2.4. Solutions for an anisotropic medium......Page 71
2.2. The elastic energy of dislocations......Page 72
2.2.1. Expression of the energy by volume or surface integrals......Page 73
2.2.3. Grain boundary energies......Page 76
2.2.4. Effect of external stresses......Page 80
2.3. The force on a dislocation......Page 82
2.3.1. Forces and couples between dislocations......Page 87
2.3.2. The mechanism of polygonization......Page 93
2.3.3. Deformation and kink bands......Page 100
2.4. Dislocation pile-ups......Page 106
2.4.1. Exact solutions......Page 107
2.4.2. The crack approximation and the integral equation......Page 110
2.5. Depression of the shear modulus......Page 114
2.6. The magnetic analogy......Page 118
3.1.1. Dislocations and disclinations......Page 120
3.1.2. Burgers dislocations in a crystal......Page 127
3.2.1. Exact and approximate solutions......Page 130
3.2.2. Application to epitaxial growth......Page 135
3.3. The model of Peierls and the basic solution of the governing equation......Page 137
3.3.1.1. Two dislocations of opposite sign......Page 141
3.3.1.2. Interphase boundaries......Page 143
3.3.1.3. Twist boundaries......Page 144
3.3.1.4. Tilt boundaries......Page 146
3.3.1.5. Barriers and pile-ups......Page 147
3.3.2. Variations and criticisms of the model......Page 150
3.3.2.1. Other laws of force......Page 151
3.3.2.2. Criticisms of the model......Page 153
3.5. Atomic models......Page 155
3.5.1. Face-centred cubic metals......Page 158
3.5.2. The effect of temperature on the width of a dislocation......Page 160
3.5.3. Alkali metals......Page 161
3.5.4. Ionic crystals......Page 163
3.6. Non-linear efiects......Page 164
3.6.1. The volume of a dislocation......Page 165
3.6.2. Grain boundaries of medium and large angle......Page 168
3.6.3. Cracked dislocations......Page 174
3.7. The Peierls force......Page 175
3.7.1. The physical significance of the Peierls force; kinks......Page 180
3.7.2. The theory of kinks......Page 187
4.1. Perfect dislocations......Page 189
4.1.1. The primitive representation......Page 190
4.1.2. Dislocation reactions......Page 194
4.1.3. Nodes......Page 198
4.2. Stacking faults and partial dislocations......Page 199
4.2.1. Stacking and stacking faults......Page 200
4.2.1.1. The face-centred cubic and hexagonal close-packed structures......Page 201
4.2.1.2. Other structures......Page 204
4.2.2. Partial dislocations......Page 210
4.2.2.1. The face-centred cubic lattice......Page 215
4.2.2.2. Other structures......Page 223
4.2.2.3. Dislocations in superlattices......Page 229
4.3. The choice of glide system......Page 230
4.3.1. Choice of glide direction......Page 231
4.3.2. Choice of glide plane......Page 236
4.3.3. Choice of dislocation line......Page 242
4.4.1. Extended and contracted nodes......Page 243
4.4.2. Networks......Page 245
4.5. The energy of a stacking fault......Page 247
4.6. Jogs......Page 252
4.7. Cross slip......Page 257
4.8. Self-trapping......Page 262
4.9. Prismatic punching and dimpling......Page 264
5.1. Elastic theory......Page 267
5.1.1. Interaction with a plane free boundary......Page 268
5.1.3. Dislocations in thin plates......Page 269
5.1.3.1. Dislocation line parallel to plane of plate......Page 270
5.1.3.2. Dislocation 1ine perpendicular to plane of plate......Page 271
5.1.4.1. Ends of cylinder clamped......Page 272
5.1.4.2. Whiskers......Page 273
5.1.4.3. Special problems of dislocations in whiskers......Page 276
5.2.1. The surface as a source of dislocations......Page 278
5.2.2. The surface as a barrier to dislocation movement......Page 282
5.2.4. The strength of thin foils and whiskers......Page 288
5.2.5. The Joffé effect......Page 292
5.2.6. The Rehbinder effect......Page 294
5.2.7. The theory of visible slip markings......Page 297
5.3. Dislocations and crystal growth......Page 302
5.3.1. Hollow dislocations and dimples......Page 303
5.3.2. Crystal growth......Page 305
5.3.3. Evaporation and etching......Page 314
5.3.4. Whisker growth......Page 322
5.3.5. Oxidation and catalysis......Page 324
5.3.6. Field emission microscopy of dislocations......Page 326
5.4. The origin of dislocations in growth......Page 327
5.4.1. The impingement of two lattices......Page 328
5.4.2. Thermal stresses......Page 329
5.4.3. Aggregation of lattice vacancies......Page 330
5.4.4. Segregation of impurities......Page 332
