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LAKE, Peter Babcock, 1942- DISLOCATION STRUCTURES IN GERMANIUM SINGLE CRYSTALS DEFORMED IN A DUAL GLIDE ORIENTATION.
The Ohio State University, Ph.D., 1969 Engineering, metallurgy
University Microfilms, Inc., Ann Arbor, Michigan DISLOCATION STRUCTURES IN GERMANIUM SINGLE CRYSTALS
DEFORMED IN A DUAL GLIDE ORIENTATION
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Peter Babcock Lake, B.S.
* * * * «
The Ohio State University 1969
Approved by
, / /' < <•■'( A Adviser Department of .Metallurgical Engineering Dedicated to my wife
Melinda
ii ACKNOWLEDGMENTS
I would like to acknowledge the assistance and advice of my adviser. Professor John Hirth. Among the
others who helped me greatly with this work, I would
like to especially thank Professor Glyn Meyrick, and
Mssrs. Ross Justus and Henry Pagean.
iii VITA
June 15. 1942 Born - Bethlehem, Pennsylvania
1964 . „ . , B.S», The Pennsylvania State University
1964 1969 . . National Science Foundation Trainee, Department of Metallurgical Engineering, The Ohio State University
iv CONTENTS
Page
ACKNOWLEDGMENTS ...... ill
VITA ...... Iv
LIST OP PLATES ...... vil
LIST OP FIGURES ...... *iv
LIST OP TABLES ...... xix
Chapter
I. INTRODUCTION ...... 1
A. Material Selection B. Dislocations in the Diamond Cubic Structure C. Previous Investigations of the Deformation Behavior and Dislocation Structure of Germanium D. Low Angle Grain Boundaries E. Studies of Equilibrium Node Angles In Iron P. Low Angle Boundaries in Face-Centered Cubic Crystals G. Conclusions Drawn from the Introductory Chapter
II. EXPERIMENTAL PROCEDURES AND TECHNIQUES .... 66
A. Introduction B. Technique Used for Producing Twist Networks C. Specimen Preparation and Deformation D. Equations for Calculating True Glide Stress and Strain
III. RESULTS— DEFORMATION CURVES AND SLIP TRACE ANALYSIS ...... 100
A. Stress-Strain Curves B, Observations of Slip Plane Rotation and Crystal Surfaces v CON'j'Er: j S ( Could , }
Chaptc r Rage
iv . d is c u s s r o:: os Mucsoscoiir c C:i/ . -r.C'i■ jri ■■ j cs oi-’ r.iv/ii:■. : defoim atio':...... 13^
A . 1 ma I T’> I i V' V c j u - R in;-1f- R 1 -t P
v. PhOCEueoSs and techie !dues for electron microscopy ...... 159
A. Tn 5 Jin :: nr J’.. Opera tin'; To ehni quc s
Vi. kesudvs—electron MrcRosco:-'Y ...... 175 _ j> A. Cry stall a Strained at 5 x 10 ‘/sec and 5 x 10 “ 3 /s e c B. Dislocation Configurations in Annealed Cry stain
VI I. DIB CUBS! C D ! ...... 2hlj
A. The Bnorrp,' of Straight Dis!ocatlons B. Application of an Analytical Method to Obtain Values of K/t(0°) and K^(60°) C. Method Used to Obtain a Working Equation for the Variation of K^C/3) i:.tth/3 D. Derivation of the Expected Node1 Angies E. The Dislocation Structure in Ar,-defamed Crystals F. Dislocation Structui^es in Annealed Crystalo v i i i .con chu s i o n s ...... 3^6
ATT EN D U E S
A...... 3^9
B ...... 39?
C...... 399
D ...... 362
E...... 365
REFERENCES...... 372 vi PLATES
Plate Page
1. Kink Bands in Single S l i p ...... 30
2. Dislocations on the (111) Twist Plane ...... 34
3. Dissociated Nodes in Silicon ...... 35
Undissociated Nodes in Germanium ...... 35
5. Dislocation Networks in I r o n ...... 53
6. Macroscopic Shapes of Specimens C-55, C-53, C-51 and C - 5 0 ...... 121
7. Slip Traces on (Oil) Pace— Set I ...... 124
8. Slip Traces on (Oil) Face— Set I I ...... 125
9. Slip Traces on (8ll) F a c e ...... 128
10. Surface Undulations on (8ll) F a c e ...... 129
11. High Magnification InterferenceContours .... 130
12. Specimen C-42, (Oil) F a c e ...... 131
13. Specimen C-33, (Oil) F a c e ...... 131
14. Specimen C-39, (Oil) F a c e ...... 133
15. Specimen C-37, (Oil) F a c e ...... 133
16. KIkuchi and Diffraction Patterns for Germanium . 170
17. Specimen C-26, Strained 4.3? l8l
18. Specimen C-26, Strained 4 . 3 ? ...... 181
19. Specimen C-26, Strained 4 . 3 ? ...... 181
20. Specimen C-29, Strained 6.2? 182
21. Specimen C-29, Strained 6.2? 182
vii PLATES (Contd.)
Plate Page
22. Specimen C-29, Strained 6 , 2 ? ...... 182
23. SpecimenC-29, Strained 6,2? 183
24. SpecimenC-55, Strained 12 . 6 ? ...... 18*4
25. SpecimenC-55, Strained 12 . 6 ? ...... 184
26. SpecimenC-55, Strained 12.6? ...... 18*4
27. SpecimenC-55, Strained 12.6? ...... 185
28. SpecimenC-55, Strained 12 . 6 ? ...... 186
29. Specimen C-55, Strained 12.6? ...... 186
30. SpecimenC-55, Strained 12.6? ...... 186
31. Specimen C-42, Strained 4.2? 187
32. Specimen C-42, Strained 4.2? ..... 187
33. Specimen C-42, Strained 4 . 2 ? ...... 187
34. SpecimenC-42, Strained 4.2? 188
35. SpecimenC-42, Strained 4.2? 188
36. SpecimenC-42, Strained 4.2? 188
37* SpecimenC- 36, Strained 8.6? 189
38. SpecimenC- 36, Strained 8.6? 189
39. SpecimenC-36, Strained 8.6? 189
40. SpecimenC- 38, Strained 16.0? ...... 190
41. Specimen C-38, Strained 16.0? ...... 190
42. Specimen C-38, Strained 16.0? ...... 190
43. SpecimenC-38, Strained 16.0? ...... 190
viii PLATES (Contd.)
Plate Page
44. Specimen C-40, Strained 16. 0% . . 191
45. Specimen C-40, Strained 16.0% , . 191
46. Specimen C-40, Strained 16. 0% . 191
47. Specimen 0-40, Strained 16. 0% 191
48. Specimen C-37, Strained 25. 4% . . 192
49. Specimen C-37, Strained 25. 4$ . . 192
50. Specimen C-37, Strained 25.4$ . . 192
51. Specimen C-26, Annealed at 850°C for 36 Hours . . 203
52. Specimen C-26, Annealed at 850°C for 36 Hours . . 203
53. Specimen C-26, Annealed at 850°C for 36 Hours . . 203
54. Specimen C-26, Annealed at 850°C for 36 Hours . . 204
55. Specimen C-26, Annealed at 850°C for 36 Hours . . 204
56. Specimen C-26, Annealed at 850°C for 36 Hours . . 204
57. Specimen C-55, Annealed at 750°C for 24 Hours . , 205
58. Specimen C-55, Annealed at 750°C for 24 Hours . . 205
59. Specimen C-55, Annealed at 750°C for 24 Hours . . 205
60. Specimen C-55, Annealed at 750°C for 24 Hours . . 205
61. Specimen C-55, Annealed at 850°C for 24 Hours . . 206
62. Specimen C-55, Annealed at 850°C for 24 Hours . . 206
63. Specimen C-55, Annealed at 850°C for 24 Hours . , 206
64. Specimen C-55, Annealed at 850°C for 24 Hours . . 207
65. Specimen C-25, Annealed at 850°C for 36 Hours . . 208
lx PLATES (Contd.)
Plate Page o 00 o o 66. Specimen C-25, Annealed at U1 for 36 Hours * ft 209
67. Specimen C-25, Annealed at 850°C for 36 Hours • ft 210
68. Specimen C-25, Annealed at 850°C for 36 Hours • ft 210
69. Specimen C-25, Annealed at 850°C for 36 Hours • ft 210
70. Specimen 0-27, Annealed at 850°C for 36 Hours ft 211
71. Specimen C-27, Annealed at 850°C for 36 Hours • ft 211
72. Specimen C-27, Annealed at 850°C for 36 Hours • ft 211
73. Specimen C-27, Annealed at 850°C for 36 Hours ft ft 211
74. Specimen C-27, Annealed at 850°C for 36 Hours • ft 212
75. Specimen C-27, Annealed at 850°C for 36 Hours • ft 212
76. Specimen C-27, Annealed at 850°C for 36 Hours • ft 212
77. Specimen C-27, Annealed at 850°C for 36 Hours • ft 212 00 t- . Specimen C-27, Annealed at 850°C for 36 Hours * ft 213
79. Specimen C-27, Annealed at 850°C for 36 Hours * ft 214
80. Specimen C-27, Annealed at 850°C for 36 Hours -• ft 214
81. Specimen C-42, Annealed at 700°C for 24 Hours ft ft 230 GO fu * Specimen C-42, Annealed at 700°C for 24 Hours • ft 2 30
83. Specimen C-36, Annealed at 700°C for 24 Hours • ft 2 31
84. Specimen C-36, Annealed at 700°C for 24 Hours •• 231 00 in
* Specimen C-38, Annealed at 700°C for 6 Hours ft ft 232
86. Specimen C-38, Annealed at 700°C for 6 Hours * ft 232
87. Specimen C-38, Annealed at 700 °c for 6 Hours • ft 232
88. Specimen C-38, Annealed at 700 °c for 24 Hours • ft 233
x PLATES (Contd.)
Plate Page
89. Specimen C-38 Annealed at 700°C for 24 Hours • . 233
90. Specimen C-38 Annealed at 700°C for 24 Hours *• 233
91. Specimen C-38 Annealed at 700°C for 24 Hours •• 234
92. Specimen C-38 Annealed at 700° C for 24 Hours *• 234
93. Specimen C-38 Annealed at 700° C for 24 Hours • • 234
94. Specimen C-40 Annealed at 700°C for 24 Hours • 235
95. Specimen C-40 Annealed at 700°C for 24 Hours *• 235
96. Specimen C-43 Annealed at 700°C for 24 Hours ** 236
97. Specimen C-43 Annealed at 700°C for 24 Hours • 2 36
98. Specimen C-43 Annealed at 700 °c for 24 Hours *• 236
99. Specimen C-37 Annealed at 700°C for 24 Hours ♦ * 237
100. Specimen C-37 Annealed at 700°C for 24 Hours •• 2 37
101. Specimen C-37 Annealed at 700 °c for 24 Hours * ♦ 237
102. Specimen C-37 Annealed at 700 °c for 24 Hours • a 2 38
103. Specimen C-37 Annealed at 700° C for 24 Hours •# 238
104. Specimen C-37 Annealed at 700°C for 24 Hours • * 238
105. Specimen C-37 Annealed at 700 °c for 24 Hours •• 2 39
106. Specimen C-37 Annealed at 700 °c for 24 Hours • a 239
107. Specimen C-37 Annealed at 700 °c for 24 Hours •• 240
108. Specimen C-37 Annealed at 700 °c for 24 Hours •• 240
109. Specimen C-37 Annealed at 700 °c for 24 Hours •• 2 40
110. Specimen C-37 Annealed at 700 °c for 24 Hours •• 2 40
xl !j],.v.i(contd.)
V 1 at c Taro
131. Specimen C-39, Annealed at y00c> C for 23 Hour;; 29l
11?. Sj-r ci ;;x-n C-39, Annealed a L YOO°C for 2 9 Hour/. 231
113. Specimen C-39, Annealed at YOG'* C for 2 3 Hours 231 lit. Specimen C-39, Annealed at 7uO°C for '. 23 Hourr. 232
119. Specimen c-39, Annealed at 70 0°C for 2 9 Hours 2 32
116 , Specimen C-39, Annealed at 700°C for 29 Hours 2 32
117. Specimen C-39, Annealed at 700°C for 29 Hourr. 23 3
4 Ilf). Specimen C-39, Annealed at 700°C for 29 Hours 233
119. Specimen C-39, Annealed at 70 0°C for 2 9 Hourr 233
120. Specimen C-37, Unsymmetrical Node ...... 239
121. Specimen C-37, Unsymne trical Node ...... 2 39
122. Specimen C-37, Unsymmetrical Node ...... 236
12 3. Specimen C-37, Unsymrnctrlcal N o d e ...... 236
129. Polygonal Dislocation in Annealed Crystal . . 321
12^. Polygonal Dislocation in Annealed Crystal . . 329
126. Stranper Dislocation Interaction ...... 333
127. Symmetrical Nodes vrbich Appear Dissociated . . 337
12 0. Sc rev; Nodes in Silicon ...... 330
129. Symmetrical Nodes with Double Inapes ...... 331
130. Theoretically Derived Dislocation Node Contrast 33 3
131. Jet-polished Dimple - Interference Mi croscopy . 35 3
132. Jet-polished Dimple - Interference Microscopy , 353
133. Jet-polished Dimple -Interference Microscopy . 353
xii PLATES (Contd.)
Plate Page
134. Jet-polished Dimple - Interference Microscopy . 353
135. Holes in Thin Foils Made by Jet-polishing . . . 355
136. Holes in Thin Foils Made by Jet-polishing . . . 355
137. Holes in Thin Foils Made by Jet-polishing . . . 355
138. Holes in Thin Foils Made by Jet-polishing . . . 356
139. Holes in Thin Foils Made by Jet-polishing . . . 356
140. Holes in Thin Foils Made by Jet-polishing . . 356
xiii FIGURES
Figure Page
1. Diamond Cubic Lattic Projected Normal to the ( I l O ) 6
2. Screw Dislocation in the Diamond Cubic Lattice . 8
3. 60° Dislocation in the Diamond Cubic Lattice . . 9
4. 30° Dislocation In the Diamond Cubic Lattice . . 10
5. Edge Dislocation in the Diamond Cubic Lattice . . 11
6. Stress-strain Curve in Compression ...... 17
7. Stress-strain Curve in Tension...... 18
8. Dislocation Density as a Function of Strain . . . 22
9. Variation in Stress-strain Behavior as a Function of Temperature-Tension 2 3
10. Variation in Stress-strain Behavior as a Function of Orientation-Tension ...... 2**
11. Variation In Stress-strain Behavior as a Function of Orientation-Compression ...... 25
12. Variation in Stress-strain Behavior as a Function of Temperature-Compression ...... 26
13. Variation in Stress-strain Behavior as a Function of Strain Rate ...... 28
1*1. Junction of Three Grain Boundaries ...... 39
15. Crossing Dislocations ...... *11
16. Reaction of Crossing Dislocations...... *11
17. Hexagonal and Lozenge-shaped Arrays ...... *1*1
18.' Twist Orientation ...... *16
xiv FIGURES (Contd.)
Figure Page
19. Burgers Vector Content of Dislocation Net .... 46
20. Intersecting Dislocations in I r o n ...... 50
21* Twist Boundary in Iron {lioj Plane ...... 50
22. Twist Boundary in Iron (l00~| Pl a n e ...... 51
23. Twist Boundary in Iron {l00} Pl a n e ...... 51
24. Schematic of Dislocation Node in I r o n ...... 55
25. Polar Plot of Orientation Dependence of K in Iron ...... 57
26. K Node In Face-centered Cubic Structure ..... 60
27. P Node in Face-centered Cubic Structure ..... 61
28. Node Types and Pairs in Face-centered Cubic S t r u c t u r e ...... 63
,29. Coordinates for Simple Compression ...... 70
30. Iso-m Contours for Single S l i p ...... 72
31. Iso-m Contours for Dual Slip ...... 74
32. [HI] Stereographlc Projection ...... 75
33. Variation in Schmid Factor with Orientation . . . 76
34. Dimensions of Compression Specimens ...... 78
35. Furnace Mounted on Instron ...... 79
36. (Ill) Oriented Slices from Deformed Crystal . . . 83
37. Ideal and Non-ideal Deformation ...... 88
38. Relationship between X0(L0) and X^(L^) ...... 90
39. Mi and mi as a Function of AL/L0 ...... 92
40. Gi as a Function of AL/LQ ...... 96 xv FIGURES (Contd.)
Figure Page
41. Glide Strain Versus AL/L0 ...... 99
42. Stress-Straln Curves - Set I ...... 101
43. Average Stress-Strain Curve - Set I ...... 103
44. Stress-Strain Curves - Set II ...... 105
45. Average Stress-Straln Curves - Set II ...... 106
46. Variation in Deformation Parameters as a Function of Strain r a t e ...... 109
47. Variation in Deformation Parameters as a Function of Temperature...... Ill
48. Orientation of Single Slip Specimens ...... 113
49. Single and Dual Slip Stress-strain Curves .... 115
50. Single and Dual Slip Stress-strain Curves .... 118
51. X^ as a Function of...... 119
52. Slip Traces on (811) F a c e ...... 127
53. Slip Traces on (Oil) F a c e ...... 127
54. Dual Slip in Z i n c ...... 143
55. Dual Slip In Anthracene ...... 145
56. Dual Slip in Silver ...... 147
57. Orientation Changes During Dual Slip In Silver . 147
58. Iso-m Contours on (111) Stereographic Projection. 151
59. Reduced Gage Length ...... 156
60. Electro-chemical Jet-polishing Apparatus .... 162
6l; Electrical Circuit-Jet Polishing ...... 165
62. Thin Foil Cross-section...... 166 xvi FIGURES (Contd.)
Figure Page
63. Klkuchl Line Pattern— Face-centered Cubic .... 169
64. Diffraction Pattern in Diamond Cubic System . . . 171
65. Histogram - Set I ...... 1 . 200
66. Histogram - Set I I ...... 225
67. Operating Reflections for Plate 119 ...... 244
68. Angle of Twist (9) Versus Glide Strain ...... 247
69. Coordinates for Screw Component of Energy .... 256
70. Coordinates for 60° Component of Energy ...... 262
71. K(0) Versus /3 as Function of v ' ...... 266
72. K(0) Versus £ as Function of v and a ...... 267
73. Polar Plot of K ( j 3 ) ...... 269
74. K(/3) Versus /3 for Copper and Gold ...... 270
75- Schematic of Node Configuration Parameters . . . 272
76. Node Parameters for Type 3 a ...... 275
77. Node Parameters for Type 3a In EquilibriumShape. 275
78. Node Parameters for Type 3 b ...... 278
79. Node Parameters for Type 3b In Equilibrium Shape. 278
80. Variation in ^K^(0) for Node Types 3a and 3b . 279
81. Variation in Angle a as Function of u ...... 281
82. Stage I work-hardening Dislocation Configurations ...... 284
83. DIpole-jog ...... 286
84. Double K i n k s ...... 286
xvil }•' 1 OU'o . (Contd.)
Fifure Pape b'j . Dislocation I rite nr ct i onr ? or. r ib ].o in Face-- Cento nod Cubic J - d t l c o ...... ?91
86. Dissociated Dislocation Barrier ...... 29 3
87 - Si npl e and Dual .Slip in Tens :i on or Coe.prea s 1 on . 3-19
8 8 . Copy or Plato ] ?'1 Show!up Coo.':.dry of Conf J purati on bscci for Calculati on of Peic-rls E n e r p y ...... 3<-2
8 9 . Anpul an Gecuatricul Approximation Sho.m in Figure 8 8 ...... 32*3
9G . Supez’p^s 11 i on of Peicrls Troey.bs on the Hnorpy C u r v e ...... 331
91. Thompson Tetrahedron ...... 380
92 . The Origin of Kikucbl L i n e s ...... 300
93. Cones of Scattered Electrons ...... 360
9*1. Dislocation Lyinp Alonp the X^ Axis ...... 368
xviil TABLES
Table Page
1. Table of Dislocations In Diamond Cubic Lattice . 12
2. Dislocation Reactions and Products ...... 64
3. Germanium Crystals Oriented for Compression in the [144] D i r e c t i o n ...... 82
4. Values of A L / L o ) ...... 97
5. Set I— Stress-strain Results ...... 102
6. Set I— Specimen Strains ...... 102
7. Set II— 6l6°C Stress-strain Results ...... 104
8. Set II— 6l6°C Specimen S t r a i n s ...... 104
9. Set II--649°C Stress-strain Results ...... 107
10. Set II--649°C Specimen Strains ...... 107
11. Specimen C-75 Stress-strain Results ...... 110
12. Stress-strain Results for Dual Versus Single Slip - 5 x 10“4/Se c ...... 114
13. Stress-strain Results for Dual Versus Single Slip - 5 x 10-3/sec ...... 116
14. Annealing Parameters for Set I and I I ...... 160
15. Twist Angle as a Function of Strain— Set I . . . 202
16. Twist Angle as a Function of Strain— Set II . . . 226
xix I. INTRODUCTION
The primary aim of the work to be described in this dissertation was to produce and study low angle twist boundary dislocation arrays In deformed and annealed crys tals of diamond cubic germanium. During deformation dis locations moving in different directions on the same glide plane can cross and interact in such a manner as to relax
Into characteristic equilibrium configurations. If two different sets of coplanar parallel dislocations interact, a planar low angle grain boundary can result. One low- energy configuration would be a hexagonal dislocation array consisting of pairs of three-fold nodes, the shape of which would depend on the Burgers vector and crystal orientation of each dislocation segment. The lowest possible energy configuration would be a pure twist boundary on the (111) glide plane (1), which consists of pure screw dislocation segments. The three-fold nodes can be characterized by equilibrium node angles, a, which are obtained theoreti cally by application of an equilibrium line tension analogy such that the vector sum of the line tensions at a node Is
zero.
Secondary goals Include: (1) characterization of dual glide in germanium crystals. It was reasoned that the simultaneous operation of two slip directions on the primary slip plane would include interaction between two sets of coplanar* non-parallel dislocations. This unique type of deformation, henceforth called "dual slip," after being imposed upon germanium by compression, is examined with respect to the stress-strain characteristics, the change in
crystal orientation during compression, the slip trace
configurations on the specimen surfaces, and the dislocation
content on the primary slip planes; (ii) the investigation
of other unresolved questions for germanium crystals, as
to whether dislocations are straight or curved in as-
deformed specimens, whether dislocations are preferentially
aligned along low index directions, and whether the stack
ing fault energy in germanium is sufficiently low enough to
allow the observation of dissociated three-fold nodes in
annealed crystals.
This study necessarily includes the examination of
thin foils of germanium using the electron microscope. This
fact leads to one of two reasons germanium was chosen for
this work. The first is because dislocations in such diamond
cubic materials have a negligible mobility at room temper
ature (2), whiclT allows retention of bulk material struc
tures in a thin foil. The second is because a close rela
tionship exists in the translation geometry of the diamond
cubic structure to the face-centered cubic structure, in
that the active Burgers vector in germanium is 1/2 <110> 3
(111), (3) and the translation lattice is face-centered cubic.
The following sections in this chapter include (A) the reasons for selection of germanium and its properties favorable to the primary goal of this study, (B) a general discussion of the properties and nature of dislocations in the diamond lattice, (C) a survey of previous work in which germanium crystals were deformed and studied with respect to the stress-strain behavior and/or dislocation structure, and
(D) a brief review of low angle grain boundaries and dislo cation networks.
Section (E) includes investigations of low angle dislocation boundaries, and equilibrium nodes therein, for body-centered cubic iron crystals. The approach used to study the node angles in iron is similar to the one used by this author in analyzing the nodes observed in germanium.
In section (P) low angle boundaries in face-centered cubic metals are discussed and the various types of twist boundary node pairs that can exist are defined.
A. Material Selection
The most important advantage of using germanium lies
In its thin foil characteristics for electron microscopy,
since the thin portions of specimen used for observation of o structure are generally of thicknesses of 1000 to 3000 A.
If one wishes to draw conclusions about the dislocation arrangement in the bulk material from the observations ob tained from thin foils, any dislocation motion either during thinning due to the image forces produced by nearby foil surfaces, or from the influence of an electron beam must be considered.
These difficulties are avoided by using a diamond cubic crystal, since plastic flow, i.e., dislocation motion, has been observed only at temperatures above about one-half the absolute melting temperature (2). Hence, at room tem perature germanium is brittle; the dislocations are com pletely frozen in, and the observations are more ideally representative of the dislocation arrangements in the bulk material.
The mobility of the dislocations in germanium has been investigated by Johnson (2) who used a spherical sapphire indenter on etched surfaces of germanium to produce damage at room temperature. After annealing the indented specimens at 550°C, slip was observed to have occurred only at the tips of the micro-cracks formed initially by the indenter.
Optical bright and interference microscopy were used in observing slip steps which Johnson (2) claimed were as small as 15 A°. He concluded from his work that germanium does not exhibit any evidence of dislocation motion below
2 00°C and only a negligible amount below 400°C. 5
B. Dislocations In the Diamond Lattice
The following description of the characteristics of dislocations in the diamond cubic lattice follows those of
Hirth and Lothe (1) and Hornstra (*0 .
Dislocations in the diamond lattice are similar in glide plane, {ill} and active Burgers vector, 1/2 a <110> , to fee metals. This can be seen by considering the diamond cubic lattice as two Interpenetrating fee lattices, one dis placed by [1/4,1/4,1/43 with respect to the first. Hence the diamond cubic lattice has a fee translation lattice with a basis of two.
The layer structure of the diamond cubic contains
(111) planes in a sequence AaBbCcAaBbCc, etc. In a pro jection normal to the (111) plane the atoms In layer Ma" project directly upon "A" as do "b" upon "B", etc. Hence dislocations which involve displacements of pairs of atoms, or pairs of layers of the same index, for example Bb, would
Involve no change in the usual four nearest-neighbor cova
lent bonds In the lattice; refer to Figure 1. Hence the packing sequence can be described as Just ABCABC, etc.
Since the Pelerls energy In germanium Is believed to be large, with deep troughs along <110> directions (4),
glide dislocations should preferentially lie along these
directions, unless other interaction forces Interfere.
The main difference among dislocations of screw, edge,
or mixed character is that in a diamond cubic lattice 6
[TTz]
ll 1 4
„+^ 5
Figure 1
A diamond cubic lattice projected normal to (lTo). 0 represents atoms in the plane of the paper and + repre sents atoms ii the plane below* The (111) is perpendicular to the plane of the paper and appears as a horizontal trace.
/ dangling bonds can exist at the core of the dislocation.
The screw dislocation should have no broken bonds in its core, as is shown in Figure 2. The 60° dislocation, as depicted in Figure 3, does have dangling bonds. Note that in Figure 1, all cut bonds on surfaces 1-5 and 4-6 are re formed after displacement, but on surface 5-6 one bond per atom site along the dislocation is left dangling. The three-dimensional projection of the row of dangling bonds is shown in Figure 3.
Bond resonance can occur between positions 1-2 and
1-3 possibly leading to a decrease in energy as compared to a rigid dangling bond configuration (4).
Other dislocations can be considered as being built up from the two simple dislocations described. The 30° dis
location can be considered as one-half the sum of a screw and a 60° dislocation, and is shown in Figure 4. The edge dislocation on the (111) glide plane can be considered as one- half the sum of a 60° dislocation and a 120° dislocation
(which is the same as a 60° dislocation but with a negative
sense vector). This is shown in Figure 5- The resulting
types of dislocations in a diamond cubic lattice are sum marized in Table 1; Table of dislocations having Burgers
vector BC * 1/2 [lioj . The axes of the dislocations are
referred to the Thompson tetrahedron in which AB,BC, and BA
capitalized pairs refer to the various £110] directions. The
Thompson tetrahedron is defined and discussed in Appendix A. 8
b
_o
Figure 2
A screw dislocation in the diamond cubic lattice, a axis, b - Burgers vector (Hornstra (*0). b
Figure 3
A 60° dislocation in the diamond cubic lattice, The numbers refer to those depicted in Figure 1. 10
Figure 4
A 30° dislocation in the diamond cubic lattice, a = axis, b « Burgers vector (Hornstra (*0). 11
Figure 5
An edge dislocation in the diamond cubic lattice with glide plane (111) a » axis, b = Burgers vector (Hornstra (*0). 12
TABLE 1
TABLE OF DISLOCATIONS IN DIAMOND CUBIC LATTICE BC * 1/2 [110]
Angle between Symbol Axis and Number of Axis of the of Burgers Glide Broken Bonds Nr Dislocation Axis Vector Plane per a cm I BC < 110> 0° 0 II AB, AC , DB and DC < 110> 60° (110) I'M III AD < 110> 90° (100) 2*83 or 0 IV BC+AC ,BC+BA, BD+BC and DC+BC < 211 > 30° (111) 0*82 V AC+AB and DC+DB < 211 > 90° (111) 1-63 VI AD+BD ,DA+BA, AD+CD and DA+CA < 211 > 730131 (311) 2 '45 or 0*82 VII AB+DB and AC+DC < 211 > 5I404 £| • (110) 1*63 or 0 VIII AC+DB and AB+DC < 100> 90° (110) 2*0 or 0 IXa AD+BC and AD+CB < 100> 45° (100) 2*0 or 0 IXb AC+BD and AB+CD < 100> 1(5° (100) 2*0 or 0
a Is the lattice constant.
For the four types of (111) glide dislocations; the screw,
30°, 60° and edge, the number of free bonds per unit length of dislocation line is proportional to sinfl , where $ is the angle between the dislocation and its Burgers vector.*
This relationship is referred to later in this dis sertation in the discussion of the general form of the energy of a dislocation line as a function of its Burgers vector and crystal orientation, which assumes a similar relationship. 'C. Previous Investigations of the Deformation Behavior and Dislocation Structure of Germanium
Initial work on the structure of germanium was con cerned with the macroscopic observation of Inherent disloca tion distributions in as-grown crystals. The density and distribution of dislocations was shown to vary as a function of the method of crystal growth, the type of pitting etchant used, and the location of the dislocation crystal (3,5 ,6 ).
Using optical microscopy on crystals grown by the
Czochralski method, networks of low energy, low angle tilt boundaries have been observed In the valleys on the surface
of the crystals. This was believed to result from
equilibrium solid-liquid Interface motion during crystal
growth (3). It has also been shown that the CP-1! etchant*
commonly used for dislocation pit counts on (111) and
(100) surfaces may only show edge dislocations which are
Inclined at fairly large angles to the surface of the
crystal. This means that the resulting biased dislocation
etch pit density for a particular crystal may be suspect
when compared to CP-*I etch pit counts of other crystals
which were deformed or treated under different conditions.
Except for localized variation in the density of
*CP-*l— a common etchant used on germanium crystals to: (1 ) remove damaged surface layers and produce a slightly undulating, mirrorlike surface, and (2 ) reveal characteris tic etch pits on either (100) or (111) surfaces. Composi tion: by volume; 3 HP (50S 5 H N C M 30S5)/3 Glacial acetic acid (70)C)/0.1 Br2 . 14 dislocations at the surfaces, Czochralski grown crystals have been shown to exhibit a more uniform distribution of grown-in dislocations (5 ,6 ).
Pfann and Vogel (3) studied the habit and etching behavior of dislocations in zone-leveled bread-loaf ger manium crystals. The etch pits were found to be concen trated in low angle boundaries or lineage boundaries, throughout the crystal. Small areas of high dislocation density were also noted by this author in a previous publi cation (6), in which the occurrence of a high density of
localized lineage boundaries was observed to be random from one crystal to the next. The types and orientations of the
lineage boundaries were found to depend upon the direction
of crystal growth by Pfann and Vogel (3). They also found
evidence of edge dislocations with a <100> line direction
and a 1/2 <110> Burgers vector. Observed tilt boundaries
consisted primarily of dislocations in either < 100> or
<211> directions. Little evidence was found for the exis
tence of screw dislocations in as-grown crystals.
Prom these studies it can be concluded that although
the attainable average dislocation density in zone-leveled
bread-loaf crystals is lower than that attainable in
Czochralski grown crystals, the extensive lineage boundaries
which have been observed in the former would not be favor
able for deformation studies of the dislocation structure,
since small deformation specimens cut from the former would 15 not have consistent levels of resistivity. Impurity content and dislocation density. The use of Czochralski crystals would provide generally a higher grown-in dislocation density, but one in which the dislocations are distributed randomly.
Some of the initial investigations of deformed ger manium were concerned primarily with the effect of strain
upon electrical parameters.
Gallagher (7) deformed single crystals of n-type
germanium in bending at various temperatures above 500°C.
Slip on {111) planes was observed, and experimental curves
showing the increase in resistivity and decrease in carrier
lifetimes were given as a function of increased deformation
and temperature.
Similarly, Pearson (8) deformed 15 ohm-cm, p- and
n-type germanium crystals in bending. He measured Hall
effects, conductivity as a function of orientation, and
carrier lifetimes, all as a function of temperature and
degree of deformation. As expected, the lifetimes were
reduced from about 300 psec. to less than 1 jusec, and the
conductivity increased several orders of magnitude, for the
crystals heavily deformed at high temperatures.
Stress-straln characteristics of germanium ~
Above about 500°C the deformation curves of germanium
single crystals deformed in tension and compression are very similar to those for face-centered crystals deformed at room temperatures, in that the stress-strain curve is separable into Stage I, II and III work-hardening ranges. Figure 6 shows a compression curve for germanium at 600°C which Is taken from the work of Alexander (9). Other workers (10,11,
12,13) have observed similar curves, with the exception that
Bell and Bonfield (10) also observed a region of zero slope before stage I hardening. A deformation curve from work by the latter is shown in Figure 7- With respect to the obser vation of the zero work-hardening slope Rezek (14) has observed that silicon crystals oriented for single slip exhibited an Ideal easy glide region, with a zero work- hardening slope, when tested at 1000°C, but not at 675°C.
At 1000°C slip occurred on only one set of planes at strains up to 14?, but at 675°C slip occurred on at least two sets
of planes as soon as the elastic limit was exceeded. The multiple slip was believed by Bell and Bonfield (10) to be
due to the limited mobility of the small number of disloca
tions originally present. Evidently any slip on a second
slip system can destroy easy glide, and result In a positive
slop for the flow stress.
The condition of instability which Is normally asso
ciated with a zero work-hardening slope does not appear in
germanium because of the relatively large Increase in stress
necessary to achieve an increase In the dislocation velocity
in such crystals. After Johnston and Oilman (15), the 17
4
N 3 E e. -i in
c K* 2 O* C 3 C C o CL tA J3 . c u CO
a nr in o o 8 16 2 4 32 4 0 A bgleitung a in %
Figure 6
Typical stress-strain curve of germanium deformed in compression (Alexander (9)). 18
2.0
N E E o» je in in «>
L_ o a> £ CO *o I 0 5 o in
% Glide Strain
Figure 7
Typical stress-straln curve for specimens deformed In tension,of Initial dislocation density 1 x 10^ cm"2 tested at 560°C and glide strain rate 2 x 10”^ sec”1 (Bell and Bonfield (10)). 19 stress dependence of the dislocation velocity in germanium is of the form:
where V is the average velocity of the total of N disloca tions per square centimeter. Velocity measurements of
Chaudlin et a l . (16) indicate that "m" for germanium is small; in the range 1.3 - 1-9- A constant stress in the zero work-hardening portion of the stress-strain curve is primarily a function of the stress necessary (1 ) to maintain the velocity 7 of a constant number of mobile dislocations
N, and (2) to overcome internal stresses which are a func tion of N. Hence a constant mobile dislocation density
implies little dislocation interaction in the zero work- hardening region.
Alexander and co-workers (9,10), in deformation
studies of germanium, also noted that secondary slip had
occurred at very low values of strain, which may explain the
lack of a zero work hardening slope in their work.
Another distinctive feature of the deformation curves
of germanium is the large yield point phenomena, as shown in
Figure 7, which appears in most stress-strain curves for
germanium (9*10,12). The initial dislocation density in most
crystals Investigated was of the order 10^ to 10^/cm^, yet
the dislocation density levels reached during yielding were
approximately 10® to 10^/cm^. The large yield drop observed 20
In germanium crystals of initially low dislocation density, oriented for easy glide, was attributed by Bell and ■
Bonfield (10) and Seitz (13) to either; (i) creation and motion of many fresh, mobile dislocations, provided that there Is not too large an Increase in the internal stress, or (11) the unpinning of dislocations pinned by Cottrell atmospheres at a stress lower than that necessary to cause dislocation multiplication. Let us consider the former explanation. Suppose with Johnston and Gilman (15)> that
the velocity V of the mobile dislocations is proportional to
the stress and that the velocity can be described by the
following equation:
where: e° * strain rate (sec~^)
N = number of mobile dislocations (cm/cm^)
b = Burgers vector (cm)
Then a large decrease in T with no change In the strain rate,
means that there has been a large increase In N. The evi
dence does point towards this explanation, since the disloca-
tion etch pit density Increased by a factor of about 10 as
specimens were strained through the yield point (9 ,10 ,11,12).
Only above 700°C does annealing lower the dislocation den
sity, Subsequent deformation of the annealed specimens re
sults in values of flow stress and yield stress which are 21 all lower than previously observed for the same crystal.
This is contrary to previous observations wherein for virgin material a lower dislocation density caused an Increase in the yield stress on deformation, but did not effect the flow stress (10), This result supports the conclusion that the stress is primarily a function of N. The value of N for the annealed crystal was lower than the value for the as-deformed crystal, but still much greater than the value of N for the original crystal. Therefore the critical value of N neces sary to achieve yielding was reached at a lower stress than in the original crystal. Figure 8 shows the increase in dislocation density through yielding (13)-
The yield point phenomena is not only a function of the initial dislocation density; the magnitude of the load change at yield was observed to increase at higher strain rates (10,11); decrease at higher temperatures (10,11);
(Figure 9) and also decrease with a change of the deforma tion axis from the easy glide orientation to a duplex and/or a multiple slip orientation (Figure 10). With a change in the orientation the value of yield stress does not increase, but rather the flow stress is increased because of the
increased amount of secondary slip in these orientations.
The general trends observed for variation in the
deformation curves as a function of orientation, temperature,
strain rate and mode of deformation are similar to those of
face-centered cubic crystals. As shown in Figures 10 and 11 IN) fO o O o o Dislocotion Density/cm; o cm O o ro 0) ai 1 5’ 13 3 0 a o ®
Dislocation density plottod as a function of strain (Fatal and Alexander (11)). 23
4
E E s 3 C l
6 6 5 6 0 0
O* 2 C 3 C C o o. V) 7 4 0 Temperatur -D £ 8 0 0 O CO
0 0 8 16 24 32 4 0 4 8 Abgleitung a in °/0
Figure 9
Variation in the stress-strain curves of germanium as a function of temperature (Alexander (9>). -2 Resolved Shear Stress kg mm
3 B Stg 3 O — — ro bi O u> O cn
_ m m «r«?
