MECHANICAL AND OPTICAL RESPONSE OF DIAMOND CRYSTALS
SHOCK COMPRESSED ALONG DIFFERENT ORIENTATIONS
By
JOHN MICHAEL LANG, JR.
A dissertation submitted in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
WASHINGTON STATE UNIVERSITY Department of Physics
DECEMBER 2013
© Copyright by JOHN MICHAEL LANG, JR., 2013 All Rights Reserved © Copyright by JOHN MICHAEL LANG, JR., 2013 All Rights Reserved To the Faculty of Washington State University:
The members of the Committee appointed to examine the dissertation of
JOHN MICHAEL LANG, JR. find it satisfactory and recommend that it be accepted.
______Yogendra M. Gupta, Ph.D., Chair
______Matthew D. McCluskey, Ph.D.
______Philip L. Marston, Ph.D.
ii ACKNOWLEDGMENTS
I would first like to thank my advisor, Dr. Yogendra Gupta, for providing me with this research opportunity and supporting my work. His guidance and leadership was invaluable to my education and training as a scientist. I am very grateful for his time and effort. I would also like to thank Dr. Matt McCluskey and Dr. Philip Marston for their advice and guidance, and for serving on my committee.
I would like to thank the administrative and technical staff of the Institute for
Shock Physics for their support of my work. Special thanks to Kurt Zimmerman and
Yoshi Toyoda for their assistance with the experimental instrumentation, to Nate
Arganbright for his help with materials preparation, to Steve Barner for machining many of the parts used in this work, and to Kent Perkins, Cory Bakeman, and Luke Jones for operating the guns. A warm thank you to Sabreen Dodson and the administrative staff of the Physics Department for their help in guiding me through the policies and requirements of the Graduate School.
I would also like to thank JiaJia Chang for her patience and support. Lastly, I would like to thank my parents, John and Anne Lang, my sister, Elizabeth, and my grandparents for their love, generosity, and encouragement as I worked to complete my degree. It was their support that enabled me to finish, and I dedicate my dissertation to them.
This work was supported by the DOE/NNSA.
iii MECHANICAL AND OPTICAL RESPONSE OF DIAMOND CRYSTALS
SHOCK COMPRESSED ALONG DIFFERENT ORIENTATIONS
Abstract
by John Michael Lang, Jr., Ph.D. Washington State University December 2013
Chair: Yogendra M. Gupta
To determine the mechanical and optical response of diamond crystals at high stresses and to evaluate anisotropy effects, single crystals (Type IIa) were shock compressed along the [100], [110], and [111] orientations to ~120 GPa peak elastic stresses. Particle velocity histories and shock velocities, measured using laser interferometry, were used to examine nonlinear elasticity, refractive indices, and
Hugoniot elastic limits of shocked diamond. Time-resolved Raman spectroscopy was used to measure the shock compression induced frequency shifts of the triply degenerate
1332.5 cm-1 Raman line.
Longitudinal stress-density states for elastic compression along different orientations were determined from the measured particle velocity histories and elastic shock wave velocities. The complete set of third-order elastic constants was determined from the stress-density states and published acoustic data. Several of these constants differed significantly from those calculated using theoretical models.
The refractive index of diamond shocked along [100] and [111] was determined
iv from changes in the optical path length along the direction of uniaxial strain. Linear photoelasticity theory predicted the measured refractive index along [111]. In contrast, the refractive index along [100] was nonlinear. The refractive indices for [110] compression were not determined, but the data showed evidence of birefringence.
The splitting and frequency shifts of the diamond Raman line were measured for shock compression along [111] and were in good agreement with predictions from prior shock work. Frequency shifts were also measured along [100] and [110] up to ~60 GPa, extending previous measurements. The anharmonic force constants determined from all shock compression measurements agree with the previous shock compression determinations.
Hugoniot elastic limits for diamond shock compressed along different orientations were determined from the measured wave profiles. The elastic limits for the three orientations were highest at ~90 GPa peak elastic stress, but decreased at the higher peak elastic stress. Shear strengths were determined from the measured elastic limits: shocked diamond was strongest for compression along [110] and weakest for compression along
[111]. The shear strength dependence on shock propagation direction was correlated with the stress magnitude normal to the slip plane, which appeared to inhibit the onset of inelastic deformation.
v TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS...... iii
ABSTRACT...... iv
TABLE OF CONTENTS...... vi
LIST OF TABLES...... xi
LIST OF FIGURES...... xiii
CHAPTER
1. INTRODUCTION...... 1
1.1 Objectives and Approach...... 3
1.2 Organization of Chapters...... 4
References for Chapter 1...... 6
2. BACKGROUND...... 9
2.1 Crystal Properties...... 9
2.2 Elastic Response of Diamond...... 11
2.2.1 Linear Elastic Response...... 11
2.2.2 Nonlinear Elasticity: Finite Strain Theory...... 13
2.2.3 Third-order Elastic Constants of Diamond...... 15
2.2.4 Application to Uniaxial Strain...... 17
2.3 Refractive Index of Shocked Diamond...... 21
2.3.1 The Optical Indicatrix...... 21
2.3.2 Photoelasticity Theory...... 24
vi 2.3.3 Application to Diamond...... 25
2.4 Raman Spectrum of Shocked Diamond...... 27
2.4.1 Strain-induced Frequency Shifts...... 28
2.4.2 Past Experimental Work...... 29
2.5 Elastic Limit and Strength of Shocked Diamond...... 30
2.5.1 Theoretical Background...... 31
2.5.2 Past Studies on Diamond Strength...... 33
References for Chapter 2...... 38
3. EXPERIMENTAL METHODS...... 43
3.1 Materials Characterization...... 43
3.1.1 Diamond...... 43
3.1.2 OHFC Copper...... 44
3.1.3 Tantalum...... 45
3.1.4 Lithium Fluoride...... 45
3.2 Mechanical Response and Refractive Index Measurements...... 46
3.2.1 Experimental Configuration...... 46
3.2.2 Target Construction...... 48
3.2.3 Projectile Construction...... 52
3.2.4 Instrumentation...... 52
3.3 Raman Spectroscopy Measurements...... 56
3.3.1 Experimental Configuration...... 56
3.3.2 Target Construction...... 58
3.3.3 Projectile Construction...... 62
vii 3.3.4 Instrumentation...... 62
3.3.5 Triggering and Synchronization...... 66
3.3.6 Spectral and Temporal Calibration...... 70
References for Chapter 3...... 73
4. NONLINEAR ELASTIC RESPONSE: RESULTS AND ANALYSIS...... 75
4.1 Elastic Wave Measurements...... 75
4.1.1 Particle Velocity Histories...... 77
4.1.2 Elastic Shock Wave Velocities...... 80
4.1.3 Elastic Wave Amplitudes...... 82
4.2 Elastic Wave Analysis...... 84
4.3 Nonlinear Elastic Response...... 87
4.3.1 Theoretical Background...... 87
4.3.2 Determination of Third-order Constants...... 89
4.4 Experimentally Determined Elastic Hugoniots for Different
Orientations...... 96
4.5 Summary of Elasticity Results...... 97
References for Chapter 4...... 99
5. REFRACTIVE INDEX: RESULTS AND ANALYSIS...... 101
5.1 Refractive Index Measurements...... 101
5.1.1 The Optical Path Length...... 102
5.1.2 Shock Compression Along [100]...... 105
5.1.3 Shock Compression Along [110]...... 108
5.1.4 Shock Compression Along [111]...... 110
viii 5.2 Nonlinear Photoelasticity...... 114
5.3 VISAR Window Corrections...... 115
5.4 Summary of Refractive Index Results...... 116
References for Chapter 5...... 118
6. RAMAN SPECTRA OF SHOCKED DIAMOND: RESULTS AND
ANALYSIS...... 120
6.1 Raman Spectroscopy Measurements...... 120
6.1.1 Analysis of Raman Spectra...... 123
6.1.2 Shock Compression Along [111]...... 124
6.1.3 Shock Compression Along [100]...... 126
6.1.4 Shock Compression Along [110]...... 128
6.2 Anharmonic Force Constants of Diamond...... 130
6.2.1 Determination From Uniaxial Strain Measurements...... 130
6.2.2 Discussion...... 133
6.3 Reanalysis of Published Shock Wave Measurements...... 139
6.3.1 Spectral Calibration...... 139
6.3.2 Mechanical Response...... 140
6.4 Summary of Raman Spectroscopy Results...... 142
References for Chapter 6...... 143
7. ELASTIC LIMITS: RESULTS AND ANALYSIS...... 144
7.1 Elastic Limit Measurements...... 144
7.1.1 Particle Velocity Histories...... 146
7.1.2 Elastic Wave Analysis...... 153
ix 7.1.3 Hugoniot Elastic Limits...... 154
7.2 Shear Strength of Diamond...... 157
7.2.1 Resolved Shear Stress...... 157
7.2.2 Shear Strengths Under Shock Compression...... 161
7.3 Role of Normal Stress...... 163
7.3.1 Resolved Normal Stress...... 163
7.3.2 Comparison with Silicon and Germanium...... 166
7.3.3 Comparison with Theoretical Calculations...... 168
7.4 Summary of Elastic Limit Results...... 171
References for Chapter 7...... 173
8. SUMMARY AND CONCLUSIONS...... 176
8.1 Summary...... 176
8.2 Conclusions...... 179
References for Chapter 8...... 182
x LIST OF TABLES
Page
2.1 Second-order elastic constants of diamond...... 12
2.2 Pressure derivatives of the second-order elastic constants of diamond...... 15
2.3 Estimates of the third-order elastic constants of diamond...... 17
2.4 First-order elasto-optic constants of diamond...... 27
2.5 Anharmonic force constants of diamond...... 29
3.1 Longitudinal and shear sound speeds for single crystal diamond...... 44
4.1 Experimental parameters for elasticity experiments...... 76
4.2 Experimental measurements for the elasticity experiments...... 81
4.3 Effective longitudinal third-order elastic constants of diamond...... 90
4.4 Third-order elastic constants of diamond...... 95
5.1 Experimental parameters for the refractive index experiments...... 102
5.2 Elastic wave and refractive index measurements for [100]...... 106
5.3 Elastic wave and refractive index measurements for [111]...... 112
6.1 Experimental parameters for Raman spectra measurements...... 121
6.2 Experimental results for uniaxial strain along [111]...... 126
6.3 Experimental results for uniaxial strain along [100]...... 128
xi 6.4 Experimental results for uniaxial strain along [110]...... 130
6.5 Anharmonic force constants of diamond...... 133
6.6 Reanalysis of published shock wave measurements...... 141
7.1 Experimental parameters for the VISAR experiments...... 147
7.2 Experimental results for the Hugoniot elastic limit measurements...... 155
7.3 Calculated resolved shear and normal stresses...... 159
7.4 Shear strength and normal stress for uniaxial strain...... 162
xii LIST OF FIGURES
Page
2.1 Conventional unit cell of the diamond lattice...... 10
2.2 General form of an optical indicatrix ellipsoid construction...... 23
3.1 Experimental configuration for mechanical response and refractive index
measurements ...... 47
3.2 Cross section, schematic view of target for mechanical response and
refractive index measurements ...... 49
3.3 Photographs of the bonded diamond samples for mechanical response and
refractive index measurements ...... 51
3.4 Photograph of the assembled target for mechanical response and refractive
index measurements ...... 53
3.5 Photographs of representative projectiles for mechanical response and
refractive index measurements ...... 54
3.6 Experimental configuration for time-resolved Raman measurements...... 57
3.7 Cross section, schematic view of target for Raman measurements...... 59
3.8 Photographs of the bonded diamond samples for Raman measurements...... 61
3.9 Photograph of the assembled target for Raman measurements...... 63
