Micro- and Macro-Properties of Solids (eBook)

D.B. Sirdeshmukh taught physics for nearly four decades, first at Osmania University and later at Kakatiya University, Warangal. He was Fulbright Visiting Professor at the University of Rhode Island, USA, and has been Emeritus Professor at the Kakatiya University. Prof. Sirdeshmukh is actively engaged in research for close to 50 years in several branches of Solid State Physics. He has authored about 120 research publications including two review articles in Journal of Materials Science. He has authored two books. L. Sirdeshmukh was educated at the Osmania University and earned a Ph.D in Spectroscopy. She taught physics for more than three decades at the Kakatiya University. Her research interests are in spectroscopy and dielectric properties of materials. She has published about 80 papers and she was the guest editor of a special issue of the Bulletin of Materials Science. She co-authored a book with Prof. Sirdeshmukh. K.G. Subhadra has been teaching physics at the Kakatiya University for over three decades. Her research interests are in X-ray crystallography and mechanical properties of solids. She has published about fifty papers and has co-authored two books with Prof. D.B. Sirdeshmukh. She has participated in a number of conferences and 3 of her papers were chosen for best presentation awards.

Preface 7
Contents 11
1 Lattice Constant – A Solid State Probe 18
1.1 Introduction 18
1.2 Experimental Methods 18
1.2.1 Principle 18
1.2.2 Experimental Techniques 19
1.2.4 Present Level of Accuracy 32
1.3 An Overview 32
1.3.1 Characterisation of Semiconductor Materials 33
1.3.2 Characterisation of Doped Crystals 34
1.3.3 E.ect of Deuteration on Lattice Constants 35
1.3.4 E.ect of Hydrogen on Lattice Parameters of Rare Earth Elements 35
1.3.5 Lattice Constants of Mixed Crystals 36
1.3.6 Mixed Valence E.ects in Lattice Constants 38
1.3.7 Temperature Variation of Lattice Constant 39
1.3.8 Pressure Variation of Lattice Parameters 40
1.3.9 E.ect of Magnetic Field on Lattice Constant 40
1.3.10 Radiation Damage 41
1.3.11 E.ect of Particle Size on Lattice Constant 42
1.3.12 Lattice Constants and Point Defects in Crystals 42
1.3.13 Lattice Constant Variations due to Dislocations 44
1.3.14 Lattice Constant as a Scaling Parameter 46
1.4 Some of our Results 46
1.4.1 Lattice Parameters – Data Generation 46
1.4.2 Lattice Constant as a Scaling Parameter 49
1.4.3 Temperature Variation of Lattice Constant 50
1.4.4 Radiation Induced Changes in Lattice Constant of NaBrO3 50
1.4.5 Lattice Constants of Mixed Crystals 52
2 Thermal Expansion 54
2.1 Introduction 54
2.2 Experimental Methods 56
2.2.1 General 56
2.2.2 Optical Methods 57
2.2.3 Capacitance Methods 57
2.2.4 Difraction Methods 58
2.2.5 Dilatometric Methods 59
2.2.6 Other Methods 63
2.3 An Overview 64
2.3.1 Some Novel Experimental Techniques 64
2.3.2 Experimental Data on Thermal Expansion of Crystals 67
2.3.3 ‘Invar’ 68
2.3.4 Thermal Expansion of Inert Gas Solids 68
2.3.5 Correlations of Thermal Expansion with other Physical Properties 69
2.3.7 E.ect of Gross Defects on Thermal Expansion 70
2.3.8 E.ect of Irradiation 74
2.3.10 Pressure Variation of Thermal Expansion 76
2.3.11 Theories of Thermal Expansion 77
2.3.12 Negative Thermal Expansion 78
2.3.13 Anisotropy of Thermal Expansion 79
2.4 Some of our Results 80
2.4.1 Coefficients of Thermal Expansion – Data Generation 80
2.4.2 USBM Inter-Laboratory Project on Thermal Expansion of MgO 81
2.4.3 Aspects of Gruneisen Theory 82
2.4.4 Studies of Some Anomalous Phenomena 85
2.4.5 Empirical Relations 88
3 Debye–Waller Factors of Crystals 94
3.1 Introduction 94
3.2 Brief Outline of the Debye–Waller Theory 94
3.3 Experimental Procedures 99
3.3.1 Measurement of Integrated Intensity 99
3.3.2 Analysis of Intensity Data 104