M H
s s A 8 * 5 oft! WH
!-»• H f 0 OH f\J U]vsB 2 *■ 2 flsSI a a> H* HO O U) -i * 5 S S ? o P ff« ~ 5' OB 5 H v A N w H» _ * H O O H M O # * | l i
H 0 2 4 6 8 10 12 14 16 18 Strain € - Percent
Figure 11
Orientation dependence of stress-strain curves at constant temperature, T * 600°C (Patel and Alexander (11)). 26 a change in orientation from easy glide to multiple slip orientations will tend to: (i) decrease the amount of stage I work hardening, (ii) increase the level of stress for any stage I work hardening, and (iii) increase the rate of work- hardening in stage II. Typical values of the slope In stages I, II, and III for the curves in Figure 6 are 20-*l0,
80 and 300 kg-mm/mm^, respectively.
As shown in Figures 9 and 12 for crystals oriented
for easy glide, an increase in the temperature of deforma
tion will: (i) decrease the length of stage I, (ii) decrease
the slope of stage II hardening, and (iii) decrease the
severity of the yield point phenomena.
An Increase in the strain rate was found to generally
raise the level of stress at any particular strain, a3 Is
predicted by equation 1. Bell and Bonfield (10) found that
a lOOJt increase in the flow stress occurred with an increase
In the strain rate from 10“^ to 10“ ^/sec. (Figure 13)*
There was a significant difference in the deformation
curves of specimens deformed In compression by Patel and
Alexander (11) as compared to those deformed In tension
(11,12,13), in that very little upper yield point developed,
and the values of the slope of stage I work-hardening were
increased. In the compression study, rectangular crystals
were used which measured about 8.5 mm by 3*5 mm by 3.5 mmj
which were deformed in an inert atmosphere between ceramic
plates of various temperatures (Figure 12). They observed 27
5 5 0 °C 06 N 7 0 0 °C
m*• 0,4 m
0.2 (VI E E 'v CP 02 0 4 0 6 0.6 10 Strain € Percent i 700°C b 800* (0 «/> 8 5 0 °C L_ co
• D [ioo]
Strain € - Percent
Figure 12
Stress-strain curves o f germanium single crystals In compression as a function of temperature (Patel and Alexander (11)). An increased strain rate produced a cooresponding increase cooresponding a produced rate strain increased An n lw tes n pe il on (el n ofed (10)). Bonfield and (Bell point yield upper and stress flow in SCHUBSPANNUNG X (KP-MM'O 0.8 0.4 .2 6 0 0 8 16 € BLIUG % ABGLEITUNG iue 13 Figure 10 24 32 -4 40 4 - 10
5 28
(1) that a significant amount of secondary slip had occurred at values of strain of about 0.2!E, and (ii) that a coarsening of the primary slip bands occurred at higher strains. They attributed the resulting kink bands, shown in
Plate 1 to the relief of lattice bending caused by the frictional constraints induced by the ceramic surfaces used in compression. This indicates that in compression more so than in tension, the expectancy of a zero slope hardening region, or of low slopes of hardening in stage X is very small.
2. Dislocation content of deformed ge rmani um cry s t alV
Alexander and Mader (17) and Alexander and Haasen (12) both used electron microscopy to investigate the dislocation structure as a function of strain, in germanium crystals oriented for easy glide. They deformed high-purity single crystals at 600°C and 520°C, respectively, in tension. A general observation made by them was that no dislocation motion occurred under the influence of the electron beam, which is consistent with the conclusions of Johnson ( 2 ).
The dislocations were observed to always be curved, of edge or nearly edge character, and generally aligned perpendicu
lar to the deformation direction on the primary slip plane.
At low strains the dislocation structure on the primary (111)
glide plane, consisted of many dipoles which were grouped in
the proximity of large bundles of edge dislocations. Even 30
P late 1 Sample of orien tation D showing fin e kink bands, deformed 14% at 800°Ct *00X. ( Patel and Alexa-ider(ll)). 31 at very low strains some dislocation Interaction and change of direction along the trace of the secondary slip plane was observed, which Is consistent with previous observations
(10,12,13) of secondary slip traces on crystal surfaces at low strains. As the strain Increased In stage I, the in crease In the dislocation density was manifested by the grouping of dislocations Into long dislocation tangles or braids, which became more closely spaced as deformation con tinued into stage I. The braids were mainly two-dimensional and were located on the primary glide plane.
In stage II work-hardening, the braid separation de creased and more often the braids were found to end abruptly on traces of a cross-slip plane. At greater strains the dislocation density increased and additional long disloca tions appeared parallel to the traces of the cross-slip planes forming complex walls of dislocations. At this point the Burgers vectors of the dislocations were varied and the numbers from each set were distributed randomly. Similarly the size of dipoles was decreased, and the number increased along the secondary slip traces.
Etch pitting studies also revealed that during defor mation braids of high forest dislocation density had passed through relatively dislocation-free areas. Hence through stage I, curved edge dislocations on the primary slip plane
formed bundles or tangles normal to the primary slip direc tion. Stage II included increased secondary slip and the 32
formation of complex dislocation walls and a somewhat cellu
lar structure.
Crystals have also been deformed In torsion (18,19) and in creep (18). In these investigations the dislocation
structure consisted of many straight dislocation segments, which were often aligned along either a <211>or <110> direc
tion. Holt and Dangor (18) determined Burgers vectors in a number of foils in various random orientations. Their
observations were categorized into configurational shapes
such as "T"-Joins, 11 elbows," "seagulls" and dipoles. Dis
locations of essentially all possible Burgers vectors were
found: screw, 30°, 60° and edge glide dislocations on fill}
planes, as well as dislocations in <211> directions on {l3l}
a n d (101> planes. This author believes that the isolated
"elbows," "T-Joins," etc., thought by Holt and Dangor (18)
to be entities continuous in themselves, were actually seg
ments of dislocation networks or interacted dislocation
lines. This may not have been obvious to them, because
their thin foils were usually not parallel to a glide plane,
and the extensions of the dislocation pieces that were
analyzed probably extended out of the foil and were removed
during thinning. Holt and Dangor observed predominantly
60° dislocations as compared to Alexander and Haasen (12)
and Alexander and Mader (17) who observed a predominance of
edge glide dislocations. There is also a contradiction in
the observations of Holt and Dangor that dislocations were aligned in <110> or <112> directions; since in other
investigations (12,13) the dislocations were curved in as- deformed specimens. Only after annealing was there prefer
ential alignment in the latter work. Holt and Dangor
attributed the alignment to a high strain rate, in that any
kinks in the glide plane may have been swept away as fast
as they were formed which can result in crystallographically
aligned dislocation lines. An example of the dislocation
structure in crystals deformed in torsion is shown in
Plate 2 from the work of Booker and Stickler (19).
Others have deformed germanium in torsion but with a
different goal; to examine possible stacking fault extension.
Aerts et al. (20) have observed evidence of node extension
in dislocation networks in silicon, which has many proper
ties similar to germanium. Using measureable node parameters,
and isotropic elasticity theory they calculated stacking
fault energies of y (intrinsic) = 50 ergs/cm and 7 (extrin
sic) = 60 ergs/cm. Calculations were taken from nodes such
as those shown in Plate 3. Similar investigations in
germanium crystals were conducted by Booker and Stickler (19)
and Arts et al. (21). The former found straight segments
but no sign of node dissociation. The latter investigation
yielded networks considered by the authors a3 both undis-
sociated and dissociated. Crystals were twisted around a
(111) axis at 8 50°C and were then annealed up to 15 hours
at 600°C. Slices were cut parallel to the (111) plane of 34
V
■ f M N
L
« 1*0* « *
Plate ___2
Dislocations In (1 1 1) s lic e o f germanium a fte r tw istin g at 600°C. (Booker and Stickler(19)) 35
P late i
Magnified view of three extended nodes used for measuring the stacking fault energy. (Aerts(2to).
P late H
Network of undissociated nodes. (Arts(2C» twist. The well-annealed specimens did show some evidence of dissociation but to a very small degree. The authors believed that most of the networks were not actually net works of continuous dislocation segments joined at the nodes, but rather superposed networks lying on different glide planes. An example from their work is shown in Plate
Only in crystals which were annealed to a temperature suf ficient for climb to occur did they feel that the disloca tions could truly interact and form networks on the same glide plane; only under these conditions could stacking fault extension occur. Rough calculations gave y (intrinsic) as approximlately 90 ergs/cm.
To summarize, Alexander and Mader (17)* and Alexander and Haasen (13) found mostly curved dislocations of edge character in the as-deformed crystals, whereas Holt and
Dangor (18) and Booker and Stickler (19) found straight dis locations often aligned along low index directions, which were of various Burgers vector characters, with a higher
frequency of 60° dislocations occurring in the work of Holt
and Dangor. Booker and Stickler found no evidence of node
dissociation in specimens deformed in torsion, whereas Arts
et al. (21) did upon annealing, find apparent evidence for
extension associated with a low stacking fault energy.
The observations of dislocation node dissociation in
silicon and germanium has been recently questioned. Booker
and Brown (22) and Shaw and Brown (23) have shown, using dynamical theory, that unextended nodes in a dislocation network can appear as if they were extended nodes with all the apparent ’’partials" visible when the thin foil is viewed under several different reflecting conditions in the elec tron microscope. They maintain that a truly extended node can only be distinguished from an unextended node if the apparent extension is greater than about one-half the extinction distance, i.e., if the partial dislocation images are well separated. This is not the case in the work of Arts et al. However, Amelinckx (2*1) disagrees with the above authors (22,23) and feels that the work of Arts et a l .
(21) is valid. The question remains unresolved,
D. Low angle grain boundaries
This section includes the development of an analogy between a three-fold grain boundary Junction and a three fold node, the derivation of a general equation for deter mining the equilibrium node angles, and a discussion of the dislocation networks which constitute the displacements in
a low angle grain boundary (Frank's criterion).
A grain boundary is defined as the interface between
two single crystals of different orientation which are con
tinuously joined. Such a boundary will tend to be planar
in order to minimize its total area and hence lower its
energy. This planar character has been observed and des
cribed for germanium in work by Wagner and Chalmers (25). 38
By analogy with the concept of surface tension of a liquid, the grain boundary tension In an isotropic solid can be defined as the reversible work of formation of the grain boundary per unit area, other variables such as temperature and volume being constant. If Y^ = specific surface energy/unit area of grain boundary, then
(2) Y^ - surface tension « dF/dA) * 9 ”
Yj_ usually depends not only upon the change in the orientation across a grain boundary, but also upon the grain boundary plane orientation. This dependence gives rise to torque terms which tend to rotate the grain boundary into a
lower surface tension orientation.
The equilibrium configuration in Figure 14 represents
three boundaries pinned at points A,B,C far removed from 0.
The equilibrium conditions can be derived by considering the
change In the total configurational energy with the virtual
displacements dx and dy. Such an energy balance of terms
yields an equation of the form:
dYI (3) 2 + (t± x s) — 3 - 0
where: = a vector normal to the plane ABC, and parallel to the line of intersection of the boundary planes t^ = vectors lying in the boundary planes normal to s_ “ surface tension of the grain boundary
* the torque term arising from the variation of Yj for each boundary plane with Its orientation 0^. 39
A
Figure Ik
Junction of three grain boundaries, A,B, and C, and a closed boundary D (Hirth and Lothe Cl))- 40
If one now considers, for example, two crossing dis locations of Burgers vectors b^ and bg (Figure 15) to
Interact and relax, the resulting configuration can be two three-fold nodes, the additional segment connecting them having been formed as the interaction product of the two original dislocations. To determine whether the formation of the additional segment is energetically feasible, a com plete analysis Involving Interaction energies Is the best ap proach (1). However, the calculation of the self-energy and
interaction energies of all of the dislocations in the array would involve a quite complicated numerical procedure. Such
an Investigation would be difficult even for symmetric
cases. A simpler method, involves the neglect of the inter
action energies and variation of the energy calculated from
equation 3 Tor the disolocation nodes. This method gives
an equally correct result as the former, more complicated
method (26).
Considering that the vectors t^ in Figure 14 are de
fined as the vectors parallel to the intersection of the
boundary i and the plane normal to s, one sees that a direct
analogy can be made between the three-dimensional case of
the surface tension of a grain boundary Y^, and the two
dimensional case of the line tension Y^ of dislocation lines
In a planar three-fold dislocation node. In doing so the
following definitions apply: Figure ______
Crossing dislocations and possible reacted configurations A3 and CO.
original
Figure
Reaction of crossing dislocations to form a pair of three-fold nodes. *12
= the longitudinal tension vector along the length of a dislocation which Is constrained to remain straight
= unit vector tangent to the dislocation line, pointing away from the node.
S4 = -1* -1 - = -3* = Normal "to the plane of the lfix -%\ \i2x -2I i-3x-31 dislocations t ^ y gg— = twist term arising from the change in the self energy * per unit length of"the dislocation line with a change in orientation = v^s 1) /
* the angle between the Burgers vector of the disloca tion line bj , and , where ~ constant + 0^, or dfl = dj? ""
The resulting equation has the form:
1 ,,
The isotropic elastic energy per unit length of a straight dislocation Is (as shown In Section VII A): * 2 (5) (Wg/L)! - j~**(l ~^vcOB2 .fll.Vj In (_R ) 1*0
or
(6) ( V D i ■ Kj g - m
where: « isotropic energy factor 1/ = Poissons* ratio U3 Substituting equation 5 into equation *1, one obtains;
b 2 (7) i C (1-^ cos2^ ) i+2ycos/91 sinfl^ ri^D * 0 fi x(£i x V where: n - =--- | R ~ x~ E (8) '-1 ~
If the glide plane for all the dislocations Is the boundary plane, then:
(9) ni * ^i x N where N is the normal to the boundary plane. %
When this equation is applied to the three-fold node, a characteristic equilibrium node angle 2 0 is established.
The angle 2a is defined such that a line bisecting the angle will always be parallel to the third and opposite segment of the three-fold note, as shown in Figure 16. Thus when a third segment b^ is formed from b^ and b2, the interaction forces that exist must be such that b^ and b^ are attractive, and b^ and -b2 repulsive. If this interaction and relaxation occurs for sets of parallel dislocations, an hexagonal net work is expected as shown in Figure 17. If interaction and relaxation is not energetically feasible the resulting
network will retain a lozenge shape as also shown In Figure
17. Ideally, the networks in pure twist boundaries should
always form hexagonal nets, unless their line directions are
only a few degrees or less from being orthogonal. A network
representing a low angle, pure twist, grain boundary would Lozenge-shaped dislocation net.
Hexagonal dislocation net.
Figure represent a raisorientatlon between two crystals in which the rotation vector is normal to the grain boundary plane, see
Figure 18. If the angle of rotation normal to the grain boundary Is 9, the net dislocation density can be calcu lated using Frank's criterion (27), which can be stated as follows,
If two vectors V and in the boundary plane are
located in each of the crystals, along the same crystallo- graphic direction, then the net displacement difference between the crystals Is:
(10) B (V) - V -V'
If there are 1 sets of parallel dislocations in the boundary
representing the crystal misorientation, then:
(11) B(V) £ i (§ 5 k-i i
where: = the number of dislocations of cut by V
Then the total Burgers vector displacement is:
(12) B ( V ) = £ n ± (V) b ± - (Vxo) = (Vxa) 2 sin 1 i 2 and
(13) w * rotation vector normal to the boundary plane
* a 2 sin — - 2 If the vector Y Is constructed parallel to one of the two
sets of parallel dislocations, as is shown In Figure 19, then
the displacement B * n2 b.2 where: b2 = Burgers vector of dislocations in set (2) k6
Figure 18
A twist boundary, where vu is normal to the boundary plane*
2 Vsin Bj
Figure 19__
Application of Frank’s criterion for the Burgers vector content of a dislocation net. 47
ri2 = the number of* disloca tion lines the vector V crosses in set (2)
Then equation 12 becomes:
(14) n 2 b 2 = (V x a) 2 sin |
If a unit vector is defined by x a) > which is normal to the dislocations in set (2) in the plane of the boundary, then the multiplication of equation 14 by Ng yields,
(15) (^2 x 3.) ' n 2^.2 “ ^ 2 x x 5.) 2 sin j
If the small angle approximation; 2 sin(9/2) = 0 is used together with the identity (N2 * V) = n2D2, where D2 Is the spacing between dislocations in set (2), equation 15 becomes:
(16) n 2 (cos 5 b 2 ) = (n2D2 cos S) 6 or bo (17) D2 = —£• S < 90° = angle between parallel sets 9 of dislocations.
Hence if any two of the three factors; the dislocation line spacing, the angle of rotation, or the Burgers vector are known, the third is determined.
The simplest low-angle twist boundary from a topo
logical standpoint would consist of two sets of parallel dislocations In which the dislocation spacing Is determined na for a given angle of rotation, G, and given Burgers vectors.
Only if the lozenge-shaped network can reduce its total energy will it relax into a characteristic hexagonal network.
Hence either network can represent a twist boundary in a crystal.
E . Studies of equilibrium node angles in body-centered cubic iron
Studies of networks which constitute low angle boun daries have been carried out on many crystal systems. This section describes investigations in body-centered cubic iron crystals in which equilibrium node angles were determined using equations similar to those developed in the previous
section. The general approach used for iron will be that used by this author to analyze the nodes observed in
annealed germanium crystals.
Carrington et al (28), using the electron microscope,
studied the arrangement of dislocation networks in body-
centered cubic iron -3% silicon crystals. Specimens were
cold-worked and then annealed at various temperatures. Many
different types of dislocation networks were observed in
cluding both lozenge and hexagonal types. For activated
glide dislocations of 1/2 <111> Burgers vectors which interact
and relax on (lio) and flOO} glide planes, the reactions
that may have taken place were of the type:
(18) (1) 1/2 a
(2) A similar reaction occurring on a £l00} plane could occur as shown in Figure 22 resulting in another low angle hexagonal dislocation boundary. A third reaction (3) and its product is shown in Figure 2 3 for a pure tWist boundary on a ^100) plane. This is a highly improbable reaction, however, because of the high energy of a <110>
Burgers vector in the bcc crystal.
The theoretical equilibrium node angles of 96-5° for
configuration (1) and 111° for configuration (2) were cal
culated by applying the condition that the vector sum of
line tensions, taken as vectors parallel to the dislocation
lines, should equal zero. The line tension of the disloca
tion segments were expressed:
(19) T d = [ ~ 1 - V ° ° - --^ ln(R/rc )
which with the condition JT^ = 0 gave the angles of 96.5°
and 111° for the two cases. The authors neglected the addi
tional forces associated with the variation of line energy
with the line orientation. Even in an isotropic material,
the dislocation will tend not only to shorten, but also to
rotate towards orientations of lower energy (screw orienta
tion), while remaining straight. Other assumptions 50
t (a) T
Lb AX. > < I00> (b) i i i
Figure 20
Intersecting dislocations (a), Join up to form a disloca tion network (b) (Carrington (28)).
IOO>
1I00
Figure .21.
Calculated pure twist boundary formed by l/2a <111> dislocations In flio) plane. 1(100) » 0.22; L(lll) 0.59; h * 0.88; w * 1.0 (Carrington (28)). 51
<\00>A
Figure 22
Calculated twist boundary formed by l/2a <111> dislocations in (100) plane (Carrington (28)).
Figure 23
Calculated twist boundary formed by a <100> dislocations in (100) plane (Carrington (28)). 52 included the neglect of differences in the (In R/r°) term between the 1/2 <111> and <100> dislocations. Nodes of the type (1) were the only ones experimentally observed and identified by Carrington et al. (28). The experimentally observed value of 2a was about 88°, about 8° lower than was theoretically calculated.
Another more recent study of annealed iron was con ducted by Ohr and Beshers (29). A crystallographic analysis of the dislocation network previously described as type (1) in the work of Carrington, was conducted by applying the kinematical theory of contrast at dislocations for a
Burgers vector analysis, together with Frank’s formula for determining the dislocation content of a low angle grain boundary. The majority of networks observed were in -{llO) planes and represented twist boundaries. The boundary net work consisted of two a/2 <111> dislocations lying in a direction about 10° from a screw orientation towards a <112> direction. The <100> segment was always of a pure screw character. The experimentally observed equilibrium angle
2 a for the type (1) case was 88°, the same as was observed by Carrington et al. (28), (Plate 5).
In deriving a theoretical value for the equilibrium
node angle 2a, a more refined isotropic calculation was used
than that by Carrington et al. (28). The energy per unit
length of dislocation included not only the Burgers vector
character of the dislocations, but also the torque term 53
os u :
i W w
(0 3 1 )
(1T0)
(refer to Figure 2*1): (20) Et= J- Aa2y esc [(1- cos2 0J/-a) ] + Aa2[ (1-v] (x-yctn a) ] where: ET = the total energy of the node; the dislocation segments pinned at points 0-A-C. A * [p/4ir(l-i>) ] ln(R/rQ ) ja = shear modulus v = Poissons' ratio a = lattice constant of iron PH * y 0M » x & = cos-1(3/3) (<£-a) = B In this analysis the equilibrium angle 2O was found to be 103°» 15° larger than that observed. Hence a difference of 25? occurred between theory and observation. More surpris ing was the fact that in using a more refined isotropic elasticity treatment, the value of 2a calculated by Ohr and Beshers was in greater error than that calculated by Carringon et al . A later paper by Chou (30) using the experimental observations of Ohr and Beshers (29), applied a more sophisticated anisotropic dislocation line tension analysis to the same dislocation networks. Chou (30) has shown 55 Pinned Points Figure 2M A dislocation node In (Oil) plane. The equilibrium angle a is determined by minimizing the elastic energy of the segments OB, BP, and BQ (Ohr and Beshers (29)). 56 that the main factor contributing to the inconsistencies between previously observed and calculated values of 2a, was the neglect of the anisotropy of the iron crystal. Iron has an anisotropy factor of A = 2 C ^ A C ^ + C ^ = 2.36. Chou (30) considered the elastic energy per unit length of a straight dislocation using the classical methods of Eshelby (30) where: (21) Ed (9) = KA (9) k-L ln(— ) u h TT r * Ka (0) depends on the orientation of the dislocation segment and the elastic constants of the crystal. In a numerical analysis using anisotropic constants, a value of KA (0) was derived of the form: (22) KA (0) = U^T(a2ncosne + b2n sin2n0) n A polar plot of the orientation dependence of the elastic energy per unit length of dislocation line lying In the (Oil) plane of a-iron, from equation 22, is shown In Figure 25 . The unit of energy is ( ln u = 1011 dynes/cm. After obtaining an accurate description of the dis location line energy as a function of its orientation, a balance of line tensions was performed using an equation similar to equation 3. From this equation a theoretical equilibrium angle of 2 a =* 81° was obtained. This was only 7° lower than the experimentally observed value of Ohr and Beshers (29). Chou also showed that proper experimental 57 [ 2 " ] ~ B .v . Figure 25 Orientation dependence of the elastic energy per unit length of dislocation line lying in the (Oil) plane of et-iron. Burgers vector: b « l/2a[lll]. Unit of energy; (Ub2/inr)ln(R/r0) (Chou (30)). analysis and determination of the foil plane, network plane, and projection plane are very important in determining the correct experimental node angles. He then showed that certain small experimental errors in the analysis conducted by Ohr and Beshers on their electron microscopic samples could make an actual node angle of 81° appear experimentally to be 88°, implying that the calculated value of 81° could be closer to the actual equilibrium angle 2a than 7°* Chou's analysis points out the importance of considering the anisotropy of a material in calculating the energy of a dislocation line as a function of its Burgers vector charac ter and crystal orientation. F . Low angle boundaries in face- centered cubic crystals, and expected node types The investigation of three-fold nodes in a face- centered or diamond cubic structure utilizes the notation and terms defined in the following section. The various possible types of three-fold nodes in hexagonal twist boundaries, whose formation is energetically feasible, are defined and briefly discussed relative to their possible existence. In describing dislocations in the face-centered transition lattice, the 1/2 <110> Burgers vector and Clll} glide planes are referred to the Thompson tetrahedron in the following discussions.^ To describe low angle grain boundary ------r------Refer to Appendix A for a description of the Thomp son Tetrahedron. 59 nodes, the two component letters of the Burgers vector are placed on either side of the dislocation line so that when read in the correct order they give b from the viewpoint of an observer looking along the positive direction of the dislocation. 1. Three-fold nodes Two types of nodes with different dissociation prop erties can be defined from the viewpoint of an observer out side the Thompson tetrahedron. The general classifications are either nK” or ”P” type nodes. A "K” node is one which when properly labeled to define the component Burgers vec tors, has one letter in each sector, as depicted in Figure 26. The "P" node is defined by two letters in each sector, each letter separated by an extension of the dislocation line segment opposite that sector, as shown in Figure 27. Each type may be further classified into Ks , Pa or categories. The subscript "s" stands for symmetri- cal, "u" for unsymmetrical. These are presented in Figures 26 and 27. A symmetrical node is one in which each dislocation makes the same angle ^ with its respective Burgers vector, i.e., one which can be rotated into an orientation in which all dislocation segments are screws. An unsymmetrical node cannot be rotated into an all screw orientation. The various nodes can be distinguished by the follow ing rule: When a boundary is viewed by an observer outside 60 A a) co m b) CO Figure 26 (a) A K node and the corresponding orientation of the Thompson tetrahedron. (b) A Ks node with unextended branches and with branches extended to form an intrinsic stacking fault. (c) A Ku node. 61 C (a) A (b) Figure 27 (a) A P3 node, unextended and extended to form an Intrinsic stacking fault. (b) A Pu node. The orientation of the Thompson tetrahedron Is also shown for reference. 62 the Thompson tetrahedron, the Ku or Ps nodes are those with the letters ABC in a counterclockwise sequence; the -Ks or Pu nodes are those with a clockwise sequence. In a material with a sufficiently low intrinsic stacking fault energy, a network consisting of alternating Ks and P3 nodes, would exhibit constricted Ks nodes and extended P3 nodes. If the nodes are not extended, then the types of networks that can be formed by the interaction of crossed dislocations consist of either Kg-Ps or Ku-Pu pairs. Other combinations are either impossible or unstable(1). The only regular hexagonal network which is complete ly free of long range stresses Is one with K3-Ps node pairs, consisting of pure screw dislocations. However, Ks-Ps type dislocation networks have been observed in which the dis locations are rotated 30° from a screw orientation and lie in <112> directions (32). These are difficult to ration alize. Boundaries rotated by 30° are expected to form from the glide systems AC(b) and BC(a), but that configuration would exhibit a long range stress field. Figure 28 shows possible interactions between crossing dislocations on the same glide plane, yielding the following node pairs In order of decreasing probability of occurrence (increasing energy): TYPE Figure 28 Node types (1) and (2) are symmetrical Ks-Ps node pairs. Node types (3a) and (3b) are unsymmetrical Ku-Pu node pairs. 64 TABLE 2 DISLOCATION REACTIONS AND PRODUCTS Reacting Dislocations Type Product AB(30°)+BC(30°) 1. Ks-Ps pairs; all screw dislocations AB(60°)+BC C 0°) 2. KS-PS pairs; all non-screw (30°) BC(60°)+AB(60°) 3- a Ku"pu pairs; (90°) (-30°) (-30°) BC(90°)+AB(90°) 3.b Ku-pu pairs; (■6o °)66o °)( 0°) G . Conclusions Drawn from the Introductory Chapter 1. It was shown that germanium is indeed a suitable material for study in that the dislocation motion at room temperature Is negligible which implies that the observed dislocation configurations are representative of the bulk crystal. Similarly it was shown that germanium does deform at elevated temperatures, in a manner similar to face- centered cubic materials. 2. It was noted that germanium crystals exhibit a large yield point phenomenon because the initial disloca tion density was small relative to that required for plastic flow at the imposed strain rate. Significant amounts of premature secondary slip have been observed in specimens deformed in an easy glide orientation, which Increases the slope of work-hardening in stage I. Some controversy was found to exist concerning certain aspects of the dislocation 65 structure in the as-deformed and annealed crystals. For ex ample, there were conflicting observations of curved, versus straight dislocations in as-deformed crystals; in the latter a definite preference for alignment along close-packed di rections was observed in both annealed and unannealed speci mens. The majority of glide dislocations in deformed crys tals were of edge character, but a few authors observed large numbers of 60° dislocations as well as others of various characters. Both lozenge and hexagonal shaped twist arrays can exist. If hexagonal networks form, a question arises as to whether they predominantly consist of the lower energy sym metrical nodes or to what extent and in what configuration unsymmetrical nodes would appear? The analysis of the node angles has been shown to be more accurate when the anisotropy of the crystal is considered, however, the agreement between predicted anistropic values and observed values In iron has been less than good. The favored orientation for symmetri cal nodes In an hexagonal twist boundary is the screw orien tation, but dislocation nodes in many boundaries have been observed to be in a 30° orientation, one of a higher energy. 5. Another unanswered question Is whether germanium has a stacking fault energy low enough for dissociated nodes to exist in annealed crystals of germanium. The validity of previous observations of such has been questioned on the basis of dynamical contrast theory. II. EXPERIMENTAL PROCEDURES AND TECHNIQUES A . Introduction 1. Experimental objectives Since the primary Interest was in the observation of low-angle dislocation twist boundaries, the main experi mental goal was to deform and anneal the single crystals of germanium such that these networks would result. A second goal was to characterize dual slip. To accomplish these, single crystals of germanium were deformed to various degrees of strain, and then examined In the as-deformed and annealed condition by means of optical bright field and Interference microscopy, X-ray analysis and electron microscopy. 2. Outline of experimental procedure The experimental procedure briefly consisted of: a) Cutting compression specimens of a rectangular cross-section In a b) compression of the crystals normal to the <14M> direction at a temperature of 6l6°C or 6^9°C,1 at strain ■'■Several specimens of set II underwent brittle frac ture when tested at 6l6°C, at the higher strain rate. To reduce this occurrence the temperature of deformation was increased to 6^9°C. 66 67 ii — ^ ? rates of 5 x 10 vsec and 5 x 10 /sec. c) Taking measurements and appropriate bright-field interference and optical photographs of slip traces on the {8lll and flio} faces, and noting dimension and shape changes in the deformed crystals, d) Cutting the deformed crystals into slices parallel to the primary {111} glide plane, and then the cutting the slices into wafers suitable for making thin foils, e) Annealing about half the wafers from each differ ent deformation group, at temperatures of 700°C to 850°C, and then performing a Laue back-reflection x-ray analysis of each differently deformed and annealed wafer, f) Electrochemical Jet-polishing of the previously prepared wafers, and g) Examining the thin foils in a lOOKv. electron microscope. Photographs were taken of the foils which were coplanar to the {"1113 glide plane, using bright field il lumination. In the cases where networks were observed, the Burgers vectors of the dislocation segments were determined using dark field (diffracted beam) Illumination techniques. 2The strain rate was increased for specimens of Set II, in order to effect an increase in the number of initially active slip planes per unit thickness normal to the primary (111) glide plane. It was hoped that this would increase the probability of occurrence of coplanar dislocation interaction. 68 3. Materials Germanium was obtained from commercial sources In the form of Czochralski grown crystals In a <100> orienta tion. The minimum resistivity was 40 ohm-cm which indicates the impurity atom fraction was below 10-9. The specifica tions included a minimum carrier lifetime of 300 p-sec., and a dislocation etch pit density between 1000 and 3000 pits/cm . The crystal was n-type, undoped, uncompensated, and essen tially intrinsic with antimony and arsenic as the major trace impurities. B . Technique for Producing Twist Networks in Germanium The type of network to be investigated was that which might form by the interaction of parallel nets of disloca tions on the same glide plane. It was reasoned that it was necessary to produce two sets of parallel dislocations at an angle of about 60° to one another, to produce dislocation interactions similar to those in Figure 28. This type of interaction would occur if two different <110> directions were equally activated on one {ill} glide plane. An orientation in which a crystal could be simply deformed in compression or tension to yield such a result can be determined by considering the resolved shear stresses developed in the various <110> {ill} slip systems. When single crystals are deformed, only the component of shear stress resolved on a glide plane and in a slip 69 direction produces a glide force. Thus among the six <110> directions on the four flllj slip planes in the face- centered cubic or diamond cubic structure, the slip systems with the greatest resolved shear stress will predominate in the slip process. Consider a single crystal under simple compression with the axis parallel to the compressive axis, as in Figure 29. The stress tensor is: (23) ij Then the resolved shear stress on any given slip system can be determined by transforming ffij to a coordinate system in which X^' corresponds to the glide direction, and X 2 ' the normal to the glide plane. If a* transforms like a second rank tensor: (24) = rn a lm where: I,j,l,m, = 1,2,3, and t = the rotation matrix for the given slip system then ai2'» shear stress on the glide system In question, (25) 0-12' ~ rn ri2 CTH = cos * 003 0 ffll = m a 11 where: 70 -O' Figure 29 Slip system coordinates for a single crystal under simple compression. 71 (26) m = cos \c o s 0 = Schmid factor (33). A = the angle between the compressive axis and the glide direction ' If, in the face-centered cubic system, the glide plane normal, [111] is oriented to be the pole (A) of a stereographic projection, and the slip vector [110], normal to the direc tion (A), is labeled point (B), then the orientation of a crystal necessary to activate solely that given glide system can be found in the following manner: In Figure 30 the Schmid factors (m-values) for deformation in compression are plotted such that the glide plane normal is the pole (A) and the direction of slip (B) is at the top of the stereo- graphic projection. By superposing these "iso-m" contours onto the similarly oriented slip system In point, the orien tation with the maxiumum "m" value of 0.5 is determined. In a face-centered cubic or diamond cubic material this orienta tion Is about the [lj^] (1). Note that the orientation Is always a maximum on this plot at M5° between the slip direc tion and the glide plane normal. If two <110> directions are to be equally activated on a {ill} plane, the orientation of the compressive axis must be such that the resolved shear stress is equally maximized In both slip directions, and each slip direction has a higher m value than any other operable <110> {ill] slip system. The superposition of the iso-m contours for 72 • I* * in Of* IN I «1 Figure IQ__ Plot of constant m contours in a stereographic projection normal to the glide plane pole. In general, the point A is the general glide plane pole (hkl) and the point B is the general glide direction [h'k'l'] . The resolved shear stress on (hkl) Ch'k* 1'j is read off at the projection of the compressive axis. As a specific example, consider an fee structure with A equal to (111) and B to [110]. Then for the compressive axis C equal to L144] , the resolved shear stress is 0.43. 73 both slip directions, the [110] and the [Toi] on the (111) plane, is shown in Figures 31 and 32. The sum of the two m-values is the greatest and they are equal for a compressive axis orientation of [1*1*1]. The net slip vector, the [211], is about *45° from the [1*1*4] and about 10° from the [Oil] axis. Figure 33 is a two dimensional plot of the sum of the m values, plotted from the [Til] direction to the [111] direction. The maximum of 2m = 0.866 occurs at [1*1*0. The dotted line on Figure 33 is the sum of the iso-m values for the slip system [101], [110] (111), the most probable slip system to be activated as secondary slip if a specimen is deformed in the [X*4*4] orientation, since in compression the axis of compression will move towards the [011] direction, and hence the [111] direction. Thus deformation of a single crystal of germanium oriented with a If dislocations of a characteristic Burgers vector orientation are activated In each slip direction, then two sets of parallel dislocations will interact at angles of 60°, and hopefully produce, on annealing, characteristic dislocation networks as previously described In Section 1.1 and depicted in Figure 28 . Figure 31 Iso-m contours for {jLOll Ij_1q ] (111) dual slip plotted on 0.11J stereographic projection. 75 101 110 ' 0(0 111 0114 110 ✓ 111 100 110 101 Figure ___ [111]stereographic projection showing the relationship of the Cl^^jdeformation axis and the iso-m contours to the major crystal axes. 1.0 m JJ.10] + m aOQ (111) 0.9 mg)ll] (111) + m [Oil] (ill) mfOll] (Hi) + m[Olj] (111) o o V=. E 1*11 144 011 144 111 0° 25° 35° 45° 70 ° Figure ^ Plot of the sum of the Schmid factors for primary and secondary slip systems along the Q.11] - 0-11] boundary for a crystal whose compressive axis is parallel to the U44] . "0 e* 77 C . Specimen Preparation and Deformation 1. Orientation for deformation The germanium single crystal was cut by means of a .015" thick diamond saw into specimens which were rectangu lar in cross-section and oriented so that their faces were normal to the [T44], [Oil] and [811] directions. The [OlT] face was chosen because it is normal to the expected pri mary and secondary slip planes for deformation, as shown in Figure 34. The dimensions were roughly 10 mm. by 4 mm. by 6 mm., respectively. All specimens were lapped on a cast iron lapping wheel in a SiC slurry, to remove most of the damaged surface layer produced during cutting, and to obtain a flat surface. The specimens were then etched in CP-4 for at least two minutes, in order to: (a) remove the remaining damaged sur face layers, and (b) to produce a mirror polish and a flat but slightly undulating surface. 