3.10 Photograph of representative projectile for Raman measurements...... 64
3.11 Triggering and synchronization diagram for the Raman spectroscopy
experiments...... 67
xiii 3.12. Timeline of the events and delays for synchronizing Raman spectroscopy
experiments...... 68
3.13 Calibration spectra from HgCd, Cs, and Kr lamps...... 71
4.1 Representative elastic particle velocity histories from experiment 1...... 78
4.2 Representative elastic-inelastic particle velocity histories from experiment 3...... 79
4.3 Shock velocity and particle velocity data for shock compression along [100]...... 85
4.4 Shock velocity and particle velocity data for shock compression along [110]...... 86
4.5 Shock velocity and particle velocity data for shock compression along [111]...... 88
4.6 Elastic response for shock compression along [100]...... 91
4.7 Elastic response for shock compression along [110]...... 93
4.8 Elastic response for shock compression along [111]...... 94
5.1 Actual and apparent velocity histories for compression along [100] from
experiment 1...... 103
5.2 Calculated and measured refractive index for compression along [100]...... 107
5.3 Actual and apparent velocity histories for compression along [110] from
experiment 14...... 109
5.4 Actual and apparent velocity histories for compression along [111] from
experiment 18...... 111
5.5 Calculated and measured refractive index for compression along [111]...... 113
xiv 6.1 Time resolved Raman spectra of diamond shock compressed along [111]
from experiment 10...... 122
6.2 Ambient and shock Raman spectra of diamond compressed along [111]
from experiment 10...... 125
6.3 Ambient and shock Raman spectra of diamond compressed along [100]
from experiment 4...... 127
6.4 Ambient and shock Raman spectra of diamond compressed along [110]
from experiment 7...... 129
6.5 Measured and predicted frequency shifts for uniaxial strain along [111]...... 134
6.6 Measured and predicted frequency shifts for uniaxial strain along [100]...... 135
6.7 Measured and predicted frequency shifts Δω1 and Δω2 for uniaxial strain
along [110]...... 137
6.8 Measured and predicted frequency shifts Δω3 for uniaxial strain along [110]....138
7.1 Particle velocity histories for synthetic diamonds shocked along [100]...... 145
7.2 Particle velocity histories for natural diamonds shocked along [100]...... 149
7.3 Particle velocity histories for diamonds shocked along [110]...... 150
7.4 Particle velocity histories for diamonds shocked along [111]...... 152
7.5 Resolved shear stress as a function of peak elastic stress...... 160
7.6 Resolved shear stress as a function of resolved normal stress for diamond...... 165
7.7 Resolved shear stress as a function of resolved normal stress for silicon...... 167
7.8 Resolved shear stress as a function of resolved normal stress for germanium....169
xv Chapter 1
INTRODUCTION
Diamond is a remarkable material that has widespread scientific and technological importance due to its unique properties. It is well known for its exceptional stiffness, strength, transparency, thermal conductivity, and chemical inertness [1].* These properties make it an ideal material for a variety of applications which include machine tooling in industry [1], generation of pressures in excess of 300 GPa in diamond anvil cells [2], use as high power laser optical windows [3], and as a proposed capsule material for inertial-confinement fusion [4].
Understanding the mechanical and optical properties of diamond at high stresses is an important need for both fundamental science and technological applications.
However, precise measurements of diamond properties at high stresses are difficult using conventional quasistatic techniques due to its exceptional stiffness and strength. Also, the availability of high quality synthetic diamond crystals has increased significantly in recent years [1,5-8], yet it is not clear how their properties at extreme conditions compare with their natural counterparts. In the present work, plane shock wave compression of natural (primary effort) and synthetic diamonds along different crystal orientations was used to investigate the nonlinear elastic response, the refractive indices, the Raman spectra, and the strength of diamond crystals undergoing large uniaxial strains.
Prior to this work, third-order elastic constants, describing the nonlinear elastic response of diamond, were estimated from a combination of available experimental data, * References cited in a chapter are listed at the end of that chapter.
1 theoretical models [9-11], and ab initio simulations [12]. The constants determined using these theoretical methods show large variations, demonstrating the need for an experimental determination of the third-order constants.
The stress and strain dependance of the refractive index of diamond was measured previously under uniaxial stress compression (< 0.3 GPa) [13], hydrostatic pressure (< 9
GPa) [14-15], and from Brillouin scattering [16]. Previous hydrostatic [2] and shock compression [17] experiments have shown diamond to remain transparent at much higher stresses, but it is not known how well the results at low stresses predict the refractive index at high stresses.
The triply-degenerate Raman line of diamond was observed to split and shift under shock wave compression along the [100] and [110] orientations [17]. From these measurements, the Raman shifts for uniaxial strain along [111] were predicted but, prior to the present work, they have not been measured.
The coupling of the brittle response, high stiffness, and large strength of diamond makes strength measurements using conventional quasistatic approaches difficult [1,18].
The ideal shear strength of diamond has been estimated using theoretical methods to be anywhere between 35 and 120 GPa [18-22]. Shock wave [23-25] and ramp wave [26] compression studies have explored the compression and phase diagram of diamond up to and beyond ~1 TPa, but the elastic-inelastic response of diamond was not well characterized in these experiments. Hugoniot elastic limit measurements at more modest stresses (~100 GPa), where the elastic-inelastic response can be carefully examined, are needed to quantify the strength and inelasticity of diamond.
2 1.1 Objectives and Approach
The overall goal of this work was to examine the mechanical and optical properties of shock-compressed diamond single crystals, and to evaluate the effects of crystalline anisotropy on these properties.
The specific objectives of this study were:
1. To determine the non-linear elastic response of diamond crystals from
experimental data; in particular, to obtain the complete set of third-order elastic
constants.
2. To determine the refractive index of diamond shock compressed along different
crystal orientations.
3. To measure the Raman spectra of diamond shock compressed along the [111]
orientation, and to compare the measured shifts with published predictions.
4. To determine the Hugoniot elastic limits of diamond crystals shocked along
different orientations, and to relate them to the shear strength of diamond.
5. To provide an explanation for the observed Hugoniot elastic limits.
6. To investigate the similarities and differences in the response of natural and
synthetic diamond single crystals under shock wave compression.
Plane shock wave experiments were used to generate uniaxial strain compression in diamond crystals shocked along [100], [110] and [111]. Peak elastic stresses ranging from 15 to 120 GPa were achieved in the diamond samples through plate impacts generated using a 4” gas gun, a powder gun, or a two stage gun, depending on the desired peak stress. All experiments were conducted at the Institute for Shock Physics at
Washington State University.
3 Two types of time-resolved measurements (~ns resolution) were used in this study: laser interferometry to obtain the mechanical response and refractive index measurements, and Raman spectroscopy measurements. For the mechanical response and refractive index measurements, a velocity interferometer system for any reflector
(VISAR) [27] was used to monitor shock wave profiles at both interfaces of the diamond samples. Analysis of the wave profiles provided the nonlinear elasticity, refractive index and elastic limit results. For the time-resolved Raman spectroscopy measurements, the experimental configuration closely followed that used in previous shock compression experiments [17], with minor changes to the design and instrumentation.
1.2 Organization of Chapters
The remaining chapters are organized as follows:
• Chapter 2 provides the background material on the mechanical and optical
properties of diamond and reviews the literature relevant to the present work.
• Chapter 3 presents details of the experimental configurations and measurement
techniques used in the shock wave compression experiments.
• Chapter 4 describes the results and analysis related to the nonlinear elastic
response of diamond single crystals.
• Chapter 5 presents the results and analysis related to the refractive index
determination of diamond single crystals shocked along different orientations.
• Chapter 6 presents the results and analysis related to the Raman spectroscopy
measurements of shock compressed diamond crystals.
• Chapter 7 presents the results and analysis related to the measurements of elastic
4 limits of diamond single crystals shocked along different orientations, and
presents an explanation for the observed behavior.
• Chapter 8 provides a summary of the main findings of this work.
5 References:
1. J. E. Field, The Properties of Natural and Synthetic Diamond (Academic, San
Diego, 1992).
2. P. Loubeyre, F. Occelli, and R. LeToullec, “Optical studies of solid hydrogen to
320 GPa and evidence for black hydrogen,” Nature (London) 416, 613 (2002).
3. M. Seal and W. J. P. van Enckevort, “Applications Of Diamond In Optics,” Proc.
SPIE 0969, Diamond Optics, 144 (1989).
4. J. D. Lindl, Inertial Confinement Fusion, The Quest for Ignition and Energy Gain
Using Indirect Drive (Springer, New York, 1998).
5. B.V. Spitsyn, L.L. Bouilov, and B.V. Derjaguin, “Vapor growth of diamond on
diamond and other surfaces,” J. Cryst. Growth 52, 219 (1981).
6. S. Matsumoto, Y. Sato, M. Kamo and N. Setaka, “Vapor Deposition of Diamond
Particles from Methane,” Jpn. J. Appl. Phys. 21, L183 (1982).
7. J. J. Gracio, Q. H. Fan and J. C. Madaleno, “Diamond growth by chemical vapour
deposition,” J. Phys. D: Appl. Phys. 43, 374017 (2010).
8. J. E. Field, “The mechanical and strength properties of diamond,” Rep. Prog.
Phys. 75, 126505 (2012).
9. M. H. Grimsditch, E. Anastassakis, and M. Cardona, “Effect of uniaxial stress on
the zone-center optical phonon of diamond,” Phys. Rev. B 18, 901 (1978).
10. E. Anastassakis, A. Cantarero, and M. Cardona, “Piezo-Raman measurements and
anharmonic parameters in silicon and diamond,” Phys. Rev. B 41, 7529 (1990).
11. C. S. G. Cousins, “Elasticity of carbon allotropes. I. Optimization, and subsequent
6 modification, of an anharmonic Keating model for cubic diamond,” Phys. Rev. B
67, 024107 (2003).
12. O. H. Nielsen, “Optical phonons and elasticity of diamond at megabar stresses,”
Phys. Rev. B 34, 5808 (1986).
13. R. M. Denning, A. A. Giardini, E. Poindexter and C. B. Slawson,
“Piezobirefringence in diamond: further results,” Am. Mineral. 42, 556 (1957).
14. W. C. Schneider, “Investigations on the Piezo-optic Properties of Solids under
High Pressure,” Ph.D. thesis, Penn State University, 1970.