3.3.3 Other Methods 109
3.4 An Overview 110
3.4.1 Earlier Work of Historical Importance 110
3.4.2 Experimental Values of Debye–Waller Factors at Room Temperature 110
3.4.3 E.ect of Choice of Atomic Scattering Factors on Measured B-values 111
3.4.4 Debye–Waller Factor for a Real Crystal 112
3.4.5 Debye Temperatures of Thin Films and Fine Particles 113
3.4.6 E.ect of Lattice Strain on B 114
3.4.7 Anisotropy of Debye–Waller Factors 115
3.4.8 Pressure Variation of .M 117
3.4.9 Temperature Variation of B and .M 118
3.4.10 Anharmonic E.ects in Debye–Waller Factors and Debye Temperature 118
3.4.11 Debye–Waller Factors from Lattice Dynamics 120
3.4.12 Debye–Waller Factors and Melting 123
3.4.13 Debye–Waller Factors and Temperature Dependence of Band-gap in Semiconductors 123
3.4.14 Debye Temperature in an Antiferromagnetic Transition 124
3.4.15 Nano Effect on Debye–Waller Factor and Debye Temperature 125
3.4.16 Energy of Defect Formation from Debye Temperature 125
3.4.17 E.ect of Electronic Environment on Debye–Waller Factor 126
3.4.18 Debye–Waller Factor of Mixed Crystals 127
3.4.19 Debye–Waller Factors of Protein Structures 127
3.5 Some of our Results 128
3.5.1 Debye–Waller Factors – Data Generation 128
3.5.2 Debye–Waller Factors and Mass Ratio 130
3.5.3 Comparison of Experimental Results with Lattice Dynamical Results 135
3.5.4 Anisotropy of Debye–Waller Factors 137
3.5.5 E.ect of Strain on Debye–Waller Factors 139
3.5.6 E.ect of Atomic Scattering Factors on B 140
3.5.7 Debye–Waller Factors and the Electronic Environment 141
3.5.8 Debye–Waller Factors in Mixed Crystals 142
3.5.10 Comparison of . from Different Methods 143
3.5.11 A modified Expression for the X-ray Debye Temperature (M) 146
3.5.12 Energy of Defect Formation from Debye Temperatures 147
4 Hardness 152
4.1 Introduction 152
4.2 Experimental Methods 153
4.2.1 General 153
4.2.2 Leitz–Wetzlar Mini-Load 2 Microhardness Tester 154
4.2.3 Shimadzu Dynamic Ultra Hardness Tester DUH 202 156
4.2.4 Nanoindentation 157
4.2.5 Relative Hardness Measurement 160
4.3 An Overview 163
4.3.1 General 163
4.3.2 Load Variation of Hardness 163
4.3.3 Solid Solution Hardening 165
4.3.4 Impurity Hardening 165
4.3.5 Dislocation Hardening 166
4.3.6 Radiation Hardening 166
4.3.7 Hardness and Chemical Bond 168
4.3.8 Pressure Variation of Hardness 169
4.3.9 Temperature Variation of Hardness 170
4.3.10 Empirical Relations with other Physical Properties 170
4.3.11 Anisotropy of Hardness 171
4.3.12 Surface Hardness 174
4.3.13 Nanohardness of Thin Films 175
4.3.14 E.ect of Magnetic Field on Hardness 175
4.3.15 Hardness of Organic Crystals 176
4.3.16 Micro-Raman Spectroscopy of Indentations 176
4.4 Some of our Results 177
4.4.1 Load Variation of Hardness 177
4.4.2 Hardness and Bonding 184
4.4.3 Radiation Hardening 193
4.4.4 Hardness of Doped Crystals 197
4.4.5 Hardness of Mixed Crystals 199
4.4.6 Empirical Relations with other Physical Properties 199
4.4.8 Surface Hardness of Crystals 204
4.4.9 Anisotropy of Hardness 209
5 Dielectric and Electrical Properties of Solids 216
5.1 Introduction 216
5.1.1 Dielectric Polarization 216
5.1.2 Dielectric Dispersion and Dielectric Loss 218
5.1.3 Dielectric Loss and Conduction 219
5.1.4 Temperature Variation of Dielectric Constant and Loss 220
5.2 Experimental 221
5.2.1 Measuring Instruments 221
5.2.2 Cell Designs 224
5.2.3 Procedural Details 227
5.2.4 Measurement in the Microwave Region 230
5.2.5 Dielectric Constants from IR Re.ectivity 230
5.2.6 Impedance Spectroscopy 232
5.2.7 Comparison of Methods 232
5.3 An Overview 233
5.3.1 Some Important Experimental Results 233
5.3.2 Temperature Variation of Dielectric Constant 236
5.3.3 Szigeti’s Theory (E.ective Ionic Charge and Anharmonicity) 238
5.3.4 Spectroscopic Aspects 241
5.3.5 Conductivity of Ionic Crystals 245