2. Apparatus for compression tests The specimens had to be deformed at, or above, at least 2/5 of the absolute melting temperature ( 2 ). Temper atures of 6l6°C+5° (1140°F) and 649°C+6° (1200°F) were arbitrarily chosen. A tubular, nichrome wiretreslstance furnace with three separately variable heating zones was mounted on an Instron constant strain rate compression testing machine, as in Figure 35. The load was applied through 1" diameter 304 stainless steel cylinders, which 78 [>4 \ End Effects Dominate Primary Slip Traces \ ^ X \ \ \ V V Secondary Slip Traces & End Effects Dominate 6 mm. Compressive Axis = [^44} Figure 3^ Dimensions of compressive specimens oriented for dual slip. 79 INSTRON Thermocouple Used For Temperofure Meosuremenf •"1 Nichrone Wound Furnace SS Thermocouples . Leading to a Rectangular. Germonium - Continuously Compensating Crystol — ' Marshall Control ler H . II . \ 0 Rings Gas Outlet C s Graphite SS = 304 Stainless /77777777T77777 Steel Figure 35 Nichrome wound tubular furnace mounted on Instron Testing machine In which germanium single crystals were deformed at temperatures of 6l6°C and 64l9°C under a slightly oxidizing atmosphere. moved through a hollow graphite cylinder. The graphite and a continuously flowing mixture of 10% H£/90% Ng gases mini mized the oxidation of samples during testing. A marshall controller maintained the temperature of the furnace within 5° of the testing temperature by means of eight alumel- chromel thermocouples extending through the furnace wall into the heating zones. The specimen temperature was measured by an Alumel-Chrome1 thermocouple imbedded in the graphite cylinder, in close proximity to the specimen. The specimen was placed between two separate stainless steel discs, which had been mechanically polished with emery papers ending with grade n U/0* A molybdenum disulfide lubricant was used to further reduce friction between the specimen and end plates. 3. Compression testing procedures _ if Two strain rates were used, viz: 5 x 10 /sec. and 5 x 10“3/sec. The specimens were heated and cooled at an average rate of about +20°C/minute and -10°C/minute, respectively, within the graphite cylinder. When the required testing temperature had been attained, and a minimum of 10 minutes had been allowed for thermal equilibrium, the samples were deformed to various degrees of strain, ranging from 1% to 30% glide strain. After cooling to room temperature, the deformed specimens were measured for dimensional changes, and the overall shapes of the specimens were noted. The specimens were 81 then etched with hydroflouric acid (50>6 concentrated) to remove the surface oxide, GeOg, which formed during deforma tion. The hydroflouric acid removed the GeOg hut did not affect the germanium. A Zeiss interferometer was then used # to examine the two lateral faces for the purpose of record ing chracteristic slip traces. Both white and thallium (wavelength =■ 0.5^n) light illumination was used to record on 35 mm. film bright field and interferometric microphoto graphs of the slip traces. Prom these photographs it was possible to obtain, for each specimen, the angle between the slip pj.ane and the compression axis, the spacing between slip planes, and other general characteristics of the deformation. Table 3 shows, for each specimen, the strain rate, the original and final dimensions and the deformation temperature. The deformed specimens were then oriented and sliced parallel to the primary (111) glide plane, with a diamond slicing wheel, to a slice thickness of about 0.025” (0.63 mm.). Four or five slices were obtained from the center two-thirds of each specimen and the ends discarded to avoid effects due to the frictional forces at the end plates (Figure 36). These slices were lapped on a cast iron wheel in a Sic slurry to produce a flat surface, and then etched in CP-*1 for a minimum of two minutes to remove damaged surface layers. Two or three of the slices from each specimen were 82 TABLE 3 GERMANIUM CRYSTALS ORIENTED FOR COMPRESSION IN THE [144] DIRECTION — ...... ------T Original -Final Dimensions (mm.) Speci Temper Strain Compressive Set men ature Rate Axis # Number °C Sec“l [144] [011] [811] ■ C-30 616 5x10-** 16.8 16.4 3.92 3.92 5.82 5-92 C-28 11 Tl 15.8 15.1 3.96 3.96 5.85 5.98 C-26 11 It 16.7 16.05 4,05 4.05 5.80 5.90 I C-29 11 ir 16.6 15.6 3.86 3.86 5.55 6.02 tl ii C-55 16.2 14.8 3.75 3.75 5.62 6.25 C-25 11 ii 15.3 13.2 4.92 4.95 5.85 6.93 C-27 It it 16.8 12.6 3.96 __a 5.87 __a L C-51 II it 16. 4 12.3 3.90 3.88 5.72 7.60 ' C-42 649 5x10-3 16.6 15.9 3.70 3.70 5.55 5.88 C-36 616 11 16.6 15.5 3.85 3.85 5.57 6.12 C-40 It 11 16.7 15.1 3.70 3-70 5.60 6.36 C-38 IV IT 16.8 14.9 3.80 3.84 5.55 __b IV ri II C-37 16.3 13.6 3.70 3.75 5.70 7.30 C-39 649 tl 16.3 13.3 3.70 3.76 5.68 7.35 C-43 If tr 16.7 13.9 3.70 3.75 5.51 __b C-50 616 u 16.6 11.8 1.85 3.89 6.46 8.80 C-53 11 ii 15-5 12. 3 3.90 3.92 5.75 7.42 C-75 750 5x10“** 16.6 12.0 4.02 5.71 11 C-33c 616 16.4 15.6 4.33 4.33 5.71 6.11 C-101 616 5x15*"^ 13.7 10.02 4.00 4.31 4.80 6.55 C-102 Tl Tl 13.6 9.05 3.80 4.11 4.65 --- C-103 II 5x10-3 13-65 9.70 3.92 4 .14 4.75 -— - aIn C-27 the specimen was deformed and twisted to such a degree that accurate measurements were not possible. The specimen fractured down the center parallel to the compression axis near the end of the deformation, so accurate measurements were not possible. cC-33, specimen whose compression axis was 5° closer to the [Oil] direction; approximately the [T66] orientation. 83 Deformed Crystal Oriented For Dual Slip [en] \ Slice Cut Parallel to Primary Slip Slice Dimensions: Traces ~ 0 6 min. x ~ 8 g mm. x ~ 4mm. ["'] [2"] [ ° |T] Figure 36 (111) oriented rectangular slices parallel to the primary slip plane were cut from the deformed crystals. 8*4 annealed at temperatures up to 850°C for up to 36 hours in a pressure of 2 x 10”^ Torr. Finally the crystallographic orientations of the slices were checked by means of x-ray diffraction. This procedure resulted in two distinct sets of crystals, each set having been deformed at a different strain rate and then annealed at a different temperature. Crystals Set I were strained at 6l6°C, 5 x 10~Vsec, from which wafers were annealed at 850°C for 36 hours.^ Crystals of Set II were strained at 6l6°C or 6*49°C, 5 x 10~3/sec from which wafers were annealed at 700°C for 2*4 hours. D . Equations for Calculating True Glide Stress and True Glide Strain from Load-Elongatlon Data More information can be obtained about the deformation characteristics of a single crystal, if the load-elongation values are coverted to the stress resolved in the slip direction, and strain per unit area of the glide plane. For a crystal in the [T*4*4] orientation, the net slip vector on the (111) plane is the vector sum of the two favored <110> slip vectors: [TlO] + [101] * [5ll]. Initially the slip plane normal is *45° from the compressive axis. The Schmid ^Except for specimen C-55, from which wafers were annealed at 750°C and 850°C for 2*4 hours. ^Except for specimen C-38 from which wafers were also annealed at 700°C for 6 hours. 85 factor for the net slip vector; [211], is a maximum, (0.50) at 45° from the compressive axis and the [811] direction. The Schmid factor m^ [211] is related to the Schmid-factors, <110> , of the favored slip directions by the equation: m I10 + m Ioi 2TCT8'6'6')— = -°50 if m^10 = .^33 m 101 = ***33 where the Schmid factor is: (28) m^ = cos0cosX = 0.500 Here: X * the angle between the compressive axis and the net slip direction and X * the angle between the compressive axis and the slip plane * 90-0. The [211] direction lies in the same plane as the(ill} and the . It is assumed that as deformation proceeds, the net slip vector is defined by: (29) [net slip vector] » fi[110]+ f_[101]+ f <110> « 3 where: f^ = fraction of slip in the [110](111) fg = fraction of slip in the [101](111) fg = fraction of slip in any other <110> (ill} systems (30) r 1 + f2 + r3 * 1.0 86 initially: f^ = f2 » **3 a 0 and then: 1/2 [110] + 1/2[101] = 1/2[211] If It is assumed that: (1) through all stages of deformation the slip occurs equally in the two directions [110] and [101], or f^ = f2 » and (2) that other slip systems operate only to a negligible degree, or f 3 < < f! - f 2 then the net direction of slip will be the [211] and 3ince (31) ([IKK] x [111]) - [211] = 0 where K = 4 initially K > U as deforma tion proceeds then: (32) and from equation 2 8 , (33) = cos sin If the true normal compressive stress Is defined as: <3*0 T i - Ai = a 0 (Ai ) " A0 ( Lc > where: = load at any instant of time t^ A^ = cross-sectional area at any Instantof timet^ La = Instantaneous length of the specimen attime t^ L0 3 original length of specimen at time tQ, A0 - original cross-sectional area at time tQ 87 assuming that there is no volume change during deformation, and that the average cross-sectional area at any time t^ is: (35) the true glide stress resolved in the direction of the net slip vector is then: (36) where the only unknown parameter is the Schmid factor which will vary as deformation proceeds. If it Is assumed that the ends of the crystal are constrained with respect to lateral motion because of the frictional forces existing between the end plates and the specimen surfaces, then slip on the [?11](111) system will result in slip plane rotation. This can be compared to the ideal case in which there are no frictional forces on specimen and surfaces, and no slip plane rotation, as shown in Figure 37. The slip plane normal [111], will rotate towards the compressive axis. This rotation was observed experimen tally. The lateral bulging observed at the center of the crystals was also indicative of constrained lateral motion of the end surfaces during compression. In this non-ideal slip condition, the variation in the angle is related to the changes In specimen length by: 88 X j - 45' (a) Ideal Deformation ( Frictionless Surface} Xj = 45 Xj > 45 tb) Non- Ideal Deformation (Lateral Constraint at End Plates) Figure 37 Ideal and non-ideal deformation of single crystals In compression. as can be seen In Figure 38* This approximation Is a very good one for lower degrees of strain. However, at higher strains when the lattice plane rotation is reduced near the ends of the specimen because of the frictional constraints, this relationship must be modified. However, equation (37) will be assumed to be true at all times for these calculations of the true glide stress, and the effect of this assumption will be discussed with respect to experimental observations, in a later section (see section II.G). From equation (37): L i COS X-£ = ( jj- ) cos Xq and since: 2 .1/2 (38) Sin X± = (1-cos X^) then equation (28) becomes: (39) = sinX^cosXi = ( ^ ) cosX0[l-cos2XQ (j^) ]1/2 The resulting equation for the true glide stress is: J1 \2 1/2 (ho) u - } cosx CO3A0LI-«Oa0 [i-cos 2x Aq^y0 (IA ) 2 ] A—lO "O O "O O O or (Ito) - P±/A 0 CMj) where the parameter M* Is defined as: 90 to) Initially at Time f0 K L( cos X0 o (b) at Anytime tj > t0; Lj < L, KLi cos Xi = tXj > X0 ) ( I— i < L 0> If R0 R| R 0 cos X0 Rj cos Xj K * — ------ cos X0 _ Lq cos Xj Li visur* 38 Relationship bstwssn ^(I^) and X^ In Figure (39). Hence the resolved shear stress can be determined completely a3 a function of (P^, PQ ,L^,L0). The equation for will be valid as long as slip occurs pri marily on the [110], [101](111) slip system. As deformation proceeds, the active [2Tl](lll) slip system will cause the compressive axis to rotate towards the normal of the active (111) glide plane, along the [Tll]- [011]-[111] great circle. As the compressive axis becomes parallel to the [Oil] direction, the value for the primary [TlO], [T01](lll) slip system will have decreased to value ideally equal to the value for slip system [110] [101] [Til]. This can be seen by referring to the [111] stereographic projection in Figure 32. This slip system shall be called the secondary slip system henceforth. At this point where (primary) = m^ (secondary) X^ is about 55°. As deformation proceeds beyond this point several possibilities exist: 1) The primary slip system can continue to dominate and the angle will continue to increase as in equation (72). 2) The primary slip system can continue to dominate beyond the value of X^ = 55° until the resolved shear stress 92 t Equal Slip on Both Planes Primary Overshoot 0.4 Equal Slip on Both Planes Primary 0.3 Overshoot 0 0.10 0.20 0,30 0.40 AL / L0 - Figure 39 Plot of Mi and mi as a function of- The value of = 55° at which this reversal would occur would, among other things, be a function of the strain rate and the amount of primary slip that had already occurred. It is difficult to predict when this reversal might occur. 3) The primary and secondary slip systems can become equally active and XA (primary) tend to remain constant at a value of about 55°, where X^ (primary) = X^ (secondary). This would represent a metastable condition in which slip Is occurring on both slip systems, and a decrease or increase in either X^ (primary) or (secondary) would be unstable. For the calculations of the true glide stress, the tentative assumption was made that beyond a AL/LQ o f — .19, or when the compressive axis became parallel to the [Oil] axis, the stress represented continued slip on the [110], [101] (111) slip system. This assumption will be discussed relative to experimental observations at a later point In this section. The true glide stress may then be calculated 94 by use of Figure 39 if AL/L0 , P^L^ ) , and AQ are known. The true glide strain is determined by calculating the unit shear in the [211] direction per unit thickness normal to the glide plane. If the engineering strain is defined: <*3> e = = < ! -LJ £ > Jo The first step is to calculate the value of e[211] from e[l44]. If the angle between the operating slip plane and the compressive axis Is X^, then for every unit of length change in the compressive direction, there will be 1/cosX^ units of length change In the [211], Hence: (44) eg l = crancj where the value of cos Xj for any given value of is (from equation (33): m i _ m i 0 5 ) cosXi - sTnXi a _cos2Xl)l/2 or (46) cosXi * [1/2 - l/2(l-4m12)]:L/2 where the value of for any given AL/LQ may be calculated from equation (39). The true glide strain may then be defined: 95 By substituting equations 44 and 46 into equation 47, the equation for the true glide strain results* 1 (48) C'gi = In (l+egl) = In ^ 1+[ 2^1/23T72 Hence from a given value of AL/L0 the value of € gj_ may be calculated. A parameter G is defined: 0 - £35x7 * which is plotted in Figure 40. A table may now be constructed in which the values of and are calculated for every AL/L0 . These values are shown in Table 4, along with the resulting values of 6 gl and the theoretical value of X^ up to 3 55°. For deformation beyond a AL/L0 of 0.19, at which point the secondary slip system becomes active, the theoretical values for M^mijGj and £ gi are computed for two different pos sible situations. The values in Table 4, column (a) repre sent case (1) described on page 91, in which the primary slip system dominates throughout deformation. This has been called "primary overshoot." The values in column (b) represent case (3) in which both slip systems are equally active. Another possible reaction at the point where the compressive axis is parallel to the [Oil], and both systems are equally favored, is that either the [110] or the [101] 96 2.0 1.74 8 Equal Slip on Both Planes CD Primary 0 0.10 0 .4 0 AL / L0— * Figure 40 Plot of parameter as a function of AL/LQ. 97 TABLE 1* Values of egl AL/LQ ) ^ g l Li AL m± Mi Q i in (l+G (4k)) Theoret Lo Lo O ical 1.00 0.00 0.500 0.500 1.414 0 45.0 0.99 .01 .499 .494 1.429 0.0141 45.4 .98 .02 .499 .489 1.445 .0284 46.2 .97 .03 .498 .484 1.466 .0428 46.7 .96 .04 .498 .478 1.481 .0575 47.3 .95 .05 .497 .473 1.492 .0721 47.8 .94 .06 .496 .467 1,504 . 0866 48. 3 .92 .08 .494 .455 1.538 .117 49.4 .90 . 10 .491 .441 1.572 .147 50.5 .88 .12 .487 ,429 1.608 .176 51.5 .86 .lit .483 .415 1.645 .206 52.5 .84 .16 .478 .402 1.681 .237 53.5 .82 .18 .472 .388 1.725 .269 54.5 .81 .19 .469 . 380 1 . 7 W .286 55.0 .80 .20 .'4(58' .469 .372 .373 1.767 1.748 .363 .293 55.5 55 .78 .22 .462 " .359 .363 1.815 .336 .325 56.6 " .76 .24 .456 .344 .354 1.862 .369 .350 57.5 " .7** .26 .448 .330 .345 1.912 .404 .375 58.5 " .72 .28 .441 " .314 .335 1.965 " .438 .398 59.5 " .70 • 30 .432 " .300 .326 2.020 " .474 .422 60.3 " .65 .35 .404 " .260 .303 2.174 " .566 .477 62.6 11 .60 .40 . 375 .225 .280 2.360 " .454 .535 64.9 " .50 .50 .325 .162 .233 2.890 ------69.3 n (a) (b) (a) (b) (a) (b) (a) (b) (a) (b) (a) assuming primary overshoot, i.e., continued predominance of slip on the primary glide plane. (b) assuming duplex slip, i.e., equal slip occurs on both the primary and secondary slip plane. (1) assuming the relationship: ^s valid cosX^ slip direction on the primary glide plane may dominate, or f^ f1 f2 in equation (29). If,for example, f^ > fg and the slip occurred predominantly in the £110] direction, the compressive axis would rotate towards the £010], However, for a rotation of 6° towards the [010] the angle X^ would only change by 1°, so that observations of the change in angle would not reveal this phenomenon. Such slip would Increase the B dimension of the crystal, and perhaps create a torque vector normal to the compressive axis. In some crystals deformed beyond about 25% (glide strain), a definite twisting was observed as well as a slight increase in the B dimension. Figure Ul shows ^ gi plotted as a function of AL/L0 , up to a AL/Lo “ 0.19. Above that value the two curves represent the values from columns (a) and (b) in Table *1. 0.60 r- 99 0.50 (a) Primary Overshoot 0.40 £ Ib) Equal Slip on £ Both Planes "•s. E E 0.30 VI/ 0.20 0.10 O 0.10 0.20 0 .3 0 0 .4 0 AL/L o Figure 41 True glide strain C^gl) plotted as a function of^L/L0 , Curve (a) for primary overshoot and' curve (b) for equal slip on both primary and secondary slip planes were plotted using values from columns (a) and (b) in Table 4. III. RESULTS— DEFORMATION CURVES AND SLIP TRACE ANALYSIS This chapter includes (A) the resulting stress-strain curves for specimens of Set I and Set II as well as for other specimens deformed in various orientations at differ ent strain rates and temperatures; and (B) observations of slip plane rotation and corresponding surface slip traces, of specimens from Sets I and II. A. Stress-Strain Curves 1. Specimens from Set I Set I includes the specimens; C-30, C-28, C-26, C-55, C-25, C-27, and C-51 which were deformed at a strain rate of 5 x 10-l*/sec. at 6l6°C. Figure b2 shows the calcu lated glide stress-strain curves for these specimens. The variation In the shape and initial values of the yield stress and minimum flow stress from one curve to the next was relatively small. The average values of the yield stress, minimum flow stress, strain at the minimum flow stress, the slope of "stage I" work-hardening, the "length" of stage I, and the approximate slope of the stage II region were calculated to be (Table 5): 100 2.0 C - 3 0 Crystals of Set I Deformed in a £744 J Dual Glide Orientation. C - 2 8 2.0 C - 2 6 10 eg 2.0 C - 2 9 o» C - 5 5 co 2.0 1.0 CO C - 2 5 2 0 a> 1.0 C - 2 7 2 0 2.0 1.0 O 0.10 0.20 0 .3 0 0.40 € gl True Glide Strain Figure *12 Specimens of Set I strained at 5 x 10"Vsec at 6l6°C. *This specimen was cut from a different crystal than the others in Set I. 102 TABLE 5 SET I— STRESS-STRAIN RESULTS Yield Stress (kg/mm2 ) 2.0 +0.5 (range) Minimum Flow Stress < " ) 1*5 + 0.2 Strain at Minimum Flow Stress (mm/mm) 0.030 + 0.010 Slope of State I (kg-mm/mm3) 10.5 + 1.0 Length of Stage I (mm/mm) 0.3 + 0.01 to 0.17 + 0.00 Slope of Stage II {kg-mm/mm ) 26 + 3 An average curve for the specimens In Set I is shown in Figure *13* Each specimen is marked jat the respective strain at which deformation was halted. The specimens were deformed to the following glide strains: TABLE 6 SET I— SPECIMEN STRAINS Specimen Number AL/L €gi __ C-30 0.006 0.008 C-26 0.030 0.0*13 C-28 0.031 0.0*1** C-29 0.0*13 0.062 C-55 0.086 0.126 C-25 0.132 0.191* C-27 0.210 0 .320* C-51 0.226 0. 3**7a aUsing the strain values assumed In column (a) in Table *». Average forcurveworking qseclmens from I, strainedSet 5 x at N Resolved Shear Stress (k g /m m 2.0 0 4 6.0 0 3 8.0 5.0 7.0 0 6 2 - C -28 C C 29 - 0.10 l Tu Gie Strain Glide True gl, € C55 - iue 43Figure 0.20 C- 25 C- 10~Vsec 10~Vsec 0.30 at 6l6°C.at 7 2 - C C- 51C- 103 101* Specimens of Set II Four specimens in Set II which were deformed at a strain rate of 5 x 10 vsec were deformed at a temperature of 6l6°C. The average values of yield stress, minimum flow stress, strain at minimum flow stress, and the slops for the stage I work-hardening for specimens: C-36, C-37, C-38, and C-40 were: TABLE 7 SET II— 6l6°C-STRESS-STRAIN RESULTS Yield Stress (kg/mm2) 6.9 +1.0 (range) Minimum Flow Stress C " ) 3.35 + 0.7 Strain at Minimum Flow Stress (mm/mm) 0.075 + 0.015 Slope of Stage I (kg-mm/mm^) 16 + 2 The Individual stress-strain curves are shown in Figure 1+1+. The average stress- strain curve Is shown In Figure 1+5. The four specimens were deformed to the following strains: TABLE 8 SET II— 6l6°C-SPECIMEN STRAINS Specimen Number AL/L C-36 0.066 0.086 C-40 0.110 0.160 C-38 0.110 0.160 C-37 0.168 0.251* Resolved Shear Stress 0 6 5.0 .0 4 3 0 3 5.0 6.0 2.0 4.0 2.0 3.0 7.0 6.0 5.0 .0 4 .0 3 m 0 6 3 C- 2 4 - C -40 4 - C 0.10 Q, re ld Strain Glide True Ql, € iue MU Figure 8 3 - C 3 4 - C 0.20 e n Cytl Strained Crystals n Set t 3/sec e s / '3 0 1 x 5 at 9 3 - C n [ 44J Dual J 4 14 [ a In ld Orientation' O Glide eomd t c ° 9 4 6 at Deformed Deformed at at Deformed 0.30 6 ! 105 6 ®c vrg okn uvs o pcmn o e I tanda 5 1"/e. Curve forAverage 10"3/3ec.curves strainedSet 5 xspecimensworkingII of at N r r Resolved Shear S fr e s s (K g /m m .O .0 0.30 0.20 O.IO 0 #3 ersns pcmndfre t 750°C. aspecimenrepresentsatdeformed gl, re ld Strain Glide True , l g € © Specimen Specimen © o Seies eomd t °C 9 4 6 at Deformed Specimens For © o Seies eomd t 1 °C 616 at Deformed Specimens For © pcmn o St H Set of Specimens iue 45Figure 75 -7 0 eomd t 750°C at Deformed 37 -3 C 5 C 7 - o 107 At the higher strain rate of 5 x 10 J specimens often fractured longitudinally during deformation. To reduce the frequency of this occurrence three specimens were deformed at 649°C. The average values of yield stress, minimum flow stress, strain at minimum flow stress, and the slope of stage I for the specimens: C-42, C-43, and C-39 were TABLE 9 SET II— 649°C-STRESS-STRAIN RESULTS Yield Stress (leg/mm^) 5*0 +0.7 (range) Minimum Flow Stress ( " ) 2 . 5 5 + 0 . 1 5 Strain at Minimum 1‘lwW Stress (mm/mm) 0.053 + 0.013 Slope of Stage I (kg-mm/mm^) 9.5 + 1 The Individual stress-strain curves are also shown in Figure 4*1. An average curve is shown in Figure 45. The specimens were deformed to the following strains: TABLE 10 SET II— 649°C-SPECIMEN STRAINS Specimen Number AL/L £gl C-42 0.039 0.056 C-43 0.160 0*258 C-39 0.171 0.256 108 Figure 46 shows the variation in yield stress, minimum flow stress, minimum strain at minimum flow stress, and the slope of stage I, as a function of the strain rate difference between Set I (5 x 10"^/sec) and Set II (5 x 10"3) at 6l6°C. The primary difference in the glide stress-strain curves for Set I and Set II was the definite increase In slope at a glide strain of about 0.18 for the specimens deformed at a lower strain rate. 3. Other single crystals deformed In compression 1. A specimen, C-75 oriented with its compressive axis parallel to the [144] direction, was deformed at 750°C, at a strain rate of 5 x 10“^/sec. The resulting glide stress-strain curve is shown in Figure 45. The specimen fractured at a strain of about 45)E, hence no observations were possible of the deformed crystal. The values of the deformation parameters were: 109 901- 18 5.0 16 E N E N E 7.0 Slope of Stage I 14 E E E 6.0 12 L 10 w ® 5.0 10 W 4.0 Yield Stress 3 3 0 "O « _> o a. «/> 2.0 4 B ® C/> DC Minimum Flow 1.0 S tress 0 J I I J —L 11 t J L J—u O 10- 3 -4 -3 5x10 5 x 10 € ( s e c ’ ) Figure 46 Variation In yield stress, minimum flow stress and slope of state I plotted as a function of strain rate at a tempera ture of 616°C. 110 TABLE 11 SPECIMEN C-75— STRESS-STRAIN RESULTS Yield Stress (kg/mm^) 1.52 Minimum flow stress ( " ) 1.20 Strain at minimum flow stress (mm/mm) 0.016 Slope of stage I (from 0.02 to 0.15 strain) (kg-mm/mm ) approx. 15 This specimen was deformed at 750°C In order to observe the effect of temperature on the yield stress, minimum flow stress, and the slope of Stage I. The results are shown in Figure *17. 2. A specimen, C-33, was deformed at 6l6°C at a strain rate of 5 x 10 /sec. The compressive axis of the specimen was oriented about 5° from the [Oil] axis, or approximately parallel to the [166]. In this orientation the initial Schmid factor for the primary [T01][110](111) slip system was about 0.486, and the Schmid factor for the secondary [101]C110](111) slip system was about 0.458. Hence the difference in the Schmid factors for the two slip systems was about 0.028. The difference in Schmid factors between the two slip systems for a crystal with Its compressive axis parallel to the [T44] is about 0.056. As deformation proceeds, and the compressive axis rotates towards the normal to its slip plane, the difference in the two Schmid factors will decrease. Ill 8.0 (M 7 0 0.10 Yield Stress 2 6.0 y> u> ® 5.0 Strain at Minimum Flow Stress w a- 4.0 O 2 0 £ gi True Glide Strain (/> a> Minimum (T Flow Stress 0.00 6 0 0 6 5 0 7 0 0 7 5 0 Temperature °C F ig u re *17 Variation in yield stress, minimum flow stress and strain at minimum flow stress plotted as a function of temperature at a strain rate of 5 x 10~3/3ec. 112 Specimen C-33 was strained to about 5%, which repre sents a rotation of the primary slip plane of 1-2°, ,and which would increase the initial angle of X^ = 50° to an angle X^ = 52°, 3° below the value of X^ = 55° at which X^ (primary) = X^ (secondary). The object in straining specimen C-33 %5 was to examine the amount of secondary slip which might occur in the specimen even though primary slip should still be favored after a primary slip plane rotation of only 1-2°. If secondary slip did occur In this specimen, It can be directly compared to that observed in specimen C-^2 which was deformed about 5*6%. Similarly the amount of secondary slip which occurred in C-33 can be compared to the amount of secondary slip which occurred In specimen C-25 which was strained 19%, and In which there should have occurred a 7° rotation of the primary slip plane to a value of about 52° = X^. The only difference then between specimen C-33 and specimen C-25 would be the amount of primary slip which had occurred before the slip plane had rotated to a value of X^ * 52°. The primary and secondary slip plane traces which were observed on the surfaces of these three specimens are shown In section III.B, and are discussed there. 3. Two specimens, C-101 and C-102 were deformed at h 6l6°C, at a strain rate of 6 x 10 /sec. The crystals were oriented such that the compressive axis was parallel to the as depicted in Figure *18. In this orientation 113 [ "S'] [i2l] Figure 48 Orientation of specimens C-101, C-102 and C-10 3 oriented for single slip. 11^1 slip should occur primarily on the [T01](lll) slip system. As deformation proceeds the compressive axis should rotate towards the [111]. As the compressive axis becomes parallel to the [012] direction, the favored secondary slip system is the [1013(111) slip system which is shown on the [111] sterographic projection in Figure 32 . This primary slip direction corresponds to one of the two slip directions active in dual slip. The work-hardening effect of the [110] "dual" slip on the [101](111) slip can be deduced from the resolved shear stress in the [101](111) slip system in both the dual and single slip oriented crystals. Figure ^9 shows the average glide stress-strain curve obtained from specimens C-101 and C-102. For comparison, the glide stress-strain curve for a dual slip specimen, C-27, from Set I is also shown, in which the resolved shear stress in the [101](111) direction is plotted. The following values were obtained from the single slip specimens: TABLE 12 STRESS-STRAIN RESULTS FOR DUAL VERSUS SINGLE SLIP-5xl0-V s E C . Single Slip Dual Slip Yield Stress (kg/mm) 1.7 + O.l(range) 1.72 Minimum Flow Stress( " ) 1.M5 +0,15 1.30 Strain at Minimum Flow Stress (mm/mm) 0.025 +_ 0.003 0.030 Slope of Stage I(kg-mm/mm3) 7.0 +0.5 7.7+1.2 CM 7 . 0 E Average Curve of Specimens £ C - 101 and C -102 Representing c-27 N 6.0 o* Single Slip, with Compressive Axis Originally Parallel to the (/> w wCD to o CD J= if) ■o CD Specimen C-27 > O Representing Dual Slip M CD with Compressive Axis (Z 1.0 Originally Parallel to The [Ta a ] 0 1 ! I J 0 0.10 0.20 0.30 0.40 0.50 0.60 £ g l, True Glide Strain Figure 49 Stress-strain curves for single slip vs dual slip for specimens deformed at 5 x 10“Vsec, at a temperature of 6l6°C, X represents the resolved shear stress on the [101](111) slip system. 1X6 In the single slip specimens, as compared to the dual slip specimens: 1) the upper yield point was less pronounced, 2) the lower yield stress was reached at lower glide strains, 3) the work hardening slope in Stage I (A'-B1) (Figure 49) for the single slip specimens was approximately the same as the slope in stage I (A-C) for the dual slip specimens, 4) the work hardening in stage II beyond B' In the single slip specimens was less than that beyond C in the daul slip specimens, and 5) the flow stress immediately beyond the minimum flow stress wa3 higher in the single slip specimens than in the dual slip specimens. Another single slip specimen, C-103* was deformed at 6l6°C at a strain rate of 6 x 10~^/sec. The compressive axis was also parallel to the [149]. The values obtained from the curve shown in Figure 5*1 were: TABLE 13 STRESS-STRAIN RESULTS FOR DUAL VERSUS SINGLE SLIP-5xlO~3/SEC. ^Single*' Slip Dual Slip™ Yield Stress (kg/mm) 4.9 5.8 Minimum Plow Stress ( " ) 2.8 2.8 Strain at Minimum Flow Stress (mm/mm) 0.040 0.075 Slope of Stage I (kg-mm/mm^) H 11.5+1.5 117 In Figure 50, the average curve for specimens C-36, C-37 and C-^JO is shown for comparison. The stress T. is the resolved shear stress in the [101](111) for both curves. In the case of a higher strain rate, the observations numbered (1), (2), (3) and (5) In the above paragraph were found to be true. However, as in the dual slip specimens In Set II, ho change in slope occurred at B' (glide strain - 0.19) or at B (glide strains 0.20). B . Observations of Slip Plane Rotation and Crystal Surfaces 1. Slip plane rotation Using equation (37) as a basis for calculating the theoretically expected angles X ^ , the observed angles X^ may be compared with the theoretical values as a function of the strain, which is shown in Figure 51. The experimental values of X^ were obtained by measuring the angle between the compressive axis and the slip traces which were observed at the center of the [011] lateral surface of the deformed crystal. This procedure should produce accurate measurements because the [Oil] direction is normal to both the slip plane normal [111] and the net slip direction [211]. The experi mentally observed angles fell along the theoretical line up to a strain of about 0.28. Above a strain of 0.28 (X^=55% compressive axis Is parallel to the [011] direction), separate theoretical lines are shown. The line labeled (1) represents the case In which "primary overshoot" occurs, and the [101](111) slip system. slip [101](111) the Stress strain curves for single slip vs* dual slip for specimens deformed at deformed specimens for slip dual vs* slip single for curves strain Stress x 0vsec t eprtr o l*. rpeet te eovdsersrs on stress shear resolved the represents T 6l6*C. of temperature a at c e s 10”v x 5 Resolved Shear Stress (kg/m m 2 ) ige lp ih Compressive with Slip Single o h [ the to Axis Specim en en Specim Originally 749 -103 C - ] Parallel Representing Representing gl € True True t iue 50 Figure ld Strain Glide vrg Wrig Curve Working Average rm Figure From ih opesv Axis Compressive with ersnig Dual Representing rgnly aall o the to Parallel Originally Slip x Experimentally observed values of X^ superposed on the theoretically derived curve derived theoretically the on superposed X^ of values observed Experimentally o h aito nX, s fnto f LL . AL/L0 of Xj, in function a as variation the for ( D e g r e e s ) 45 50 0 6 .0 .0 .0 40 0. 0 .5 0 0 .4 0 0.30 0.20 0.10 0 C-30 C-26 28 -2 C C- 29 C- 0.10 40 -4 C 25 -2 C L/ L0 / AL gl £ iue 51 Figure (Corrected) Overshoot - Overshoot Primary f C-37 C-39 0.20 C-51 ulx Slip Duplex 0.30 \ Overshoot Primary 120 slip occurs primarily on the [101], [110] (111) slip system, and (2) represents no "primary overshoot, in which case both slip systems operate equally such that X^ * 55° - constant for strains beyond 0,285. The dotted lines in Figure 51 represent possible combinations of "primary overshoot" fol lowed by reversal of the slip plane rotation which would result if secondary slip became dominant in the deformation process. Plate 6 shows to lateral views of the specimens C-55, C-53, C-51 and C-50 strained to 12JC and which were deformed to strains greater than 30%. The observed values of for the three latter specimens vary significant from the expected values. Specimens which were deformed beyond about -.25 strain were often observed to be twisted normal to the com pressive axis, whereas those strained less than 25X exhibited only lateral expansion and bulging in the [811] direction. Specimen C-50, which was strained to about 37%, more than any other specimen, exhibited an experimentally observed angle of X^ « 58° which was close to the expected angle for pri mary overshoot. This was 5° lower than the measured angle of X^ = 63° for C-51, which was strained to about 35J; 2J£ less than specimen C-50. Similarly specimen C-50 was the only crystal which exhibited little twisting, and a more uniform lateral expansion. Evidently the two observations are related to one another. This behavior has been partly rationalized in a later section: IV. 121 Plate Magnification: 3X Specimen C-55 which was strained to 12%. View of (Oil) and (811) faces. Specimen C-53 which was ■trained to 30%. View of (Oil) and (811) faces. Specimen C-51 which was ■trained to 35%. View of (Oil) and (811) faces. Specimen C-50 which was •trained to 37%. View of (Oil) and (811) faces. 122 Specimen C-101, which was deformed to about 395C In easy glide, exhibited an angle of about 58°, which would fall on the theoretical line for primary overshoot in Figure 51. 2. Crystal surfaces Each deformed specimen was examined by means of optical bright field and interference microscopy. Photo graphs were taken of the two lateral faces; the (OTl) and the (811), with white light illumination for bright field observations, and both white light and green thallium light illumination for interference patterns. Thallium light has a half-wavelength of A/2 = 0.27 microns, which yields for the distance from the center of one dark fringe to the center of the next a value of 0.27 microns. The objectives of this examination were to obtain the following information if possible: i) from the (OTl) face: to obtain measurements of the angle X^ for slip plane rotation, as mentioned previously; to note any change in the angle X^ along the length of the specimen; and to obtain an estimate of the average spacing between slip traces, ii) from the (811) face: to obtain an estimate of the spacing between slip steps; the height of slip steps, and to observe any deviation from ideal "dual" slip behavior. If one of the two operating <110> slip directions was 123 dominating the slip system, the (8ll) face would show any slip plane bending normal to the [211] direction. The height and spacing between slip steps on the (811) face were not measured because good interference pat terns were difficult to obtain from the distorted (811) face. The slip trace spacing on the (Oil) faces generally decreased as the strain increased. However, above about 3% strain the change in the spacing was small. This change in spacing can be observed in Plate ?. C-30 (1!6 strain) (a) exhibited an average spacing of 6u-80 microns. The remain ing two specimens (b) and (c), had an average spacing of about 30-50 microns. All measurements were made on inter ference patterns at 51X, and included all visibly discernible slip traces, although the fine structure was not clearly visible at this magnification. A shortening of the 3lip trace lengths as well as coarsening at higher strains is also observable in Plate 7. The average spacing between slip traces on the (Oil) plane for all the specimens from Set II was about 15-35 microns, a value about 30% lower than for the specimens deformed at the lower strain rate. There was only a very gradual trend toward a smaller spacing at higher strains, as can be seen in Plate 8. Examination of the (8ll) faces of all the crystals revealed that there was slip plane rotation or bending normal to the [211], Except for specimens C-30 (3°) and Plate 7 View of (cTl) face using bright and interference microscopy. Set I (a) Specimen number: C-30, strained 0.8*. (b) Specimen number: C-55, strained 12.6X. (c) Specimen number: C-25, strained 19.&*. Plate 8 View of (Oil) faces using bright and interference micro scopy. Set II (a) Specimen number: C-*»2, strained 5 - 6 St. (b) Specimen number: C-^0, strained 16.056. (c) Specimen number: C-50, strained 37%. 126 C-28 (2°), most specimens exhibited slip plane traces in tersecting at angles of 6-8° , regardless of the amount of strain. Figure 52 show3 an idealized drawing of a typical (811) face. Plate 9 shows changes in the (811) face as a function of strain. The plane rotation or bending was most pronounced near the (Oil) lateral surfaces of the specimens. If frictional restraints were reduced near the perimeter of the specimen base, the outer portions could have permitted freer and perhaps more ideal slip to occur near the (Oil) surface. At larger strains such as In specimens C-50 and C-43 as shown In Plate 10, grosser bending of the lattice planes, or "kinking'* was noticed. This was accompanied by large macro-slip steps and undulations normal to the (811) face. Plate 11 shows specimens C-29, C-25, and C-37 which were 3trained to 6%, 19? and 25?, respectively. The photo graphs were taken at 113X and enlarged to 304X. These interference micrographs of the (Oil) faces reveal the fine slip structure, and the greater step heights observed at higher strains. Plates 12 and 13 show specimens C-M2 and C-33oriented for compression parallel to the [lM] and [166] axes, respectively. Each was deformed to about 5? strain. Each underwent a slip plane rotation of 1 or 2° during deforma tion. A significantly larger amount of secondary slip occurred In specimen C-33 as compared to C-42. The value of 127 6- 8' [oi [8ll] Figure 52 Typical slip traces on (811) face of dual slip specimens C. A > Areo Shown in Plote _ i 4 (Oil) Foce Figure 53 Bending of slip planes resulting from non-uniform slip because of end constraint. Plate 9 ... Slip tracts on the (811) face. (a) Spec loan nunber: C-30 Strained: 0.8% (b) Spec lean nuaber: C-42 Strained: 5.6% (c) Spec lean nunber: C-40 Strained: 16.0% 129 Plate 10 (a)Specimen number: C-43 Strained: 23.87. (b)Specimen number: C-50 Strained: 37 % View of (811) face. <•) l- M O > I Plate View o f (O il) face using thallium light interferometry. (a) Specimen number C-29 strained: 6.2% (b) Specimen number C-25 stra in ed :19.4% (c) Specimen number C-37 stra in ed :25.4% 131 Plate _12__ Specimen number: C-42 Strained: 5.6%. View of the (Oil) face of the specimen deformed in compression, with the deformatlog axis oriented near the C144] pole, about 10° from the C0113 pole. Plate 1.3_ Specimen number: C-33 Strained: 5.0%. View of the (Oil) face of the specimen deformed in * . V ** ’ ■ compression, with the J 7 V ^ deformation axis oriented *■ P ■ H ‘ ‘ '* ' near the C166 1 p o le , about , -s*1' 5* from the [O il] pole. 4‘ >"• - / * s “ ^ 7 Note the large amount of :,•' *7- secondary slip on the -*•'.