5.4.5. Crystals free from dislocations......Page 338
6.1. Dislocations, vacant sites, and interstitial atoms......Page 340
6.1.1. Point defects and dislocation climb......Page 341
6.1.2.1. Kinetics of climb......Page 349
6.1.2.2. Creep at a rate proportional to the stress......Page 355
6.1.2.3. Creep rates varying as a power of the stress......Page 357
6.1.2.4. Dekinking of whiskers......Page 358
6.1.3.1. Geometry of loops, helices, and tetrahedra......Page 360
6.1.3.2. Equilibrium of helices and loops......Page 368
6.1.3.3. Growth kinetics of dislocation loops......Page 371
6.1.3.4. Loops as sources of dislocations......Page 374
6.1.4.1. Dislocations formed by irradiation......Page 377
6.1.4.2. Crowdions and focusing collisions......Page 378
6.1.5.1. The experimental evidence......Page 381
6.1.5.2. Mechanisms of vacancy production......Page 382
6.1.5.3. Jog dragging......Page 384
6.2. Dislocation locking......Page 391
6.2.1.1. Interaction with the hydrostatic stress......Page 399
6.2.1.2. Electrostatic interactions......Page 404
6.2.1.3. Binding of vacancies and interstitial ions......Page 406
6.2.1.4. Interaction with shear stresses......Page 418
6.2.1.5. Extended dislocations......Page 421
6.2.2. Kinetics of dislocation locking......Page 424
6.2.2.1. Stationary dislocations......Page 425
6.2.2.2. Slowly moving dislocations......Page 431
6.2.3. The theory of yield phenomena......Page 435
6.2.3.1. Yield in single crystals......Page 436
6.2.3.2. Yield in polycrystals......Page 449
6.2.3.3. Repeated yielding......Page 458
6.3.1. Short-circuit diffusion......Page 464
6.3.1.1. Short-circuit difiusion along isolated dislocations......Page 467
6.3.1.2. Small-angle boundaries......Page 469
6.3.1.3. Large-angle boundaries......Page 471
6.3.1.4. Bulk diffusion......Page 472
6.3.2. Dislocations as nuclei of precipitation......Page 474
6.3.2.1. Theory of nucleation on dislocations......Page 476
6.3.2.2. Experimental observations......Page 480
7.1.1. The elastic field of a moving dislocation......Page 482
7.1.1.1. The elastic model......Page 483
7.1.1.2. The model of Frenkel and Kontorova......Page 489
7.1.1.3. The model of Peierls......Page 491
7.1.2. Supersonic dislocations......Page 492
7.1.3. The equation of motion of a dislocation......Page 493
7.1.4. The electromagnetic analogy and the force on a moving dislocation......Page 496
7.2. Resistance to the motion of dislocations......Page 505
7.2.1. The experimental evidence......Page 506
7.2.2.1. Eflect of the Peierls force......Page 510
7.2.2.2. Changes in structure of the dislocation core......Page 513
7.2.2.3. Acoustic dispersion......Page 516
7.2.3. Drag by dissipation of elastic strains......Page 517
7.2.3.1. Thermoelastic damping......Page 518
7.2.3.2. Phonon viscosity......Page 520
7.2.4. Scattering of phonons and electrons......Page 522
7.2.4.1. Phonon scattering......Page 523
7.2.4.2. Electron scattering......Page 529
7.2.5. Drag by the displacement of dissolved atoms......Page 530
7.3. Dislocations and internal friction......Page 532
7.3.1. Damping by fixed dislocations......Page 537
7.3.2. The Bordoni peak......Page 538
7.3.3. Damping by pinned dislocations......Page 546
7.3.4. The model of Granato and Lucke......Page 552
7.3.5. Damping at high temperatures and at large amplitudes......Page 557
7.3.6. Damping by the displacement of dissolved atoms......Page 559
8.1. Development of the linear theory......Page 562
8.2. Dislocated lattices and the geometry of generalized spaces......Page 565
8.2.1. The geometry of generalized spaces......Page 567
8.2.2. The geometry of dislocated lattices ......Page 676
8.2.2.1. Bilby’s geometry of the lattice ......Page 577
8.2.2.2. Special examples ......Page 579
8.2.2.3. Surface dislocations ......Page 581
8.3. Stresses in the non-linear theory ......Page 582
8.4.1. The dynamics of non-holonomic systems ......Page 585
8.4.3. Time-like dislocations......Page 588
8.4.4. Dislocations in zoology, botany, and geophysics......Page 590
9.1. Electrical efiects in ionic crystals......Page 595
9.1.1. Charged dislocations......Page 596
9.1.2. Charged points of emergence and jogs......Page 599
9.1.3. Atmospheres of charged point defects......Page 602
9.1.4. Plastic deformation and charge transport......Page 604
9.1.5. Mechanical effects of large electric fields......Page 609