15. N. M. Balzaretti and J. A. H. da Jornada, “Pressure dependence of the refractive
index of diamond, cubic silicon carbide and cubic boron nitride,” Solid State
Comm. 99, 943 (1996).
16. M. H. Grimsditch and A. K. Ramdas, “Brillouin scattering in diamond,” Phys.
Rev. B 11, 3139 (1975).
17. J. M. Boteler, “Time resolved Raman spectroscopy in diamonds shock
compressed along [110] and [100] orientations,” Ph.D. thesis, Washington State
University, 1993.
18. A. L. Ruoff, in High Pressure Science and Technology, edited by K. D.
Timmerhaus and M. S. Barber, (Plenum, New York, 1979), p. 525.
19. W. R. Tyson, “Theoretical strength of perfect crystals,” Phil. Mag. 14, 925 (1966).
20. A. Kelly, W. R. Tyson, and A. H. Cottrell, “Ductile and brittle crystals,” Phil.
Mag. 15, 567 (1967).
21. A. L. Ruoff, “On the yield strength of diamond,” J. Appl. Phys. 50, 3354 (1979).
22. D. Roundy and M. L. Cohen, “Ideal strength of diamond, Si, and Ge,” Phys. Rev.
7 B 64, 212103 (2001).
23. D. K. Bradley, J. H. Eggert, D. G. Hicks, P. M. Celliers, S. J. Moon, R. C. Cauble,
and G. W. Collins, “Shock Compressing Diamond to a Conducting Fluid,” Phys.
Rev. Lett. 93, 195506 (2004).
24. M. D. Knudson, M. P. Desjarlais and D. H. Dolan, “Shock-Wave Exploration of
the High-Pressure Phases of Carbon,” Science 322, 1822 (2008).
25. R. S. McWilliams, J. H. Eggert, D. G. Hicks, D. K. Bradley, P. M. Celliers, D. K.
Spaulding, T. R. Boehly, G. W. Collins, and R. Jeanloz, “Strength effects in
diamond under shock compression from 0.1 to 1 TPa,” Phys. Rev. B 81, 014111
(2010).
26. D. K. Bradley, J. H. Eggert, R. F. Smith, S. T. Prisbrey, D. G. Hicks, D. G. Braun,
J. Biener, A. V. Hamza, R. E. Rudd, and G. W. Collins, “Diamond at 800 GPa,”
Phys. Rev. Lett. 102, 075503 (2009).
27. L. M. Barker and R. E. Hollenbach, “Laser interferometer for measuring high
velocities of any reflecting surface,” J. Appl. Phys. 43, 4669 (1972).
8 Chapter 2
BACKGROUND
This chapter presents background material on the mechanical and optical properties of diamond relevant to the present work. In Section 2.1, the crystallography of diamond is summarized. Section 2.2 reviews elasticity theory and previous studies regarding the elastic response of diamond. Section 2.3 discusses general aspects of the strain-induced refractive index changes or photoelasticity theory. In Section 2.4, the theory and experimental background regarding strain-induced splitting of the Raman spectrum are summarized. Section 2.5 reviews the theory and past work on the elastic limits and shear strength of diamond.
2.1 Crystal Properties
Diamond is a covalently bonded, monatomic crystalline form of carbon [1]. The diamond lattice, shown in Figure 2.1, consists of a two atom primitive unit cell with atoms located at (0 0 0) and (1/4a 1/4a 1/4a), repeated over a face-centered cubic lattice
[2], where a is the lattice constant. The diamond lattice can also be viewed as two interpenetrating face-centered cubic sub-lattices, with one displaced relative to the original by (1/4a 1/4a 1/4a) along the body diagonal. Each carbon atom is at the center of a tetrahedron with its four nearest neighbors forming the vertices of the tetrahedron.
The lattice symmetry is cubic, space group Fd3m, and the point group symmetry of the atoms is Td [1]. The lattice possesses three high symmetry directions: [100], with
9 a
Figure 2.1: Conventional unit cell of the diamond lattice, where a is the lattice constant.
Each atom is at the center of a tetrahedron of four nearest neighbors. Adapted from
Ref. 2.
10 four-fold symmetry; [111], with three-fold symmetry; and [110], with two-fold symmetry.
3 The lattice constant (a) is 0.357 nm [1] and the density (ρ0) is 3.515 g/cm [3].
Diamond crystals are classified according to the type and amount of the chemical impurities present in the crystals [1]. Only Type IIa crystals were used in this work.
Diamonds of this type have very low nitrogen impurity concentrations, typically 4-40 parts per million, and no measurable boron impurities [1]. They are typically colorless and have a very high thermal conductivity.
2.2 Elastic Response of Diamond
The extreme stiffness of diamond has resulted in considerable scientific and technological interest [1]. In this section, a phenomenological model for describing the nonlinear elastic response of cubic crystals, and how it may be applied to the results of shock wave compression experiments, is presented. Many aspects of the material presented here are similar to that presented by Boteler [4].
2.2.1 Linear Elastic Response
In the linear regime, the relationship between stress and strain tensors σij and εkl is given by Hooke’s Law [5]
σij=C ijkl εkl , (2.1) where Cijkl is the second-order elastic stiffness tensor. The symmetry of the stress and strain tensors reduce the independent components of the stiffness tensor from 81 to 36 constants. The stiffness tensor also includes the material symmetry. For diamond, a cubic crystal, the number of independent components reduces to three constants. In Voigt
11 notation, the second-order elastic constants are given by C11, C12, and C44 [5].
In anisotropic materials, pure mode elastic waves propagate only along certain crystallographic directions [6]. For a cubic crystal, such as diamond, these directions are
[7]: [100], [110] and [111]. The longitudinal and shear elastic wave velocities for these specific directions are related to the elastic constants by [7]:
C C [100]: C = 11 , C = 44 (2.2) l √ ρ s √ ρ
C +C +2C C −C C [110]: C = 11 12 44 , C = 11 12 , C = 44 (2.3) l √ 2ρ s √ 2ρ s ' √ ρ
C +2C +4C C −C +C [111]: C = 11 12 44 , C = 11 12 44 . (2.4) l √ 3ρ s √ 3ρ The shear wave velocities for propagation along [100] and [111] are degenerate, but for the [110] direction they are nondegenerate.
McSkimin and Andreatch [8] measured the longitudinal and shear wave velocities along the [110] direction in diamond using an ultrasonic technique and calculated the values of the second-order elastic constants, given in Table 2.1.
Table 2.1: The second-order elastic constants of diamond, determined by McSkimin and
Andreatch [8].
Modulus Value (GPa)
C11 1079 ± 5
C12 124 ± 5
C44 578 ± 2
12 2.2.2 Nonlinear Elasticity: Finite Strain Theory
At larger elastic strains, a material may no longer obey the linear relationship in
Eq. (2.1) and the elastic stress-strain relation becomes nonlinear. Following the approach of Thurston [9], finite strain theory details are summarized and used to describe the nonlinear elastic deformation of diamond.
Consider a solid body subjected to elastic deformation. A particle is a point within this body that always moves with the material being deformed. Prior to deformation, the position of a particle is given by the vector ai, which can also be used as its Lagrangian coordinate. After deformation of the body, the position of the particle is given by the vector xi, which can also be used as its Eulerian coordinate. The displacement of the particle following deformation is given by [9]
αi = xi−ai . (2.5)
Strain is a tensor quantity describing the change in the relative positions of the particles in a deformed body. The Green strain tensor is defined by comparing the distances between particles in the undeformed and deformed states and is written as [9]
1 ∂ xm ∂ xm η = −δ . (2.6) ij 2 ( ∂ a ∂a ij) i j
Using this strain definition, the internal energy per unit mass (U) can be expressed as a function of entropy and strain. A Taylor series expansion of the internal energy at constant entropy about zero strain gives [9]
1 1 ρ U ( S,η ) = ρ U ( S ,0 ) + C η + C η η + C η η η + ... , (2.7) 0 ij 0 ij ij 2 ijkl ij kl 6 ijklmn ij kl mn where
13 ∂U Cij = ρ0 , (2.8) ( ∂ ηij )S, η=0
∂U Cijkl = ρ0 , (2.9) ( ∂ ηij ∂ ηkl )S ,η=0
∂U Cijklmn = ρ0 . (2.10) ( ∂ ηij ∂ ηkl ∂ ηmn )S ,η=0
At zero strain, the stress is zero, requiring that the right-hand side of Eq. (2.8) vanish.
Eq. (2.9) and Eq. (2.10) define the second- and third-order isentropic elastic constants, respectively.
The thermodynamic stress, defined as negative in compression and conjugate to the Green strain, is given by [9]
∂U τij = ρ0 . (2.11) ( ∂ ηi j )S
Substituting Eq. (2.7) into Eq. (2.11) and taking the derivative provides the nonlinear elastic stress-strain relationship for an elastically deformed solid
1 τ = C η + C η η + ... . (2.12) ij ijkl kl 2 ijklmn kl mn
The experiments in this work involve the Cauchy stress, which is related to the thermodynamic stress by [9]
1 ∂ xl ∂ xk σij= τkl . (2.13) J ∂ ai ∂ a j
For uniaxial strain loading, the focus of the present work, the Jacobian of the coordinate transformation J reduces to [4]
∂ x ρ J = i = 0 . (2.14) ∣∂ a ∣ ρ j
14 2.2.3 Third-order Elastic Constants of Diamond
Third-order elastic constants may be used to describe the nonlinear elastic stress- strain response of a crystal. For cubic crystals, the lattice symmetry reduces the number of independent third-order elastic constants to six. Written in Voigt notation [5], they are:
C111, C112, C123, C144, C166, and C456 [4].
Third-order elastic constants are typically determined from sound speed measurements under hydrostatic pressure and uniaxial stress [10]. These experiments provide the pressure and stress derivatives of the second-order elastic constants for a total of six independent experimental measurements. The pressure and stress derivatives are related to linear combinations of the third-order elastic constants [11], and together provide sufficient information to determine the six constants uniquely. However, the extreme stiffness and brittle behavior of diamond under uniaxial stress conditions have precluded such measurements. Only the pressure derivatives of the second-order elastic constants of diamond have been measured [8], and they are listed in Table 2.2.
Table 2.2: Pressure derivatives of the second-order elastic constants of diamond, measured by McSkimin and Andreatch [8].
Pressure derivative Value dC 11 5.98 ± 0.7 dP dC 12 3.06 ± 0.7 dP dC 44 2.98 ± 0.3 dP
15 These pressure derivatives are related to the third-order elastic constants by [11]
dC C M +2C M = − 11 (C +2C )−3C −2 C , (2.15) 111 112 dP 11 12 11 12
dC 2C M +C M = − 12 (C +2C )+C , (2.16) 112 123 dP 11 12 11
dC C M +2C M = − 44 (C +2C )−C −2C −2C , (2.17) 144 166 dP 11 12 11 12 44
M where Cijk are the mixed isentropic-isothermal third-order elastic constants. The mixed third-order constants are converted into purely isentropic constants by [12]
S S M T S S S M M M ∂C ij Cijk = Cijk + (C k1+C k2+Ck3) α (C ij1+Cij2 +Cij3 ) α− , (2.18) ρ C ( ( ∂ T ) ) 0 P P
S M where Cijk and Cijk are the isentropic and mixed third-order constants, respectively,
S T is the temperature, CP is the molar specific heat, Cij are the isentropic second-order constants, and α is the thermal expansion coefficient.