- i (111) plane. I 2 0 0 * I 132 the Schmid factor for secondary slip in C-33 was higher initially than it was for secondary slip in C-42. The secondary slip which occurred in C-33 (Plate 13) was com parable to that which occurred in specimen C-25 (Plate 7). Plate 14 3hows an example of the localized bending of the slip planes which occurred at higher strains; usually noticeable to some degree above 25% strain. This phenomena is another direct result of the large frictional constraints placed on the crystal at the end plates, as depicted in Figure 53* This localized bending of the slip planes always occurred along the band of slip traces which intersect a corner of the crystal base. The degree to which this localized bending occurred varied from crystal to crystal for the same amount of strain, but in general was a function of the amount of strain beyond 28%. The occurrence of the localized bending of slip planes was often accompanied by a twisting of the crystal normal to the compressive axis. Specimen C-50, deformed to 37? in strain, was an deception. Plate 15 is a photograph taken near the constrained end of specimen C-37* In the upper portion of the photograph secondary slip has occurred. However, it can be deduced that the secondary slip occurred before primary slip had been activated to any large degree, since the secondary slip traces are bent and Jagged where they meet primary slip traces, and the primary slip traces are relatively straight. Plate 15 Specimen number: C-37 Strained: 25.4 % View of (O il) face Secondary slip has occurred before primary slip. P late m Specimen number: C-39 Strained: 2$.6 % View of C011) face near the constrained end of the specimen. Note the localized bending of the slip planes. IV. DISCUSSION OF MACROSCOPIC CHARACTERISTICS OF SPECIMEN DEFORMATION The stress-strain curves for crystals of germanium -9 compressed In a dual slip orientation showed a definite yield point phenomenon (Figures 43 and 45). The decrease in flow stress after yielding for the dual slip specimens of both Sets I and II which were deformed at 6l6°C was 25$ and 50$, respectively. By comparison specimens C-101, C-102, and C-103 oriented for single slip exhibited decreases of 15% and 45$, respectively. Patel and Alexander (11) who compressed germanium single crystals oriented for single slip observed no yield point, as was shown in Figures 11 and 12. Their Initial flow stresses were, however, within the present range of values of minimum flow stress for the dual slip specimens of Sets I and II. Several possibilities exist for explaining this dif ference. In the first case Patel and Alexander (11) used crystals with a dislocation density of approximately 5000-pits/cm^ (CP-4 etchant) as compared to an initial dislocation density ofl000-3600 pits/cm^ (CP-4 etchant) in these crystals. Both sets of crystals had a minimum resistivity of 40 ohm-cm, were undoped, and were n-type 134 135 semiconductors. The flow stress in germanium has been shown (15) to be governed by the equation where N is the number of mobile dislocations, € is the strain rate and m is a number. This relationship indicates that the greater the original dislocation density, i.e., the fewer the number of dislocations that must be generated dur ing yielding to reach some critical value of N, the lower the stress necessary for plastic flow to be initiated. This may account in part for the lack of a well-defined yield point, but another difference existed; that of the specimen size and shape. It has been shown (3*0 that in compression, the greater the ratio of the length to the square root of the cross-sectional area, the more ideal the plastic deformation behavior, and the lower the volume fraction of the crystal affected by the end plate frictional constraints. The 1/2 ratio (L/A ) for specimens in the work by Patel and Alexander (11) was about 2.4 as compared to a ratio of about 3-3 in the present study. This indicated that the approximate fraction of the crystal volume affected by end constraints was about 4llE In their work as compared to about 25% in these experiments. It has been observed in this study that end constraint does lead to the production of secondary slip on another crystal plane in the affected volume during yielding, as is shown in Plate 15 and 136 discussed briefly In section III.A. Therefore Increased end effects may prematurely increase the dislocation density at the specimen ends, and tend to reduce or subdue a large yield point. Furthermore the slope in stage I of the stress-strain curve up to about 105E (155C glide strain) strain varied be tween 2 3 and 27 k g - m m / m m ^ for their work as compared to values of 7-11 k g - m m / m m 3 for the specimens oriented for easy glide in this study. Patel and Alexander observed significant secondary slip at strains as low as 0.2% which explains the greater hardening rate In their specimens. Bell and Bonfield (10) deformed in tension, at a h ^ strain rate of 2 x 10 /sec and a temperature of 560°C, germanium crystals oriented for single slip, in which a decrease in the flow stress of about 35% occurred after yielding (Figure 7). Because In their Investigation the p Initial dislocation density was about 1000 pits/cm and because they used careful alignment In the tension tests, the amount of secondary slip Introduced by specimen end- constraints was minimized. The resulting deformation curves exhibited an initial region from 2 to 3% strain, with a zero work-hardening slope and a subsequent region of stage I work-hardening with a slope of approximately 10 k g - m m / m m 3 . In a similar study by Alexander (3) to that of Beil and Bonfield (10), single crystals oriented for single slip, which were deformed at 600°C at a strain rate of M x lO^/sec in tension, exhibited little yield point 137 phenomena, however, the Initial dislocation density was greater than 20,000 pits/cm2 . In the work by Patel and Alexander (11), Bell and Bonfield (10) and Alexander (9) as well as in this study, a definite correlation was observed between temperature, strain rate and the degree to which the yield stress exceeded the subsequent minimum flow stress. The drop in stress from the yield point to the minimum flow stress is now defined as A (50) v - ~ V -*. x 100* y In the present work A 49 and 50. Alexander (9) observed a 4 x 1 0 ~ ^ / s e c . This behavior also follows from the disloca tion dynamics as described in Equation (1). An increase in the strain rate indicates an increase in the stress is necessary to obtain the critical value of N for initiation of plastic deformation, A higher temperature of deformation was also found by all investigators (9,10,11), to decrease the values of A ^ y and the minimum flow stress. Haasen (35) has described this quasi-viscous flow in germanium by an equation of the form: 138 (51) V a (constant) "t m * exp( - E/KT) where the experimentally determined activation energy E for dislocation motion in germanium is about one-half that for self-diffusion. Experimentally the value of "in" in equa tion (1) was found to vary from 1-7 (35) » primarily as a function of the stress, and the logarithm of the stress was found to vary inversely with the temperature. Bell and Bonfield (10) observed in single slip specimens, strained at -3 5 x 10 /sec. a change in yield stress and minimum flow 5 2 stress of - O .65 kg/mm^ and -O.lOkg/mm per 100°C, respec tively, as compared to values obtained in this study of -03-7 kg/mm^ and -1.3 kg/mm^ per 100°C, respectively, for specimens oriented for dual slip,as shown in Figure **7. The slope of the initial stage of work hardening was found to increase with the strain rate as shown in Figure 46 for both dual and single slip specimens. In the work by Alexander (9) and by Bell and Bonfield the increased slope coincided with an increased amount of secondary slip. In this study no observable Increase in the secondary slip was observed by slip trace analysis over the Initial linear portion of the work hardening curve for dual slip specimens from Set I or Set II, although more secondary slip was observed to have occurred Initially at lower strains In Set II. This suggests that the higher slope of work harden ing at greater strain rates may have resulted from a greater amount of secondary slip being activated during yielding, 139 which occurred at a higher relative stress which may have exceeded the resolved shear stress necessary for Initiation of both primary and secondary slip. Once the mobile dislo cation density reached the critical value of N° necessary for plastic flow, then primary slip would dominate in sub sequent plastic flow at lower stresses, as demanded ideally by Schmids law. The greater strain rate was also observed to produce a longer initial'linear stage in the work hardening curve for both the dual slip and single slip specimens (Figures *19 and 50). It has been observed in face-centered cubic metals that a greater strain rate does promote primary slip, and subsequent primary "overshoot" (36); i.e., it tends to hinder the massive occurrence of secondary slip before and beyond the point where the compressive axis has rotated into an orientation where the resolved shear stress for secondary slip is either equal to or greater than the resolved shear stress for primary slip. This point occurs ideally at strains of about 28% in the specimens oriented for dual slip, relative to net [?11] slip rotation towards the [0113, and at about 28% strain for specimens oriented for single slip, relative to [101} slip rotation, towards the [012], An increase In the length of the initial stage I work hard ening portion with Increased strain rates or with decreased temperatures of deformation was also observed in the work by Alexander (9), and Bell and Bonfield (10). Alexander 140 ' observed linear stage I work hardening up to strains of 40? in specimens deformed in single slip at 600°C with a strain rate of 4 x 10~^/see. A. Dual Slip Versus Single Slip Direct comparison of the work hardening characteris tics of specimens oriented for single slip and dual slip is possible if the glide stress and glide strain for dual slip are resolved into components parallel to one of the two dual slip vectors.* This has been done for specimens from Set I and Set II which were deformed at 6l6°C. The values of the yield stress and minimum flow stress for both the dual slip and single slip specimens for each strain rate were approximately equal, as were the shapes of the curves for each strain rate, similar up to strains of 30?, with the exception of the values of the strain at the minimum flow stress, which were greater for dual slip. Beyond the minimum flow stress the slopes of the stage I work hardening regions were from 0 to 10? higher in the dual slip specimens. — 4 For specimens strained at 5 x 10 /sec, as shown in Figure 49, the slope of stage I was about 7.5 + 0.6 kg-mm/mm up to a strain of about 21?, at which point the *Since the angle between the [211] and [101] vectors Is 30°, the glide stress In the [211] is resolved by multi plying through by the cos 30°, and the glide strain is sim ilarly resolved by dividing through by cos 30°, which means for example a slope of the work hardening stage I in a dual slip specimen, C-27 which Is 10 kg-mm/mm3 is multipled by cos 30° to obtain the slope of 7.5 kg-mm/mm3 in-terms of the resolved single slip stress and strain components. m ‘ slope Increased rapidly to a value of about 25 kg-mm/mm^. For single slip the slope was 7-0 +0.5 up to a strain of about 19$ beyond which the slope varied between values of 7 and 15 kg-mm/mm^. The most obvious explanation for an in crease in the work hardening at strains of 1955 and 21$, respectively, would be the activation of massive secondary slip, especially in the dual slip specimen in which the slope increased rapidly. However, ideally a rotation of the compressive axis of about 10° is necessary before the resolved shear stress for slip on the secondary (Til) slip plane exceeds that for primary slip on the (111) plane. A rotation of 10° corresponds to a strain of about 28$ for the single slip, and 32$ (for the resolved components) for dual slip. Nevertheless secondary slip did occur to a noticeably greater extent in specimen C-25 strained to 19$ (22$ for resolved components), as shown in Plate 7- Specimens strained at a higher strain rate exhibited similar linear slopes of 11 kg-mm/mm^ within 10$, over all recorded strains. This would superficially indicate, as previously mentioned, that latent hardening occurred in the secondary slip system. Dual slip has been previously investigated in zinc by Edwards et a l . (37), in anthracene by Robinson (38) and in silver by Jackson (39). Edwards conducted shear tests on hep zinc single crystals which were oriented and constrained in a manner originally described by Parker and Uashburn (40) in which slip could only occur on the basal plane. The experimental apparatus allowed the stress to be resolved parallel to one of two slip directions, or in any intermediate direction. Figure 54 shows the results obtained by Edwards et al. (37) for various shear orientations. The specimens constrained to produce equal slip in the two slip directions exhibited a much greater slope of work hardening than specimens oriented for single slip. No secondary slip was observed in the specimens of either orientation over all strains. Because the hep structure has a limited number of possible secondary slip systems, Edwards reasoned that the increase in the rate of work-hardening should represent ideally, the initial stage I hardening for dual slip specimens of the face-centered cubic structure. He concluded the hardening was the result of progressive formation of dislocation sub boundaries produced by the interaction of glide dislocations in coplanar slip planes. The extension of this work to the face-centered or diamond cubic system is difficulty however, because similar constraints to those imposed upon zinc, when placed upon the deformation of such crystals would conceiv ably activate appreciable slip on another slip plane before significant amounts of coplanar slip and glide dislocation interaction could occur. Robinson (38) investigated the deformation of anthracene single crystals which were constrained to deform 143 180 160 140 (A 120 (A 100 (A UJ 80 ttT f="---- 40 20 0 0 0.1 0.2 0.3 0.40.5 0.6 0.7 SHEAR STRAIN Figure 54 Stress-strain curves for zinc crystals showing that the shape of the strain hardening curve for simple shear is influenced by substructure and by the number of slip systems operating. Test temperature was 25°C (Edwards et al. (37)). in shear in the (001) basal plane, and parallel to the [010], [110] or any net slip direction between these two. Anthracene was used because it exhibits a work hardening behavior which parallels that of similarly oriented hep crystals, and because the type and extent of the work hardening could be easily controlled. Robinson observed three-stage work-hardening when the crystals were deformed such that equal amounts of slip occurred in the two basal slip directions, which is shown in Figure 55 along with results obtained for single slip in one of the two slip directions. His conclusion after observing this behavior in which work-hardening occurred at low values of strain with no secondary slip on other planes, was that the forma tion of sessile dislocation locks between the two primary glide dislocations was extremely effective in increasing the work hardening through stage I. He concluded that stage I represented slip primarily in one of the two dual slip directions, and that stage II would not begin until a cer tain stress level was reached at which the critical resolved shear stress for massive slip on the second co planar slip direction was exceeded. The subsequent dual slip and the resulting Increase in the number of sessile dislocations, produced the large increase in the rate of work hardening called stage II. Jackson (39) Investigated the simultaneous operation of two coplanar slip directions in cylindrical silver 145 70 60 E 50 -S. D> 40 30 [mq];;s a s i n g l e s l i p . 10 0 0 0.01 0.02 0.03 0.05 0.060.04 0.07 SHEAR STRAIN Figure 55 Shear stress vs. shear strain surves for anthracene single crystals deformed by slip in the basal plane in the [010] and [110] directions and in a direction bisecting these two, showing single stage work-hardening when slip occurs in a single direction and three-stage work-hardening when dual slip occurs (Robinson (38)). 1M6 crystals, which were strained In tension at room temperature with the tensile axis oriented approximately parallel to the [133 3 axis. Similar crystals were oriented about 5° from the [110]-[111] boundary towards the center of the orienta tion triangle such that single slip occurred in only one of the two dual slip directions. These tensile experiments on face-centered cubic crystals were an example of "uncon strained" slip when compared to that in the work by Edwards et al. (37) and Robinson (38), because of the low degree of end constraint inherent in these tensile tests in which the 1/2 ratio of (L/A ) was about 10/1, and the specimen grips were free to move laterally during deformation. As deformation proceeded, the change in the orienta tion of the tensile axis with respect to the active slip plane, was recorded by means of a Laue back-reflection camera. The results obtained from single slip and dual slip are shown in Figure 56, in which the glide stress and glide strain for dual slip is resolved into its components of glide stress and glide strain parallel to the single slip direction. Of several specimens deformed in a dual slip orientation, in only one specimen did the tensile axis remain on the [110 3-C1113 boundary, and maintain a net slip direction of [2113* The others tended to favor one of the two slip directions as shown in Figure 57. This type of metastable behavior in unconstrained drformation was to be expected because any deviation of the tensile axis from the 1*17 3.0 * 2.0 Ul 0 10 20 30 40 50 RESOLVED SHEAR STRAIN % Figure 56 Stress-straln curves for single slip and dual slip In tension (Jackson (39)). 1(112 '101 o o 110 Figure 57 Changes In the orientation of the tensile axis for single slip (S - S ’) and for dual slip (D - D1)- (Jackson (39)). 148 [111]— [110] boundary Increased the resolved shear stress for a particular slip direction, and hence favored further deviation of the same kind. The dual slip work-hardening curve in Figure 56 represents the one crystal In which the tensile axis remained on the [110]-[111] boundary. The same work hardening slope was observed for both single slip and dual slip specimens in stage II, but the initial stage I portion of the curve In Jackson's work ex tended to 6? as comparedto 3% for the dual glide specimen. He concluded that in the case of unconstrained deformation of silver, dual slip does increase the effectiveness of the hardening process in stage I, but not in stage II, because stage I hardening represents primarily Interaction between the elastic strain fields of individual dislocations, whereas stage II is primarily controlled by forest interactions with glide dislocations resulting in sessile dislocation forma tion, rather than coplanar glide dislocation interactions. The tests by Edwards et al. (37) on the latent hard ening in zinc have shown that the flow stress In latent coplanar systems is approximately 30£ higher than on the operative system. If this was so for the silver as deformed by Jackson (39),unconstrained dual slip would not have occurred. The conclusion Is that the work-hardening results for dual slip in the constrained hep crystals cannot be directly compared to relatively unconstrained slip in face- centered cubic metals, or to large amounts of constrained 1*19 slip in diamond cubic or face-centered cubic crystals in which a multiplicity of alternate non-coplanar slip systems would favor secondary slip at relatively lower stress levels. 1. Dual and single slip in germanium The deformation of germanium in a dual slip orienta tion in the present study is similar to the deformation of zinc (37) and anthracene (38) in that by using compression with rectangular specimens as described in Figure 3^, the system is somewhat constrained as compared to the cylindri cal specimens deformed in tension by Jackson (39), but on the other hand germanium is also diamond cubic, which has the same translation symmetry as face-centered cubic silver. The similar stage I work hardening slopes observed for both single slip and dual slip in germanium specimens would indicate little effect of a second slip vector upon the primary slip as suggested by Jackson (39), however, the apparent stage II hardening in the dual slip speciments of set I was much greater than In single slip specimens deformed at the same strain rate. This was similar to observations by Robinson (38) where the slope of the stage II hardening for dual slip was more than twice that for single slip (dual slip was resolved in to its single slip components of glide stress and strain). The observation of 6 to 8° rotations of slip traces on the (811) faces of the deformed crystals may indicate that the dual slip specimens had suffered, at some time dur ing deformation, slip predominantly In one slip direction. If one slip vector did dominate, the compressive axis may have effectively rotated in a direction such as A-A' in Figure 58- Such rotation might occur, because in the initial deformation stages, the frictional effects upon the ends of the crystals will probably affect deformation very little, I.e., the deformation may coincide with Jackson's observa tions that In dual slip the tensile axis often rotated away from the [110]-[111] boundary when one slip direction was favored. At greater strains, the rotation normal to the [211] net slip direction did not increase above 6-8°, which may Indicate that the end constraints did effect deformation at higher strains, and cause additional slip. From this point on the deformation may have been similar to the con strained deformation in the work by Edwards et al. (37) and Robinson (39) in that the additional constraint may have caused dual slip and the observed Increase In the work hardening slope In set I specimens at strains greater than about 19/5. A tilt of 6 to 8° toward the [010] pole caused by [TlO] slip would not change the angle XI as measured on the (Oil) face, by more than one degree from the angle pre dicted Ideally for rotation resulting from [211] net slip. A rotation of the compressive axis parallel to the vector bisecting the great circle between the [ T M ] and [T9*0 poles, and the great circle between the [lM] and the [Oil] pole, m D 5. ' 0 1,° Part of a (111) stereographlc projection showing iso-m • controus for primary slip In the dual slip systems [Toi] [TlO] (111). A-A" represents dual slip along the [111]- [011] boundary; A-A' represents a possible deviation of the compressive axis from ideal dual slip rotation, and A-A'-A"* represents a possible glide trajectory of the_ compressive axis if the secondary slip system, [101] (111), interacts significantly with the [101] (111) primary slip system. 152 as shown in Figure 58, would result In a net rotation of the [2ll] such that X^ would Increase from 45° to 50°, which corresponds to a strain of about 18$, at which the slope Increased. Another possible explanation of the increase in the work hardening rate at a strain of about 19$ in the sped- _ ji ments deformed at a strain rate of 5 x 10 /sec, is that one slip vector dominated slip in the initial deformation stages, as described above, but by rotating in a direction such as A-A’, the Schmid factor for the resolved shear stress on the secondary slip system [110](Til) increased such as to favor secondary slip. The rotation A-A' effectively increases the resolved shear stress on the [110] secondary slip direction from an initial value of about 0.37 to about 0.43 , In which the difference in Schmid factors for primary and secondary single slip is reduced to 0.02 . If Ideal dual slip had occurred, and the compressive axis had rotated 5° along the [011]-[111] boundary to A!t, the re solved stress on the [110](111) slip system would have in creased from an initial value of 0.37 to 0.40, In which the difference between the Schmid factors for primary and secondary slip would still be 0.20 yet the resolved shear stress of the secondary slip will be lower by 0,03 . From this it can be concluded that single slip In a dual slip orientation would favor secondary slip at lower values of 153 strain relative to strain parallel to the Ideal [ Back Reflection Laue x-ray photographs were taken normal to the (111) primary slip plane in deformed dual-slip crystals. As the strain increased, the asterism of the spots Increased in the [2*11] net slip direction. At higher * strains in stage I for specimens of set I, the direction of the tails on the diffraction spots remained parallel to the [211] but the spot intensity had shifted slightly towards one of the <110> slip directions. This would indicate that at least locally, one slip direction did dominate slip to some degree. In stage II, the spots broadened In width, evidence of secondary slip plane activation as well as some stress relaxation. In the as-deformed specimens, and especially in the annealed specimens definite polygonization was indicated parallel to the direction of deformation at higher strains which would result from relaxation of lat tice bending strains, and the formation of sub-boundaries. It can be concluded from these observations of repre sentative x-rays that both slip directions were activated, and continued to operate throughout deformation, although in any one localized slip plane one slip vector was dominant, At higher strains, In unannealed specimens, the asterism of 15^ the spots associated with the deformation direction were often split (weakly) Into two or more points of Increased Intensity suggesting polygonlzation and cell wall formation. This effect occurred to a greater extent in specimens from set II. If secondary slip had occurred to any significant degree during yielding or in the subsequent deformation process, the forest dislocation interactions between secon dary and primary glide dislocations may have formed areas of high dislocation density, which through dislocation annihilation and interaction relieved the internal stresses resulting from piled-up glide dislocations on the bent slip planes, and resulted in a polygonized structure consisting of low angle boundaries parallel to the Intersection of the primary and secondary slip planes. Further discussion of the dislocation structure Is reserved for section VII . E. In Figure 51, specimens C-53, C-50, C-51, and C-27 all had measured angles of X^ which were well above the theoretically expected values for either primary overshoot or duplex slip. This apparent anomaly can be explained by referring to Figure 53 In which the localized lattice bend ing resulting from end constraints Is depicted for highly strained specimens. The effect of 1his behavior upon the stress-strain work-hardening characteristics, and upon the assumed equation (37) defining Xi»f(X0,L0 , L^) Is related to a reduction In the effective gage length within which 155 the majority of deformation occurs. At higher strains the crystal rotation occurs over a reduced gage length i which can be roughly described by equation (52) in which Is replaced by L^': (52) L±' = L± -W^ctn X1 i53)°) where L (^£=0.15); X ^ ^ (^0.15) (Figure 59), at a AL/L greater than 0.15. In using equations 52 and 37 the reduced length effect is assumed to be negligible below a strain of about 23% > and to increase linearly above this value. This procedure will effectively place the specimens which were strained 23% or more at even higher strains for their respective values of dL/L which were originally measured over the entire length, and will more accurately represent the strain to which the center portions of the crystal were subjected. This correction decreases the dif ference between observed and theoretical values of X^ for C-27, C-37, C-39, C-53 and C-51 such that they fall closer to the theoretically predicted angle for primary overshoot. Another possible reason for the remaining difference between the experimental, and the predicted values of X^ may lie in the method used for experimental measurement of the angle X^, which was obtained by measuring the angle between the compressive axis (normal to the end surface) and the slip traces on the (Oil) faces. If different portions 156 - L j - wj ctn Xj (01 I ) (811) Face Compressive Axis) Figure 59 Geometry used for defining a reduced effective gage length 1. 157 across the cross-section of the crystal suffered varying stress conditions, as suggested by the slip traces observed on the (811) face, then the exterior (Oil) surfaces may reflect localized domination by slip in one direction. If such occurred, and the "effective" compressive axis rotated along a path such as A-A' in Figure 58, then the plane normal to the axis of slip plane rotation was not the (Oil) but rather might have been a plane rotated for example 15° towards the (811], The projection of the (Oil) slip traces upon this plane would then produce the correct angle X^, Such a calculation would decrease the angle on the (Oil) face by approximately 1.0° for strains of about 30^, and about 2° for strains of ^056 in which secondary slip occurs to a large degree. From this second approximate correction, that single slip in one direction did dominate locally at the (Oil) surfaces in deformation and would cause a compressive axis rotation as described by A-A', the theoretical curve in Figure 51, plotted using equation 37 would be adjusted upwards, as shown by the dotted line. The resulting correlation between corrected theoretical and experimental angles is then acceptable, if it Is assumed that primary slip on the (111) plane dominates at all strains in the deformation process whether it was in the form of single slip, dual slip or some combination of these. Specimen C-50, as was discussed briefly in section II,E, was exceptional In that after being heavily deformed more than any of the other of the dual slip crystals, It exhibited very little twisting normal to the compressive axis, and very uniform deformation over the entire length of the specimen. There was essentially little localized bending as shown for specimen C-39 in Plate 1*1. The reason for the obvious lack of end constraint on this specimen alone was not apparent because all specimens were prepared and deformed using identical procedures.^ Specimen C-50 did have a final angle X± = 58° (Figure 51) which fell on the original theoretical line representing equation (37). Such behavior may also have resulted in a more equal divi sion of slip between the two dual slip vectors, since the twisting effect can be generally correlated with the domination of slip at higher strains by only one slip vector on each of the primary and secondary slip planes. For these reasons the two adjustments made for for specimens C-51, C-53, C-*47, C-39 and C-27 were not applied to specimen C-50. 1 Except for specimen C-25 which was lubricated on only one surface. V. PROCEDURES AND TECHNIQUES FOR ELECTRON MICROSCOPY A. Thinning Specimen preparation for thinning From each specimen about five slices had been cut parallel to the primary (111) slip plane. The slices from each specimen were then cut by means of a diamond slicing saw into 3 mm* by 2.5 mm. by 0.5 mm. rectangular wafers which were etched in CP-1! for a minimum of two minutes to remove surface damage and contamination. They were then separated into two equal groups. One of the groups of wafers was annealed in a vacuum furnace at approximately — 5 2 x 10 Torr as depicted In Table I1!. Note that the specimens from Set I were annealed at 850°C for 36 hours1 and specimens from Set II were annealed at 700°C for 2k 2 hours. The purpose of the different annealing temperatures and times was to produce annealed structures that would vary in the degree to which local equilibrium between ^Except for specimen C-55 from which wafers were annealed at 850°C and also at 750°C for 2k hours. 2Except for specimen C-38 from which additional wafers were annealed at 700°C for 6 hours. 159 160 TABLE ANNEALING PARAMETERS FOR SETS I AND II Additional Wafers Annealed at Speci Anneal Time Time men Strain T° °C Hours T® °C Hours f C-30 0.003 850 36 If C-26 .043 If C-28 .044 tl ft ff If Set I C-29 .062 C-55 .126 If 24 750 24 ir C-25 .194 36 L C-27 .312 M 36 r c -36 .086 700 24 C-40 .160 H II C-38 .160 tl 11 700 6 11 Tl C-37 .254 Set II C-42 .056 11 II 11 11 C-43 .238 IT If L C-39 .256 dislocations was obtained. This Is discussed to a greater extent In Section VI.B. 2. Eletrochemical Jet-polishing The method used to thin the 0.5 mm. thick rectangular wafers was an electrochemical Jet-polishing technique developed by Heisz and BJorling (4l) for germanium, and modified for the present experimental conditions. A problem encountered in thinning germanium Is the inherent brittleness of the material. Specimens less than four or five thousandths of an Inch in thickness are too fragile to handle without frequent fracture. Hence a method had to be used in which only part of the specimen was 1 6 1 thinned, so that bulk was retained for secure handling and mounting in the electron microscope specimen holder. The method which solved this problem is now described. The wafers were Placed in an apparatus which is shown in Figure 60, and were dimpled (the polishing action is localized on the surface of a wafer to form a polished depression or ’’dimple'') on one side. The wafer was then reversed and dimpled on the opposite side until perforation occurred. Using this particular technique, after being dimpled on one side, the wafer could have been mounted in the electron microscope specimen holder for the second stage of dimpling, and then transferred directly to the microscope without further handling. However, in this work the wafer was not placed directly into the holder for final thinning. To hold the wafer in the apparatus, a silver- aluminum-acetone paste was used, which provided weak bonding and electrical contact between the wafer and the specimen holder of the apparatus. The paste then had to be thor oughly and completely removed from the wafer after thinning, to avoid subsequent contamination of the electron microscope. Once the wafer was placed in position in the apparatus the Jet polishing system was activated. With the wafer as the anode, the electrical circuit was completed via an electrolytic solution which underwent gravity flow from the header tank through a glass tube, and out through a 0.030” diameter glass tip as a focused, smooth stream onto the — rr Vblve for White Adjusting Flow Light Source Rote----- Light Fiber ~ / _ Gravity Flow'low JLL 7 of Electrolyte 7 / uHeader , Tank ■Photocell Germanium Focusing Specimen Lenses f l Disc Anode) Electrolyte Pump. Restrveir^ Glass Jet O P h Silver-Aluminum* Acetone Paste for Securing Specimen | Drain L Adjusting Screws for Specimen Alignment & <22 DC Power Microammeter Relays D.C. Power Supply Supply Retcy 0 - 1,000 V , 0 - 1 0 0 ma 1700 V (Variable) Figure 60 162 Schematic Diagram of electro-chemical jet-polishing system. center of the wafer. A light-fibre extended back up through the glass tip to a light source, which allowed the light beam to Impinge directly on the dimpled area of the wafer. The light beam was used after the wafer had been reversed, to activate at hole perforation a photocell, located behind the specimen, which then closed a microammeter relay. Then, through a series of relays, the primary polishing circuit was interrupted, and a buzzer signaled completion. The mlcroammeter-relay connected to the photocell was adjustable so that one could vary the amount of transmitted light necessary to activate the relay. Because germanium is somewhat transparent to white light below thickness of about 3000 A°, a very sensitive photocell could be activated, in principle, before the hole had formed. In practice, only once was the polishing Interrupted In such a manner. This was an exce-lent foil for observation purposes, and 13 shown In Appendix B. Following perforation the wafers were rinsed imme diately with a weak glacial acetic acid solution, to halt any corrosive action of the caustic electrolyte, and then Immediately immersed in warm alcohol, and cleaned of paste. Finally it was either placed directly in an electron micro scope holder or in an evacuated storage disiccator. Speci mens could be stored many days without discernible degen eration. The electrolyte was a 0.07 N.KOH solution Impinged at a flow rate of about 30 cc/min. The polishing current 1614 was usually between 6.0 and 7.5 milliamperes at a voltage of about 600 V. The average polishing rate through the wafer was about 0.015 mm./min. Figure 6l is an electrical circuit diagram of the polishing apparatus, which was built by the author. The dimples produced on either side of a 0.5 nun. thick wafer were fairly shallow, with an outer diameter of about 1 mm. At perforation a wedge-shaped thin foil, circular in cross-section, resulted as idealized in Figure 62. The shape of the surface of an average dimple was calculated from the interference fringe spacing of photo graphs taken using a Zeiss interferometer with thallium light illumination, which had a half-wavelength of 0.27 microns. An example of such an interference photograph is shown in Plate B-3 in Appendix B, as well as a discussion of thinning techniques employed, and examples of thin foils obtained. The nearly flat surfaces of the foils were consis tently within 3° of the wafer plane in all cases. This jet-polishing method of thinning resulted in acceptable foils,which were consistently of the same thickness and shape,about 90% of the time. The general advantages of this method of thinning were the speed of producing a foil, the handling ease of the bulk foil specimens, the consistently acceptable foils produced, the complete automation of the process, and the 130 v 0C Pg#«r £76 B Mircury Tub* D*loy J .t - Pol iinmg oft I apoaroiul SO kQ r t Circuit Ctttm? i!3'43« Cb«k* Ptiltur* - ! iSOO 2 H -0 $*n*iiw*S*iteb £ 1 f a ' 7^) & (0-251 Micro-Amm«t#r ,71 Micro- S» i 10 H I 0 " | | B ,I lit | M«lti * i r ? r / J i »- t-o-f'»rf i -0—1 7TinTimor - 0-9 9 9 T r Q W Q f t n t r i£iHlUirvjtoll i50»n: —-uv-rf\- M5/+200V 0 - 1200V DC [ Circuit *nl»rrupt*r) Pb»*r Supply ( CepOCtor Troncf«rm*r 'IP2J * liltffJ ,ii”' l*r'*P20"> j j r Phi>»oc*n A * > * « (50*12 J u H 1200 V _,_f.3t I ' OH ’I ! M' ; ; ’P r 1-* ___ i L . - o Prtilur* Svri.itiut P(»V 1 \ .X^f V T'i /iSOkfl Swicn Micro-Suite* ,*r 150.n 7” \ \/ Mormoliy Ck>**d ! ■ J i_ ] Bripg* iSOkQ 1200 V DC Pbu*r Supply FIGURE 61 ELECTRICAL CIRCUITRY - JET-POLISHING 120 V. AC 165 O Microns a cluae fo nefrne microscopy. interference from calculated was rs-etoa ceai ve fdmldseie, which specimen, dimpled of view schematic Cross-sectional 0 4 60 0 6 0 4 20 20 0 8 y Factor a By - Nt Ta the That Note + f 50. of en Contracted Been Ai Has Axis Y 0 80 6 0 20 40 60 80 100 0 8 0 6 0 4 0 2 O 20 0 4 6 0 8 0 100 iue 62 Figure I ra oihd Away Polished Area i r _ — _ ro e for sed U Areo A 0 0 0 4 a _ _ ra sd for Used Area ° X ( A° Observation Observation i 0'2) I0 166 167 small amount of individual attention needed onee the second stage of dimpling was started. 3. The electron microscope The thinned germanium wafers were studied using a Hitachi HU-11A 100 KV Electron Microscope. All foils were observed using an operating voltage of 100KV. With proper alignment the maximum resolution limit was about 6 A°. The microscope was equipped with a tilting stage which allowed a maximum tilt of 10° in any direction. The maximum resolution with the stage in place was reduced to about 20 A°. A liquid nitrogen "cold finger" permitted indirect cooling of the specimen while it was under observa tion. Observed electron transmission images were recorded on Kodak 3 1/2" x 4 1/4" Electron Image glass plates. B . Operating Techniques for Electron Mlcroscope^ 1. Orientation determination The deformed germanium crystals were sliced parallel to the primary {ill} glide plane, electropolished to a suitable thickness which would allow the transmission of a beam of electrons, and then mounted in the electron micro scope for observation. Initially the general crystallographic orientation of the sample was obtained by means of selected area diffrac tion patterns. A crystal-foil oriented within three degrees of a <111> pole will produce a characteristic diamond cubic (Ill) diffraction pattern. The only difference in the diffraction pattern within 3° of the (111) would be a change in the relative intensities of the various diffraction spots. To accurately record the orientation, use was made of the Kikuchi line structure in the diffraction patterns, which resulted when the crystal was moved such that the electron beam passed through thicker portions of the foil. Figure 63 shows the Kikuchi (**2) pattern for a variation in tilt of 8° around the {ill} pole for face-centered cubic or diamond cubic structures. Plate 16 shows a Kikuchi pattern and diffraction pattern obtained for a range of 3° around the [ill] pole for germanium. Figure 6H shows a corresponding [ill] diffraction pattern for the diamond cubic structure. A brief treatment of the origin of Kikuchi lines is given in Appendix C. The value in using Kikuchi lines is that when the specimen is tilted, the spots will not change position in the diffraction pattern, only the intensity of the spots will vary. However, the Kikuchi lines will move as though rigidly fixed to the crystal, so that the direction and magnitude of movement reveals the orientation change with a high degree of accuracy. Once the orientation had been determined, and the {ill} crystal plane oriented normal to the direction of the elec tron beam, a photograph was taken of any dislocation array of interest. Assuming that the dislocation array was located on the {ill} plane, this procedure assured that the 169 L001] pel* 133 052 025 153 [011] pole Figure 63 The Index Kikuchi line pattern for 8 degrees around the [111] pole, for the face-centered cubic structure (Hirsch etal. (1*3)). Plate 16 Kikuchi and diffraction patterns for (111) plane in Germanium single crystal. 171 422 O 224 202 220 242 o o o o 022 000 022 o o o 242 22 0 202 224 ° O o o 422 O 4 = + 224 a * + 4£2 Q = ± 242 a = ± h 11Q h = ± h oil s . = + h 101 F igure Diffraction spots observed for a diamond cubic structure, and notation adopted in this study for the low index directions on the (111) plane. 172 photograph taken was perpendicular to the array. 