5.3.6 Dielectric Constant and Polaron Conduction 246
5.3.7 Dielectric Constant and Additivity of Polarizability 248
5.3.8 Dielectric Behaviour of Proteins Dielectric Properties and Protein Hydration 248
5.3.9 Irradiation E.ects 251
5.4 Some of our Results 252
5.4.1 Dielectric Properties – Data Generation 252
5.4.2 Analysis of Temperature Variation of Dielectric Constant 252
5.4.3 Application of Szigeti’s Theory 258
5.4.4 Spectroscopic Aspects 259
5.4.5 Polaron Conduction in Garnets 262
5.4.6 Dielectric Constant and Additivity of Polarizability 263
5.4.7 Ferroelectric Behaviour in NaCIO3 and NaBrO3 264
5.4.8 Analysis of Conductivity Data 265
5.4.9 .-Irradiation Studies 268
5.4.10 Dielectric Properties and Protein Hydration 271
6 Theoretical Evaluation of Some Crystal Properties 274
6.1 Introduction 274
6.2 Elastic Constants of Ionic Crystals 274
6.3 Coefficient of Thermal Expansion from Interatomic Potentials 276
6.3.1 Thermal Expansion Coeffcient of Crystals with Fluorite Structure 276
6.3.2 Thermal Expansion Coeffcients of Some Anisotropic Elements 277
6.4 Debye Temperatures from Elastic Constants 278
6.4.1 General 278
6.4.2 Debye Temperatures from Single Crystal Elastic Constants 279
6.4.3 . from Polycrystalline Elastic Data 282
6.4.4 Brief Review of Earlier Work 283
6.4.5 Some of Our Results 285
6.5 Gruneisen Parameter 289
6.5.1 Gruneisen Parameter from Interatomic Potentials 290
6.5.2 . from Pressure Variation of Debye Temperature 292
6.5.3 Evaluation of . from Pressure Derivatives of Elastic Moduli 294
6.5.4 Mode Gruneisen Parameters of Fluorite-Type Crystals 299
7 The Physics of Mixed Crystals 302
7.1 Introduction 302
7.1.1 General 302
7.1.2 Earlier Reviews on Mixed Crystals 303
7.1.3 Theoretical Models 304
7.2 An Overview 305
7.2.1 Molar Volume and Lattice Parameters 305
7.2.2 Debye–Waller Factors 309
7.2.4 Hardness of Mixed Crystals 311
7.2.5 Dielectric Constant 315
7.2.6 E.ective Ionic Charge in Mixed Crystals 316
7.2.7 Colour Centres in Alkali Halide Mixed Crystals 317
7.2.8 Defects in Mixed Crystals 319
7.2.9 Melting Temperature 320
7.2.10 Pm3m < ->
7.3 Some of our Results 323
7.3.1 Lattice Constants of Mixed Crystals 323
7.3.2 Debye–Waller Factors 327
7.3.3 Debye Temperatures of Mixed Crystals 329
7.3.4 Hardness of Mixed Crystals 332
7.3.5 Dielectric Properties 334
7.3.6 E.ective Ionic Charge 337
7.3.7 Colour Centres in RbCl–RbBr Mixed Crystals 339
7.3.8 Defects in Mixed Crystals 341
7.3.9 Melting Temperatures of Mixed Crystals 342
7.3.10 Pm3m.Fm3m Transition in NH4Cl–NH4Br System 344
8 Elastic Properties of Solids – A Critical Analysis 348
8.1 Introduction 348
8.2 Experimental Methods 348
8.2.1 Piston Displacement Method 348
8.2.2 Shock Wave Method 350
8.2.3 X-ray Diffraction Method 351
8.2.4 Optical Interferometric Method 351
8.2.5 Ultrasonic Method 353
8.2.6 Other Methods 354
8.2.7 Relative Merits and Limitations 354
8.3 Discrepancies in Elastic Properties 354
8.4 Consistency Checks for Bulk Moduli 355
8.4.1 Phenomenological Relations as Consistency Checks 355
8.4.2 Theoretical Consistency Checks 360
8.4.3 Empirical Relations as Consistency Checks 369
8.2.5 Ultrasonic Method 353
8.2.6 Other Methods 354
8.2.7 Relative Merits and Limitations 354
8.3 Discrepancies in Elastic Properties 354
8.4 Consistency Checks for Bulk Moduli 355
8.4.1 Phenomenological Relations as Consistency Checks 355
8.4.2 Theoretical Consistency Checks 360
8.4.3 Empirical Relations as Consistency Checks 369
8.5 Consistency Checks for Single Crystal Elastic Constants 373
8.5.1 Cubic Crystals 374
8.5.2 Tetragonal Crystals 374
8.5.3 Trigonal and Hexagonal Crystals 377
8.6 Conclusions 378
References 380
Chapter 1 380
Chapter 2 382
Chapter 3 386
Chapter 4 392
Chapter 5 397
Chapter 6 403
Chapter 7 406
Chapter 8 410
Index 416