Prior to this work, insufficient experimental data were available to determine the complete set of third-order elastic constants. Instead the third-order elastic constants of diamond have been estimated from valence-force-field models [13-15] and from ab initio simulations [16].
The valence-force-field models attempt to relate harmonic and anharmonic interatomic force constants to the second- and third-order elastic constants [17-18].
These interatomic force constants describe the bond-stretching and bond-bending parameters of the first- and second-nearest neighbors of an atom in a diamond lattice.
The pressure derivatives of the second-order elastic constants were inputs to the valence- force-field models. A model using three anharmonic parameters was developed by
16 Keating [17-18] and used by Grimsditch et al. [13] to estimate the third-order elastic constants. Both Anastassakis et al. [14] and Cousins [15] utilized five-parameter extensions of Keating's model and obtained considerably different results. Nielsen [16] performed ab initio simulations of diamond in different strain states, calculating the stress at a given strain. Fits to the ab initio stress-strain curves provided estimates for the third- order elastic constants.
The third-order elastic constants estimated from the valence-force-field models
[13-15] and the ab initio simulations [16] are listed in Table 2.3. These calculated results show large variations with each other, demonstrating both the inconsistency of the models and the need for experimental data to determine the third-order elastic constants.
Table 2.3: Estimates of the third-order elastic constants from valence-force-field models and ab initio simulations.
Modulus Ref. 13 Ref. 14 Ref. 15 Ref. 16
C111 ‒6260 ‒7367 ‒6475 ‒6300 ± 300
C112 ‒2260 ‒2136 ‒1947 ‒800 ± 100
C123 112 1040 982 0 ± 400
C144 ‒674 186 115 0 ± 300
C166 ‒2860 ‒3293 ‒2998 ‒2600 ± 100
C456 ‒823 76 ‒135 ‒1300 ± 100
2.2.4 Application to Uniaxial Strain
The elastic stress-strain relations for uniaxial strain along three specific directions for wave propagation in diamond are presented in this section. These are the [100],
17 [110], and [111] directions [7]. The stress-strain relationships for uniaxial strain along these directions were calculated using the finite strain theory relations summarized in
Section 2.2.2.
The finite strain theory relations were expressed relative to the principal cubic axes, in the crystallographic coordinate system, with the x1-axis along [100], the x2-axis along [010], and the x3-axis along [001]. A laboratory coordinate system, denoted by primed variables, can be defined with the x1'-axis oriented along the direction of shock wave propagation or uniaxial strain. The tensor transformations [5]
τ' ij = aik ajl τkl (2.19)
C 'ijkl = aℑ a jn ako alp C mnop (2.20)
C 'ijklmn = aio a jp akq alr ams ant C opqrst (2.21) relate second-, fourth- and sixth-rank tensors in the crystallographic system (unprimed) to the laboratory system (primed).
For uniaxial strain along [100], the axes of the laboratory and crystallographic coordinate systems are aligned. The strain tensor is expressed in both systems as
η11 0 0 η̃ = 0 0 0 , (2.22) [ 0 0 0] where
2 1 ρ0 η = −1 . (2.23) 11 2 (( ρ ) )
Substituting Eq. (2.22) into Eq. (2.12) and using the lattice symmetry, the thermodynamic stresses to second-order in strain are expressed in Voigt notation as
18 1 τ = C η + C η η (2.24) 1 11 1 2 111 1 1
1 τ = τ = C η + C η η (2.25) 2 3 12 1 2 112 1 1
τ4 = τ5 = τ6 = 0 . (2.26)
Using Eqs. (2.13) and (2.14), the Cauchy stresses are determined to be
ρ 1 σ = 0 C η + C η η (2.27) 1 ρ ( 11 1 2 111 1 1)
ρ 1 σ2 = σ 3 = C 12 η1 + C112 η1 η1 (2.28) ρ0 ( 2 )
σ4 = σ5 = σ6 = 0 . (2.29)
The lateral stresses are degenerate and the off-diagonal terms vanish.
For uniaxial strain along [110], the crystallographic and laboratory systems are no longer aligned. The laboratory system is defined with x1', x2', and x3'along the crystallographic directions [110], [110], and [001], respectively. The transformation matrix ã for this configuration is
1 1 0 √2 √2 ã = −1 1 . (2.30) 0 √2 √2 [ 0 0 1]
Transforming the strain tensor to the crystallographic system using the inverse transformation ηij = aki alj η'kl gives
η' η' 0 1 11 11 η̃ = η' η' 0 . (2.31) 2 11 11 [ 0 0 0]
From Eq. (2.12) and Eq. (2.31), the thermodynamic stresses to second-order in
19 strain, for uniaxial strain along [110], are expressed in the laboratory system as
1 τ ' = C ' η' + C ' η' η' . (2.32) ij ij11 11 2 ij1111 11 11
Applying the tensor transformations Eqs. (2.19)-(2.21) and using Eqs. (2.13)-(2.14), results in the following expressions for the principal Cauchy stresses in Voigt notation:
ρ 1 1 σ ' = 0 (C +C +2C ) η' + (C +3C +12C ) η' η' (2.33) 1 ρ ( 2 11 12 44 1 8 111 112 166 1 1)
ρ 1 1 σ ' 2 = (C 11+C 12−2C 44 ) η' 1 + (C 111+3C112−4C166 ) η' 1 η' 1 (2.34) ρ0 ( 2 8 )
ρ 1 σ '3 = C 12 η' 1 + (C112+C123 +2C 144 ) η'1 η'1 . (2.35) ρ0 ( 4 )
Note that in this strain state, the lateral stresses are nondegenerate.
For uniaxial strain along [111], the laboratory system is defined with x1' along
[111], x2' along [110], and x3' along [112]. The transformation matrix ã is
1 1 1 √3 √3 √3 −1 −1 2 ã = (2.36) √6 √6 √6 1 −1 0 [ √2 √2 ] and the resulting strain tensor in the crystallographic system is
η' η' η' 1 11 11 11 η̃ = η' η' η' . (2.37) 3 11 11 11 [η'11 η' 11 η'11 ]
Thermodynamic stresses to second-order in strain, for uniaxial strain along [111] in the laboratory system, are also given by Eq. (2.32). Using Eqs. (2.13)-(2.14), Eqs. (2.20)-
(2.21), and Eq. (2.36), the principal Cauchy stresses expressed in Voigt notation are found
20 to be
ρ 1 σ ' = 0 C +2C +4 C η' + 1 ρ ( ( 11 12 44) 1 3 (2.38) 1 (C +6C +2C +12C +24 C +16C ) η' η' 18 111 122 123 144 166 456 1 1)
ρ 1 σ ' 2 = σ ' 3 = (C 11+2C 12−2C44 ) η' 1 + ρ0 ( 3 . (2.39) 1 (C +6C +2C −8C ) η' η' 18 111 112 123 456 1 1 )
As in the case of uniaxial strain along [100], the lateral stresses are degenerate.
The Cauchy stresses derived for uniaxial strain along [100], [110], and [111] in this section will be used in subsequent chapters to analyze the shock wave data obtained in this work.
2.3 Refractive Index of Shocked Diamond
Diamond is a cubic crystal and is optically isotropic: it has a single dielectric constant for all propagation and polarization directions, and a refractive index of 2.426 at
532 nm [19]. However, the application of uniaxial strain reduces the symmetry of the crystal and it becomes optically uniaxial or biaxial [5,20]. The optical indicatrix and photoelasticity theory used to calculate the optical properties of the crystal following the application of strain are summarized next.
2.3.1 The Optical Indicatrix
The development of the optical indicatrix in this section follows that presented by
Nye [5] and Yariv and Yeh [20]. Consider the propagation of electromagnetic radiation in
21 a homogenous, nonabsorbing, magnetically isotropic medium. The electric displacement
Di within the medium is related to the electric field Ej by the constitutive equation
Di = ε0εijEj [5, 20], where ε0 is the permittivity of free space and εij is the dielectric tensor.
The dielectric tensor is diagonal when expressed in the principal coordinate system and, in general, has three independent components [20]. Crystal symmetry may further reduce the number of independent components.
The optical indicatrix, shown in Figure 2.2, is a construction used to find the two velocities (corresponding to the refractive indices) and respective polarizations for electromagnetic waves propagating in an arbitrary direction in a dielectric solid. The indicatrix is a surface of constant electric energy density in Di space, taking the shape of an ellipsoid. If the principal axes of the dielectric tensor are x1, x2, and x3, the indicatrix has the form [5, 20]
2 2 2 x1 x2 x3 2 + 2 + 2 = 1 , (2.40) n1 n2 n3
where nk = √εkk are the refractive indices along the principal axes. The optical indicatrix can be written equivalently as [5, 20]
Bij xi x j = 1 , (2.41)
( −1 ) where B i j = ε ij is the dielectric impermeability tensor.
To determine the polarization and velocity of electromagnetic waves propagating in a crystal, consider the indicatrix shown in Figure 2.2. The propagation direction of the wave is given by the line OP and is perpendicular to the shaded elliptical cross section.
The semi-axes of the ellipse are shown as OA and OB. The polarizations of the wave are along the semi-axes of the ellipse, and the refractive indices for the wave are given by the
22 Figure 2.2: The general form of an optical indicatrix ellipsoid construction. The principal axes of the dielectric tensor are denoted by x1, x2, and x3. Point O is the origin.
The line OP is perpendicular to the shaded ellipse. The segments OA and OB are the semi-axes of the ellipse, and represent the polarizations and refractive indices for electromagnetic waves propagating along OP. Reproduced from Ref. 5.
23 radii of the ellipse, namely, OA and OB. The wave velocity is related to the refractive index by v = c/n. Propagation directions for which the refractive index is the same for both polarizations are called optical axes.
In cubic crystals, the dielectric tensor has only one independent component and the indicatrix is a sphere. Therefore all propagation directions are optical axes. This means that cubic crystal are not birefringent; the refractive index is the same for all propagation directions and polarizations of light.
For hexagonal, tetragonal and trigonal crystals, the dielectric tensor has two independent components so two of the principal radii of the indicatrix are the same. Only one optical axis is present and these crystals are said to be uniaxial. The dielectric tensor of orthorhombic, monoclinic and triclinic crystals has three independent components making the indicatrix a triaxial ellipsoid. These crystals have two optical axes and are called biaxial [5, 20].
2.3.2 Photoelasticity Theory
When subjected to an arbitrary deformation, the dielectric properties of the crystal may change. The deformation may change the shape, size and orientation of the
( −1 ) indicatrix, and is expressed as a change in the coefficients Bij = ε ij using the phenomenological model developed by Pockels [5, 20-21].