2. Burgers vector determination The second objective was to determine the Burgers vectors for each segment of the dislocation array. A brief discussion of part of the kinematic theory of contrast necessary to understand the method used to determine Burgers vectors, can be found in Appendix D. If the dislocation line lies in the plane of the foil normal to the electron beam, and the glide plane of the dis location is approximately parallel to the foil plane, then (53) £ ■ b = 0 where £ = reciprocallattice vector of the perfect lattice plane from which the beam is reflected. b = Burgers vector of the dislocation line from which the Burgers vector of the dislocation may be determined. The method used to view the Image of the specimen using a diffracted beam was relatively simple. After a selected-area-diffraction pattern was obtained of the por tion of crystal in which a dislocation existed, an aperture was Inserted into the microscope column which could isolate any one of the diffracted spots. Therefore an observed dislocation network which consisted of three different dislocation segments was analyzed in dark field, by moving the aperture over each of three different [22^1 diffraction 17 3 spots, and recording for each spot its location in the dif fraction pattern and the dark field image. The C221?] spots were those most used; since [22^T]*[110] = [2^2] *[101] * [Tf22]*[011] = 0. One of the main disadvantages of this method besides inherent spherical abberation in the dark field image, was that often a lack of intensity in certain spots of interest could not permit observation of that image, and the deter mination of a particular Burgers vector. For example to find out if a dislocation had a Burgers vector of + a/2 [110], the diffracted spot which had to be observed in dark field was the + [22^f], since for this spot there is only one case of invisibility— b = + 1/2[110]. Again the Kikuchi patterns were advantageously used. If for example the [22^] diffraction spot was necessary for analysis, the crystal was tilted and the tilt direction rotated until essentially a "two-beam1’ case resulted, which occurred when one line of a + [22^4] Kikuchi line pair ex tended through the [000] spot and the second line of the pair extended through the [22^] spot. In this orientation, these two spots, the [000] and [22^f], were more intense than any others in the pattern. The resulting dark field image for the [227] then exhibited a significantly higher Intensity than when viewed without the "two-beam condition," By successive rotations of the tilt direction by 60° around the [111] pole, each of the three <22*t> diffraction 17*1 spots needed for analysis was successfully brought into a two-beam condition. In this manner all Burgers vectors of the type + 1/2 <110> in the (111) plane were determined, although the determination of only two in a three-fold node was necessary since the third could be determined by the well-known relationship (4*0: L=n (5^1) b-n = "( S ^i) (at an n-fold node) 1*1 The third was checked when possible to avoid any possibility of error. VI. RESULTS— ELECTRON MICROSCOPY A. Crystals Strained at 5 x lCT^/sec and -5- x To-T/Bec" Plates 17 through 50 are photographs of (111) ori ented thin foils of specimens from Set I (C-26, C-29, C-55) and Set II (C-42,C-36, C-38, C-lJO, C-37) in the as-deformed and furnace-cooled condition. Each plate Is labeled with respect to the crystal orientation and the net deformation direction. The <112> net deformation direction In each case implies that the Burgers vectors of the dislocations will be primarily parallel to one of the two <110> slip directions which vectorlally add to give the individual <211> net deformation vector. In the specimens (C-26, C-29, C-*t2) strained M.3S* 6.2% and b.2%, respectively,into the initial part of Stage I deformation, in Plates 17 through 23, and Plates 31 through 36, the dislocations were found to always be curved, many as dipoles. The dislocations, that were oriented into dipoles and which were joined at one end by possibly a Jog, were often oriented normal to one of the two primary slip vectors. The dislocations observed in these three specimens, whether they existed as Individual curved dislocations or as a 175 176 dipole pair, were usually fairly long and continuous, and more widely spaced than those observed in crystals subjected to higher strains. Some of the dipoles such as those in Plate 21 exhibited a definite contrast change between the surrounding crystal area and the area between the disloca tions, suggesting that in these particular dipoles, a com ponent of the Burgers vector of the two dislocations may have been normal to the plane of the foil. The length of the dipoles, and the separation between the two component dis locations decreased as the strain increased. Specimen C-29 (strained 6.2%) contained areas of varying dislocation density. Some areas contained many longer dislocations which were oriented parallel to one another, and normal to the deformation direction, in which smaller dipole segments ending in a Jog were normal to one of the two slip directions (Plate 22). Plate 23 shows many dislocations from 1 to 5 microns in length which are ori ented normal to the [OlT] slip direction, but which are con centrated in or near tangles of dislocations which are separated by areas of a lower dislocation density. The braids were usually oriented parallel to a <110> direction, but the most dense braids were parallel to the [101]; the trace of the Intersection of the (111) secondary slip plane with the (111) primary slip plane. Specimens C-55 and C-36 which were deformed to 12.6 and 8.6?, respectively, in Stage I, are shown in Plates 2*1 177 through 30 and Plates 37 through 39, respectively. In these crystals the dislocations were definitely oriented in tangles or braids often normal to the net deformation direc tion, which Is best shown specifically in Plates 25 and Plate 38. The braids were composed of curved and tangled dislocations, dipoles of much smaller length and separation than were previously observed, and dislocation or dipole loops which varied In size and were believed to be edge dipoles joined at both ends by jogs. The parallel braids were often connected by curved dislocations in various orientations. Plate 37 shows a dislocation tangle in the upper corner which Is oriented normal to the deformation direc tion C, and which is separated from another similar but less dense braid in the lower corner by dislocations oriented normal to a probable b slip vector. Plates 38 through 39 show typical braids formed in C-55 strained 12.6$. The dislocations definitely favor a direction normal to the slip vector and are grouped parallel to one another as part of the cellular structure. Specimens of set II which were deformed at a higher strain rate exhibited banding which was more often normal to the deformation direction, than in set I specimens. Crystals from the latter set also contained a larger pro portion of dislocations which were grouped parallel to one another, and normal to one of the slip vectors. 17 8 Some interaction was noticed in Plate 38 which re sulted in a small hexagonal array of dislocation segments. Specimens C-40 and C-38 from set II which were de formed to 16% strain, are shown In Plates 40 through 47. Here the braids were found to be more dense and closer to gether. Also the number of dislocations between the braids had increased, and had assumed roughly the form of low dislocation density braids. A two-dimensional network of high dislocation density cell walls appeared to be forming, with relatively low dislocation density areas between the braids. Specimen C-37, of set II, which was deformed to a strain of 25.9% is shown in Plates 48 through 50. At this higher strain the tangles had formed continuous two- dimensional cells, with dislocation-free areas located be tween the braids. In Plate 48, the two-dimensional cell structure is clearly shown. The thicker portion of the crystal was at the bottom of the photograph which explains the increase in dislocation density down the plate. The braids consisted of very short dislocation lengths, short dipoles, and many small dislocation loops. The areas be tween the dense braids usually Included a few long curved dislocations, suggesting that additional dislocations had been generated in the relatively dislocation-free areas. The dislocation density increased with strain as expected for specimens from both Set I and Set II. The 179 primary difference in the two sets was the preferred orien tation of the braids in Set II which tended to be parallel to the trace of the secondary slip plane. Tangles in Set I were found normal to the deformation direction but many were oriented parallel to other <110> directions. The dislocations in Set I were often grouped in small parallel sets oriented normal to one of the slip directions, and near to or forming part of the dislocation braid. Beyond a strain of 15% comparison was not possible because speci mens in the as-deformed condition were not available from specimen C-25 (strained to 19. **%) • A few small hexagonal or lozenge-shaped networks con taining straight dislocation segments were observed, such as Is shown in Plates 28 and 43. Braids were always found which were normal to the <112> net deformation direction, and parallel to the trace of the secondary slip plane (111), even In specimens strained less than 1 or 2% beyond the lower yield point. Some of the less dense braids as In Plate *J3, were oriented approx imately normal to either the [10T] or the [110] slip direc tions, but the high density braids were fairly straight and were normal to the <211> deformation direction. The dislocation lines which often were observed to form small lengths of dipole, were usually normal to one of the two primary slip vectors, as demonstrated in Plate 22. Similarly, observations made for annealed specimens C-29 and C-55, shown in the next section, also indicated primary slip dislocations with a Burgers vector of primarily edge character, oriented normal to one of the slip vectors. Specimen Number C-26 Strained 4 * 336 Deformation direction: A Deformation direction: A Deformation direction: C 1 8 2 Specimen Number C-29 Strained 6.2% Deformation direction: B Deformation direction: B De formation direction: B Plate SPECIMEN C-29 IN THE AS DEFORMED CONDITION Magnification: 4 8 0 0 X Deformation direction: £ Specimen Number C-55 Strained 12.6% ■ 2 . 0 a i P L a te Deformation direction: C Deformation direction > \ Plate 26 Deformation direction: C 185 Specimen Number C-55 Strained 12.6X Deformation direction: A Specimen Number C-42 187 Strained 4.2% Plate 31 Net deformation direction; C t Plate 32 Net deformation direction; C ■ « Plate 13 Net deformation direction; A 188 Specimen Number C-42 Strained *t.6* Plate 189 Specimen Number C-36 Strained 8,6% P L a t e Net deformat ion direction: C 190 Specimen Number C-38 Strained 16.0% PLate ^0 Plate Ml Deformation direction: A Deformation direction: A PLate M2 Plate M3 Deformation direction: A Deformation direction: A 191 Specimen Number C-40 Strained 16.OS Plate Jtl Dark field photograph of Pl&te *16 using the [422]diffraction spot. 192 Specimen Number C-37 Strained 25.^% Plate Deformation direction: B Deformation direction: B Deformation Direction: C B . Dislocation Configurations In Annealed-Crystals 1. Introduction On each plate the orientation of the crystal is shown in one of the corners. Each foil was oriented parallel to the primary{ill} slip plane. In dislocation networks analyzed for their Burgers vector content, the appropriate dislocation segments have been labeled either "a," "b>" or "cn where a = + 1/2[110]; b = + 1/2[011]; c = + 1/2 [10l] The directions "A,11 "B," and frC" are parallel to <112> (111) directions, where direction "A" is normal to "a,11 B is normal to "b" and "£M is normal to "c,11 A = + [112]; B - + [211]; £ = + [121] In reference to low index directions on the (111) plane these abbreviations are often used. The crystals from Set I wereannealed at 850°C for 36 hours, with the exception of several foils of specimen C-55 which were annealed at 750°C for 24 hours. The specimens from Set II were annealed at 700°C for 2 4 hours, with the exception of a wafer from C-38 which was annealed at 700°C for 6 hours. The crystals from Set I were annealed only 100°C below the melting point in order to obtain structures which approached a state of quasi-equilibrium; such that the 19 11 node angles would represent the lowest energy configuration. The reason for annealing crystals of Set II at a lower temperature for a shorter time will be discussed in Section VI.B.3. 2. Results for annealed crystals of Set I Observation of foils from annealed specimens C-26, C-29, C-55, C-25 and C-27, which were strained to U .3%, 6 .2%, 12. 6*, 19.4% and 31.2%, respectively, at a strain rate of 5 x 10 /sec., are Included in this section. Elec tron micrographs from each of those specimens are shown in Plates 51 through 80, from which specific plates will be chosen to illustrate changes in the annealed structure as a function of strain. There are several micrographs included from each specimen, taken at magnifications of 5000X to 50,000X, in order to fully characterize the changes in dislocation structure in 850°C annealed specimens deformed -4 In dual slip at a strain rate of 5 x 10 /sec. No examples are shown of observed dislocation struc tures in foils from annealed wafers from specimen C-30 which was strained to 0.8%. A small number of angular dislo cation segments infrequently dispersed through the foil were revealed in these lower dislocation density crystals. No regular networks of dislocation nodes were observed. Plates 51 through 56 show representative structures observed in foils from C-26, strained to 4.3% before annealing. The dislocations were generally straight, and often aligned close to either a <110> or a <112> direction. Plate 52 shows several parallel dislocation lines which end adjacent to a roughly lozenge-shaped network of dislocations which are thought to be the result of the interaction, of one parallel set with another set of parallel dislocations. Plate 53 contains a number of dislocations, parallel to the [211] direction, which could lie on another plane. There is very little visible interaction between the disloca tions; they could be of an edge character in which case * rearrangement into symmetric nodes would require a drastic change in orientation. Plate 5^ includes two photographs used in the analysis of the first dislocation array in the plate. The nodes appear to be in a symmetrical orientation, but the Burgers vectors of two of the three segments of the three-fold nodes lie in the (111) cross-slip plane. Plates 55 and 56 show symmetrical nodes rotated 4° and 5° > respec tively, from a screw orientation. They did not appear to be a part of a more extensive network of similar nodes. Plates 57 through 60 show typical arrays observed in foils from specimen C-55, which was annealed at 750°C for 2H hours. The array in Plate 57 is a regularly spaced, hexagonal dislocation network in which the symmetrical nodes were oriented about 5° from a screw orientation. Plate 58 shows one of the dark field views of the array in Plate ^7. The nodes in Plates 59 and 60 are also symmetrical, but the 196 orientations of the nodes vary across the networks; 20-30° from a screw orientation in the first, and from 0-15° from a screw orientation in the second. Note that the crystal was annealed at a lower temperature for lower times, which may explain apparent non-equilibrium structures. Plates 61 through 64 are photographs from a foil of specimen C-55 which was annealed at 850° for 24 hours. A number of curved dislocations as well as straight disloca tion segments were observed. The straight segments often ended at three-fold nodes. This was the only annealed speci- ment of Set I in which such curved dislocations were observed to such a large extent. The dislocations oriented parallel to the [Oil] direction in Plates 62 and 63 are normal to the deformation direction, and parallel to the trace of the secondary (111) slip plane. This suggests that they could have a 60° Burgers vector character, either "b" or M£," or more probably have a Burgers vector with a large component normal to the (111) slip plane. Arrays shown in Plates 65 through 69 were observed in foils from specimen C-25. Plate 65 shows one of the most extensive uniform hexagonal dislocation networks observed in this study. The dislocation segments in the symmetrical nodes were oriented about 30° from a screw orientation. This boundary represented a twist of about 2.1 minutes. Plate 66 shows another extensive hexagonal network. It was rotated about 3° from a screw orientation. It represented a low 197 angle twist boundary of about 1.8 minutes rotation. Plates 67 and 69 show hexagonal dislocation networks of symmetrical three-fold nodes oriented about 25° and 29°, respectively, from a screw orientation. The foils from specimen C-25 con tained both regular arrays as shown in the previous plates, as well as many smaller scattered straight dislocation seg ments, which were often parts of an isolated three-fold node configuration. Very few dislocations were observed that did not end in a symmetrical three-fold node. The angles 20 between node segments were always within 3° of * 120° . Plates 70 through 78 are photographs of dislocation arrays observed in specimen C-27. Plates 70 through 74 show examples of the different types of dislocation configura tions most often observed. Plate 70 taken at 6800 X shows a typical grouping of small hexagonal dislocation networks separated by straight, long dislocations. In an array such as this the orientation of the small hexagonal, symmetrical networks within the array often varied from one network to the next. Most networks were hexagonal in shape, but often a small lozenge-shaped network composed of two crossed parallel sets of dislocations were observed adjacent to a hexagonal network. An example of such a lozenge-shaped net work is shown in Plate 71. In the upper right corner a small number of nodes exist in a hexagonal configuration. Plate 72 and Plate 73 each shows two networks of different 198 character; an apparent tilt boundary oriented at an angle to the foil plane, meeting a hexagonal network of twist charac ter. These arrays were not analyzed for their respective Burgers vector contents because in each case the arrays were located in thicker portions of the crystal, which limited the intensity of the diffracted spots, and hence limited the observation of the dislocation array in dark field. Plates 75 and 76 are photographs, taken in dark field, of the same array that is shown in Plate 7^* The nodes labeled "a," "b,'! "c" are oriented about 30° from a screw orientation. In an array such as this, the line directions were often difficult to ascertain with accuracy because of the large contrast effects involved. An enlarged photograph of the sane array Is shown In Plate 78. Plate 77 shows a symmetri cal hexagonal dislocation array in a screw orientation. It was generally observed that: (i) the number of hexagonal dislocation arrays ob served, and their size with respect to the number of three fold nodes In the network, Increased with strain, (ii) the most extensive networks observed were In specimen C-25» as was shown in Plates 65 and 66. C-25 was strained 19.4JG to the end of stage I in the work hardening portion of the stress-strain curve. (iii) in specimen C-27 which was strained 31.2!f, Into stage II,the development of a cellular structure was observed as shown In Plate 70. High dislocation density 199 areas consisting of mixed boundaries, hexagonal, and lozenge-shaped networks with small dislocation spacings were separated by areas of low dislocation content which contained straight dislocation lines, and random three-fold dislocation nodes of relatively larger dislocation spacings. An example of the cellular or banded structure is shown in Plates 79 and 80. The photographs are of the same area, but the second photograph was taken after the specimen had been tilted about one degree in the [121] direction. (iv) In all specimens, except specimen C-55 annealed at 850°C, the dislocation lines were straight and often aligned along <112> or <110> directions. (v) All nodes analyzed were of the symmetrical kind. The Burgers vectors were always within 30° of the line direc tion. The angles 2a between the dislocation segments of the three-fold nodes were always with 3° of 120°, with the exception of the nodes observed in specimen C-55 annealed at 750°C. (vl) The orientation of the three-fold nodes in the different networks which were analyzed, varied between 0 and 30°. There was no apparent orientation preference as a function of strain. The dislocation segments did, however, favor alignment parallel to either a <112> or <110> direc tion. This Is shown in the histogram In Figure 65 for the 13 networks analyzed for Set I. The histogram is a plot of the frequency of occurrence of a specific alignment of an 1.4 g 12 o z 1.0 Q> in .O ° 0 8 S' 0 6 c 4) f 04 02 I 1 I I I I H - r t 0 2 4 6 8 (0 12 14 16 18 20 22 24 26 28 30 32 34 | I Figure 65 Histogram for symmetrical node orientation variation in annealed crystals of Set I. r\j o o observed network of dislocation nodes versus the angle between the Burgers vector and the line direction. The symmetrical nodes favored orientations of 0 to 6° or 24-30°. None were observed with an angle of $ that was greater than 30°. In preparing the historgram, a maximum experimental error of + 2° was assumed in the measurement of the angle Q . If for example, every array analyzed was given a weighted value of 1.0, then an observed angle of $= 29° for array " X 1* was plotted as if the observed angle was 29+2°, or was observed over a range of (27 to 31°). Every value In this range (27-28) ,(28-29),(29-30),(30-31) was then given a weight of (1.0V4 or 0 .25 , and plotted. If another array contained nodes varying in fi from 0-6° (a range greater than 4°) then the values (0-1), (1-2)..... (5-6) were each given a weighted value of 1.0/6 or 0.167, and added to the histogram. Once this was done for all 'n' dislocation arrays, the total values for the frequency of occurrence were averaged over every 2°, which resulted in the histogram In Figure 65 for crystals from Set I. The total area under the histogram should equal (n x 1.0) (number of specimens - degrees). The histogram shows that 76% of the nodes analyzed were oriented with (0-6°) or (24-30°) of a screw orientation. (vii) Of the networks which were analyzed, all contained symmetrical nodes with coplanar Burgers vectors. Such 202 networks when characterized as low angle twist boundaries, can be described by the twist angle 9. Using the approxi mate formula: where: b_ (Burgers vector) ' _io 0 (radians) = DlDisTocatioh- spacingT " deters (1) the four specimens in Set I were compared using an average value of the spacing between dislocation nodes times ( 3 ) 1 / 2 ag ^ e v a i u e 0 f d, The results as shown in Table 15 indicate a definite change in the angles of twist as a function of strain. TABLE 15 TWIST ANGLE AS A FUNCTION OF STRAIN SET I Specimen Strain Average Angle 9 Number Spacing ______(microns )______(minutes ) C-26 *4.3 1. *1 + 0.4 0.35 C-55 12.6 1.2 + 0.3 0.40 C-25 19.4 0.28 + 0.2 1.6 C-27 31.2 0.15+0.1 3.1 203 Specimen Number C-26 Strained 4. 371 Annealed at 850°C far 36 Hours Plates 51 General dislocation structure ■ 1-0* » Plate __ ^2 Interaction between parallel sets of dislocations Plate 53 $ 20*4 Specimen Member C-26 Strained 4.3% Annealed at 850*C for 36 Hours « 1-0* l Plate su M agnification: 23300X : 18800X :1900QK Bright field Dark field Dark ri*ld $ - CM01 $ - C 022 ] d 1103, « -SC0113 ■- V f Plate Symmetrical nodes rotated Symmetrical node rotated ,(bout 4° from a screw about 3° from a screw >r icnt.it ton. orlentat ton. 205 Specimen C-55 Strained 12.67. Annealed at 750°C for 24 Hours I.AOIS—I Plate Plate ^8 Magnification: 11600X Dark field photograph of Bright field photograph of the dislocation array eyemetrlcal nodes rota ted about 5° froe g scr«wA P1,t* Plate RQ Plate 6q Syometrlcal nodes rotated Symmetrical nodes rotated atxxit 3 0 ° t 1 0 from a screw from 0° to 15° froe a screw orientation. 2a ■ 1 2 0 ° ± 5 orlentatIon. 2 a varies from 110° to 135°. 206 Specimen Number C-55 Strained 12.4% Annealed at 850°C for 24 Hours P late 6 l Deformation direction: C I: P late 62 Deformation direction: B w \w P late 63 207 Specimen Number C-55 Strained 12.6T Annealed at 850°C for 24 Hours Arrows denote curved dislocations with Burgers vectors which lie In the (111) plane. 208 Specimen Number C-2S Strained 19.4% Annealed at 8S0°C for 36 Hours Symmetrical nodes rotated about 30° from a screw orientation. 2a * 120° + 3 209 Specimen Number C-25 Strained 19.4% Annealed at 850°C for 36 Hours Symmetrical nodes rotated about 3* from a screw orientation. 2 « * 120° + 3 210 Specimen Number C-25 Strained 19.47, Annealed at 850°C for 36 Hours Plate 67 Symmetrical nodes rotated about 25 from a screw \ orientation. 2a = 120° + 3 Plate M. The Burger vectors were not analyzed. 2a - 1 2 0 ° + 3 Plate __ 69 Symmetrical nodes rotated about >5* from a screw orientation. 2« * 120° ± 3 Specimen Number C-27 Strained 32.0% Annealed at 850°C for 36 Hours General view of a high Typical lozenge-shaped dislocation density portion dislocation array. of a typical foil observed. L UQH P late 1 2 - P late A small twist boundary An array similar to that merging with a boundary in Plate 72 of probable tilt character. Specimen Number C-27 Strained 32,0% Annealed at 850°C for 36 Hours Plate J± Bright field photograph of .a larger angle twist boundary. Symmetrical nodes are rotated 30 from a screw orien ta tio n . 2 a appears to be about 120? s j P late 75 Dark field photograph of array in Plate 7 U . 8 [224 ] Plate JL Dark field photograph of array in Plate 7H 8 C 422 3 *i«te — 77 Sysnetric nodes in a screw orientation. 2« ■ 120 v j 1-0 ■ ■ > Vi x 213 Specimen Number C-27 Strained 32.0% Annealed at 850dC for 36 Hours Symmetrical nodes rotated about 30° from a screw orientation. Enlarged photo graph of Plate 7^. Notice heavy contrast effects. Plate __22 General view of cellular structure* The areas of high dislocation density which consist of law angle tilt and twist boundaries* are located along the change of contrast parallel to the 0.0U direction* Plate __80 The same area that is shown in Plate 79 after the crystal was tilted about 1* In the J.2TT direction. 215 3. Results— Annealed crystals of Set II Observations of foils from annealed specimens C-M2, C-36> C-38, C-40, C-43, C-37» and C-39 are Included In this section. The specimens were annealed at 700°C for 2H hours. The hexagonal dislocation arrays observed in Set I contained only symmetrical nodes, most of which appeared to be in a state of quasi-equilibrium in that the symmetrical nodes usually exhibited angles 2a of 120° + 2°, and usually were aligned close to low index directions. It was anticipated that with the decrease in temperature and time of annealing, the resulting dislocation configurations would not reach the state of quasi-equilibrium that was attained by anneal ing at 850°C for 36 hours. Instead, the dislocation con figurations should reach only a very localized state of partial equilibrium, a condition in which metastable, higher energy unsymmetrical nodes may exist. Typical dislocation arrays and hexagonal dislocation networks from crystals of Set II containing symmetrical nodes are shown in Plates 81 through 119. Plates 81 and 82 show dislocation configurations observed in specimen C-42 strained to 5.6if. The long dis location lines parallel to the [Oil] are oriented normal to the [211] direction of deformation. The Burgers vectors of these lines are probably parallel to one of the two defor mation slip vectors, b or c_, which would make the Burgers vector character of the dislocation about 60°. In many cases the shorter, straight dislocation segments are oriented close to either a <112> or <110> direction. Many of the shorter segments end in a three-fold node. This type of structure in which many three-fold nodes exist among both straight and curved dislocations of longer lengths, was representa tive of the structure in the annealed C-42 specimens. Note that the angles 2a for the three-fold nodes often vary greatly from 120°. A dislocation segment is often observed to extend away from a three-fold node such that its exten sion through the node bisects the two opposite segments, but at a distance in the order of about one micron from the node, the dislocation line will curve away. This type of behavior appears to indicate that a local equilibrium is preserved within a distance of about a one micron from the node, whereas a meta3table configuration exists at greater dis tances . Plates 83 and 84 represent foils from specimen C-36 strained to 8.65E. The first is a typical structure in which symmetrical nodes are rotated from 0 to 25° from a screw orientation. At one point the crossing dislocations with Burgers vectors £ and b have interacted to form a segment a, but an adjacent intersection of dislocations c and b have not interacted to form a segment a. The latter constitute a small lozenge-shaped network. Whether the latter lie on different (111) planes cannot be ascertained. Plate 84 is an isolated dislocation node in which the angle' 2a between 217 the segments with Burgers vectors c_ and a is 66° within + 6° depending on the orientation of foil and plane of node. Ideally the equilibrium angle 2d should be 120°. Plates 85 through 87 are typical observations of the dislocation structure of specimen C-38 strained to l6£, and annealed for 6 hours at 700°C. Specimens were annealed only 6 hours in order to compare these dislocation structures with those observed in C-38 specimens annealed for 24 hours. Plate 85 reveals many straight dislocation segments crossing one another with little interaction and relaxation into three-fold nodes. Plate 86 shows a dislocation array In which the Burgers vectors were determined to be oriented in symmetrical nodes existing in an orientation from 10 to 20° from a screw orientation, but curved dislocations were also present. Plates 88 through 93 are dislocation arrays observed In crystals from specimen C-38 which were annealed for 24 hours. These dislocation networks can be directly compared with the arrays observed in specimens annealed 6 hours. In Plate 88 some of the dislocations which exhibit a darker contrast appear to lie In a plane above that of the dislo cations with a lighter contrast. The crystals observed generally consisted of areas of alternately high and low dislocation density. The Burgers vectors were determined for the arrays shown in Plates 89 and 90. The symmetrical nodes were rotated about 6° from a screw orientation In the first, and from 17 to 24° from a screw orientation in the second. Both hexagonal networks exhibited uniform spacings between nodes, however, the spacing varied from about 0.12 microns in Plate 89 to about 0.30 microns in Plate 90. The angles 2a were within 3° of 120° for both dislocation net works. The segments with Burgers vector b between the sym metrical nodes shown in Plate 91 are good examples of the interaction product which results when a dislocation line of Burgers vector a crosses a parallel net of dislocations of Burgers vector c_. The dark spots in the background re sulted form contamination on the foil. Plate 92 shows a large hexagonal network of symmetrical nodes within 3° of a screw orientation. There is an example in this disloca tion array of two crossing dislocation lines with Burgers vectors b and c_ which do not interact to produce a third segment, where such an interaction and relaxation would ideally reduce the overall energy of the configuration, even though the majority have relaxed into a lower energy, three fold node configuration. Plate 93 shows a hexagonal network of symmetrical nodes near a screw orientation. This network is unique in that the dislocation segments with Burgers vectors b_ and c are on the average rotated about 5° from the screw orientation, such that the average angle between the two segments is 130°. The shorter segments of Burgers vector a are in a screw orientation. Plates 9*t and 95 show dislocation arrays observed in 219 crystals from specimen C-40 strained to 16% before annealing. In this specimen both straight and curved dislocation lines were observed, but no regular hexagonal networks. The reason for this was not apparent since both specimen C-38 and specimen C-40 were deformed and annealed similarly. Plates 96 through 98 show three hexagonal networks of symmetrical nodes observed In crystals from specimen C-43, strained to 23.8?. The nodes were rotated from 0 to 11°, 15 to 25° and 0 to 18°, respectively, from a screw orienta tion. In the three networks the nodes varied in orientation from 11 to 18° across only a few hexagons. Similarly in each case the angles 2a varied up to 5° from 120°. The average spacing between nodes was about 0.24 microns. Ideally this represents a twist angle of about 2.0 minutes. Various dislocation networks observed in specimen C-37 strained to 25.4% are shown In Plates 99 through 110. Plate 99 shows a hexagonal network rotated 15 to 20° from a screw orientation. Plate 100 shows a typical variation Ir the dislocation density across a crystal from C-37. Areas of high dislocation density were separated by areas of a lower dislocation density which consisted of longer straight dislocations and scattered angular pieces of dislocations. A typical foil also consisted of more dense dislocation areas which included low-angle tilt boundaries as well as lozenge- and hexagonal-shaped networks with spacings between dislo cations of less than 0.1 micron. A cellular structure, which was dominated by parallel bands of high dislocation density, was observed. A Burgers vector analysis was iconducted for the array in Plate 101 but the results were not conclusive. The dislocation segments have been designated on the photo graph as having coplanar Burgers vectors, but an equally plausible, alternate analysis yielded Burgers vectors with components normal to the glide plane. Plate 102 shows symmetrical nodes in a hexagonal network, which are rotated from 0 to 5° from a screw orientation. Again many disloca tion interactions are observed not to occur where interac tion and relaxation resulting in a third dislocation segment of Burgers vector £ would ideally lower the energy of the configuration. The symmetrical nodes in Plate 103 present a unique and abrupt change in dislocation orientation. The nodes with dislocation segments of Burgers vectors a°, b° and c° are oriented 0 to 5° from a screw orientation. In the same network the nodes of Burgers vectors a,b,c, are rotated 25 to 30° from a screw orientation. The change in dislocation orientation across the network occurs abruptly. One set of node angles between segments of Burgers vectors £ and b are only about 95° * This type of change in line orientation lends support to observations made on crystals of set I that symmetrical nodes may reduce their energy by alignment parallel to either a <110> screw or a <112> 30° direction. Plate 10*1 shows an irregular hexagonal network of symmetrical nodes rotated from 5 to 1*4° from a screw 221 orientation. Plates 105 and 106 show hexagonal dislocation arrays of symmetrical nodes rotated from 0 to 3°, and 15°, respectively, from a screw orientation. Plate 107 shows a hexagonal dislocation array with symmetrical nodes In a screw orientation. Plates 108 through 110 show the dark field Images of the dislocation array in Plate 107 obtained using the [422^], [22^] and [2^2] reflected spots, respec tively. The average dislocation spacing between nodes was about 0,12 microns. Plates 111 and 112 both show the same hexagonal dislocation array of symmetrical nodes rotated from 8 to 17° from a screw orientation. The operating primary reflections differ for each photograph. The differ ence in contrast effects will be discussed in Section VII.F. Plate 113 shows symmetrical nodes in which two dislocation segments of the nodes are rotated 17° from a screw orienta tion, but the third segment with a Burgers vector a is only rotated 12° from a screw orientation. The third segment is also the longest segment in the dislocation network. Plates 11^ through 119 are dislocation arrays observed in crystals from specimen C-39, which was strained to 25.6? before annealing. These crystals also exhibited a roughly cellular and banded distribution of dislocation density, similar to that observed in crystals from specimens C-27 and C-37. Plates 11*4 and 115 show two sets of symmet rical nodes rotated 5 to 7°3 respectively, from a screw orientation. The angles 2a were very close to the ideal 120°. Plate 116 shows a typical grid of crossing disloca tions, with little or no interaction such as to form a third segment of dislocation. Plate 117 shows both symmetrical nodes rotated from 0 to 10° from a screw orientation, and crossing dislocations with little interaction. Plate 118 contains a dislocation network with essentially three dif ferent sets of symmetrical nodes. The nodes in the set labeled "X" on the plate were rotated from 7° counterclock wise, to 10° clockwise from a screw orientation. The aver age spacing between dislocation nodes is about 0.24 microns. The set labeled MY" contains symmetrical nodes In a screw orientation, with an average node spacing of about 0.12 microns. The third set labeled "ZM contains symmetrical nodes rotated 10° counterclockwise from a screw orientation. The average spacing between nodes Is about 0.16 microns. The three sets appear to be continuously connected across the plate; however, changes in contrast effects do suggest that the networks can exist on different (111) planes. Plate 119 shows a small hexagonal network of symmetrical nodes rotated about 15° from a screw orientation. The following are general observations concerning the annealed structures observed in crystals from set II. (i) The primary observation made of the specimens from set II annealed at 750°C was that the resulting dislo cation structures were not as symmetrical or regular as in crystals from set I annealed at 850°C. For the most part 223 the dislocation lines in set II crystals were straight, but the three-fold nodes often exhibited node angles, 2« , that varied as much as 10° from 120°, in contrast to the symmetrical node angles observed in crystals from set I, which were usually within 3° of 120°. (II) It was generally observed that, similar to crystals from set I, the number of hexagonal dislocation arrays observed, and their size with respect to the number of three-fold nodes in a network, increased with strain. In set II, below about 8£ strain, the crystals contained few if any regular symmetrical networks, and often a large per centage of the dislocation lines were curved. Ciii) In specimens strained beyond about 20%, a cellular or banded distribution of low and high density dislocation areas were observed. The areas of high dislo cation density consisted of low angle hexagonal and lozenge shaped networks, and often what appeared to be low angle tilt boundaries. The larger areas of a low dislocation density typically contained dislocation nodes of larger node spacings Joined by long straight dislocations. Areas were also observed which contained only fragments of dislo cations, or none at all. (iv) All hexagonal networks of more than one hexagon, in cases where the Burgers vectors were determined, con sisted of symmetrical nodes with coplanar dislocations. In describing the crystals from set II, the term ,Tsymmetrical" 22*1 was used in the sense that the node angles, 2 of , would be 120°, in the ideal state of quasi-equilibrium, even though actual angles differed appreciably from 120°. In contrast, an unsymmetrical node would ideally exhibit node angles unequal to 120° in quasi-equilibrium. (v) A histogram, similar to that described in section VI.B.2 for the 23 networks containing symmetrical nodes observed in specimens from set II, is shown In Figure 66. The most favored orientation appeared to be one close to a screw orientation, but no preference was observed for a 30° orientation, as In crystals from set I. A large fraction of the nodes were oriented with angles of @ between 0 and about 18°. The total variation in frequency of occurrence for various node orientations for crystals of both set I and set II is also plotted in Figure 66. It was calculated that In set I 6% of the nodes were oriented within 6° of the screw orientation, and 3055 were oriented between 2k and 30° of a screw orientation. For nodes from set II the figures were **5% and 756 , respectively. In the latter set 22% of the nodes were oriented within 3° of £ = 15°, as compared with less than 2% for crystals from set I. For all networks of symmetrical nodes analyzed, 5SJE of the symmetrical nodes were oriented with 6° of either a screw orientation or a 30° orientation. If the Burgers vectors were randomly oriented between an edge and screw orientation, the last figure would be 2055. If they were randomly oriented between Histogram for oil Specimens from Set I and Set H I, ■' 30 . r I.___ 0) in .a O 20 — Histogram for Set H T u> 0)c D cr 01 Cl i o I =n I I I I i r i_____ t___ I i____ i 10 20 30 < It0> /9 ( Degrees) < II2 > Figure 66 Histogram of node orientations from specimens of Set II which were annealed at 700°C. Dotted line represents histogram for all node groups analyaed In both sets; 13 in Set I, 23 In Set II. 225 226 a screw and 30° orientation this figure would be itOjE - (vi) As in Set I, the angles of twistjO,calculated from average dislocation spacing between nodes for the vari ous networks observed in crystals from Set II, were found to be a function of strain, as shown in Table 16. TABLE 16 TWIST ANGLE AS A FUNCTION OF STRAIN SET II Specimen Strain Average Angle 6 Number Spacing (*> (microns) (minutes) C-36 8.6 1.0 + 0.5 0 . **5 C-38 16.0 0.32 + 0.2 1 .1*0 C-MO 16.0 C-**3 23.8 C-<37 25.