9.1.6. Dislocations in piezoelectric and ferroelectric crystals......Page 611
9.2.1. The electrostatic field of a dislocation......Page 613
9.2.1.1. The edge dislocation in linear elasticity......Page 616
9.2.1.2. Non-linear elasticity......Page 617
9.2.2.1. Electrical resistivity of deformed metals......Page 618
9.2.2.2. Theory of the contribution of dislocations to the electrical resistivity......Page 627
9.2.2.3. Magnetoresistivity and the Hall effect......Page 631
9.2.2.4. Thermoelectric power......Page 633
9.2.2.5. Electronic properties in normal metals at low temperatures......Page 634
9.2.2.6. Dislocations and superconductivity......Page 635
9.2.2.7. Dislocations and the electrical resistivity of transition metals and alloys......Page 639
9.3. Electrical effects in semiconductors......Page 641
9.3.1.1. The equilibrium charge distribution......Page 643
9.3.1.2. Electron transport......Page 645
9.3.1.3. Recombination of electrons and holes......Page 647
9.3.1.4. Electronic noise and junction breakdown......Page 648
9.3.2. Walls of dislocations......Page 650
9.3.2.1. Recombination at a wall......Page 651
9.3.2.2. The wall as a blocking layer......Page 652
9.3.2.3. Mobility within the wall......Page 653
9.3.2.4. Formation of etch pits......Page 654
9.3.3. Clean and dirty dislocations......Page 655
9.3.4. Electromechanical effects......Page 656
9.4. Emission of electrons from the free surface......Page 657
10.1. Dislocations in ferromagnetic crystals......Page 658
10.1.1. The approach to saturation......Page 660
10.1.2. The initial permeability......Page 663
10.1.3. The coercivity......Page 667
10.2. Dislocations and electron spins......Page 670
10.2.1. Moment-bearing stacking faults and dislocations......Page 671
10.2.2. Paramagnetic and ferromagnetic resonance......Page 674
10.3. Widths and shapes of nuclear magnetic resonance lines......Page 675
10.4. Conductors with long free paths for electrons......Page 681
11.1.1. The energy and free energy of a dislocation......Page 683
11.1.2. The dislocation theory of melting and of the liquid state......Page 688
11.2. The storage and dissipation of energy during plastic deformation......Page 691
11.2.1. The energy stored......Page 692
11.2.2. The heat generated: jerky flow......Page 694
11.2.3. The mechanisms of energy dissipation......Page 699
11.2.4. The thermodynamics of plastic deformation......Page 701
11.3. Thermally activated processes......Page 704
11.3.1.1. The original model of Becker and Orowan......Page 706
11.3.1.2. The expansion of a loop of dislocation......Page 707
11.3.1.3. A dislocation crossing a linear potential barrier......Page 711
11.3.1.5. Forest cutting......Page 715
11.3.1.6. Jog dragging......Page 716
11.3.2. The statistics of activation processes: creep......Page 717
11.3.2.1. Exhaustion creep......Page 719
11.3.2.2. Creep limited by work hardening: Andrade creep......Page 723
11.3.2.3. Steady-state creep......Page 725
11.3.2.4. Experimental analysis of the activation process......Page 727
11.3.3. The thermal activation of kinks......Page 735
11.3.4. Quantum effects and the influence of the zero-point energy......Page 741
11.4. Dislocations and thermal conductivity......Page 746
11.4.1. Stationary dislocations......Page 748
11.4.3. The experimental evidence......Page 749
12.1.1. Photoelastic patterns......Page 752
12.1.2. Light scattering by dislocations......Page 753
12.1.3. Dislocations, optical absorption, and excitons......Page 756
12.2. Photoplasticity......Page 760
12.3. The scattering of X-rays, electron and neutron beams......Page 761
12.3.1. The image of a dislocation......Page 763
12.3.1.1. The geometrical optical theory......Page 764
12.3.1.2. The kinematical theory......Page 766
12.3.1.3. The dynamical theory......Page 771
12.3.1.4. The experimental observation of dislocations by X-rays......Page 773
12.3.2. Diffraction by a single dislocation......Page 777
12.3.2.1. The geometrical theory......Page 778
12.3.2.2. Diffraction theory......Page 779
12.3.2.3. Small-angle scattering of X-rays and neutrons......Page 780
12.3.3.1. Analysis of line broadening......Page 783
12.3.3.2. Special techniques and methods of analysis......Page 786
12.4. Dislocations in moiré patterns......Page 792
APPENDIX......Page 794
AUTHOR INDEX......Page 795
SUBJECT INDEX......Page 817