The change in the dielectric impermeability tensor ΔBij is related to strain by the elasto-optic tensor pijkl [5, 20]:
Δ Bij = pijkl ηkl . (2.42)
The elasto-optic tensor reflects the crystal symmetry and the symmetry of the dielectric
24 and strain tensors. Therefore, in the absence of crystal symmetry considerations [5, 20],
pijkl = p jikl = pijlk , (2.43) giving 36 possible independent components. After crystal symmetry considerations are incorporated, the independent components will be reduced accordingly. Pockels' phenomenological model has been extended to non-linear strain contributions by Vedam and Srinivasan [22]. To second-order in strain, the change in the dielectric impermeability tensor becomes [22]
Δ Bij = pijkl ηkl+ pijklmn ηkl ηmn . (2.44)
The deformed crystal will have the symmetry elements that are common to both the undeformed crystal and the applied deformation [23]. Similarly, application of photoelastic theory results in a new indicatrix having the symmetry of the deformed crystal. The dielectric properties of the deformed crystal, such as the principal axes, principal refractive indices, and the optical axes, may be determined from the new indicatrix.
2.3.3 Application to Diamond
Diamond is an optically isotropic crystal. However, following uniaxial strain compression, the crystal symmetry is reduced according to the direction of the applied strain. Photoelasticity theory is used to predict the dielectic properties of the strained crystal. The dielectric response of diamond with respect to strain, to the first-order, is described by the three independent elasto-optic constants p1111, p1122, and p1212. Nine additional elasto-optic constants are required to describe the response to the second-order in strain [22].
25 Uniaxial stress [24-25], hydrostatic pressure [26-27] and Brillouin scattering [28] measurements have been carried out on diamond to determine the elasto-optic constants.
Uniaxial stress measurements of the stress induced birefringence provide good determinations of (p1111 ‒ p1122) and p1212 [24]. Determining p1111 and p1112 individually requires measuring the absolute change in the refractive index, which is more difficult and susceptible to error [24]. Hydrostatic pressure measurements of the change in refractive index provide (p1111 + 2 p1122) [26]. In Brillouin scattering measurements, the intensity of the scattered light is related to the elasto-optic constants. The relative intensity of the different scattering modes provides ratios of the photo-elastic constants
[28].
To date, the best available and consistent set of first-order elasto-optic constants are the ones presented by Grimsditch and Ramdas [28]. These constants were derived from the uniaxial stress measurements to 3 kbar by Denning et al. [25] and the hydrostatic pressure measurements to 7 kbar by Schneider [26]. Brilloiun scattering measurements [28] and more recent hydrostatic pressure measurements up to 9 GPa [27] are in good agreement with these studies. The first-order elasto-optic constants of diamond [28] are listed in Table 2.4. The higher-order elasto-optic constants of diamond have not been reported in the literature. One part of the present work has partially addressed this issue.
26 Table 2.4: First-order elasto-optic constants of diamond [28].
Elasto-optic constant Value
p1111 ‒0.249
p1122 0.043
p1212 ‒0.172
2.4 Raman Spectrum of Shocked Diamond
Time-resolved spectroscopy measurements in shock compression experiments can provide an understanding of the microscopic changes in the shocked state, complementing continuum measurements [29]. Raman spectroscopy, in particular, can probe the atomic/molecular changes occurring in shocked crystals [30]. Diamond is well suited for time-resolved Raman spectroscopy measurements under shock wave compression because of its large Raman cross section [31].
The Raman spectrum of diamond under ambient conditions consists of a single peak at 1332.5 cm-1 [32]. This peak is triply-degenerate and corresponds to the relative motion of the two face-centered cubic sub-lattices that make up the diamond lattice. The normal modes of these vibrations are along the [100], [010], and [001] directions [33].
Application of uniaxial strain in shock compression experiments changes the symmetry and interatomic spacing of the lattice, which alter these vibrational modes. Depending on the loading direction of the applied strain, the lattice symmetry is lowered and the degeneracy of the vibrational modes may be partially or completely lifted.
In this section, the theoretical background for predicting the strain-induced frequency shifts is summarized, and results from past experimental work are presented.
27 The Raman spectroscopy results of the present study are based heavily, and expand, on the work done by Boteler [4] and Boteler and Gupta [34-35], and much of the material summarized here is also presented in Boteler's thesis [4].
2.4.1 Strain-induced Frequency Shifts
Ganesan et al. [36] have developed a microscopic theory for predicting strain induced changes in the Raman spectrum of crystals of the diamond structure. The quasi- harmonic approximation was used to determine the lowest-order changes in the lattice vibrations. Atoms in the strained crystal were assumed to vibrate harmonically about new equilibrium positions in the strained lattice. The new vibrational frequencies were determined by the effective anharmonic force constants in the strained diamond lattice, denoted as p, q, and r. First-order perturbation theory was used to calculate the strain- induced frequency shifts. The frequency of the vibrational modes in the strained lattice,
ω, are calculated by solving the determinant [36]
pη11 +q ( η22+η33 )−λ 2r η12 2r η13
2r η12 pη22 +q ( η11 +η33)−λ 2r η23 = 0 (2.45)
∣ 2r η13 2r η23 p η33+q ( η11 +η22)−λ∣ with
2 2 λ = ω −ω0 , (2.46) where ηij are the components of the strain tensor in the crystallographic system, and ω0 is the frequency of the ambient Raman mode. The phonon polarization directions of the new vibrational modes are given by the eigenvectors corresponding to Eq. (2.45).
28 2.4.2 Past Experimental Work
Studies of the splitting of the diamond Raman spectra have been performed under uniaxial stress and uniaxial strain conditions. These measurements provided the values of the anharmonic force constants p, q, and r, presented in Eq. (2.45).
The splitting of the diamond Raman spectra was first examined by Grimsditch et al. [13]. Uniaxial stresses up to 1 GPa were applied along the [100] and [111] directions of diamond crystals. The diamond Raman peak split into single and doublet peaks for both these directions of applied stress. The splitting of the Raman peak at these low stresses was not sufficient to resolve the split peaks, so polarization techniques were used to resolve the frequency shift of each peak, individually. The anharmonic force constants were determined from the stress dependence of the frequency shifts, and are listed in
Table 2.5.
Table 2.5: Anharmonic force constants of diamond.
Constant Grimsditch et al. [13] Boteler and Gupta [36] p 2 ‒2.81 ± 0.19 ‒2.55 ± 0.21 ω0 q 2 ‒1.77 ± 0.16 ‒1.70 ± 0.11 ω0 r 2 ‒1.9 ± 0.2 ‒1.96 ± 0.07 ω0
Gupta et al. [30] demonstrated the feasibility of time-resolved measurements of the Raman spectrum of diamond undergoing shock wave compression. They performed two experiments on [110] oriented crystals shocked to a final state of 12.1 GPa. Uniaxial
29 strain along this direction was predicted to lift the three-fold degeneracy completely. Due to the low spectral resolution of the instrumentation, the three singlet peaks were not resolved. However, the observed shift of the aggregate peak was comparable to the expected splitting predicted by theory.
Boteler and Gupta [4,34-35] performed shock wave uniaxial strain experiments on diamond crystals, oriented along [110] and [100] to 45 GPa. For uniaxial strain along
[110], the three-fold degeneracy was lifted completely and the shift of the three singlet peaks provided the experimental data to determine all three anharmonic force constants, listed in Table 2.5. These constants were used to predict the shift of the singlet and doublet peaks for uniaxial strain along [100]. The results of the experiments performed along [100] were in good agreement with the predictions. The details of this work are given in Boteler's thesis [4]. The anharmonic force constants determined in these studies
(uniaxial stress and uniaxial strain) are summarized in Table 2.5.
Although no experimental work was reported for [111] experiments, Boteler also calculated the expected frequency shifts and phonon polarization directions for uniaxial strain along [111] [4]. Here, the experimental measurements for shock compression along [111] are reported.
2.5 Elastic Limits and Strength of Shocked Diamond
Measurements of the Hugoniot elastic limit (HEL) of a material undergoing plane shock wave compression are useful for determining the material strength and understanding the yield response under rapid dynamic loading [37]. These measurements
30 are likely ideal for determining the strength of diamond crystals because the brittle nature of diamond makes it difficult to obtain strength results under quasistatic loading conditions [1, 38]. The theoretical background for using Hugoniot elastic limit measurements to determine material strength, and the past work related to the elastic limits and dynamic yield strength of diamond are presented in this section.
2.5.1 Theoretical Background
The Hugoniot elastic limit of a crystal corresponds to the amplitude of the longitudinal elastic wave in a shock compressed crystal undergoing elastic-inelastic deformation. For isotropic solids, the elastic wave amplitude can be related to the maximum resolved shear stress, a measure of the material strength. In single crystals, the elastic wave amplitude or the Hugoniot elastic limit can vary significantly with crystal orientation [39]. Hence, a more meaningful measure of material strength for single crystals is the maximum shear stress (corresponding to the elastic wave amplitude) attained for the relevant slip systems operative under the loading conditions of interest.
When the elastic limit of a crystal is exceeded, slip often (but not always) occurs on the widest separated planes in the direction of the shortest Burgers vector [40-41].
Brar and Tyson [41] used a simple Peierls model of a dislocation to show that dislocation mobility is greatest on the {111} plane along a 〈〉110 direction for diamond structure crystals. This result is consistent with the experiments of Evans and Wild [42].
They performed three-point bending tests on diamond plates heated to 1800 °C. The samples were observed to bend up to an angle of 20°, and the direction of the slip lines were consistent with the operative slip system being {111}〈〉110 . The accurate
31 determination of relevant slip systems, either theoretically or experimentally, remains a challenging issue.
Under plane shock wave or uniaxial strain loading, the maximum resolved shear stress for a slip system may be calculated from the elastic wave amplitude (Hugoniot elastic limit) following the theoretical developments of Johnson et al. [37]. In Section
2.2.4, the crystallographic coordinate system, denoted as unprimed, was defined as having the axes xi along the principal cubic axes. The laboratory coordinate system, denoted as primed, was defined as having the x1'-axis oriented along the direction of shock wave propagation. A third coordinate system, denoted as double-primed, is defined such that the x3''-axis is normal to the slip plane and the x1''-axis is in the direction of slip.
To obtain the resolved shear stress for the relevant slip system of interest, the longitudinal stress (σ1') corresponding to the elastic wave amplitude is measured in a shock compression experiment. Next, the lateral stresses (σ2' and σ3') corresponding to σ1' may be calculated using finite strain theory and the elastic constants as presented in
Section 2.2.4. These three quantities provide the complete stress tensor in the laboratory coordinate system since the other stress components are zero. The stress tensor is then transformed from the laboratory system to the coordinate system of the relevant slip plane according to the orthogonal transformation xi' ⇒ xi'' connecting the two coordinate systems [37]. The resolved shear stress applicable to the slip system is given by the element τ13'' of the transformed stress tensor. The resolved shear stress τ13'', corresponding to the elastic wave amplitude in the primed state, provides a measure of the elastic limit or material strength which may be compared for compression along
32 different crystallographic orientations.