** 0.22 + 0.1 2.1 C-39 25.6 Figure 68 shows the angle of twist derived from average node spacings of strain for all the specimens from both sets. Only one node was verified by a Burgers vector analysis as being unsymmetrical, of the type 3(a) referred to in section I. G. The node was observed in a foil from specimen C-37, which had been annealed at 700°C for 2H hours. Two different photographs of the node are shown in Plates 120 and 121. It does not exist in a regular hexagonal net work, but is in an array composed of various lengths of h vrain n h age f ws ascae ih the with associated twist of angle the variation- in The ewrs fnds s fnto f h strain. the of function a as nodes of networks Angle of Twist Q ( Minutes ) 4 0 3 2 0 10 pcmes f Set of ens Specim pcmes f Set of ens Specim % iue 68 Figure Gie Strain Glide € 0 2 % % 0 3 227 dislocation segment and differently oriented nodes. The nodes in Plate 120 labeled "a0 ," "b0," and "c°" are symmet rical nodes rotated about 29° + 1° from a screw orientation. Two dark field photographs of the images of the array from diffraction spots [22'*n and as well as a correlation of the contrast changes, and Burgers vector contrasts from Plate 120 and Plate 121 were used to determine the Burgers vectors: a ° , , c° , a , b and c for this array. The measured angle between the segments with Burgers vectors a and c was about 2 cK =» 82° + 0.5°. The segment with Burgers vector a was rotated 8° from a screw orientation, whereas the segment with Burgers vector c was oriented about 14° from a screw orientation, and gradually rotated to within 8° of a screw orientation at the connecting symmet rical node. The segment with a Burgers vector b was ori ented within 2-3° of an edge orientation, and gradually curved away from the edge orientation as it extended away from the unsymmetrical node. Plates 122 and 123 show an enlargement of the node under two different contrast con ditions (In bright field), In which the angles described are more clearly observable. The only doubt to the validity of the analysis of the Burgers vectors of the "apparently" unsymmetrical node arises from the observation of a triple contrast Image of segment b in Plate 122. If the Burgers vector of this segment has Instead, a large normal component, then this 229 contrast may reflect the (be x V term In the equation for the displacement field around an edge dislocation.1 How ever, this author observed many widely varied contrast effects which were a function of the bright field operating <’220> reflections, which may indicate the triple image is only a random contrast effect. Other nodes which may have been unsymmetrical were observed during the study of crystals from Set II, but any appropriate Burgers vector analysis or complimentary study of contrast changes at the node were not possible, usually because the node was oriented In a thicker portion of the foil. Other nodes which appeared to be of the unsymmetrical type proved to be In a symmetrical orientation despite a variation in 2c< of as much as 54°, for example as in Plate 84. ^Refer to Appendix D. 230 Specimen Number C-42 Strained 5.6% Annealed at 700°C for 24 Hours Specimen Number C-36 Strained 8.6% Annealed at 700®C for 24 Hours Isolated dislocation node with an angle of 66° between the dislocation segments with Burgers vectors £ and £. 2 32 SptclMn Number C-38 Strained 16.0% Annultd at 700°C for 6 Hours Plate __85 Plate 86 Even In this specimen that mas annealed only 6 hours, symmetrical nodes exist with rotations of 0 to 30° from screw orientations. 2 ■ - 120 ± 15° PUta __87 233 Specimen Number 0 3 8 Strained 16.0% Annealed at 700°C for 24 Hours P late 88 A typical view of random dislocation arrays. Some of the dislocations that exhibit a darker contrast seem to li e above the other lighter arrays in the thin foil. Plate 89 Symmetrical nodes rotated about 6° from a screw M orientation. 2c * 120° i 3 ■ 1 - 0 * I Plate s s l Symmetrical nodes rotated from 17 to 24 from a screw orientation. 2a - 120° ± 3 The (111) glide olane is tilted in the COIU direction about 4° to the foil plane. 234 Plate 2 1 Symmetrical nodes rotated 27° from a screw orien ta tio n . 2 a * 120° ± 2 L J L B — i P late Syometrical nodes within 3” of a screw orientation, 2 e 120° ± 2 Plate .21 Symmetrical nodes. Dislocation segments with a Burgers vector ^ are in a screw orientation. The other two segments with Burgers vectors b *nd £ meet at an angle of 130° 1 5. Bach Is rotated about 5° from a sscrew orientation. Specimen Number C-40 Strained 16.0% Annealed 700° for 2 4 hours Typical general area of foil Plate 95 236 Specimen Number C-^3 Strained 23.8% Annealed at 700°C for 2k Hours PLate Qfi Symmetrical nodes rotated from 0 to 11° from a screw orientation. 2« = 120° + 6 P late 97 Symmetrical nodes rotated from 15 to 25° from a screw orientation. 2 a - 120® *, 5 Plate 3Z. Symmetrical nodes rotated from 0 to 18° from a screw orientation. 2 a = 120° + 2 237 Specimen Number C-37 Strained 25. Annealed at 700°C for 2^ Hours Typical photograph of high dislocation danslty araa of the thin foil. Specimen Number C-37 Strained 25*4% 2 38 Annealed at 700°C for 24 Hours P late 102 Symmetrical nodes rotated from 0 to 5° from a screw orientation. 2«* 120° + 3 Plate i n q Symmetrical nodes. Those nodes described by Burgers vectors a.*, t * , and s.* are rotated from 0 to 5° from a screw orientation. Those described by Burgers vectors a > k > And C. are rotated from 25 to 30° from a screw orientation. P late Symmetrical nodes rotated from 3 to 14* from a screw orientation. 2 a * 120* ± 4 239 Specimen Nixnber C-37 Strained 25.4% Annealed at 700°C for 24 Hours Plate 105 Symmetrical nodes rotated from 0 to 3*' from a screw orientation. 2 a « 120° ± 2 Plate IQ6 Symmetrical nodes rotated about 15* from a screw orientation. 2a ■ 120* ± 2 Weeber C-37 llvtlMd t9.lf in— ltd *ft TOO*C f*r II B«nn M agnification: XSSOOK Sjm trlctl nodtt in a iem orientation. 2a • 120* ± 3 _ MagnificatIon: MOOK Mgnlflcatlon: MOQX S?duliSa?lSS0SS£l Photograph of Dark field photograph of Dlalooactona with a Dta location. *Uh a _ , DUlocZcTSTwltha t f ' t i f f ! * " . a ’? ’# * araZZrJZiZZX? lnvleabia. h" ± t 0 l T J «r.®ur*7"«v*f?or lmrleabLe. & -*[lTo3 SSJSare lnvleabia. iSSo?“ - rfioii Specimen Number C-39 Strained 25.6% 241 Annealed at 700°C for 2 4 Hours Syometrlcal nodes rotated Same array as is shown in from 8 to 17° from a screw Plate i n - The opergtlnt orlantatlon. 2a ■ L20° ♦ 3 reflections_are the (220) The primary operating and the (022). refLection Is the <2Z0). Plate H i Symmetrical nodes. The dislocation segments with Burgers vectors b and g are rotated 17° from a screw orientation. However segment & is only rotated 12» from a screw orientation. 2 a - 115°, 120° and 125° 242 Specimen Number C-39 Strained 25.6% Annealed at 700°C for 24 Hours Plate i Hi 74' Symmetrical nodes rotated 5° from a screw orien tation , 2 a * 120° + 2 V ■ l-0a > \ Plate U S Symmetrical nodes rotated 7° from a screw orien tation , 2 a * 120° ± 1 * * i * i Ufift i Plate 114, Crossing dislocations with littla interaction or relaxation such as to form a third segnent of dislocation. 2 JJ 3 Specimen Number C-39 Strained 25.6>t Annealed at 700°C for 2U Hours I Plate 117 Syrnnetrical nodes rotated from 0 to 10° from a screw orientation. 2* - 120® ± 5 P late 118 Symmetrical nodes. Array X: The dislocation segments in this area are rotated from 7° counter clockw ise to 10° clockwise from a screw orien ta tio n . Array Y: The dislocation segments are in a screw orientation. Array 2 : The d islo ca tio n segments are rotated 10° counter-clockw ise from a screw orientation. The angle between segments described by Burgers vectors Ji and £. is about 135°. 2HH Specimen Number C-39 Strained 25.6% Annealed at 700°C for 2** Hours Plate US,,-. Syrnnetrical nodes rotated about 15° from a screw orientation. Note that in this photograph the nodes appear to be d isso cia ted into partial dislocations, The operating reflections are snown in Figure 67 below. b / V : / . \ •( 66(0 Figure _ 67 _ (247) >(54*) The in te n s itie s of the primary diffraction spots were relatively (0 0 0 ) the same. '(606) . / *(524)\ I / • * '\ 245 Specimen Number C-37 Strained 25. Annealed at 700°C for 24 Hours liacont lmiltv - B u r g e r s v e c t o r Plate 120 changes Symmetrical nodes of Burgers v e c t o r s a® , b® , a n d a® a r e rotated about 35° from a screw orientation. The node described by Burgers vectors 3.* * b’ » And £* is an unsym- m etrical node of the type 3(a). The measured angle 2d i s 8 2 ° . P l a t e 121 Same dislocation array as in P l a t e 1 ?n but this photograph was taken at a different operating higher mmagi agnification and reflections show a under different operating contrast change at reflections. the discontinuities in the Burgers vector Plate 122 Magnification of enlargement: 78200X P late 123 Magnification of enlargement: 96600X VII. DISCUSSION The first section (A) includes a discussion of the energy of straight dislocations in isotropic and anisotropic media, from which expressions are developed to determine the anisotropic values of the energy factor KA . The application of these expressions to dislocations in germanium Is dis cussed in Section B. Prom the values of KA for & = 0° and 'S = 60° evaluated in section B, a working equation for the variation of K^(^) from 0° to 90° Is derived in Section C. Once the values of are determined, the equilib rium node angles for symmetrical nodes of the type 1 and 2, and unsymmetrical nodes of type 3(a) and 3(b)^, can be de termined using the line tension theory and the resulting equation (4) developed In Chapter I, section D. The calcu lations and results are shown in Section D. Section E is a discussion of the dislocation struc tures observed in the as-deformed crystals, and F the annealed crystals. *As discussed in Chapter I.G. 248 A . The Energy of Straight Pislocations 1. Isotropic media Suppose that In a right-handed orthogonal coordinate system (x^x^x^) , a screw dislocation lies along the x^ axis, f_ = b^. Here £ = unit vector parallel to dislocation line b = Burgers vector of screw disloca- line Then Its displacement field Is, (55) u(r,0 ) - b (e/2 'TT ) = (b/2Vr ) tan“1 (x2/x1) 2 2 2 where In polar coordinates; x^ + x2 = rc 6 = arc tan (x2/x^) The corresponding strain field Is, where: I = (1,2) J - (1,2) and the stress field is, (57) ^"93 = ^ — = the shear stress associated with the 2Tfr dislocation per unit length, normal the 9 direction, acting in the X3 direction where: = shear modulus The self energy per unit length of dislocation In the region r0 to R Is: /rQ\ ,r/T ^ 6 3^ jub2 ,R . (58) W/L * 2}i • 2 tt r-dr - In (jr O 2*19 Similarly the strain energy of an edge dislocation (f -b - 0) is: ,R 2 IT C59) (W/L)s .J rdr /d0 tjL ^ ^ tr33 >] where E = elastic modulus = poisson’s ratio and where: / 2 2 \ iib xl'xl "x2 > (60) ub x2(3x12tx22) °11 " " 2tT(l->l) (x12+x22 p 2 2 rt-22 _" nrr-i^y *2(xi7—2— -x2 lfr~ ) (X1 x2 ' ^33 - <°ll + ®22> In a similar form to equation 5S» the self-energy of the edge dislocation Is: (61) (W,/L)e • jJttU-s) l n * r ^ The energy of the straight dislocation segment of arbitrary Burgers vector can be additively composed of its screw and edge components; 250 (62) (63) Cws/L)m = C sin2 + H ln where P Is the angle between C and b. Equation 63 simpli fies to the well known equation: , , jub2 f l-^cos?l - /R \ (64) Ws/L ** Tprr *- 1-^ J ro If an energy factor Kj is defined, which Is only a function of the independent isotropic elastic constants ( n> ,jjl) and the angle ^ between £ and b, then equation 64 becomes: 2 (65) (Ws/L) = (Kj) jpjf In (^-) o where (66) Kl - HU-^icogLj. Hence In an isotropic media the energy factor In equation 66 for the self-energy for a straight dislocation line is a function of (67) ()i» V j ^ ) , where (shear modulus) (68) V =* ^/2(u +A) » C12/2 (Ct|l4 + C12) (PolssonsT ratio) 2. Anisotropic media The assumption of elastic Isotropy generally, used to determine the energy of dislocations often involves errors of 20 to 30% (1). It is therefore useful to use, when possible, anisotropic elastic constants for the material involved, and the appropriate anisotropic theory for straight dislocations. The anisotropic elasticity theory as applied to straight dislocations was developed Initially by Eshelby (31) and extended by Foreman (^5). A brief treatment of the theoretical development of Eshelby as expanded byHIrth and Lothe (1) is given in Appendix E. The equations developed from this treatment are given in the following paragraphs. Let a dislocation be placed on the X3 axis In a right-handed orthogonal coordinate system (x1 ,x2 ,X3), where subscripts at 9 £ = 1,2 and i,J,k,l = 1 ,2,3 for the matrix C which represents the matrix of elastic constants. The ijkl quantities D(n),p(n),Ak(n), B^^Cn) and = x^ + p(n)xj (Appendix E) are parameters of the analysis, and Re means "the real part of" while Im means "the imaginary part of." In terms of these parameters, the following results were obtained from Appendix E. 3 252 (70) (71) W,/l = In 5_ Im B1Jk(n)Ak(n)D(n)] 3° Kl- J (72) KAb2 = bi Im [ £ B12k(n) Ak (n)D(n)] n 1 Then the self energy per unit length of dislocation line - can be described in an equation similar to equation 65: (73) (w8/l = $nr ka ln (P^> The main effect of applying anisotropy to straight disloca tions lies in the factor in equation 73, since the logarithmic term is relatively dependent only on R and r0 , and the core term is only a small fraction of the total energy and relatively independent of anisotropy (l). 3. Simple solutions to the anisotropic equations If a dislocation is oriented such that the screw component gives rise to only a displacement u^, and similarly an edge component gives rise to only u^ and u2 > then the solution for the energy factor, KA> will consist of separate pure screw and pure edge parts. If this is the case, then using the previously derived equations for the general anisotropic case to find the stresses and energy of the dislocation, the problem can be solved by separating the solution into: 1) a screw part involving a second order polynomial In p and 2 x 2 matrices, and 253 2) an edge part involving a fourth order polynomial in p and 2 x 2 matrices. If the dislocation Is parallel to x^; the are the elastic constants referred to this particular x^xgjX-j system of Cartesian coordinates. If the x^X2 plane is a ‘ reflection plane of the crystal, or In another words, the x^ axis is of evenfold rotation symmetry, then the simple solutions are applicable. A further development of the simple solutions for the edge and screw components Is given by Hirth and Lothe (1), Chapter 19- The following are the resulting equations, as applied to the accompanying reduced matrices and coordinate systems Pure Edge Let bel = (b ,0 ,0) £ = (0,0,1) b^2 = (0,b,0) and assume that Cjg* ** c26* = 0 in the reduced elastic matrix in equation Jh. Then, the resulting energy factors Ke^, and Ke2 for the two possible edge components of the total Burgers vector are: c66 (cll “ c12*^ (75) Kel - ( C u 1 + C12>) C 22 ^ ^ n +c 12 * +2Cgg *~) 25*1 (7*1) cll' C12 1 0 0 0 cl6 ' ' C12 c22 0 0 0 c26' ( c : \ = 0 0 0 0 0 0 0 0 0 Ci| J4 1 Cll5 ' 0 0 0 0 Ci451 C55' 0 C16' c26 1 0 0 0 c66 * m (77) fc 1 Ci2 * 0 0 Ci6 ’ ^11 C131 C2l' C22' C23’ 0 0 0 a ' } - C 31f C32' c33’ c3v C35' 0 0 0 c43' C44' C55T 0 0 0 C53 ' C5H' C55 0 cl6* C26' c36’ 0 0 c66’ 2. Pure screw With b = (0,0,b) (0,0,1) and with the reduced matrix in equation 77, £ CS 'J » the value of K_ for the screw component of the Burgers vector is: 1/2 (78) Ks = (<;„„' C55* -C45 ) Now the previous solutions for the screw and edge components are applied to the face-centered cubic transla tion lattice in which the important glide planes are fill} and the Burgers vectors 1/2 O - 1 0 > . In the face-centered 255 cubic structure, if the dislocation line is parallel to <110> then the simple separation into screw and edge parts is valid. This means that the anisotropic elastic energy factor K can be computed analytically for a screw A dislocation or for a 60° dislocation. In each case, both the direction and the Burgers vector lie in a <110> direc tion on a fill) six-fold rotation symmetry glide plane. For the case of the screw component, use is made of a dislocation in the coordinates shown in Figure 69« The axe3 of the coordinate system are: (79) i' = 1/ V6 [121] k' = 1/ v/2 [101] J = 1/ v/3 [HI] The elastic constants for this orientation can be derived by transforming the matrix fCg ’J tofcs"} by the matrix: 1 +1 -2 + 1 (80) T « " s/6 f 2 J 2 /5 --J3 0 J 1 where if: <81> W - TKra T ln Is the (9 X 9) transformation matrix then C 11 a Q f1 * "H ijkl ghij ghmn mnkl or (82) (C" } - ^Q) (C'l {Qj 256 k* Figure __ 69 • Projection normal to the (111) plane in a face-centered cubic crystal with axes: i’ = 1/^6 [121] , k' = 1/V2[l0l]. j - l/v3 [lli] points out of the page. 257 Using this type of transformation, the final form of ^C" ^ s for the coordinate system described in Figure 69 is': 0 0 0 0 (83) ^11 C12 c13 C16' cl6 ' C12 C22 c12 0 0 0 0 0 0 C13 c12 C11 0 0 Cl6 ’ 0 . 0 _cl6' 0 0 0 0 0 M Ci|l|* ■-C 16’ Cun' ci6 ’ 0 0 0 0 -*Cl6 ' C55* -C16* c55 0 Cl6 0 -C16 0 0 ci| 1+' 0 0 0 0 0 CH*|' --Ci6 ’ 0 Ciin ~C16' 0 0 0 0 ■-Cl 6 * c55' 0 -Ci6 f C55 0 0 0 0 CliU1 0 0 cim' Cl6 _ci6 where: (8*0 >c55 1 = 1 C n * ~ c13 where in terms of the standard elastic constants referred to the cube axes , (85) Cll’ C11 + 1/2 H Cl2 ’ = Cl2 ~ 1/3 H - C13* - C12 1/6 H c22’ Cll + 2/3 H =3 ClU - 1/3 H =3 C55’ c*t^ - 1/6 H C161 — z M H where (86) H — 2Ctyty + C^2 “ ^11 258 The result for the screw component, from equation 78, is . 2 ^ / 2 (87) K = (Cl|i|' C 55 * - Cl6' ■=) For the case of the edge components, the applicable coor dinates to satisfy a simple analytical solution are found by rotating the normal elastic matrix by around the i axis of the cube. This can be accomplished by applying a transformation matrix: (88 ) 1 0 0 1 0-v/I J2 ij 2 0 J2 71 And if: Qmnkl Tmn Tln then: (89) fcE"} - {q H (90) Ci i * C12* ® 0 0 C12’ C22 1 C23' 0 0 0 C. 0 (c e "3 ’12' °23’ C22* 0 0 0 0 0 CjjV 0 0 0 0 0 0 C, ? 0 '55 0 0 0 0 0 c55; where: in terms of the elastic constants referred to the cube axes, 259 cll' = C11 C12' = C12 (91) 055’ - C44 C22’ - C1X + 1/2 H C23’ = C12 * 1/2 H Cj|i| - 1/2 H The results for the edge components using equations 75 and 76 are: r c5 5 ,(cll’ + c12'> 1 1/2 (92) Kel - (Cxl'+ C 1 2 ') L C22* {C'i i ' + C12,+ 2 C ^ tTJ K . ,C22',l/2 (93) *2 ' ■XI el Hence using the elastic constants for a given face- centered cubic material, and using equations (32), (33), (3*0, and ( 38), (39), (^0), the energy factors KA (0°) and K a ( 6 0 ° ) for a screw and a 60° Burgers vector dislocation, respectively, can be calculated analytically. Since Ke i / and because C n ' / C22T> the cross term in the energy between the two edge components is zero only for the coordinates used in the preceding section. If the edge component of a dislocation lying along a [110] direction were decomposed Into one component in the (111) plane and one component perpendicular to the (111), a cross term would result in the energy of the edge component. Also for the edge and 30° glide dislocations lying on {ill} 260 planes, the direction of the dislocation would have to be <'112> , so that the edge and screw components would, not separ ate such that a simple analytical solution is possible. Hence only the energy factors for the screw and 60° dislocations can be calculated directly. B. Application of an Analytical Method to Obtain ' Values of K^CT6) and KA CfrO*Q The values of the anisotropic energy factors KA of a straight dislocation line in germanium are now calculated as a function of the line orientation and the Burgers vector. The values of KA (0°) for a dislocation line in a screw orientation, and KA(60°) for a dislocation line with a value of - 60°, are calculated using the analytical method described in section A. 3* The values of Kj^< 0°) and K^(60°) are the only ones that can be obtained using the "simple solution.” The values of the elastic constants for germanium are (46): (94) C 1X = 12.89 X 1011 dynes/cm.2 C12 ^ 4.83 " " C44 = 6.71 " " and H = 2C44 + C12 - cn = 5.36 X 1011 dynes/cm.2 1. ka (o°) Using the coordinates in Figure 69 and the elastic matrix fCg} of equation 83, the resulting elastic constants from equation 85 are: (95) C jh! * - Cm, - - » 4.92 X 10" dynes/cm*2 c55' = - | = 5.82 X 10" dynes/cm.2 Cl6' = __= lt25 X 10" dynes/cm.2 6 H These values when substituted Into equation 87 give for the value of KA (0°): (96) Ka (0°) = 5.20 X 10" dynes/cm. 2. KA (60°) To obtain K^fSO0 ) for germanium the coordinate system in Figure 70 is used, where the glide plane is the (llT), and bgQo = a/2[110] and fg0° ~ C101]. The Burgers vectors b f, 6', b* and Bio also shown where: —s -e J - e 1 ^ = Tf [101] “ |[12‘1] = h^x + b^2 = §[010] + §[101] Vectorally adding the component energies, Kb2 ; (97) KA(60)b602 - Ksb / + Kel b2x + Ke2b^2 where, 2 Z II [101] 262 Y II [ io i] x 11 [010] (111) Figure 70 Coordinate system used for determining K A (60°). 263 and by substituting the values In equation 98 Into equation 97, the resulting equation defining K^(60°) Is; (99) KA<60°) = Ks(l) + Kelc|) + Ke2(.|); Ks = KA<0°) The value of KA (0°) has already been calculated In sec tion 1, as 5.20 X 10" dynes/cm.2 The values of the elastic constants as defined in equation 91 yield for ger manium : 3 Cll* C11 12.89 x 10" dynes/cm. 11 C12' - c12 - 4.83 = 3 It c55 * Cl, 4 6.71 (100) = 3 11 c22* cn + \ 15.57 3 If C23 * = 13 - | 2.15 " = = 11 c44’ o„k - | 4.03 * where, (101) 11 which are used for the determination of the values of and Ke2 as defined in equations 75 and 76. For germanium the values are: rc55'^' -c* ) 1 1/2 (102) Kel = (Cj, +C’ ) --- — — 3^------12 LCi0(Cnn+C' +2C1-) J 22' 11 12 55 = 6.69 X 10" dynes/cm. C' 1/2 (103) “ ( * Ke2 = 7.36 x 10" dynes/cm. C11 26*4 Substitution of the values of Ks ,Kel and Kq2 from equations 96, 102 and 103 into equation 97 yields as the value of Ka (60°); (10*4) Ka (60°) = 6.*49 X 10" dynes/cm. which is about 25% greater than KA(0°). c * Method Used to Obtain a Working Equation for the Variation of K^/9}- with g An exact variation of the energy factor KA with the orientation (fS) of a dislocation line in germanium with respect to its 1/2 <110> Burgers vector can be calculated using anisotropic elasticity by means of a numerical analysis. Such an analysis however is very complicated and lengthy, and has not yet been done in general form for face-centered or diamond cubic systems. Chou is working on a solution to this problem at the present time (*49). Two exact values of KA have been calculated, using an analytical approach, for KA(0°) and KA (60°) in the preceding section. If the isotropic form of the energy factor, r 1 — ^ q o s 2 -I = ji (_ j— g J is used, where the value of ji is taken as the value of KA (0°), and the value of N) is arbi trarily varied to obtain at ■? - 60° the same value of Ka (6o°) as was obtained by the exact analytical calculation in the preceding section, a very close fit to the probable anisotropic curve for the variation of with ? can be obtained. 265 The equation which results from an arbitrary vari ation of and which describes the variation of the energy of a straight dislocation line in germanium as a function of the angle $ , is henceforth called the "modified isotropic equation" for K^. The form of the equation Is: (105) Km (£) = [ l - ^ ’cos2 ^ ] where = .an arbitrary value to be determined jx = 5*20 X 10” dynes/cm.2 = KA (0°). Equation 105 Is plotted in Figure 71 for several different values of ’ as a function of The value of ^ 1 - 0.2*19 results In an equation which has a value of KA (60°) = 6.49 X 10 dynes/cm.2, Identical to the value obtained analytically. Equation 105 is plotted again in Figure 72 as a function of P , along with other possible functions of K^(^) which were determined using in equation 105: (1) Voigt anisotropic average values of = 0.200 and ju = 5.64 x 10" d y n e s / c m . ^ and (2) Reuss anistropic average values of '■J = 0.214 and y = 5.30 x 10” dynes/cm.2 .* Note that the greatest difference in the values for KA (0°) and KA (90°) i3 obtained using the modified isotropic values of N) f and ja. *Voigt and Ruess average anisotropic values of V and p were obtained from HIrth and Lothe (1)^Appendix. s 0 lte s function aas plotted 0 vs.K K (10 dynes cm 6.8 6.6 7.0 6.4 6.0 6.2 5.2 5.6 5.8 5.4 30 (degrees) $ iue 71 Figure of''). 60 90 M M = V = V V 0.220 = 0.280 9 4 2 . 0 0 6 2 . 0 0 4 2 . 0 0.200 266 267 7.0 • • 6.8 -(modified isotropic) = 0 .2 49 K = 5.20 + 1.73sinp *4 = 5.20 6*6 _(Voigt) M = 0.200 V = 5.64 - (Reuss) 6.4 Ai = 0 . 214 >) = 5.30 6.2 CN I oB w(U 6.0 c ■o 5.8 5.6 5.4 5.2 0 3060 90 0 (degrees) Figure K plotted as a function of 3 using (i) the modified isotropic equation, (ii) Voigt average elastic constants, and (iii) Reuss average elastic constants. 268 Figure 72 shows that the KA factor is indeed sensi tive to the value of "0 used, so that a proper choice will make a significant difference in the value of KA(90°). As discussed in section I.B. Hornstra (4) noted that the num ber of broken bonds per unit length in the core of a straight dislocation in germanium, was proportional to sinff. A relationship of the type: (107) K a (£) = Ka (0) + [KA(90°)-KA(0°)] sin/? was plotted in Figure 72 for comparison. The shape of the curve suggests that the other correlations are better. In Figure 73 the modified isotropic equation for K a (^) is plotted in polar coordinates, where the 0° axis is oriented parallel to the O l O ^ Burgers vector, and the line orientation relative to the Burgers vector Is varied by 360°. In order to obtain some check of the validity of using a "modified" curve, KA was plotted as a function of f* in Figure 74 for the energy variation for a straight dislo cation line In the face-centered cubic metals copper and gold. The points shown are values of K^C?5) which were either computed analytically by this author, or were de rived by Foreman (45) using a numerical approximation for these metals. The continuous curves are plots of a modifi- fled isotropic equation calculated for gold and copper by this author, In a manner Identical to that used in the this 269 ENERGY FACTOR KA 5.0 I 6.0 7.0 0 ° BURGERS VECTOR <5l0> 330° 270 Figure _ JJJ__ Variation in the energy factor K^(P) ( X 10“^^dynes cm-^) as a function of orientation assuming the modified - isotropic equation in which ju = 5.20 X 10-^dynas cm"^ and V = 0.249. 270 K(Cu) = 7.25 - 3.06 cos-0 = 0.421 7.0 - CURVE F r CTEO TO DATA USING K(0°) AND K(60°) 6.0 E u 5.0 w CJ c TJ ■-J i-H o 4.0 3.0 / CURVE FITTED TO DATA USING K(0°) AND K(60°) K(Au) = 4.96 - 2.47 cos^p si = 0.500 2.0 0 6030 90 0 (degrees) Figure 7*t K vs. 0 for copper and gold. The curves were computed analytically using a modified isotropic equation and the values of K(0°) and K(60°). The solid points were computed numerically by Foreman (*15)). section for germanium. The values of ji and 'J ’ used in the isotropic modified equation for gold and copper were 2. *47 x 10 dynes-cm. _^/0.500 and *4.21 x 10 dynes-cni-^/ 0.*I21, respectively. The fit is good, but it appears that a flatter curve would better fit the actual values. The best fit using the modified equation is obtained if the value of V ’ used for copper is increased from 0.*421 to 0 .*426 , and 1 for gold V is increased from 0.500 to 0.510, in each case an increase of less than 2%. These good correlations of the isotropic modified equations of to the actual values of Ka in gold and copper provide reasonable support for the validity of applying modified values of KA (^) to germanium. D . Derivation of Expected Node Angles Each type of three-fold planar (111) node which was described in section I.I.2 and depicted in Figure 29 is now investigated to theoretically determine its probable equilibrium node angle, 2 &. The general configuration to which calculations refer is shown in Figure 75, where I 1 - - 1 (108) ^2'il = oos“ 5.3 - ^ = -sin c* -3*—1 cos* 5l2 ’-1 ~ +sint* 272 - 2 '-'2 00 - - Figure 75 Node configuration and parameters used in calculations for determining equilibrium node angles. 273 Depending upon the Burgers vector and orientation of each dislocation line, the final equilibrium angle 2 « will be reached by a variation in the orientation between the two dislocation lines labeled (2) and (3), such that the exten sion of line (1) through the node will always bisect the angle . This is valid if the coordinate system is always constructed such that the x axis is parallel to dislocation {1) and the dislocation array lies in the plane normal to the z axis. Similarly, the Burgers vectors of the dislocation lines (2) and (3) will always have equal values of P , as shown in Figure 75 as long as the three Burgers vectors are coplanar, and equal in magnitude. In the calculations involved in determining the angle 2 « , the value used for the energy of dislocation line as a function of the angle , was that derived in section C; equation 106 where 'O' - 0.249. When this value ' is substituted into equation 109, using equation (4) an equation of the form; 3 (110) 0 « 2 [(1-V'cos2^ ) £i+2V» cos^sini^n^] is obtained, where n* - ($ x N) —X — 4 — This equation is now used to determine the equilibrium anglecx for each of the types of nodes; 1,2,3(a) and 3(b) 274 described in section I.F. The type (1)KS-PS node is one in which the three dislocation lines are oriented parallel to their respective Burgers vectors, i.e., /^=0. The application of equation 110 yields a value of ^ = 60°, which was expected because each dislocation is oriented in its lowest energy state, the energy factor for each dislocation line was the same, and the three-fold node lies along the sane three low index directions on a plane which exhibits three-fold rotational symmetry. The type (2)K3-P3 node is one in which all segments are oriented 30° from their respective Burgers vectors, i.e., 7^1 = 30°. The resulting value for cK of 60° was as expected for reasons similar to those for the type 1 node, however in this orientation the node is subject to a torque vector which tends to rotate the node into an all screw orientation. Hence the type (2) node should be unstable with respect to rotation into a type (1) node configuration. The type 3(a) Ku-Pu node is one in which one disloca tion is in an edge orientation =90°) and the two remain ing dislocation lines are in approximately a 30° orientation (t^ j = = 30°). In the application of equation 113 the edge dislocation is oriented to be dislocation (1), as Is shown in Figure 76, where; \ 275 A Figure 76 Orientation of node parameters for node type 3a. A Figure 77 _ Equilibrium configuration for node type 3a, a =-41.5°. 276 b x = CB CA + AB = CB b 2 = AB b 3 =* CA (111) - 90° 2 ~ ^3 =ct- 30° where ^is the angle between the >)' = O.2M9 dotted exten sion of dislo- n^ - 1 cation (1) and b? or b 3 n 2 = n$ = -sin c< Substituting these values into equation 110, results in the equation: (112) 0 = -1.000+2.000 {cos* [1.000-0.2^9 cos2(« - 30)] - 2.000[(0.2M9)sin« sin( <*> - 30) cos (e*-30)]} The value of c* which satisfies this equation is cX (3a) = Ml.5°. Figure 77 shows the resulting node configuration, in which one dislocation is in an edge orientation, and the two others are oriented 11.5° from a screw orientation. By rotating towards a screw orientation, the dislocations (2) and (3) have reduced their energies, and the total energy of the configuration to a minimum. Any further rotation of dislocations (2) and (3) would again increase the total energy because of the further increase in energy of the con figuration associated with the term (1- ^ 'cos2^ ) for each segment. The type 3(b) Ku-Pu node is one in which one disloca tion line is in a screw orientation, and the two remaining dislocations are oriented approximately 60° from their 277 respective Burgers vectors. For analysis this node is oriented such that the screw segment is dislocation (1 ), i.e., - 0°, as shown in Figure 78, where b 1= AC AC = AB + BC b 2 = BC b 3 = AB (113) = 0 v?2 = ^3 = C^-tX) = 120° \)' = 0.249 — 2 ~ ~ + sin c* Substituting these values into equation 110 results in the equation: (114) 0 = -1.000 + 0.249 + 2.000 f (1.000-0.249cos2(120-^)cosK + 2.000(0.249) sin sin (120- oO cos (120-ot) } The value of 81.1°. The two dislocation lines (2) and (3) will rotate 21° towards a screw orientation until the minimum total energy is obtained for the node configuration, which is shown in Figure 79. It may be noted that the node described as 3(b) may be rotated 30° into a node of type 3(a), but to do so would require an increase in the total configuration energy, as shown in Figure 80. The total energy of node 3(a) per unit length of dislocation is about 2.7% greater than node 3 (b), when each is compared in Its lowest energy state. On this basis alone, the 3(b) type node may be 278 -2 A .. t.J S3 Orientation of node parameters for node type 3b A Figure 79 Equilibrium configuration for node type 3b, a = 81°. 279 17 . 50;- oE « co> >% ^ 17 .0 0 - IU degrees (SEGMENT 1) CCV7 ROT AT 101' E = Wa/L (segment i),where Wsi/L ■ 6.92-1.73 cos2 (10" dynes/cm2) Figure 80 Variation in the energy of an unsymmetrical three-fold node, per unit length of dislocation segment as the node is rotated from the 3a position 30° to the 3b position, 30° to 3a position, and so forth. 280 -S / -L * expected to occur more often than 3(a)* Figure 81 shov/s the calculated equilibrium angles, for node types 3(a) and 3(b) plotted as a function of the value of ’ assumed in the equilibrium node equation 110. The resulting values of« are also shown for calculations in which the anisotropic average values of derived by Voigt and Reuss (1), as well as the value of "0 j = 0.208 obtained from isotropic elasticity, were used in equation 110. In each unsymmetrical node, the largest deviation of from 60° was predicted by use of the anisotropic "modified isotropic" value of >)1 = 0.249. E . The Dislocation Structure in As-deformed Specimens 1. Dislocation configurations in strained face-centered cubic materials The following discussion concerns various theories and observations on stage I and stage II work hardening in face-centered cubic and diamond cubic metals. a) Easy glide region The easy glide phenomena refers to that Initial por tion of the work-hardening curve for face-centered cubic crystals oriented such that the deformation axis is located near the center of the orientation triangle close to the pole, and deformation involves slip on only one plane Not including effects of Peierls-Nabarro Energy. o ( D egrees) Variation in the calculated values of the equilibrium node equilibrium the of values calculated thein Variation angles for node types 3a and 3b plotted as a function of function a as 3b 3aplotted and types fornode angles h vle of the value 2 4 4 4 0 8 82 84 76 76 JL 78 0 4 38 200 0220 0. 260 0. 30 .3 0 0 8 .2 0 0 6 .2 0 0 4 .2 0 0 2 2 0 0 0 .2 0 = 43.7 = - a V era = 78.3 a r e . 44. ° .7 4 4 ; a .0 7 =7 a ° 7 . 7 7 * a X J 44 2° v iue 81 Figure dfe Isotropic odified M Constants Constants us vrg Elastic Average euss R V stoi Elasticity Isotropic og Aeae Elastic Average Voigt v Constants Constants 0. 8 0 .2 0 = 0. 9 4 .2 0 = - v v 200Constants 0 .2 0 = 0.214 281 with a negligible contribution from the dislocation forest to the work-hardening rate. The action of long range stress fields of the glide dislocations must then account primarily for the positive slope of hardening in stage I. Although the existence of well defined slip lines on crystal surfaces show that the dislocations may be generated in groups of at least ten to twenty, internal stress calcula tions (*l8) indicate that in the easy glide region the stress field of individual dislocations, and not piled-up groups must be considered. Seeger et al. (^9) have postulated that if a source such as a Prank-Read source is assumed, the dislocations will initially be held up by obstacles grown into the crystals, such as lineage sub-boundaries. If the density of the grown-in obstacles is negligibly small, the source will emit a large number of loops under a small stress, creating steps on the crystal surface and only a few piled-up groups at obstacles. The hardening then re sults from the long range stress interaction of disloca tions on the primary slip system in parallel and neighboring planes. The following are various configurations that have been invoked to explain this hardening. (1) Taylor Model. This model was developed by Taylor (50) in 193*1 to explain hardening in easy glide. He assumed the dislocations which were generated by N sources/ unit volume would slip through the entire crystal, of length L, with an average distance h between the glide 283 planes, (115) h = N/L2 He assumed that the dislocations became somewhat stuck somewhere In the crystal because of the elastic interactions between dislocations on adjacent planes. Then if new loops are formed at the source faster than they disappear at the surface, an increased stress is necessary to overcome the elastic interactions, and the distance "ln between disloca tions on adjacent crystals planes will decrease until the dislocations can pass through the crystal. His model is depicted in Figure 82. It has been shown however that the distance "h" between slip planes is so small for equation 155 that the elastic interaction is very large relative to the applied stress (51). Another fault with this model is that the dislocations are assumed to remain straight and parallel to one another, which could only occur in struc tures with very high Peierls-Nabarro forces (51). More complicated arrangements are expected and usually observed. (2) Dipoles. Two dislocations of opposite Burgers vectors slipping in neighboring planes of several hundred angstroms separation, attract at long range until they are almost on top of one another. If the distance between the two glide planes is small compared with the length of the dislocations, even If the dislocations are not parallel, they tend to form a dipole over some definite length. 284 [\ (a) T AY LOP LlOtn-L X.L Tx \ (b) DI FOLKS XX r x x X X (c ) SLIP / \ / \ / \ POLYGON I?-AVIOI! ^ Nx / X Figure 82 Various dislocation configurations that may contribute to Stage I work-hardening. (Freidel (51)). It hcs been shown by Tetelman (52) that the dipole is stable for most levels of applied stress in the easy glide region, therefore dipoles are expected to be formed In the easy glide range. Dipoles have been observed by many investiga tors as a characteristic feature of deformation in metals (Alexander and Mader (17) in germanium). The orientation of the dipoles should be random, but they are more often observed with an edge Burgers vector character in which one end Is connected by a Jog. One explanation of the formation of a dipole-Jog pair is as follows (52). If In Figure 83 the dipole bend3 , the portion XYX'Y1 is rotated into a screw orientation, which will annihilate while building a jog 1 - i' by a small amount of cross-slip, the dipole will then reduce Its line tension by rotating into an edge orientation. Dipoles do not produce appreciable long range elastic stresses that could harden the crystal because the two dislocations with opposing Burgers vectors will compensate at long range. The hardening associated with a dipole is related (i) to its large local stresses at a distance of about nh" (dipole separation), which will tend to capture moving dislocations and form complex dis location arrays, and (ii) to the fact that the formation of a Jog will prevent motion of the dislocations from which the dipole was originally formed. (3) Slip polygonlzatlon. When two nearby pure edge dislocations of the same sign slip in parallel planes and 266 b t /i Figure 83 Possible mechanism of production of dipole-jog dislocation configuration (Freidel (51)). Figure 84 Double kinks in compression testing. (Freidel (51)). 287 are forced, by dislocations generated by the same source, to pass near one another, then they may take a more stable position by arranging themselves on top of one another (Figure 82). This type of interaction will show up more frequently If a bending stress is involved in the deforma tion process. This type of array of n dislocationshas a long range elastic stress field proportional to a disloca tion with a Burgers vector of nb^. They should stop other dislocations in neighboring planes and build sub-boundaries consisting of edge dislocations. The distance between the dislocations will vary in separation however because little climb is allowed at the stress levels involved, and there fore the effect of the hardening will vary. (*l) Deformation bands. Deformation bands can form if, for example a wall of dislocations as described in the previous paragraph becomes unstable and meets a wall of opposite sign moving in the opposite direction (5 3)> or it may form by the successive addition of dipoles (51). The mutual attraction of dislocations of opposite sign would result in a metastable deformation band often seen in easy glide in cubic crystals (5*0. The lattice curvature is formed perpendicular to the slip direction as expected for a band of edge dislocations. The formation of such bands would cause local obstacles to slip, but little long range hardening. 