2.2.5 Dilatometric Methods (p. 42-43)

Push-rod dilatometry is one of the techniques of measurement of thermal expansion. Several push-rod dilatometers are commercially available. Fused quartz dilatometers are simple and comparatively inexpensive. In a typical dilatometer the sample is enclosed in a fused quartz tube. The end of the sample is in contact with a sensitive dial-gauge. The composite holder (quartz tube and sample) is placed in a heater and length changes are directly read on the dial-gauge. Janson and Sjoblom [2.42] describe such a dilatometer. A push-rod dilatometer was constructed and was used by Rao [2.43] for studies of some crystals. The instrument is described in some detail since it is simple in design, inexpensive and made from indigenously available components. The main parts of the push-rod dilatometer and the procedure for its use are discussed in this section.

The Push-Rod Assembly

The rods used in the present experimental set-up are made of fused silica. They are cylindrical in shape and their surface is satin-glazed. The rod assembly is made up of two parts. Three rods of equal length form the base of the assembly and a set of four short rods makes its upper part. The three base rods rest on two metal stands fixed to a wooden base. The metal stands are provided with horizontal projections on either side. The two projections facing each other have three grooves made in each of them to hold the rods. The exterior projections are flat so as to support the dial-gauge micrometers.

The rod assembly as seen from the front is shown in Fig. 2.1a (not included in the extract). The long rod, designated BR is one of the base rods fixed to the metal stands. The two shorter rods placed on the base rods rest lengthwise in the space provided by the adjacent base rods. The sample is sandwiched between the two short rods wherein the rod marked FR is the fixed rod and MR the movable rod. The shaded portion marked S shows the position of the sample.

The movable rod MR is in communication with the dial-gauge DG. The position of the heater is shown in the figure by H. The differential arrangement of the rod assembly, along with the positions of the specimen and the standard reference material, as seen from above is presented in Fig. 2.1b (not included in the extract). ,The shaded portions, S1 and S2, represent the two specimens, i.e., the sample and the standard reference material. FR1 and FR2 are the two fixed short rods, fixed to the base rods by means of an adhesive applied at the end away from the heater. MR1 and MR2 are the two movable rods which are in communication with the two dial-gauge indicators DG1 and DG2 kept on the two exterior projections of the metal stand.

The position of the heater with respect to the two samples is shown by H. When heated, the specimen under study communicates its dilation to the micrometer kept on the metal stand on one side of the assembly while the standard reference material communicates its dilation to the micrometer on the other side. The positions of the two thermocouples, T1 and T2 are as shown in the figure.