2.5.2 Past Studies on Diamond Strength
A. Theoretical Studies
The shear strength of diamond has been estimated from theoretical calculations, and was first calculated by Tyson [43]. Slip was assumed to occur on the the most widely separated lattice plane in the direction of the shortest Burgers vector, namely the {111}
〈〉110 slip system. The shearing force required to move a single atom in the slip system over the potential energy barrier of the lattice was calculated. The maximum shearing force per unit area was used as the ideal shear strength of diamond, which was calculated to be 91.6 GPa. Kelly et al. [44] revisited these calculations and revised the shear strength to be 121 GPa.
Using a von Mises yield condition, applicable to isotropic solids, the yield strength of diamond was estimated by Ruoff [45,46]. He calculated the yield strength of
Si and Ge (both crystals possessing the diamond structure) from the Hugoniot elastic limit measurements. He found that the ratio of the yield stress to shear modulus had nearly the same value (0.0656) for both materials. Assuming the same ratio for diamond, he estimated the yield stress of diamond to be 35 GPa. Because Ruoff's approach is valid only for isotropic solids, its validity for single crystals is questionable.
Using molecular dynamics simulations, Roundy and Cohen [47] calculated the stress-strain curves of diamond, silicon and germanium for shear deformation along the
{111}〈〉 112 slip system. The shear strength of each crystal was taken as the maximum
33 stress in the stress-strain curve, and they reported a shear strength for diamond of 93 GPa at a strain of 0.3. They observed a large instability in diamond beyond the shear strength, as the crystal relaxed into a graphitic structure. In contrast, silicon and germanium showed a much less dramatic instability, likely due to their inability to form strong π bonds and form a graphitic structure [47]. It should be pointed out that an elastic shear strain of 0.3 may be unrealistic.
Using molecular dynamics simulations, Umeno et al. [48] calculated the critical resolved shear stresses as a function of normal stress for diamond, silicon, germanium and silicon carbide for shear in the {111}〈〉〈〉 110 and {111} 112 slip systems. Their results complemented the results of Roundy and Cohen [47]. The critical resolved shear stress of diamond was found to increase with compressive stress normal to the slip plane. In the other materials, the critical resolved shear stress decreased with compressive stress normal to the slip plane. They suggest that the difference between the behavior of diamond and the other materials that were studied is due to the extremely strong convalency of the carbon-carbon bonding [48].
B. Experimental Studies
Shock wave measurements of the elastic limit of diamond are extremely limited and show very disparate results. Pavlovskii [49] reported the earliest shock compression measurements on single-crystal diamond. [100] oriented crystals were compressed up to
600 GPa and the shock velocities were recorded using electrical self-shorting pins. The shock velocity-particle velocity relation was found to be D = 12.16 + 1.00 up, mm/μs
[49]. For small particle velocities below the elastic limit, this relation predicts a wave
34 speed that is significantly slower than the elastic longitudinal sound velocity of
17.52 mm/μs [8]. Instead, the relation corresponds to an inelastic wave in diamond, without any consideration for an elastic precursor [50].
Kondo and Ahrens [50] measured the shock wave and free surface velocities of shocked diamond using the inclined mirror technique and electrical self-shorting.
Diamond samples were oriented at an angle of 15.4 ± 0.6° from the [111] direction in the
{110} plane and shock compressed to peak stresses of ~200 GPa. Their measurements suggested a two-wave structure in the diamond samples, with the first and second waves having wave velocities of ~20 mm/μs and ~15 mm/μs, respectively. They also reported that their electrical pins failed to respond to the first wave, suggesting this as the reason that Pavlovskii [49] did not detect it either. They reported Hugoniot elastic limits of
62.0 ± 5.3 and 63.6 ± 27.7 GPa. They calculated a shear strength of 30 GPa by assuming an isotropic response using the offset between the Hugoniot elastic limit and a
Murnaghan-Birch equation of state calculated from ultrasonic data. This method assumes shearing along a plane 45° from the loading direction and the validity of their equation of state at high pressure. As noted earlier, the maximum resolved shear stresses along relevant slips systems are a more appropriate measure of the shear strength of a single crystal.
Knudson et al. [51] reported the first observations of a two-wave structure and the
Hugoniot elastic limit in diamond. Two [110] oriented diamonds were shock compressed in plate-impact experiments to stresses of ~2 and ~3 Mbar. Diamond samples were sandwiched between a tantalum driver and a LiF window. Particle velocity histories at the diamond-LiF interface were obtained using velocity interferometry (VISAR). Two-
35 wave structures were observed: a sharp elastic wave followed by a decay and the arrival of a second wave. The amplitude of the first wave, the Hugoniot elastic limit, in both experiments was ~90-100 GPa. The VISAR diagnostics used in these experiments had a time resolution of ~1-2 ns. This resolution was inadequate to fully resolve the details of the particle velocity history and prevented an accurate determination of the Hugoniot elastic limit [51].
The most recent work on the elastic limits of diamond was performed by
McWilliams [52] and McWilliams et al. [53]. Diamond crystals were laser shocked along the [100], [110], and [111] directions to peak stresses that were estimated to range from 100 GPa to 1 TPa. Shock waves produced in these laser-driven experiments were of short duration (<6 ns) and were inherently unsteady, resulting in an estimated 7.6% variation in stress at the shock front during transit [53]. Line VISAR measurements provided free-surface particle velocity histories and shock wave velocities in the diamond crystals and in reference quartz samples. From these shock wave velocities, the stress in the diamonds were determined using impedance matching calculations. The uncertainties in the shock velocity measurements were estimated to be ~2-4%, resulting in particle velocity uncertainties of ~3-15%, not including uncertainties due to shock wave unsteadiness. Even so, they reported elastic limits which corresponded to the weakest elastic wave amplitude observed for compression along each orientation. The reported elastic limits were 80 ± 12 GPa, 81 ± 6 GPa, and 60 ± 3 GPa for uniaxial strain along the
[100], [110], and [111] directions, respectively [53].
To determine the strength, lateral stresses were calculated using finite strain to third order using the elastic constants of Anastassakis et al. [14]. The resolved shear
36 stresses for the {111}〈〉 110 slip system were calculated to be 25 ± 6 GPa, 29 ± 4 GPa, and
16 ± 4 GPa for uniaxial strain along the [100], [110], and [111] directions, respectively
[53]. The von Mises criterion was used to evaluate strength beyond yielding, although this yield criteria is not applicable to single crystals. They claim that diamond retains strength when compressed along [111], but the [100] and [110] orientations lose strength.
However, the improper use of the von Mises yield criterion, and the large uncertainty in their strength calculations (25-50%) [53] cast doubt on their conclusions. Additionally, analysis of the time-dependent response observed beyond the elastic wave requires numerical simulations of the time-dependent inelastic deformation in diamond. Such a material description is currently unavailable. In view of the experimental and conceptual difficulties with this work, it is difficult to draw firm conclusions about the findings reported in Refs. 52 and 53.
In summary, the reported shear strengths of diamond from theoretical calculations
[43-47] and experimental measurements [50,53] are significantly different. The scarcity of experimental data and difficulties with existing data demonstrate the need for precise measurements in well characterized, shock wave experiments.
37 References
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39 25. R. M. Denning, A. A. Giardini, Edward Poindexter and C. B. Slawson,
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40 Phys. Rev. B 66, 014107 (2002).
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41 48. Y. Umeno, Y. Shiihara and N. Yoshikawa, “Ideal shear strength under
compression and tension in C, Si, Ge, and cubic SiC: an ab initio density
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42 Chapter 3
EXPERIMENTAL METHODS
In the present study, three types of measurements were carried out to examine the anisotropic behavior of shocked diamond single crystals: mechanical response, refractive index changes, and Raman spectra. Because the first two measurements involve the same experimental method (laser interferometry), details regarding the two types of experiments (laser interferometry and Raman spectroscopy) are presented in this chapter.
Relevant information about the diamond samples and other materials used in this work is summarized first.
3.1 Materials Characterization
In addition to diamond single crystals, three other materials (OFHC copper, tantalum, and lithium fluoride crystals) were used in the impact experiments.
Characterization of these materials and their properties are summarized in this section.
3.1.1 Diamond
Type IIa, natural or synthetic, diamond single crystals were obtained from
Dubbeldee Harris Diamond Corporation and Element Six. Diamonds of this type are characterized by their low nitrogen content (< 40 ppm), as determined from the infra-red absorption spectrum [1]. All diamond samples were transparent, optically polished circular or square plates 0.25-1.0 mm thick and 1.0-4.0 mm wide. The crystal orientation
43 of each sample was measured using Laue x-ray diffraction, and only samples with clean diffraction spots and an orientation within 3° of the [100], [110] or [111] directions were used in the present work. Small sample sizes prevented accurate measurements of the density and sound speeds. Literature values for the density (3.515 g/cm3) [2], and sound speeds (listed below) were used in the present work.
Table 3.1: Longitudinal and shear sound speeds for single crystal diamond from Ref. 3.
Propagation Longitudinal sound speed Shear sound speed direction (mm/μs) (mm/μs) [100] 17.52 12.82 [110] 18.32 11.66, 12.82* [111] 18.58 12.06 * Shear sound speeds are nondegenerate for the [110] propagation direction.
3.1.2 OFHC Copper
Oxygen-free high conductivity (OHFC) copper (99.995% pure) was used as an impactor and buffer material in most experiments. The mechanical properties of each sample were measured to ensure consistency in these experiments. Sample densities were measured using the Archimedean method and ranged from 8.91 to 8.95 g/cm3, in good agreement with the published value of 8.93 g/cm3 [4]. Longitudinal and shear sound speeds were measured using the pulse-echo technique. Measured longitudinal sound speeds ranged from 4.75 to 4.78 mm/μs and measured shear sound speeds ranged from 2.28 to 2.35 mm/μs, comparing well with the respective published values of 4.76 mm/μs and 2.33 mm/μs [4]. The linear shock velocity-particle velocity (D-up) Hugoniot used for calculations of the shock wave response at the stresses of interest was [5]:
44 D=3.933+1.5u p . (3.1)
It should be noted that at a particle velocity of zero, the shock velocity calculated from
Eq. (3.1) is not the longitudinal sound speed. Because the response of copper is elastic- plastic at the stresses of interest, the shock velocity calculated at zero particle velocity corresponds to the sound speed calculated from the bulk modulus.
3.1.3 Tantalum
Annealed tantalum (99.9% pure) was used as an impactor in a single experiment to check the reproducibility of the diamond results using a different impactor material.
The Ta sample was characterized identically to the Cu samples. The density, and longitudinal and shear sound speeds of the Ta sample were 16.68 g/cm3, 4.18 mm/μs, and
2.12 mm/μs, respectively. The measurements were in good agreement with the respective published values of 16.66 g/cm3, 4.16 mm/μs, and 2.09 mm/μs [4]. The linear shock velocity-particle velocity (D-up) Hugoniot used for calculations of the shock wave response at the stresses of interest was [5]:
D=3.293+1.307u p . (3.2)
It should be noted that at a particle velocity of zero, the shock velocity calculated from
Eq. (3.1) is not the longitudinal sound speed. Because the response of tantalum is elastic- plastic at the stresses of interest, the shock velocity calculated at zero particle velocity corresponds to the sound speed calculated from the bulk modulus.