288 (5) Double kinks In compression testing. In compres sion because of the end constraints, and a lack of lateral deformation at the ends, the specimen will exhibit plastic bending. To compensate the bending of the lattice planes, the dislocations will reorganize in the form of kink bands, as shown in Figure 84. These have been observed by Rosi (55) in face-centered cubic materials. (6) The coplanar interaction of two non-parallel dislocations. If two dislocations cross on the same glide plane, or in two neighboring planes located very close to one another, the dislocations may interact as described in section I.I. to produce a third coplanar dislocation seg ment with a Burgers vector which equals the vector sum of the two original Burgers vectors. The stability of such an array and its effectiveness in providing an obstacle to slip would depend upon the reduction in energy associated with the formation of the third dislocation segment In the three-fold pair. Other less significant configurations which may con tribute to hardening have been suggested, but these are not discussed because few have ever been observed. b ) Stage II work-hardening Stage II work-hardening generally has a constant slope several times greater than the slope of the easy glide stage I region. Different mechanisms of hardening 289 have been proposed for this stage, the more probable ones can be separated Into three groups: (1) The pile-up theories in which hardening primarily results from long range internal stresses from piled-up groups of dislocations which interact with glide disloca tions (^9). (2) The forest models in which work-hardening results from a decrease in the mean free path of the glide disloca tions, either by the formation of high density dislocation braids, or by secondary slip (56). (3) Jog formation in which the motion of dislocations is hindered by the formation of sessile Jogs. The three different classifications all have in common the postulate that cross-slip in a secondary plane, slip in an additional direction, or other obstacles present in the material are continually increasing in their capacity to hinder slip in the primary slip system. The observation of decreased lengths of slip traces on crystal surfaces, and the smoothing out of slip traces on crystal surfaces indicate this increasing effect of hardening resulting from a decrease in the mean free path of the glide dislocations, and the subsequent increase in the local stress fields near the obstacles. The following discussion includes the various observed intercrystalline barriers. 290 Of the twenty-four possible glide systems In the face-centered cubic structures which could interact with a primary slip system, slip on twelve of them would promote work-hardening by providing strong blocks to primary glide propagation. The types of possible barrier interactions are discussed thoroughly by Hlrth (57) for materials with a face-centered cubic structure. If the core energies of dislocations and the ani sotropy of the crystal are neglected, and if the criterion for the interaction of dislocations on different slip planes Is that the energy of the product dislocations is less than the reactant dislocations (equation (2) In Hirth (57)), then the most probable reactions between glide dislocations on one secondary slip plane (a) and a primary slip vector BA on plane (d) (Figure 85) to produce sessile Jog-jog barriers to further glide on either plane are: (i) BA(d) + BD(a); these dislocations repel but if they are forced to Intersect under an applied stress, will lead to jog formation in both reactant dislocations. (ii) BA(d) + DB(a); these dislocations attract and will form a DA segment which is coplanar to the (b) and (c) planes. (Ill) BA(d) + BC(a); these dislocations repel but can be forced to intersect and lead to the formation of a kink In the primary glide plane and a Jog In the secondary. 4 Thompson *s tetrahedron uni qto; Figure 85 Thompson tetrahedron showing the 2*t possible dislocation interactions of other slip systems with the primary slip system, BA(d). (+) or (-) indicates that long range forces are attractive or repulsive. Stresses, , are given as fractions of the maximum resolved shear stress on the primary system, and finally, the most likely Interaction between the primary and secondary slip systems is listed under the secondary slip system. (Hirth (57)). 292 Civ) BA(d) + CB(a); these dislocations will attract and form the segment CA(d), or in some cases a Jog in the primary slip plane and a kink in the secondary plane. (v) BA(d) + DC(a); these dislocations are attractive, if they cross at large angles to one another, and would form the dislocation BA + DC (a<100>), which if formed would be an effective Jog in both planes. In materials which have low stacking fault energies, the product dislocations In reactions numbered (ii),(iv) and (vi) may split into partials which would increase the effectiveness of these barriers. The most effective barriers to cross-slip are formed when dislocations on two slip planes interact to form partials which are extended on each slip plane, and which are connect ed by a sessile "stair rod'1 dislocation located along the Intersection of the two planes (1) as shown in Figure 86. This type of barrier includes the Lomer-Cottrell (58) bar rier as well as several other effective hardening configu rations of dissociated dislocations described by HIrth (57). Once a barrier is formed In the glide plane, subse quent glide dislocations emitted from a dislocation source will pile up against the obstacle, each at some distance from one another. It has been calculated (59) that at large distances from the obstacle these piled-up groups of n dislocations should exert the same stress on the obstacle as one dislocation with a Burgers vector of nb. However it has also been shown (60) that this configuration is only 293 Stair rod dislocation which is sessile on both slip planes Figure 86 View normal to the intesection of the 111 primary and secondary slip planes. Dislocation dissociation provides a strong barrier to dislocation motion. effective in producing long range stresses up to a value of n = 5 for ductile face-centered cubic metals, since greater values of n would give a stress which would exceed the elastic limit, and hence relaxation of the piled-up groups occurs as soon as the stress approaches a value in the order of the elastic limit. This relaxation in Itself, by interaction with the secondary slip dislocations can produce a stable configuration not easily destroyed, and therefore becomes a larger obstacle to further slip. If the relaxa tion of the long range stresses associated with the piled- up dislocations is accomplished by the building up of an average Burgers vector of -nb by emission of dislocation loops and interactions with secondary slip dislocations, the relaxed group will cause little hardening, but the large cloud of compensating dislocations which have increased the local dislocation density will harden the material, c) Dual slip in deformed crystals Curved dislocation lines were observed In all crystals of deformed germanium; the dislocations were gen erally oriented normal to one of the slip directions Indi cating that the glide dislocations were of mixed or edge Burgers vector character. These observations are similar to those of Alexander and Haasen (12), and contradict findings by Holt and Dangor (18), and Booker and Stickler (19) who observed primarily straight segments In deformed germailum. The few straight segments observed were found in small 295 hexagonal arrays, such as is shown in Plate 28, and were oriented close to a low index direction in most cases* The crystals deformed to low strains such as specimens C-26, C-29 and C-*J2 strained *1.5 , 6.2 and 5.6%,respectively, ex hibited many of the characteristic features of easy glide, such as dipoles, edge-oriented dipole-Jog pairs (Plates 27, 22 and 32), and groups of glide dislocations roughly paral lel to one another, and perpendicular to one of the slip vectors (Plate 28). The latter groups included dipoles, and al3o what appeared to be dislocations that may have been grouped as predicted by Taylor (50)- The latter were not ex pected to occur to a large degree, except in materials with a large Peierls-Nebarro energy, but germanium which is expected to have a larger Peierls-Nebarro force (*0 than most metals, would tend to allign dislocations along low index directions over greater lengths of dislocation. Definite examples of slip polygonization or double kinks were not observed at the lower strains, although the asterism tails observed on diffraction spots in the Laue x-rays photographs, which were parallel to the deformation direction, indicates that some of the kink bands and slip polygonization expected in specimens deformed in compres sion did occur within the specimen. This was consistent with the observation of bending of the slip traces and kinking on the (Oil) face of the dual slip crystals. 296 Small hexagonal networks were observed in specimens C-55 (12.6%) and C-37 (25.*U) which may have been of the type discussed in section I.P. They are shown in Plates 26, 27, 28, 29 and 43. The banding or braid formation which increased with respect to dislocation density with increased strain remains to be characterized. The banding could have been a result of walls of dislocations of opposite sign meeting to form crude deformation bands, as described previously. Alter natively they could be examples of slip polygonization since they are oriented normal to the net deformation direction. The tangles can also be explained by the occurrence of secondary slip on other {ill} planes which would produce bar riers to primary slip which is also feasible because most of the high dislocation density tangles were oriented roughly parallel to <110> directions. In specimens C-38, C-42, C-4o, C-36, and C-37, strained 16.0, 5.6, 16.0, 25.4, and 8.6£, respectively, it was especially noticeable that at the higher strain rate (in Plates 42 through 50) the densest banding was oriented nor mal to the deformation direction. This along with the observation of many dislocation loops and very short lengths of dislocation with abrupt changes in direction has been shown to be an Indication of the occurrence of secondary slip (61 ,12), in this case even at very low strains in specimens of set II. This supports the observations made 297 in section IV, where it was suggested that the higher strain rate favored the initiation of secondary slip during yield ing. If some secondary slip had occurred during yielding, this would represent the primary hardening mechanism In set II crystals. The observed pile-up of dislocations at tangles par allel to the trace of the secondary slip plane could have been enhanced by a tendency toward kink band formation and slip polygonlzation, which would partially relax the inter nal stresses In regions of cross-slip. The dislocation lines In specimens from set II did not favor any particular slip direction, which indicates that both dual slip direc tions were equally activated. However, the overall harden ing process appears to have been dominated by the forest interactions with secondary slip dislocations rather than by glide Interaction of the two dual slip vectors. This is consistent with the observation of similar work-hardening rates In both single slip and dual slip specimens strained at the higher rate. At Increased strains the dislocation density of the tangles normal to the deformation direction Increased, but the average separation between tangles remained at about 2 to 3 microns for strains of 5*^ (C-^2) to 25? (C-37)* In specimens C-37 the areas between the braids did include a small number of long dislocations in various orientations which were probably generated after the dislocation tangles 298 had developed. The tangle observations Indicate that little additional secondary slip was activated on additional planes at the higher stresses, up to 25% strain. The observations of work-hardening in specimens of set II deformed at the higher strain rate, can then be explained by the initiation of secondary slip during yield ing and subsequent restricted primary glide, which produced dislocation Interaction between primary glide dislocations and secondary slip segments which resulted In the production of sessile Jogs and subsequent pile-up of the glide dislo cations, The observation and analysis of many dislocations with Burgers vectors normal to the (111) primary plane in the annealed crystals, which are discussed In the next section, confirm this. The dislocation clouds which are part of the dislo cation tangles will tend to reduce the long range stresses occurring In compression, and in so doing will produce a slightly polygonized structure as indicated by the x-ray patterns. Specimens C-26, C-29 and C-55 which were strained *1.3, 6.2 and 12.6%, respectively, at the lower strain rate exhibited a dislocation structure different from that found in set II crystals. As Is shown in Plate 23 the disloca tion tangles tended to form parallel to other <110> direc tions as well as normal to the deformation direction (parallel to the trace of the secondary slip plane). In the tangles not parallel to the trace of the secondary slip plane, there were not as many short dipoles, dislocation loops, or sharply curved small dislocation segments which would suggest that these tangles were not direct results of a secondary slip forest interaction. Most of the disloca tions were normal to one of the primary slip vectors and often parallel groups of primary glide dislocations were observed to be piled up at a tangle. A good example of this is shown in Plate 28 for specimen C-55. The work- hardening of set I crystals up to strains of 125E, at least, was apparently caused by glide dislocation pile-ups at tangles which consisted of mainly glide dislocation inter action products. Arrays in Plates 23, 26, 27, 28, 29, and 30 for specimen C-55 showed that one slip vector probably dominated deformation on slip planes in close proximity to one another, although the formation of several planar hex agonal groups in Plates 27 and 29 indicated that the second dual slip vector was also active. That some secondary slip occurred in these specimens was evidenced by a small number of straight, relatively narrow, dense dislocation tangles parallel to the secondary slip plane trace, as in the right hand side of Plate 23. These tangles had very few primary glide dislocations extending through them. It appears that hardening In stage I for crystals of set I was caused by two processes; primarily that of glide dislocation interactions and long range elastic stress field interactions, and secondarily that of glide disloca tion interaction with forest dislocations on the secondary slip plane. The appearance of secondary slip and sessile dislocation formation at low strains was also observe d by Alexander and Haasen (12) in crystals deformed in tension at a strain rate of 5 x 10-5/sec, in an easy glide orienta tion, even though none should have occurred according to a critical resolved shear stress law, because their crystals were oriented for single slip only. The slip traces of the secondary dislocations were spaced approximately 5 to 10 microns apart on the primary slip plane, and the spacing did not decrease until stage II hardening occurred at strains of 24? or more in their work (12). The spacing be tween secondary slip traces for the set I crystals was approximately 4-8 microns. Alexander and Haasen (12) ob served a work-hardening rate of about 5-6 kg-mm/mm^ in stage I up to a strain of about 24? as compared to a work- hardening rate of about 7.0 kg-mm/mm^ for specimens of set II and single slip specimens deformed at the lower strain rate, up to strains of about 20?. This suggests that the hardening In stage I was at least partially a function of the strain rate and therefore indirectly a function of the amount of secondary slip which occurred during yielding. Many investigators have observed the occurrence of secon dary slip before the compressive or tensile axis reached an 301 orientation favoring secondary slip (12,17,51,62). This fact, as well as observations by others (51) of an incuba tion period of continued easy glide after secondary slip had formed slip barriers, suggests that secondary slip in the dual or single slip specimens of set I could have been initiated at an even lower strain than 19%. Because specimens deformed at the lower strain rate for both single and dual slip had similar values for the work hardening rate in stage X, for similar increments of strain, this author believes that the effect of a second dual slip vector in easy glide is relatively small at lower strains as compared to single slip. However, the work-hardening rate in set I crystals oriented for dual slip was much greater beyond 2056 strain than that in the single slip specimens. No electron micro scopic observations were made of specimens C-25 (19.**%) and C-27 (32.0%) in the as-deformed condition, but assuming the increase in the work-hardening for the single slip speci mens was caused by activation of secondary slip; i.e., an increased number of Jog barriers to primary slip, then the greater increase in the work hardening for the dual slip specimens must have been related to the dual slip which occurred in stage I, or which continued to occur through stage II. When the micrographs of the dislocation structure of specimen C-55 strained to 12% are compared with the dis location structure in a specimen (number 25) strained In tension to 13% in easy glide in the work of Alexander and Haasen (12) , the main difference besides the orientation o f the secondary slip traces, was the formation of the two dimensional structure in dual 3lip such as is shown in Plate 2*J. The structure in the single slip specimen consisted o f bundles of long parallel edge dislocations with essentially no two-dimensional structure. If it is assumed that strain ing either the single slip specimen (C-101, C-102) or the dual slip specimens of set I, 5% beyond the initiation o f stage II would initiate the same amount of secondary slip, the cellular network of glide plane tangles would probably cause considerably more Immediate hardening of the crystal, than would the long bundles of dislocations in the easy glide specimen. The greater Increase in stage II hardening for dual slip as compared to the single slip in this work can be explained in terms of the difference in testing techniques* In this experiment the specimens were deformed in compres sion, as compared to the work of Jackson (39) in whose exper- ments face-centered cubic crystals of silver were deformed in a dual slip orientation In tension. The deformation axe3 for dual slip and single slip In tension will move as shown in Figure 87, from T(d) to T(d)' and T(s) to T(s)*, respectively. Similarly the rotation of the deformation axis for dual and single slip in compression will rotate from C(d) to C(d)* and C(s) to C(s)1, respectively. As can 303 110 Ck O- 111 Figure 87 The deformation axis for dual and single rilip In tension w ill move from T(d) to T(d)' and T(s) to T(s)*, r e s p e c tively. Similarly, the compression axis w ill move from C(d) to C(d)* and C(s) to C (s)f , respectively. 304 be seen from the stereographlc projection of the (111) plane, the active secondary slip planes will differ in the cases of tension and compression. If the primary (111) slip plane is defined as plane (d) in the Thompson's tetrahedron (Appendix A), and if the active slip vectors are each oriented to be the active vector BA(d), then for the cases of dual or single slip, in tension or compression, the type of secondary slip-primary slip glide barrier formed for each slip vector can be determined. The primary slip vec tors, as labeled in Figure 87 for single or dual slip in com pression or tension are the a/2[XlO] or/and the a/2 [101]. For a/2 [110](111) single slip in compression the secondary slip vector will be the a/2 [110] (111) and for a/2 [101] (111) it will be the a/2 [101](111) because during deforma tion the compressive axis will rotate towards the [111] pole, until it reaches the [010]-[011] or the [001]-[011] boundary. At that point both primary and secondary slip vectors, as described above, will be favored. The two coplanar secondary slip vectors a/2 [110] (111) and a/2 [101](lII) each would interact with its respective primary slip vector as a dislocation with the Burgers vector DC(a) would interact with the primary slip \ector BA(d); the product of the interaction is a strong glide barrier (Jog-jog), as is shown in Figure 85. Dual slip in compression would then activate two secondary slip vectors which would interact with the two primary slip vectors to 305 give two barriers of the type associated with DC(a) (strong jog-jog) and two barriers of the type associated with DB(a) (jog-jog)» or four strong "jog-Jog” combinations which would provide very effective hardening barriers to any primary glide. For single slip in tension, the secondary slip vector occurring with a/2 [110](110) primary slip would be a/2 [001](111), and for a/2 [101](111) or the [001]-(lII) boun dary, respectively. At that point secondary slip would occur in the [011](111) and [011](111) systems. In each case the types of barrier formed would correspond to the Interaction of the BA(d) vector with a secondary slip vector DA(b) and a "Jog-jog" type barrier would result. However in dual slip in tension, the hardening would only be pro duced by one slip vector, the [011], on two different planes, or effectively there would be produced two barriers of the type Jog-jog (DA(b)) and two weak barriers of the type Jog-kink (DA(c)). Therefore in tension the Increase in the number of strong glide barriers formed by secondary slip in going from single slip to dual slip Is one to two, whereas the increase in the number of strong barriers In going from single slip to dual slip in compression is one to four. It is believed that this is the reason the work- hardening rate in stage II for dual slip in specimens from set I, is greater than that observed for single slip under 306 the same conditions (Figure ^9). This reasoning should be valid for specimens of set I even if one primary slip vector did dominate during deformation, because the study of slip trace orientations, Laue back reflection x-rays, specimen dimensional changes, and electron micrographs all indicate that dual slip did occur to some degree within the crystals. Further evidence is presented in the next section (section 2)which includes the electron microscopy of annealed speci mens, and in which are shown many observations of hexagonal dislocation arrays which also indicate the operation of two slip vectors during deformation. F. Dislocation Structures in Annealed Crystals This section Includes a discussion of the dislocation structures of annealed crystals of set I (section 1), set II (section 2) and of the symmetrical and unsymmetrical nodes found therein. Section 4 is concerned with the observed polygonization of dislocation lines In specimens C-55, strained to 12.5^, and annealed at 750°C, which Is related to the high Peierls energy along the <110> and <112> direc tions. Calculations are made of the Peierls energy In these directions. The last two sections include (section 5) the analysis of the probable Burgers vectors of "stranger" or forest dislocations which are observed within symmetri cal dislocation node arrays of set II, and (section 6 ) a discussion of observations of apparent node dissociation. (1) Annealed crystals— set I In crystals of set I annealed at 850°C the symmetri cal dislocation nodes were found to decrease In average spacing and Increase In number per unit area as the strain increased. This can be considered as a direct consequence of a larger number of activated glide dislocations at higher strains, but the electron micrographs of the dual slip specimens in the a3-deformed condition revealed very little interaction such as to yield hexagonal arrays. The majority of the dislocations were within the dislocation tangles. Therefore the resulting hexagonal nodes resulted not directly from crossing grids of dislocations, but rather indirectly from the operation of two different slip vectors. During deformation these systems caused a rotation of the (111) slip planes towards the compressive axis, and also caused the compressive axis to rotate away from the [011]-[111] boundary because of the inherent metastability (39) of the axis on this boundary. If adjacent slip planes favored one of the two slip directions in different degrees, the result would be different amounts and directions of net slip on each plane, and consistent with this, a different amount of rotation normal to the [011]-[Ill) boundary, for each plane. If this did occur, annealing would reduce the misorientatlon between the two planes to a planar twist boundary, with coplanar hexagonal three-fold nodes consisting of dislocations with coplanar Burgers vectors similar to those observed in crystals from set I. The number of boundaries as well as the amount of twist associated with a boundary would vary to an extent dictated by the amount of previous strain. The observed slip traces on the (Oil) face for dual slip crystals when compared with slip trace on the (112) face of single slip crystals revealed that the slip traces for single slip were finer, i.e., more closely spaced with less relief or smaller slip step heights. These greater slip heights and undulations on the face which was oriented normal to the net slip direction in the dual slip specimens indicate some local variation in the net slip direction from one slip plane to the next. As shown in Table 15» the angle of twist associated with the symmetrical hexagonal dislocation arrays did increase with strain, which was also true for crystals deformed into stage II. All the nodes observed In set I with the. exception of those in specimen C-55, were symmetrical, with node angles within 3° to 120°, By annealing within 100°C of the melting point, a state of quasi-equilibrium should have been attained, by a mechanism Involving either glide dislocation rearrangement or thermally activated climb. This would be one reason why none of the higher energy unsymmetrlcal node configurations were observed. The specimen which exhibited the most uniform and extensive symmetrical hexagonal dislocation networks was C-25. This crystal was strained 19%„ to the end of the 309 stage I work-hardening range, and therefore would suffer little secondary slip to disrupt the dual slip action on the primary slip plane. Another reason for the larger prepon derance of networks, is related to the fact that this was the only specimen to the end of which a lubricant (moss) was not applied before deformation. The result was that the specimen did not show much twisting normal to the compres sive axis, although there was a considerable amount of lateral bulging in the [811] direction. This tended to con strain the compressive axis to rotate along the [Qll]-[111] boundary to a greater degree during deformation, than for other specimens, and thereby reduced the net amount of slip plane rotation normal to the [211] direction. Specimen C-27 strained to 3 *2SC into stage II, showed evidence of secondary slip in that the hexagonal networks were reduced In overall size and in the average dislocation node spacing, and various configurations of dislocations were observed which contained dislocations which had not Inter acted (Plates 67 through 7*0. Plates 69 and 70 show some small hexagonal node arrays adjacent to sets of parallel dislocations which end along a trace of a <110> secondary slip plane. This apparent dislocation pile-up Is ascribed to forest Interactions between glide dislocations and secondary slip dislocations. Evidence of the operation of a secondary slip plane, and the resulting low angle boundaries between adjacent areas of the crystal Is shown 310 In Plates 79 and 80, which are views of the same crystal area before and after the crystal was tilted about one degree in the electron microscope. The boundaries are parallel to the trace of the secondary slip plane, and represent (i) a decrease in the long range stresses created during deformation, and (ii) secondary slip which would also relieve stresses, and in relaxation of the piled-up glide dislocations, create clouds of dislocations which would on annealing, result in a localized band of high dislocation density. 2. Annealed crystals— set II Crystals which were deformed at a higher strain rate, and annealed at a lower temperature (those of set II) ex hibited some marked differences in dislocation structure as compared to crystals from set I. The primary differences in the crystals of set II were: (i) the higher strain rate pro duced a structure which included some secondary slip at all strains, and (ii) the annealing temperature (700°C) and time (24 hours) were 150°C and 12 hours lower than the annealing parameters for set I crystals. This yielded dis location arrays which had not reduced their energy to the degree of those in the annealed crystals of set I. This combination of secondary slip, and lower annealing tempera ture and time produced differences in the annealed struc tures, such as; (i) smaller more irregularly spaced hexagonal arrays, 311 (11) dislocation arrays with symmetrical nodes of widely varying node angles and wide ranges of orientation, (ill) dislocation arrays with non-coplanar Burgers vectors, anc twist boundaries which contain "stranger" dis locations (with Burgers vectors not coplanar to the (111)) extending through them, (iv) lozenge-shaped networks with little interaction between the crossing dislocations such as to produce a third dislocation segment, and (v) unsymmetrical nodes of which one was analyzed. The annealed structures of specimens C-42 and C-36 (strained 5.6JC and 8.6JC before annealing) exhibited no regular arrays, but rather some curved dislocations, and smaller straight segments which often ended in a three-fold node configuration. The apparent decrease in the disloca tion density was small as compared to annealed specimens from C-26 strained to 4.3JC and C-29 strained to 6.2Jf. This seemed to indicate a state of metastable equilibrium did exist, in which some local relaxation of higher energy con figurations had occurred, but the lower temperature and time of annealing did not permit large scale dislocation rearrange ments, which would include some cross-slip if secondary slip had occurred to an appreciable degree. The parallel dislo cations in Plate 8l are probably of a mixed character be cause they lie in a direction approximately normal to the net deformation direction. In order for the dislocations In 312 this type of arrangement to lower their energy by decreasing their respective values of /3 , large rearrangements would be necessary. A similar structure including small straight dislocation segments and curved dislocations was observed in specimen C-38 strained to 16.0% and annealed for 6 hours at 700°C. The specimens C-38 which were annealed at 700°C for 2*1 hours exhibited small groups of hexagonal dislocation arrays containing symmetrical nodes which varied in crystal orientation from 6° to 2 *1° from a screw orientation, and dislocation node spacings which varied from 0.12 to 0.30 microns. In every case the surrounding crystal areas con tained dislocations of widely varying orientation, length and Burgers vector indicating local variations in the effect of annealing upon the dislocation structure. Similar dis tributions of symmetrical dislocation nodes were also ob served in specimens C~*13, C-37 and C-39 strained to 23.8%, 25.*1% and 25.6% before annealing. As the strain increased the dislocation arrays were found to Increase in number while the size and spacing between nodes decreased. The arrays of symmetrical nodes were very Irregular In that the orientation of the nodes often varied 10 to 20° across one array, even though the Burger vectors remained the same. An example of a drastic change in the node orientation is shown In Plate 103 for specimen C-37. The two sets of nodes were rotated 25° across the array. This may have represented a 313 tendency for the segments to align themselves along <110> and <112> directions because of the expected higher Peierls-Nebarro energies along them, but other arrays con tained variations in the angle/3 , of the dislocation node segments, of 3° to 15° over the whole range of values of (0 to 30°). This as well as the variation in the spacing between nodes across the arrays also indicated a metastable equilibrium in these crystals. The variation in the node orientations is shown in the histogram in Figure 66, in which the screw orientation is the only orientation defin itely favored. 3. Unsymmetrlcal nodes The primary reason for decreasing the time and temper ature of annealing for crystals of set II, was to increase the probability of obtaining higher energy dislocation arrays, more specifically ones which might include unsyra- metrical nodes. The only node arrangement which was analyzed for its Burgers vector content and which included an unsymmetrlcal node pair is shown in Plates 120 through 12 3. The node described by a,b, and c, was of the type 3(a) as depicted In section I.F. The value of a measured from the photographs in Plates 122 and 123 wa3 82/2 » Ul° * a(3(a)). Using the modified Isotropic equations described in section VII.C. the predicted angle of a for this type of node was ^1.5°. The agreement between the value predicted by the modified Isotropic equations was the best when com 311* pared to values of a predicted by the other values of KO). (Figure 81) which were: (i) for an isotropic C3),- a= M.2°, (ii) for using Voigt average elastic constants, a » 4^.7° and (iii) for using Ruess average elastic constants, a “ ^3*7°. The best value was obtained by considering the anistropy of the crystal. Even though the method was not an exact one, calculations of the type used to arrive at a value of a (unsymmetrlcal) must include the anisotropy of the crystal involved. For example the difference in germanium between the modified isotropic value and the isotropic value of a was 2.7°, not a large difference, but in other systems, such as in iron, the differences may be much greater as shown by Chou (30), Although the reliability placed upon measurements taken from only one node cannot be great and taken as abso lute proof that the assumptions used in deriving the modi fied isotropic variation of KA(0) as a function of orienta tion are exactly correct, the fact that the observed node angle of ^1° is lower than the predicted angle of 41.5° lends further credulence to the use of crystal anisotropy where possible. The three dislocation segments of the unsymmetrlcal node, as shown in Plates 122 and 12 3 remain straight within a radius of about 0.30 microns, but beyond this value the dislocation lines bend towards lower energy orientations. This cut-off radius of 0.30 microns is consistent with that assumed in calculations used to derive 315 a value of the Peierls-Nebarro energy In section VII.E4. The connecting nodes shown in Plates 120 and 121 are about 0 .7 and 0.9 microns away from the unsymmetrlcal node, hence the effect of these nodes upon the unsymmetrlcal node singles should be small. The node at the upper left-hand corner of Plate 123 would correspond to an unsymmetrlcal node of the type 3(b), however, the short segment of Burgers vector c was only about 0.1 micron long, which did not permit under the oper ating contrast effects, a good measurement of its line direction. The angle between the extension of the segment with a Burgers vector a° and the adjoining segment with a Burgers vector b_ is about a= 78° * as compared to a predicted theoretical value from modified Isotropic theory of 81°. Because the third segment was sosmall it may have disrupted the equilibrium conditions at the node, hence the value of 78° may be unreliable although It agrees well with the pre dicted value of 81°. From the contrast effects it appears as If the short segment has Interacted with a dislocation on another plane, and Its continuation has abruptly changed to a direction leading out of the crystal. The lack of more than one observation of nodes with angles of 2a close to 83° or 162° in configurations as depicted in Figures 77 and 79 indicates that even In the partial equilibrium state of the annealed crystals of set II, the unsymmetrlcal node Is Indeed a higher energy configur ation not expected to be observed In well-annealed struc tures. The long range stresses associated with the unsym- metrical node (1) as well as the fact that in their equilibrium configurations only one of the three disloca tions of an unsymmetrical node will lie along a low index direction with a high Peierls-Nebarro energy tends to diminish the probability of their existence as compared to a symmetrical node which would have little (£=30°) or no ( 0=0°) long range stresses associated with it. Also, for the latter nodes in an equilibrium configuration, all three dislocation segments will lie along low index directions in Peierls troughs. **. Dislocation line orientation ahd Peierls energy calculations Most of the dislocation lines in the well-annealed specimens of set I were straight, as was observed by Arts et al. (21), Aerts et al. (20) and Holt and Dangor (18) for annealed germanium crystals. The orientation of the straight segments which were not a part of a regular symmetrical network of dislocations, were usually aligned close to a low index direction and within 30° of a screw orientation. The exceptions to this were specimens from C-55 which were annealed at 850°C in which the curved dis locations were observed to run normal to one of the two slip vectors between the dislocations which were oriented parallel to the trace of the secondary slip plane. The interpretation is that the set of dislocations labeled with arrows in Plate 64 were active (glide) edge dislocations during deformation and were strongly pinned by dislocations in the secondary slip plane, so that the curvature is the result of the lowering of the self-energy of the disloca tions by partial alignment along a low index direction (assuming a high Peierls-Nebarro*s energy), or by rotation such as to reduce the angle 3 between the dislocation line and its Burgers vector which would also reduce the self energy per unit length of the dislocations. The same ex planation can be applied to the array observed in Plates 61 and 62 in which the group of parallel dislocations are also parallel to the trace of the secondary slip plane. In the same foil, three-fold nodes were observed with short lengths of straight dislocation lines (Plate 63). The various curved dislocation segments observed in specimen C-55 which was strained to 12.6?£ and annealed at 750°C are be lieved to be the result of three different forces that will affect dislocations In an array of three-fold nodes. The first is that force resulting from the interaction of the three dislocations that meet at a node in that the three segments meeting at a symmetrical node, will tend to main tain an orientation such that each dislocation segment is straight and will bisect the angle between the opposite two segments. This force, resulting from the equilibrium conditions at a node as defined by equation *1, will however decrease quite rapidly as the distance from the node in creases. The second force is that of line tension, in that a dislocation line will tend to maintain an orientation and configuration which minimizes its length and therefore its free energy, i.e., a dislocation line will act like a "stretched string." The third force is that related to the anisotropy of the crystal, and to a possible decrease in the energy of a dislocation line if it is oriented along a low index direction, i.e., in a Peierls trough. As calcu lated in section VII.D the dislocation line will lower its energy if it minimizes the angle £ between Its Burgers vector and its line direction. This contribution will be important especially if a dislocation line is greatly rotated from an orientation In which 0. Otherwise the tendency of dislocations to align themselves along a <112> or <110> direction will be a function of the Peierls- Nebarro energy. Hornstra (4) has shown by examination of the bonding characteristics of the dislocation core in diamond cubic structures that germanium should have a higher Peierls-Nabarro energy along the low Index directions because rotation of a dislocation cut of a Peierls trough by kink formation would Involve the breaking of bonds. Observations by Dash (63,7*0 of a strong alignment of dis locations along <110> directions in silicon, as well as similar observations by others (18,19,21) in germanium indicate that the Pelerls-Nebarro energy is high in diamond cubic materials. Values of the activation energy associ ated with double kink formation for germanium (65) are, as calculated from internal friction measurements,about 1.1 eV (50) which i3 an order of magnitude greater than that observed for face-centered cubic metals gold, silver and aluminum. This activation energy may represent the oscil lation of a dislocation line from a straight orientation to a double kink and back, but may also include another com peting process such as the unpinning of dislocations by point defects, so that the calculated value of 1.1 eV is only another possible indication of a high Pelerls-Nebarro energy in diamond cubic structures rather than a proof. Hence the curved dislocations in specimen C-55, reflect the competing crystal forces upon the dislocation line orienta tion. All three forces will tend to straighten the disloca tion lines, but the first two are such that the direction of resulting dislocation line is not dependent upon the crystal orientation, but rather on the dislocation config uration. The third force neglects the configuration and reflects only the crystal orientation of the dislocation. Specimen C-55 annealed at 750°C also provided several in stances of dislocation arrays in which the dislocations were exposed to these forces in competition with one another. Examination of the resulting quasi-equilibrium dislocation configurations yields an approximate value of 320 the Pelerls-Nebarro energy relative to the self-energy of the dislocations. Two arrays from C-55 have been enlarged (Plates 12*1 and 125) and are now analyzed. These calcu lations involve an energy balance between the line tension and the Pelerls-Nebarro energies, so that an assumption must be made concerning the first force and its effect upon a dislocation line which sends in a three-fold node. In each case the dislocation segment is assumed to be free of the effect of the node requirements at distances greater than about 0.30 microns. In Plate 124, the dislocation line labeled X curves radically from the orientation favored by line tension alone. The quasi-equilibrium configuration has been approx imated by a geometrical configuration ACDB which will be used to calculate Its energy. This configuration is de picted In Figures 88 and 89. The sides AC and DB are paral- let to the B and c directions, respectively. The Burgers vector of the dislocation line Is the a/2tl01]. Hence 6(DB) =0°, 6 (AC)* 30° and the measured value of ft(AB) * 0(CD) * 17.5°. The following are the measured parameters shown in Figure 89: L(AB)» 4.10 microns L(CG) » 0.70 microns L(AF)* 3.04 microns L(GD) * 0.40 microns L(FB)* 2.06 microns L(EF) « 0.65 microns L(AE)* 3.11 microns L(GF) * 0.50 microns (116) L(AC)* 2.395 microns 6(0) - 17.5° L(FB)» 2.16 microns 6(30) * 12.1° L(DB)= 1.66 microns L(Alr) * 2.34 microns L(DB*) - 1.58 microns Plate 124 Specimen from C-55 deformed 12*6% and annealed at 750°C* Figure 88 Reproduction of Plate 12 4 showing geometrical approximate configuration. <$13 L{ AB) L ( AF) L(FB) L{ Ac') M O B 1} 9 (30) VO^ ©* _^,eA . o . ^ ? Angular gcomotricax uratlon in Plate 12k» dicular to segment AB, s?.’i The total energy of the line- segment AB is: W(AB) x'- B(AB) - l.'