3.1.4 Lithium Fluoride
Ultra-pure, [100] oriented lithium fluoride single-crystals were used as optical
45 windows in all mechanical response and refractive index measurements. LiF samples were discs, with both large faces parallel and polished with an optical finish. Crystal orientations were verified to be [100] using Laue x-ray diffraction, and crystals with multiple or elongated diffraction spots were not used in this work. Sample densities were calculated from volume and mass measurements and ranged from 2.60 to 2.67 g/cm3, comparing well with the published value of 2.64 g/cm3 [6], which was used in the analysis in this work. Measured longitudinal sound speeds ranged from 6.47 to
6.61 mm/μs and measured shear sound speeds ranged from 4.88 to 4.95 mm/μs, comparing well with the respective published values of 6.57 mm/μs and 4.91 mm/μs [7].
The linear shock velocity-particle velocity (D-up) Hugoniot model used for calculations of the shock wave response was [6]
D=5.148+1.353up . (3.3).
3.2 Mechanical Response and Refractive Index Experiments
3.2.1 Experimental Configuration
Figure 3.1 shows the experimental configuration used to measure the mechanical response and the refractive index of shocked diamond crystals. A powder gun or a two- stage gun launched a projectile, which impacted the target to produce a shock wave in the diamond sample. A multi-point velocity interferometer system for any reflector (VISAR)
[8-10] recorded two particle velocity histories, one at each interface of the diamond sample. The arrival times of the shock wave at each interface were used to determine the shock wave velocity in diamond. The particle velocity history measured at the diamond-
46 Copper Impactor Diamond LiF Window Buffer
PZT Pin
To VISAR
PZT Pin
Projectile Target
Figure 3.1: Experimental configuration for mechanical response and refractive index measurements of shocked diamond. Laser interferometry was used to record the particle velocity histories at both diamond interfaces, and PZT pins provided a measurement of the shock wave tilt. Dimensions shown are not to scale.
47 LiF interface was used to determine the nonlinear elastic response and Hugoniot elastic limits of diamond. The particle velocity history measured at the Cu-diamond interface provided the change in the optical path length through the diamond, which was used to determine the refractive index.
3.2.2 Target Construction
A schematic, cross-sectional view of the target is shown in Figure 3.2. A diamond sample was mounted on a Cu buffer and backed by a LiF optical window. Using a single lens tube, two fiber and lens coupled VISAR probes monitored the velocities of both interfaces of the diamond. Lead zirconate titanate (PZT) piezoelectric pins positioned around the diamond sample provided a measure of the shock wave tilt at the Cu-diamond interface. The VISAR and diagnostic instrumentation were triggered by an optical cutoff trigger.
The Cu buffers were 1" diameter and 1-1.2 mm thick discs prepared with a 3 μm final polish. LiF optical windows were 4-8 mm in diameter and 3-4 mm thick, depending on the available samples. Thin aluminum mirrors were vapor deposited at the centers of the diamond samples and LiF optical windows. Mirrors deposited on the diamond samples were typically 2-3 mm in diameter, and < 1 mm in diameter on the LiF windows to simplify alignment when bonding.
In each target the diamond was bonded to the center of the Cu buffer with the aluminum mirror against buffer. The LiF optical window was bonded to the diamond with the mirror toward the diamond. The LiF window was positioned such that the mirror on the window was centered on the diamond and that the mirror on the diamond
48 Target Ring Optical Fiber to Photodiode
Optical Trigger Target Plate Collection Optics and Mount PZT Pins Focusing Lens: 18 mm Focal Length Cu Buffer
Diamond Optical Fibers Sample to VISAR
LiF Window
Collimating Lens: PZT Pin 24 mm Focal Length Plate
Cut-off Beam Lens Tube
Lens Tube Bracket
Insulating Optical Trigger Spacer Laser Diode and Mount Laser Diode Power
Figure 3.2: Cross section, schematic view of the target used in mechanical response and refractive index measurements. Dimensions shown are not to scale.
49 was not completely obstructed by the mirror on the LiF window. All bonds were made using 815 epoxy and samples were pressed for 24 hours to allow the bond to cure before proceeding to next bonding step. Figure 3.3a shows the Cu buffer, diamond sample and
LiF window after bonding.
The bonded samples were then epoxied to a plate containing four PZT pins positioned in a square around the sample, with opposing pins 0.625-0.675" apart. Figure
3.3b shows the samples bonded to the PZT pin plate. The sample and the PZT pin assembly were mounted on a target plate, which was mounted to a target ring. The target surface was aligned parallel to the face of the target ring using three sets of bolts and
Belleville washers placed between the target ring and the plate.
Laser light was brought to the sample and returned to the interferometer using 200
μm diameter optical fibers. A fiber connector with two optical fibers, 1 mm apart was attached to a threaded lens tube. The lens tube contained a 24 mm focal length collimating lens and a 18 mm focusing lens, both 12 mm in diameter. This lens configuration produced a demagnified image of the fiber tips on the mirrors in the target, with spots 0.75 mm apart and 150 μm in diameter. The lens tube was mounted in a bracket attached to the target plate. The spots were aligned so that one spot was focused on the mirror at each interface of the diamond, and that the returning beam intensity from both spots was maximized.
An optical cutoff trigger was attached to the impact side of the target plate (Figure
3.1). It consisted of a 650 nm or 670 nm laser diode and 4.6 mm focal length lens on one side of the target plate, focused over the center of the Cu buffer and collected by a 4.6 mm focal length lens and a 400 μm diameter optical fiber. The distance between the laser
50 (a) Cu Buffer
LiF Window
Diamond Al Mirror
(b) PZT Pin
Diamond LiF Window
Al Mirror
PZT Pin Plate
Figure 3.3: Photographs of the bonded samples (a) before and (b) after bonding to the
PZT pin plate. The square near the center was the diamond sample, mounted on a Cu buffer and backed by a LiF optical window. The small circle nearly covering the diamond was the mirror deposited on the LiF window. The mirror on the diamond was present but not clearly shown in these photographs.
51 beam and the Cu buffer was measured and was typically 15 mm. A photograph of the assembled target is shown in Figure 3.4.
3.2.3 Projectile Construction
For the powder gun experiments, depending on the projectile velocity, either 3.4" or 2.25" long lexan projectiles were used. A 1" diameter by 3 mm thick Cu impactor was prepared with a 3 μm final polish and bonded into a pocket in the face of the projectile using 815 epoxy. After curing for 24 hours in a press, the tilt of the projectile face was measured and only projectiles with a tilt of 0.5 mrad or less were used.
For the two stage gun experiments, a Cu or Ta impactor was machined into a ‘top hat’ nominally 1.5 mm thick and with the impact face 0.84" in diameter. After preparing the impactor with a 3 μm final polish, a heated press was used to mold the impactor into a lexan cylinder. The molded part was machined to the proper dimensions and a face cut was made on Cu impactors. A polyethylene boot was attached to the rear of the projectile. Projectile and impactor concentricity and tilt, and impactor planarity were checked after assembly. Photographs of representative powder gun and two-stage gun projectiles are shown in Figure 3.5.
3.2.4 Instrumentation
The instrumentation for these experiments consisted of a multi-point VISAR [10], optical trigger, PZT pins and relevant diagnostics. Optical and electrical signals were transmitted between the target and the instrumentation using optical fibers and coaxial cables.
52 Figure 3.4: Photograph of the assembled target. The VISAR optical fiber was connected to the lens tube near the top of the picture. The optical trigger diode on the right focused a laser beam across the target to the optical fiber on the right. PZT pin cables in the background and foreground were connected to PZT pins at the center of the target.
53 (a) (b)
(c)
Figure 3.5: Photographs of representative (a) long projectile for powder gun, (b) short projectile for powder gun, and (c) two stage gun projectile used in the experiments.
54 As shown in Figure 3.1, particle velocity histories at both interfaces of the diamond were recorded using a multi-point VISAR. Light from a 532 nm Verdi laser was transmitted to and reflected from the mirrors and was collected by two graded-index optical fibers, one for each of the mirrored interfaces. The two optical signals were passed through a beam-splitter, producing four optical signals, two for each interface.
Pairs of signals from different interfaces were sent to two VISAR interferometers that were set up with different velocity per fringe (VPF) constants. VPF constants of 1.5789 mm/μs and 1.0226 mm/μs were used for most experiments; other VPF constants near these values were used when the details of the interferometer setup were slightly different. This dual-VPF configuration was used to provide unambiguous interference fringe counts when the instrumentation bandwidth was exceeded [10]. The propagation delays through the optical fibers connecting the target to the digitizers of the multi-point
VISAR were measured. This information was used to accurately correlate the particle velocity histories at the the two interfaces.
All instrumentation was triggered by the fall-off of the intensity of the light collected by the optical cutoff trigger when the projectile blocked the beam. The optical signal was converted to an electrical signal by a photodiode and amplified using a fast signal amplifier. The amplified signal triggered a Stanford Research Systems DG535
(SRS) digital delay generator, which in turn triggered the instrumentation according to the programmed delays.
A Tektronix TDS-694C digitizer (3 GHz, 10 GS/s) recorded the signals from the four PZT pins in the target. The propagation delays through the coaxial cables connecting the PZT pins to the digitizer were measured and used to correlate the arrival
55 times of the PZT pin signals. For diagnostics, a second digitizer recorded the optical cutoff trigger signal, the trigger signal for the Verdi laser and the VISAR digitizers, and the total intensity of light returned to the VISAR. When an experiment did not go as planned, the information from the diagnostic digitizer was used to troubleshoot and helped to determine what had gone wrong so it could be corrected for the next experiment.
3.3 Raman Spectroscopy Experiments
3.3.1 Experimental Configuration
Figure 3.6 shows the experimental configuration designed to measure the time- resolved, Raman spectra of diamond single crystals. Boteler [11] reported on shock measurements along [100] and [110] orientations. The focus of this work was to measure the Raman spectra of diamond shocked along [111], and to extend the published measurements [11] to higher stresses. A gas or powder gun launched a projectile, which impacted a target and drove a shock wave through the diamond sample. A pulse from a dye laser excited the Raman spectra in the diamond. Light from the target was recorded using a spectrally and temporally resolved detection system which consisted of a spectrometer, streak camera and CCD. The data recorded by this system showed the evolution of the Raman spectra as the shock wave propagated through the diamond. The experimental techniques used in this work to measure the Raman spectra of diamond were based heavily on those used by Boteler [11] and Root [12].
56 Copper Copper Impactor Diamond Impact Delay Buffer Buffer
PZT Pin
From Dye Laser
To Detection System
PZT Pin
Projectile Target
Figure 3.6: Experimental configuration for time-resolved, Raman measurements in shocked diamond. A pulsed dye laser was used to excited the Raman spectra which were recorded by the detection system. PZT pins provided a trigger for the detection system and a measurement of the shock wave tilt. Dimensions shown are not to scale.