(AE)/L (117) W (AB) ^ 'J.10 • V;(AB)/L which represents the energy of the dislocation lino if onDy line tension governed its orientation. The energy of thc equi libi'iuni configuration that was approximated by the geometrical configuration is: (118) W(ACDB) = L(AC) *VJ(AC)/L + L(CD) • YI(CD)/L + L(DB) * V/(DB)/L = 2.395 W (AC) /L + 1.18 V;(CD)/L + 1.66 W(DIi)/I. If it is assumed that: (119) W(AC)/L + W(CD)/L = W(DB)/L then, (120) W(ACDB) = 5 . 235 • VJ/L This provides one with an isotropic ratio of energies; ( . W(ACDB) 5.235 _ , 0 „ 7 C121) TKAH5T = 57T0U - 1-027 The energy increase of 2.7% can be called an isotropic esti mate of the Pelerls-Nebarro energy associated with changing the orientation of segment l.(AC’) £-17 50 to L(AC) £ = 30° and L(DB') = 50 to L ( D B ) ^ =q. If the segment L(CD) Is excluded because it has essentially the same orientation it would have had in the segment AB, the actual ratio may be redefined as: y^ s I W orr , _ = -v:(0r)/l, + L(AO)-V.’CAC:)/!, n (122) Wtomvi on J-'O.'h.) (cc^O 7 . 5° ) V; Ul A V ' t ^ 00:02 ■1 ’ ] j ? . xO'iV.'CAO/i: + 3.66 v:(p-,)/b _ n r, 't ti = 2 ( a ::0 7 i / + 3.52 v: ( a 0 /j, A 3-^f i l j f-1-o an e in t h r energy rrpn- r cn t s the decrease ir: the Pelt'rIs-Nob;jrro energy of the two scgments which werc rotated, into a <13 G> or < 1 1 ? > &li gnmcnt. liov:ovcr this calculation does not include the effect of crystal anisotropy, and the change in the energy per unit length of the dislocation liner, as a function of their respective values of (3. Using the values of the energy factor K/t(/3) as derived in section VII. C, and shown in Figure 7?> the change in the energy for a change in the dislocation orientation from a line tension configuration to an equilibrium configuration can be cal culated more exactly. Assuming, no change In the value's of the term (b^ In ) v.’ith £ , and if from Figure 72, bn ° W(0 °) = 5.20 (b2/bn In -) x 10" dyncs/cm2 (123) W(30) = 5.65 ( " ) W(17.5) = 5.36 ( " ) then WeqTn „ 2.393(5.65)+ 1.66 (5.20) = , n,R {12M> Cnsion'lnlo. ------1 ‘°j8 Considering the anisotropic variation of the energy with j8 , the total energy is increased by 5.82 In assuming the "equilibrium" configuration instead of the"line tension" 320 configuration. If it is a^su::.od thct the- equii brl urn con figuration is genuinely in complete equl litriurn, then the 5-88 increase in energy Must equal the decrease in the Peicrls-rJcberro energy for the- segments AC and Db ty align ment along the increase in the elastic; energy between the equilibrium conf J gurat i on and the hypothesized 3 i ne tension configu^at ion will represent a difference in the Peierls~Nabarro energy for the two possible configurations in Plate l?Jl, which is reflected by a cheapo in' the effective core radius rc, in the term (In H / r 0 ) . For the configuration in Plate l?fl the ratio of energies for segments AC and AC" is; W(AC) _ Ka (AC) In (H/rD)Ac. L(AC) VTTXC j ) K A ( A 0 ' T In £ K/r cT) AC / L '(TcT T 5.65x10" dynes * crii~^ln( H / r 0 ) ( 2 . 398 )inicron.c; (1291 = ------— ------5 . 3 ^x1 0 ” dynes • cm”^ln ( H/i\r0 ) (2 . 3^0 )microns = 1.08't ln(R/ro) whcre. v = r°' I n T H / l ^ ) wllcro- r\ r— or there is an 8 .JJ£ difference in the elastic energy between the two orientations. Because the equilibrium configuration is favored over the line tension configuration, the total energy of the segment AC must be at least equal to if not lower than the energy of the latter. If the two configura tional energies arcmade to equal one another, then a limiting value of = r^ / r 0 can be obtained for a dislocation with a 0 =1 7 .5 ° as compared to a dislocation with 0 = 30 ° (<112>). 3.'-'7 UldVly) If i - - (l.oBn) V'o and if it Is assumed that (R/r0) = 1 0 the vr]i:e of K that i v, oh fa! no d is: (12f> 1:'<112> = I‘o/l’o = ° - E0 where ro' - core radius of a dislocation v;ith 0 = 17. 5 ° rQ - core radius of a dislocation v.'ith 0 - 3 0 ° <112>. A similar analysis for the sc rev: segment Dh v;hen co'uparccS to segment Db1 ( 0 =17.3°) yields a value o' , W(I)I0 _ , nic, AnIn */roR/r, (1?7) WCUJU - 1>01J TnR/^ro- (120) K,<110> - - 0.93 o where r ’ = core radius of a dislocation v:ith 0 = 1 7 .0 ° o r*0 = coi^e radius of a dislocation with 0 - 0 ° This is a lower litnit for the effective difference in the core radii between a screw dislocation and a dislocation with 0 = 17.5°. In this case the difference is reduced with respect to the <11?> dislocation because the screw orientation is also favored elastically over an orientation with 0=17.5°, In that KA (17.5°) is greater than K^fO3 ), whereas KA (17.5°) Is less than KA (30°) for a <112> oriented dislocation. 3 ? 8 A similar analysis of the dislocation curvaiyrc X-Y in Plate 1 25 yields a ratio of tho c:ncrj;ififi of the two straight segments AC/AC1 and DP/DU1 of: WC-VP.™).! . _ = ylAC^LjLCACl_+_W(l?D)/T.iLC])Iil = 1.0A8 ( 12 9 ) V: f T°115 1 ^ W(AO )/L‘L(AO) V*< Dr,’ )/ L ■ I( D H 1) which is a 20f lower difference then for the configuration in Plate 12 4. The ratio of VJ (AC) (li°) vf( AC*1”)’ K 1.109 yields a lov/er limiting 'value of: (131) \ - 112> = r0 V r 0 h 0 .7 0 , if I” = 103 where rQ ' = core radius of dislocation with 3 ” 11.9° rc - core radius of dislocation v.’ith 3- 30° Similarly for a screw segment DB, tl32> W r y - 1-008 which yields a value of (133) K* - 1=Li 0.97 o where r^ = core radius of dislocation with 3 = H* 9 ° rQ = core radius of dislocation with 3 “ °° From Plates I P 1! and 125 the net change in energy AW achieved by alignment along a <110> or <112> direction was -5.8# and -4.8#, or an average value of abo\it -5.3£, which 329 Specimen C-55 annealed at 750°C for 2U hours* The Burgers vector of the dislocation line Joining the nodes X and Y Is c. 330 corresponds to un r'0/r0 0,87, This v.ould correspond to an avorapt; decrease in the energy per unit length of V//Jj( r ) /cor, 0, wlu i’c b ~ 15°. Hence the average decrease in tht; fciioi'pv p r r unit length of dislocation line oriented alonf; < 1 10> or <112> wan about 5.4f, lower than predicted by clastic conr.ioerat:i one. . Labusch (66) has theoretically treated gcrrnan 1 urn by methods similar to that of DJetze (67) and obtains for the Peierls energy of a screw dislocation, 0.225 eV. Teichler (68), using known pseudopotentials of germanium, has cal- * culated the interaction between any pair of atoms in a diamond structure and a screw dislocation. He obtained 0.230 eV for the Peierls energy of germanium. Suzuki (6 9 ) obtained a value of about 0,265 eV. Assuming an energy of 5.6 eV for a screw dislocation, their predictions of a decrease in the self-energy per unit length, of *1.055 (66), 4,1£ (6 7 ) and 4.75S (68), respectively, are in good agreement with the value of 5.4£ derived from the experimental con figurations in Plates 124 and 125. The superposition of a 5-4# lower self-energy of a dislocation oriented parallel to cither a <112> or <110> direction on to the curve in Figure 72 showing Ka (jQ) , yields a curve which is shown in Figure 90, which includes the effects of the Peierls energy on the dislocation orientation in germanium crystals annealed at 750°C. uepsto o te eel togs n h eeg curve. energy the on troughs Peierls the of Superposition CM i Ka (B) b^/4 ln(R/r) lO^dynes u B 5.0 .8 S 5Ai 6.2 6.6 7.0 iue 90 Figure (degrees) P 30 60 90 331 332 It appears that the; 1’eIcrls-Mubar.ro energy does play an Important part in the alignment of die 1 ocation segments in annealed german3 urn. This is also evidenced by the fact that G 7 % of the; cVj a local- i on segments in the symmet rl cat th re o - f old nodes in annealed crystals of s o t 1 were or:i onte d witbi n 6° of either a <110> C^3-0C’) or a index direction as chov.’n in the histogram in Figure Gb. This is the most logical explanation for observing node segments oriented along the <112> direction, because this dislocation line direction has an anisotropic self-energy which is 8;i greater than the ctlf-cncrfyv of a screw dislo cation, and about a l\-G% greater energy than a dislocation line oriented between /3 = 10tJ and 20° in which range very few dislocations wore observed to exist in the crystals annealed at 8b0°C. 5. Stranger dislocations in Hex agonal networks It was also observed that as the amount of strain before annealing, increased the number of ’’stranger" dislo cations extending through the hexagonal arrays of set II, annealed crystals also increased, thereby decreasing the uniformity of the arrays. Most of the strangers did have Burgers vectors which were non-coplanar with the array, a direct result of the Increased amount of secondary slip which had occurred during deformation at the higher strain rate. For example in Plate 9 3 in a foil specimen from C-38, which the: s t r a np er dialoc as proven by VC'C toi deteiminntion to have a liurf.ort. vector other than a,b or c id is shown in Pi ate IP 6 which has been laheied with ror.poct to the throe coplnnnr Burners vectors and throe node internet\ ons which involved the stranper dislocation and the throe different dislocation segments with Burners vectors a,b_, and c. The probuble Burners vector of the stranger dislocation is b(str.) - a/P [101] which will piide on the (111) and the (111) sc con- i dary slip pianos. It is not an unreasonable assumption since the stranper dislocation could therefore p;l i do or cross-slip in a manner which would allov.1 it to obtain an orientation as shown in Plate 126. The interaction labeled (1) would be: (13J0 | [101] + I [110] -5 | [Oil] c' ( I l D a (111) b' (III) (111) (111) in which the resulting third segment with Burners vector b" is parallel to the [101] direction, and also parallel to the trace of the (111) plane on which the dislocation can p.lide as it is formed after interaction, i.e., its formation would not involve climb. The interaction labeled (2) would be: \ \ Trace of (111) plane Trace of (111) plane of secondary slip \ of secondary slip with possible Burgers with possible glide vectors of; ------dislocations of Burgers vectors; 1/2 C101] - c' 1/2 [110] - a* 1/2 [101] - o' 1/2 [Oil] - b \ 1 / 2 [llo] - a 1/2 [Oil] - F" P late 126 i (13b) -*7 I -1 0a :j + £ [Oil] - § [lie j c 1 (Til) b (ill) a f (ill) (ill) ' (ill) in which the third segment, a_' would be parallel to the LlOl] dlruction, and hence parallel to the trace of t\ie (ill) plane on which thin dislocation cun also glide, hence as in interaction (1) the formation of the third segment could occur by dislocation glide. The interaction labeled (3) would involve a reaction of the type: (136) | [101] + | [101] = a [001] C' (111) C (111) -010? (ill) -aicr- f The resulting a [001] vector is a highly improbable burners vector in germanium, although it has been observed in face- centered cubic metals (1). The u[001] burgers vector, if formed, could glide on either rl00] or -110' planes, hovicver the formation of the a[001] segment on any of these planes would require climb except for the (010) plane. Motion on the latter would require glide by both dislocations £ and c' on slip planes non-coplanar to the array in order to form a segment of Burgers vector a[001] which would also have a component of its direction normal to the plane of the array. Both the formation of a higher energy Burgers vector and the increases in length associated with its forma.tion 33(> v.'ould tend to prevent interaction to produce a third seg ment, v/hich jr. observed in Plate 1?6. St ranr;!.- r dislocations appear in most phot op raphe of arrays in crystals of set II, A similar'set of s t ran re r d3 s locat.i ons v:i Vh a Pur per vector b_’ ~ >>[0313 traverse the cip/strl in Plate 102 disrupting the cord anar dislocation array described by Burgers vectors a >li anc'* £. The third type of interaction in which the formation of a third dislo cation segment of Burners vector a <100> is not cncrpctically favored nay also explain' the observation of arrays such as that in l3late 116 in which three areas of widely differing node orientation and spacing exist. In the latter, stranger dislocations v.’hich run between the groups, X,Y, and Z may have caused a variation in the amount of slip within each area through local interactions which occurred either during deformation or during annealing, 6. Mode d1ssociation In only one dislocation array, in specimen C-39, was there observed a hexagonal dislocation network v/hlch appeared to have dissociated dislocation nodes. This net work is shown in Plates 119 and 127. The symmetrical nodes were rotated about 15° from a screw orientation with the exception of one or two segments which wore rotated about 30° from a screvr orientation; one is labeled by an arrow in Plate 127. Figure 67 depicts the relative intensities of 337 Ida PI*t* J £ Z __ Enlargement of Plata H Q . Magnification: 117000X Symmetrical nodea obaarvad In specimen C-39 which was annealed at 700*C for 24 hoars. 31-iO the- vurious di f f r ; - ciod c pots which v:c re opore it vo v :h o ii the photor.raph was taV.cn, Tn:i y fi gure in nljo\.n be c an sc appLa*-* ent disc oc 1 a t :> on of the notion in germanium hen been sh own to depend upon the open a. Li np. j'efh'ct.l on:^ (1)(2), as well as actual splitting of' the nodes into partial d j s i.ooat .1 oris . The question of whether the- nodes such a:: those shov:ri in Plate 127 observed in penman 1 urn are actually dissociated, has boon discussed by several authors, v.’ith little agreement. Hornstra (Jt) suggested solely on theoretical grounds that dislocations in diamond cubic perman!urn or silicon * could dissociate into partial dislocations. Aerts et a1. (20) investigated hexagonal networks in silicon sinple crystals v;hieh were twisted at 1200°C, and observed ex tended dislocations ribbos and nodes, which had ribbon widths of 8[i A° and node radii of 125 A° . From this in trinsic stacking fault enerpy of about 50 ergs/cm. v:ac cal culated. Hooker and Stickler (19) conducted similar experi ments and observed no dissociation. Then Arts et al. (21) conducted experiments that paralleled those of Acrts et al. using germanium single crystals, and observed "extended” dislocation nodes with an average node radius of about 90A°. They observed that the dissociated nodes only ap peared in woll-annealed crystals (annealed at 600°C for 15 hours), and concluded that the longer annealing times at higher temperatures was a necessity for their occurrence. Booker and Brown (22) and Shaw and Brown (23) studied 339 the contra-'it offsets at die location nodes in dl.air.omi c ub 1 c structwvK . bool:or and Brown concluded that the apparent Bp .1 It t i) jr. of dislocations and the go one t ry of the disloca tion node were artifacts depending upon the operating re flections during observati on and also on the depth of the dislocation node:; in the thin foil. They showed that in many casco, in which it appeared as if two of the three parti ale wore vis ah le, that these we re contract effects v:hich depended solely on the v£iri ous <220> spot;; operating in di ffrac t i on ; for example in silicon, for a screw dislocation oriented between two symmetrica.! nodes, the two configurations in Plate 128 arc* possible if the the crystal is titled several decrees between observations. This appears to be the case for the dislocation array shewn in Plate 129, in which an apparent ribbon of partial dislocations is observed which has a spacing of about 190 A". Booker and Brown (22) showed that in silicon an apparent dissociated node radius of 190 A° can be generated by contrast effects alone, and that the value of Bfnode radius)/r(ribbon width)) for con trast effect is not as high as the same ratio that lias been calculated for truly dissociated nodes (70). They c o n cluded that in silicon, and probably germanium also, because the dissociation of the dislocations into partials involved the creation of broken bonds for a screw dislocation, or awkward bond rearrangements in other 30° or 60° dislocations w the dislocations in silicon and germanium were not truly extended. Plate 128 Screw nodes in silicon. (220) reflection* s=0. Note double image of the screw dislocation with g * b * - 2.(23) 9 220 (220) reflection, s>0. Note the change in the double image and the region of no contrast at the nodes. 9 220 A three-beam image of a network in silicon. The two reflections are (220) and (022), with s>0. 022 2 2 0 Plata 129 it froa Plata ______69 ____ . Magnification: 134000X ric nodas in a acraw oriantat ion obaarvad in apaelaan C-27 Which waa artnaalad at 850*C for 36 hours. 3'i2 oh'.‘" wid brown (1’3) used a linonr c 1 a t :V c ity approx imation of' a throe-fold node, v;h.1 cb included the: s uperp os i- tion of tv:o anrpilai1 dis 1 ocatIors after Yoffe (71) to a p p r o x imate theoretically the contract effects -at a dislocation node. Their caloul atioii:- resulted in the sane cone Inal one as v;ore reached by booker and Urov.ui (2?) . They computed that a node vicv.'cd un be r the condition of six symmetrica] oper ating reflections would yield an apparent dissociation at an undi ssoci a te-d three--fold node fv;rn ch is sho.vn in Plate 130; conditions whi ch wore similar to the operating conditions under which the array in Plate 119 v j as photographe d. Plates 111 and 11? show a dislocation array which was photographed under tv;o different operating bright field reflections. The contrast effects make the dislocation nodes in Plate 111 appear to have two of three partials visible, and each node appears to have dissociated to the same decree. In Plate IIP the nodes appear to be alternately constricted and extended. It can be seen that small variations in the crystal orientation can lead to erroneous interpretations If the large dislocation contrast effects in germanium are not considered. Shaw and Brown (2 3) concluded that unextended nodes viev?ed under random or uncontrolled operating bright field reflections cannot be distinguished from truly dis sociated nodes, unless the apparent node radius of the truly dissociated node is significantly greater than one-half the contrast extinction distance. Plate 130 Theoretically derived dislocation node contrast, obtained using dynamical contrast theory (Booker and Brown (22;). In sj.dtt of the ovjdMicf.' that contrast cffecU- may well ex])l:j in the splitting arid node c31syociia.it :i on on l'ori:if;.ni ui;i and r-i 1:1 con, Amell ricx (2Ji) believe:; the work by Aerts ct ai_. (10) and Art s ot _el. (21) was si gnu ficent in that true node d 1 s s ocl at i on did occur1. This author believe:; that the nod or. observed in Plate 12 6 were not truly dissociated and that the apparent splitting and extension is caused by contrast effects in the crystals, because, (i) the reflecting conditions, as shown in Figure 67, wore unusual, and were also similar to those predicted by Shaw and Brown (23) to five an apparent node dissociation, (ii) in the fifty or more annealed foils which wore investigated in this study, only that one dislo cation array in Plate 126 showed any indication of possible node extension, and (iii) according to Arts ct al. (21) the extended nodes were expected to occur moi^c often in well- annealed crystals. Those extended nodes observed by Arts were in crystals which were annealed at 600°C for lb hours whereas all crystals in this study which contained hexagonal dislocation arrays were annealed at 700°C, 75Q°C or 850°C for tines of 2*1 or 36 hours, yet only one example of lfdis- sociation" was observed, in a crystal annealed 2*1 hours at 700°C. Further observation of the array in Plate 126 under various reflecting conditions would have proved one way or the other whether it was dissociated. The array was analyzed for its Burgers vector content by observation of 3 ’;rj the u-i n'jMcliou r.pol-s A and b wn:! C *’J ooyymita utth hurperu vectuf:; a and b_ were invisible; no indication of dissociation van observed unde;' these daub field conditions. i'hls autiior believes hove; w r tiiat until a thorough dark field il 1 *ins 1 nat * on study Is conducted on several dislo cation arrays that exhibit an apparent node- dissociation, t It e q ue s t i on i a n ot d o f 1.n i t e 1 y re s o 1 v e d . v i i i . cow cuts ions Fro:n the work in this dissertation as wo 11 as from related studies, the* following conclusions can be made: 1) Dual slip in compression favors the activation of tv:o slip directions on the primary glide plane. Continued deformation vrill bo manifested by continued slip in the two <110> directions, more so than in tension, because slip in each direction favors the motion of the compressive axis along the same <211> - <011> - <111> groat circle, whereas in tention they do not. 2) The yielding phenomena in germanium especially at higher strain rates, favors the premature activation of slip on slip planes other than that oriented for easy glide, thereby yielding higher values of flovi stress In stage I work-hardening. 3) The dislocations activated in easy glide are pri marily of an edge character, and are usually curved without particular alignment along low index crystal directions, in crystals of germanium deformed between 6l6°C and 6J19°C. D u r i n g annealing the dislocations tend to become straight and aligned along the low index directions. 4) The reliability of only one analysis of an unsym- metrical node is not sufficient to. conclude that the 3^6 Modified inotropic Kotbod, which too!-: into account the cry stal aniiui j'oj);; , accurately predicts valid equilibrium node an<;k::-; «. However the predicted node angle for uusymmcti*l- cal nodes of type 3a v;as l\ ] .3° v.’hi c h agrees v;o 11 with the observed angle of tl° +_ ]f . Vbo value of a predicted using isotropic elasticity v:as Vi. 2°; 8f lovmr than was observed. A general observation in tiiat the frequency of occur rence of an unsymmotiri cal node in diamond cubic structures (and probably face-centered cubic structures) is very lev;. 4 They vjore* not observed in regular networks * and only to a limited decree in other random dislocation configurations. 5) The formation of symmetrical nodes during anneal ing is greatly favored over the unsymmetrical nodes because the former (i) exhibit lov:er configurational and line ener- (ries, and hence Jiave lower Ion p. ranpc stresses associated wit}) them, (ii) all three of the node segments will lie in Peierls troughs. 6) The symmetrical nodes should be of type 1 (0=0°) or of type 2 (0- 30°) because; (1) both exist in a sym metrical configuration (2a- 120°) and consist of disloca tion segments with energy factor values, (K), 1 lj% and '(%, respectively below the K value associated with the average dislocation orientation of 45°, and (ii) both lie along either <110> or <112> directions in the {ill} planes. 7) The I'eit-rls trour.hs are auffie:1 ontly deep to dom inate the d.i :,3 oc;itIon orientation after anneal! nj -. The Peierls trouphc h; vu been calculated to be about of the yelf-enerpy of the- dislocation line. The effect is loupe enouph to nop.ale the chi fferonce in K value's for1 dislocations oriented v; 1thin 10° of the <11?> direction. 8) Node din::, oci alii on doe:: not occur in annealed permaniun crystals to a depree that the extension is visible in the electron i:iicroscopc. The problem of discern!np con trast effects from true node extension remains an obstacle to obtaininp conclusive evidence of their existence. APFK1IMX A THOai'SOM 'S ThlhAFKPhOb In dosordking d:1 s locot i onr in the f a c o~eontc i ■; .-d cubic lattice, or in the diamond cubic lattice which hcs a facc-centerod cubic translation lattice, the 1/? <13 0> Burgers vectors and (111] glide pianos are often defined using the notation introduced by Thompson (72). The basis of the no tation is the Thompson tetrahedron. The tetrahedron is. formed by joining the atoms of a 1/C unit cell of the face-centered cubic structure by straight lines. The faces of the resulting tetrahedron represent the four possible (111) glides planes, and the edges correspond to the six <110> glide; directions of the face-centered cubic structure. The atom at the origin is labeled "D" and the others are labeled ABC in the clockwise order as shown in Figure 91- If the tetrahedron is opened up at D, it can be folded out into the planar arrangement also displayed in Figure 91* The {ill} planes arc represented by letters a,b,c,d and the perfect 1/2 <110> dislocations arc represented by pairs of Roman letters BA, etc. The vector algebra for the addition and subtraction of the Thompson vectors is defined by the following set of 3*9 z * A one-eighth unit cell of edge length 1/2 of an fee l a t t i c e , showing the Thompson tetrahedron ABCD. (C) n (in) i (d) D D F ig u re 91 A Thompson tetrahedron opened up at corner D. Both the Thompson notation [(a) for glide plane, AB for Burgers vector of perfect dislocation, and A5 for Burgers vector of partial dislocation] and one possible set of Indices for the same planes and directions are presented. The notation [llCfl is used, Instead of the usual notation [llo], to Indicate the sense of the direction. 3‘>1 re \r> t J on:; : I’O = -Of HJ/K3 = -Kb/l'e PQ/Rh -- QP/hR = PO/bR *- QP/fR FQ+QR *■- I‘R PO+KR - PR/OR The Greet letter:; a , /3 , y , 5 rcpi’ow e n t points 02i each t;l.i Go plane a,h,c,cl, respocti vely, which are equidistant frum the three vertices of each triangular face of the tetrahedron. 1 A vector of the type A/3,'By, Ba , etc., represents iiU <11?> vector In the pertinent (ill) plane. A combination A a , B/3, CV, etc. represents a vector ^ <111> normal to the pertinent '111} plane. a p p e n d i x b EIJ^CT kOChKIlL CAli JKT-l-'OLIBll.Ei, i"' P AR/i;!’'T;\Pid. Hocausr the dir.tsmcc between intcrfcroncc1 fringes in the Zoi r.r inter Ter gm;1 ter is 0.2'/ riicrons, the sharc of trie Jet-polished dimple could be determined quite accurately except right at the center of the dimple. When a smooth curve was fitted to the measurements taken from the inter ference photograph,the cross-sectional view in Pi pure 6? re sulted. Plates 131 through 13^ are a scries of dimples produced on the same piece of Germanium. The dimples shovm in Plato 131 end Plato 132 wore underpolishcd, the surfaces wore slightly etched. The dimple in Plate 133 was one which would produce a suitable thin foil for observation in the electron microscope. The radius of curvature was large and the surface roughness was minimal. Plate 13^ shows a dimple which is very slightly over-polished. It exhibited the lowest degree of surface roughness, but the radius of curvature was not as large as it was for the dimple in Plate 133. The surface roughness for the last two dimples was, within the limits of optical resolution of the inter ference fringes ,belov; 1/ 1 0 ( A/2) or below 270 A°, from peak to valley. This author believes that it was often below 352 J«C -EUctropollshtd dtuples on 353 (111) orlanttd Gtnanlui M f c r at various polishing currants Plata 0 2 1 , Plata 132 Magnificat Ion: 304X Magnification: 304X Polishing ccurrant: u m 5.5 ua. Polishing currant:curr 6.0 A slightly atchlng condition A polishing condition with etching action Plata 133 MagnificatIon: 304X Magnification: 304X Polishing currant: 6.Ssa, Polishling currant: 7.2 as, Optlaua condition; a Large Good condition;onol the lowest radius of eurvatura at the surface roughness, but has cantor; no atchlng action a lower radius of curvature 3 5 100 A° In some foi la , based on observutJ ons In the electron microscope. Plates 13b through 1/tO shov; several photo graphs taken In the electron microscope of various foils. The size of the hole visually varied between 10 and 20 microns in diameter. The area around the hole, whl ch v.’as thin enough to allov: visual, observation of the transmitted electron beam, was usually between 3000 and 6000 square microns. These dimensions have* been used to shov: in Figure 62 that the maximum thickness of foil useful for observation of the structure, was about ^000 A° . The experimentally useful area was usually in the range of 2000 to 5000 square microns however, since the thinnest areas of foil adjacent to the hole were always too thin to yield any useful infor mation. These areas often exhibited few or no dislocations even in very high dislocation density material. This is apparent in Plate 136, a higher magnification view of Plate 135. The optimum procedure for producing an acceptable thin foil included first, obtaining perforation in the center of the wafer, which permitted a larger volume of germanium to be removed from either side of the wafer. If 1/3 mm. misalignment of the center of one dimple with respect to the opposite dimple occurred a suitable foil, within 4° of the wafer plane, would still result, however the closer the alignment of the two dimples, the larger the resulting area of useful thickness in the foil. Also any damaged surface 355 Plate 135 Thin foil in which preforation did not occur, Plate 136 Same thin foil as is shown In Plate 135 at a higher magnification. V Plate O i l - Slight ly overpolished thin f o i l . 4 3b7 I\ second optix.i sinp {'i-ocoduvc was that of starling that tlj :t t hel: t ic.-a 1losed oiT,y }iand l;i n<" w i th H :i 11le charice of bra at; j fy the wafer. It w also found that usl nr; a vru.fcr of a X h :I ch) w s less than abort 0 . raro lad to a decrcauin," rate: of po 1 i sh:1 na The need for slower poll edit itf, rate at the tine of n o t i c o d v: iitii 111.1 n n o r v; a f e r were observe be consistently over-polished (the thin arenr, around the holt’ had been drastically reduced). The reason for tin c can bo shov;n by examining the proe: ss of circuit Interrup tion. If, for* example, the rate of polishing is 0,0?0 m,m/mi r or 0.33 microns/sec, then tlic polishing action had to bo Interrupted within about 1 / J1 of a second or leas after initial perforation to obtain a suitable foil. If the rato is slowed by a factor of two then the1 critical time Interval would be 1/2 second or less . Thi s reasoning vjas based on the assumption of a shape of the dimple similar to that shown in Figure 82. The time interval Is actually smaller than the polishing rate and foil geometry predict because of some inherent disadvantages of the automatic photocell interrup tion apparatus which result in an additional time factor. First, the photocell has a definite minimum level of sensi tivity to the light used which must be exceeded to produce the diecci'iri hle current output to activate .inter ruption, Secondly, there ;ia a cnial. 1 but definite- tine delay which occurs ho tween photocell activation and polish ing circuit 11 i terruption ar:i r-i np; froir. the interval:! rip rel ay tuber. involved. hence a decreased ]>ol 1 sh I ny. rate- at pc-r- foratlon v:ar» an important facto:” in producing conci rto:itly pood foilrj. Arm:n;rx c KJKtlClil LlkKB Klkuchl lines consist of a structure- in the diffuse bacif.round of a diffraction pattern. The origin of the Klkuchl linos involves both elastic and inelastic scatter!up,. Tiie lines arise from the subsequent elastic s cat te ring of electrons vrhloh suffer an inelastic collision involving only small energy loss. If inelastic scattering occurs at P (refer to Pi pure 92) so that P be conies the origin of a spherical wavelet, the Inelastically scattered electrons are peaked generally in the direction of the incident beam OP. Local variations in the inelastic scatterLiif, occur because rays traveling in certain directions from P are at the Bra pip; angle 0 for reflection by a set of crystal planes. The r;iy P Q v:ill be deflected into direction QQ r , and the ray PH v.'ill be deflected into the direction RR' . Since the intensity of the inclastically scattered electrons is greater along PQ than along PR, the Bragg reflections cause more electrons to be transferred to the direction PR than are lost from PR, and the background intensity Is therefore Increased along Pfi and decreased 359 36ft Figure 92 The origin of Kikuchi lines (Hlrsch eta l. (^3)). T plan* Figure 93 The cones o f rays due to elastic scattering of the diffusely scattered electrons (Hlrsch et al. (*3)>. ------301 a 1 cup HQ. Y.'hen all possible- directions T o t 1 the rc’fUt.'ciiioti^ from a given not of crystal plnnc:, arc- cons! derod, the di j't'cti ons a 3 on:; v/hi cb pa3 nr or 3 or see of back ground inten sity occur are pi vcn by two cones or rays with sci,:J-angles of 90-0, The; intersection of these cone;/ of ray:; with a screen pit-cod rone ell stance away from the Rpeetnon (normal to the incident bean) describes two hyperbolae- which very closely apnrox.liiiatc tec straight lino.-, becanr.e of the small values of G, as shown in Figure 93. 'These represent, a pair of Kikuchi lines. Similar pairs exist for all possible reflecting pianos in the crystal, so the complex array in Figure 6 3 is obtained. If the specimen is tilted, the spots will not change position in the diffraction pattern, only the Intensity of the spots will vary. However the Kikuchi lines w .1 11 move as though rigidly fixed to the crystal, so the direction and magnitude of movement reveals the orienta tion chanp.o with a high degree of accuracy, Tills discussion has been a simplified version similar to that originally sug gested by Kikuchi (M2), A revicv: by Hirscli e_t_ a 1. (M3 ) of the work by Kikuchi was the source of this brief discussion. APPKNDTX D USh Oh K.'i dhhAV.i CAb fOhV]; fOT 'TiibOhY FOR bUJlGbKO v k c tu r i.'a t t g i ; Conr.i uc.r v. di f bract e (D-l) ^ = j ® e 2"15* dz v.’h c r e : (D-2) a = 2 rr (fV H) and a = the phase angle betvjeen the incident and diffracted beam of electrons £ = the reciprocal lattice vector of the perfect lattice plane from which the beam is reflected R = the displacement vector associated v.ritb a unit cell located at a depth z in the crystal eiai= the additional phase factor introduced into the transmitted beam by a unit displacement vector R in the crystal s = the deviation parameter for the perfect reference ~~ lattice, where s is parallel to z i2rsz = term giving variation in the intensity of the trans e mitted beam because of the crystal orientation and change in crystal thichness 3^2 t - tli'l el: no:s of thin f o 1 .1. crystal Cori;-'l .:cr now tho contrast of foe cs associs tod v:j th a sc row arul an edge cl in) ocati on : 1 . fcj'C'.y cl.) :■ot .1 on Kor a no cl t a 1 oc a t ;i on lying in tho plan-' of tin foil, froai equation B--H : CI j— 3) a ~ 2 v f ‘ If. v.’here, = tho Burpors vector of tho s c ro w d i s 1 oc a 1 1 on (D--»i) K* = y s = a scalar constant Hence, (l)-b) a = 2 tt In tho case v:hero b^ lien in the roflecting plr.no g, the condition * bs ) = 0 results. Hcnco tho die location line which wais visible (in tho transmitted Image beam) because of tho contrast produced by many reflected beams, will now be invisible in the linage obtained us inf, the dark field linage of the spot g. Hence, from the condition that <£ ’ ^s> = 0 , and knowing tho value of £ for the diffraction spot used, the Burners vector can be determined for a particular segment of dislocation. 2 . Similarly for an edge dislocation (D -6 ) a = 2 77 ( E. * He ^ but the displacement vector for an edge dislocation includes an additional term: 3 G'! (11-7) - irjtv + i:i. ' £ c X £> * c where: ~ Ecaliii' constant a or functions b the Purpors vector of the o(;(y dislocation £ ~ tht- positive vect or aloof; the di r;1ocati on (b x £) ~ a component of hi :■ pie cement normal to the plidc plane I f 11 e :.i 1 n tli e re f 1 e c t :l n t ■: p 1 a no p, then: (d-B) (n * Kg) ~ Ki(^e \r.) + K^r. * (be x £ ) however If the dielocation line lies in the plane of the foil normal to the electron hear:, and thr.' plide plane of tho dislocation in approximately parallel to the foil plane, then; (D-9) (1^ x f) * n = 0 and effectively for an edpc or mixed dislocation the (p; * b) criterion for determining the Burgers vector also applies. APPENDIX K kjjj-,} IIP AM 13 OTP OP IC IE U l A Hookes J.av: reads. (B-l) - C1Jkl * kl where (H-?) * kl = 1/2 (C'u£ +e!--) dxq dxk ' and tho equilibrium condition is, v 1 --- =T_ = n 1=12 1 dx ^ u 1 If the dislocation is along the axis, and the matrix c ^ k-^ -*-s referred to this system, then all displace ments, strains and stresses are independent of X3 . Hence, subscripts a , (3 arc either 1 or 2 and ijkl are 1,2,3. Then equation E-3 can bo written: (E-Jl) d ° ia W ~d " 0 i “ 1,2,3 From E - l , a, , = c, . _ _dy k h j ' “i-kp dXj3 Inserting E-5 into E-JJ, one finds, ( K - 6 ) - i r t r - 1 > 2 ’ 3 ciak/3 K a ft = 0 1 P 365 3 ft Equuti on K-C three simulfamous ccjus t ions; i ~ 1 ,2,3 for tl^rcc functions of ; hence the solution of K~6 .1S O f t he; f G I w (h-7 ) u k ” A k ~ (17 ) v.'here 11 - - B) >7(1) = x 3 + p and where p and Ay are cone fan to. Sub a t i tut i on of E-8 info I’’-6 yields P ,i 2 (E-9) c CJL11; l + jZ,2 , After cancellation of the common factor d ^ 2 ’ equation E-9 become: E-10) ^ikAk = 0 v.'here r.-ll) alk = cijk^ + Cciik2 + Ci2ki)p 4 ciPk2 p? Equations D-ll have a non-aero solution for Ak only when the deform!nant: (E-12) = 0 { a ik } Equation E-12 is then a sixth order equation in p with roots Pn , n - 1,2, 3, Jl, 9, C , Thus for each pn there exists an Ak (n) set v:hich sat isfies E-12, tho Ak (n) appears in ratios piven by sub - determinants, where if A^(n) = 1 arbitrarily, 36 7 aj^Cn) *133(11) A-j(n) = a22(n ) a.^^Cn) ailC n) aj 3 (n) a2 1 ^a) a;i;i ( n ) a1 1 (n) £i 3_ 3 < 11 ) A^(n) = a2 i(n) ay 3 (n ) & 1 1 (n ) 3. jl 3 C j j j a21(n) ayy(n)i A^Cn) ^ 1 If ay shovni by Eshelby (31), the roots pn are never real, then since the coefficients in the polynomial are real, roots must occur in pairs of complex conjugates; P4 " Pi* P 5 = P 2 " P 6 88 P 3 Then : A u ^ D * A, (1)« (15-15) a£(5) - At(2)* n |5 “ n2 Ak (6) - Ak (3)* n6 = n3* and for the analytic function: f{f}) (15-16) Ak (l )/(*?1) = [Ak(4) /( nA )3#, ... Since ju (equation E-7) must be a real function it K must be composed of complex conjugates such as (K-1 7 ) l/2[(Ak (l)/(ni) + Ak (H) /(T)i|)] = Re rAk (l)/(r/i) j J>-I' U fi v.rljl’?*o _Rr* - " real part of" VIic.-ii equation b-7 : 3 (L-1 8) -*-’k - he [ ^ A k (l‘) / p t 1? )] n:| where fn flVf- three ai'I'jirai'y .'malyt-i c funct:i.on:: . The d.Ispl accin 'ii ts o k ere multiple valued, in the re pp on c o: i ta 1 n in;; the d i s local 1 on . x2 Vipure 9^ A cut cncire11 rip; tho dislocation lyinp; alonp; the Xq axis. £ points into tho pane (Mirth and Lot lie (1)). The boundary conditions require to be analytic and single-valued, outside the cut, and to have a discon tinuity across the cut, (E-19) Au„= ( o k < x ,. ° n - M k (x,,o-) = bk *.>£} Considering now the form of / (n>> since the stresses involve the functions ancj the stresses are single.- d fj valued and continuous except at the origin, the functions are so also. _ d 71 £ The most pencral function forj (r^) is of the form 36‘J (Jllrth and (3.) pare J *: -O (K_?0) y (rj) -■ In tj(1 + Y, a " 170 From E-39 (.1), (}’-- 2 J ) A ir. n - :t_ 2 T6 i- and f;u : K-19 Cl), (K-2 2 ) a / - - d * giving _ T-P-3|n » 0 ' . - 2 i ) ^ n 2 f.r Kqunti ons 10-lR f K-IQ, E--2? combine to form tho ex pression involving D(n): j (}•'.-?t ) Ko_ [ Y ± - - bk h = 1,^,3 Equation H~2Jl is a sot of three equati ons containing six unknowns : (A { n) , D (11), h=- .1,2,3) . Tho other three equations necessary to solve tho equation for D(n) result from tho requirement that there; be no not forco 011 the dislocation core. The net force per unit length on the surface of a rod parallel to the dislocation, and containing it is (E-2 9 ) Fjl = f S * (ell x o 3) where X = stress tensor (13-26) dl - £pdxp + e^dx^ 37 0 If V = 0 7) /•< O ' a J x z "• -ii f,0 ) - 0,1 - 1,?,3 This contain:- the nocos ae j-y throe cquafi on:' . In::t r~ tlon of equation (]■,-( ) into E-?Y r.iver / d jUw + c l?k? Pn>M"> d> 1 1 v.’here ;((n } = >. j -1- p n x^ The iiitoj'i';’.]:: in E -?8 con be <1:1 recti-/ integrated for the doformed path of integration; d f ;(n ) Cr-29) f dxj - A.r n = + b(n) J r\dx y q Tho result of equation K-?8 i r, then: 3 (h'-30) lie L ^ t B i?hCn) Ak (ri) D(n)J = 0 j 1 = 3,?, 3‘ n -1 where (1,-31) = cijk 1 + cijk? Pn Tho six equations: E-?9 and E-31 will then completely determine D(n), hence one knov/s all of the paramo tors in tho expression for .uk ; 3 (K-32) u k = He ][,Ak (n) ln i11 The stresses are obtained from E-l, E-?, and E-3H as; (K-33) a ±i = Hc[- rrrm 2 Hl1kAk (n) D(n) *)n *"} £ ~ 1 The energy can be determined from the work done in forming the dislocation by cutting and displacing the plane x^> = 0. If ^ n =xl) i 3 C !'■-3 J0 a* * ~ - ~ 1" L Y I'M i (n ) Ai; (n) D(n) ] j-j / 'i; / * " ~ J J n =: fi'i1e v;ork done on the of tho cut between xj_':r0 and x ^ K per un.it length oJ‘ di h location an it in formed in; R (K- 33) W/b - *-1/2 f 3 (l:~ 36) W / L = -% — In [ Irn y (n )/.p(n )U(n) 3 4 iV o - ^ The onei’py factor K is defined, (is-37) Kb? = bjLlm [ y Bi2k(n) Ak (n)l>(n) ] n • a So that equation E-33 becomes; (E-3S) W/L - K A a o );'}•'I' h h\.X:\6 1. 11 ■ l11 r i, J. 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