57 3.3.2 Target Construction
The cross-sectional, schematic view of a Raman spectroscopy target is shown in
Figure 3.7. The Cu impact and delay buffers were bonded in a stack and the diamond sample was bonded on top. A lens tube focused the light from a pulsed dye laser onto the diamond sample. Two additional lens tubes collect the scattered light which was sent to the detection and diagnostic instrumentation. PZT pins positioned around the diamond sample provided the arrival time of the shock wave at the interface between the two Cu buffers to trigger the streak camera. An optical trigger provided an additional signal to trigger the dye laser and the diagnostic instrumentation.
Cu impact buffers were 1" in diameter and 1-1.2 mm thick discs, prepared with a
3 μm final polish. Cu delay buffers were 0.3" or 0.4" in diameter and 3.0-3.6 mm thick discs. The face of the delay buffer adjoining the diamond sample was polished using a
0.06 μm colloidal silica suspension, resulting in a mirror finish. The opposite face was prepared with a 3 μm final polish. The thickness of the delay buffers varied between experiments, depending on the delay needed to synchronize the streak camera with the shock wave arrival in diamond. The two Cu buffers were bonded in a press using 815 epoxy.
Due to the laser intensity, any contaminate, including epoxy, between the diamond sample and Cu buffer can cause burning and an increase in noise during the experiment.
To avoid having any epoxy between the diamond and the Cu buffer, the diamond sample was first held in place on the Cu delay buffer with a press. A small amount of five minute epoxy was applied around the diamond sample edge on the buffer face. Five minute epoxy is relatively viscous and very little crept between the sample and the buffer. It
58 Target Ring Optical Fiber to Photodiode
Target Plate Optical Fiber Optical Trigger to Photodiode Collection Optics and Mount PZT Pins Lenses 24 mm Focal Length Cu Impact Buffer
Diamond Sample Optical Fiber Cu Delay Buffer from Dye Laser
PZT Pin Lens Tube Plate
Cut-off Beam Optical Fiber to Detection System
Lens Tube Bracket
Insulating Optical Trigger Spacer Laser Diode and Mount Laser Diode Power
Figure 3.7: Cross section, schematic view of the target used in Raman spectroscopy experiments. Dimensions shown are not to scale.
59 served to seal the interface between the diamond and Cu from contaminates. Then, 332 no-shrink epoxy was applied around the sample to firmly hold it in place. The epoxy was allowed to cure for 48 hours and the sample was examined under a microscope to check for contaminates and air gaps. Samples having bonds with contaminates or air gaps, indicated by interference fringes, were rebuilt.
The bonded samples were then epoxied to a plate containing four PZT pins positioned in a square around the sample, with opposing pins 0.6" apart. Figure 3.8 shows the bonded samples before and after bonding to the PZT pin plate. The target assembly was mounted to a target plate, which in turn was mounted to a target ring. The target surface was aligned parallel to the mounting surface of the target ring using three sets of screws and Belleville washers placed between the target plate and ring.
The transmission and collection optics at the target consisted of three lens tubes.
The inside and outside diameters of the lens tubes were centered about parallel axes offset by 0.025". This offset allowed for fine adjustment of the laser spots on the small diamond sample by rotating the lens tubes. All lens tubes were built for 1:1 imaging, using the same collimating and focusing lenses: 24 mm focal length and 12 mm in diameter. Lens tubes were mounted in an optics bracket, with one tube at normal incidence to the sample and the other two 45° from normal. Continuous wave 543 nm and 633 nm HeNe lasers were used to focus and overlap the three lens tubes on the diamond sample. The lens tube at normal incidence brought the dye laser pulse to the sample and the lens tubes at 45° from normal collected the scattered light for transmission to the detection and diagnostic systems.
The optical cutoff trigger was the same as that used in the VISAR experiments. A
60 (a) Cu Impact Buffer PZT Pin Plate
Cu Delay Buffer
Diamond
(b) PZT Pin
PZT Pin Plate
Diamond Cu Delay Buffer
Figure 3.8: Photographs of the bonded samples (a) before and (b) after bonding to the
PZT pin plate. The circle near the center was the diamond sample, bonded to the top of the Cu impact and delay buffer stack.
61 photograph of the assembled target is shown in Figure 3.9.
3.3.3 Projectile Construction
For the 4" gas gun experiments, standard 4" diameter by 8" long aluminum projectiles were used. Two holes were drilled in the projectile face for an impactor mount. The projectile face was lapped flat, and squared to the projectile side to within
0.05 mrad. A 1" diameter by 3 mm thick Cu impactor was bonded into an aluminum impactor mount using 815 epoxy. After curing for 24 hours, the back face of the impactor mount was hand lapped so that it was parallel with the front face of the impactor. The impactor mount was bolted to the front face of the projectile. The bolt heads on the inside of the projectile were covered with 5 minute epoxy to prevent gas from the rear of the projectile from leaking through to the front. A representative 4" projectile is shown in Figure 3.10.
For experiments using the powder gun and the two stage gun, the projectile construction was the same as used in the interferometery measurements, and is described in section 3.2.3.
3.3.4 Instrumentation
The excitation pulse for the Raman measurements was provided by a Cynosure
LFDL-8E flashlamp-pumped dye laser. The lasing medium used was 0.28 g of Exciton
Coumarin-504 dye dissolved in 18 L of a 50/50 mixture of distilled water and HPLC methanol. Burn spots were made on exposed photographic paper to check the alignment of the laser cavity. Distortion or clipping of the burn spot was corrected by adjusting the
62 Figure 3.9: Photograph of the assembled Raman spectroscopy target mounted in the 4" gas gun.
63 Figure 3.10: Photograph of a representative projectile used in Raman spectroscopy experiments on the 4" gas gun. A Cu impactor was bonded in the impactor mount on the projectile face.
64 high reflectivity mirror in the laser head. The laser cavity was tuned to emit a pulse centered at 514.5 nm. A 4-5 cm-1 spectral width peak with little time-dependent variation over the ~2 μs pulse duration was attainable after proper alignment.
At the output of the laser head, a pair of lenses expanded the laser beam and passed it through a 514.5 nm bandpass beam cube, which improved the spectral width of the beam. Apertures were used in the beam path to reduce scattering between optical elements and to improve beam uniformity. A turning prism directed the beam through a lens which focused the beam into a 400 μm diameter, step-index optical fiber. The other end of this optical fiber was connected at the target to a lens tube at normal incidence to the diamond (Figure 3.7). The laser pulse energy at the output of the optical fiber was set to ~75 mJ.
Scattered light from the sample was collected by the two lens tubes at 45° angles from the diamond (Figure 3.7) which were connected to 400 μm diameter, step-index optical fibers. One optical fiber was connected to a photodiode for diagnostics. The other optical fiber was connected to an optics rail just outside the target chamber of the gun. The optics rail imaged the output of the fiber through a holographic notch filter centered at 514.5 nm and focused the transmitted light into a 400 μm diameter, step-index optical fiber connected to the detection system. The notch filter removed the elastically scattered portion of the collected light and the inelastically scattered light was transmitted to the detection system.
The detection system consisted of the following instruments in sequence: a Kaiser
Optical Systems HoloSpec f/1.8i spectrometer, a Hamamatsu C7700 streak camera, a
Photek MCP-140 image intensifier, and a Princeton Instruments SCX 1300B CCD
65 detector. The spectrometer had a 150 μm slit and a Kaiser Optical Systems HDG-549.7 high-dispersion grating. This grating was centered at 549.7 nm with a coverage from 700 cm-1 to 1600 cm-1 from 514.5 nm. The spectrometer was lens coupled to the streak camera photocathode with an 85 mm focal length lens. The streak camera dispersed the spectrum in time and was set to a sweep range of 200 ns. The image intensifier amplified the spectrally and temporally dispersed image before it was recorded by the CCD detector. The CCD consisted of an 1300 by 1340 pixel array with a 20 μm pixel size.
With this detection system configuration, the spectral and timing calibrations were about
0.72 cm-1/pixel and 200 ps/pixel, respectively. However, because of the binning, the spectral and temporal resolution were typically 5.4 cm-1 and 4 ns, respectively.
3.3.5 Triggering and Synchronization
A successful experiment required the synchronization of three events at different time scales: the dye laser pulsing on (~2 μs), the streak camera sweeping (~200 ns), and the diamond in the shocked uniaxial strain state (~20 ns). Because of the uncertainty in the expected projectile velocity, the optical cutoff trigger time could vary by hundreds of ns. Therefore, this trigger alone was not sufficient to synchronize the streak camera with the shock wave entering the diamond. To achieve the requisite synchronization, the signal from the PZT pins triggered the sweeping of the streak camera, while the optical cutoff triggered the dye laser and the remaining diagnostics. Figure 3.11 shows the instrumentation setup used to trigger the Raman spectroscopy experiments. A time-line of the experimental synchronization is shown in Figure 3.12.
When the projectile blocked the optical cutoff trigger, the intensity fall-off was
66 Target
Optical Trigger
Signal
PZT Pin Signals
Optical Trigger Photodiode and Amplifier r e z
i Power Diode t i g
i Divider Combiner
D Trigger Input
SRS Pulse Generator Delay Generator Tilt Timer Cable Delay Outputs Main Trigger Backup Trigger
Diode Combiner
Cable Delay
Trigger Gate Gate Begin Input Input Input Readout Dye Laser
Image Spectrometer Streak Camera Intensifier CCD
Figure 3.11: Triggering and synchronization diagram for the Raman spectroscopy experiments. Diagnostic digitizers are not shown.
67 Events at Instrumentation Observe PZT Observe Raman Pin Signal Shift on CCD Observe Main Trigger Observe Optical Trigger Trigger Falloff Camera
Generated Delay Tilt Cable for Testing Timer Delay Streak Camera D D I 1 I1 2 2 Optical Trigger Fiber PZT Cable Raman Fiber
F1 F2 F3 Optical Trigger Offset Cu Front-buffer Cu Mid-buffer
T1 T2 T3
Optical Impact Shock Wave Shock Wave Trigger Enters PZT Pins Enters Sample Blocked Events at Target
Figure 3.12. Timeline of the events and delays for synchronizing Raman spectroscopy experiments. T1, T2, and T3 were delays at the target. I1 and I2 were the tilt timer and streak camera insertion delays. F1, F2, F3, and F4 were propagation delays in optical fibers and coaxial cables. D1 = T1 + T2 + F2 ‒ F1 was the delay for the fake PZT signal used in ambient testing. D2 = T3 + F3 ‒ I2 ‒ I1 ‒ F2 was the delay set on the second cable delay box to synchronize the arrival of the shock wave in diamond with sweeping of the streak camera.
68 sent through an optical fiber to a photodiode and fast signal amplifier. The SRS delay generator was set to trigger when the intensity dropped below 50% of its initial value.
The SRS sent trigger pulses with the appropriate output delays to trigger the dye laser, to gate the streak camera and image intensifier on, and to begin the readout from the CCD.
The signal from the four PZT pins was used to trigger the streak camera sweep.
The four signals were split into three sets with a power divider. One set was recorded on a digitizer and was used to determine the impact tilt. The second and third sets provided the main and backup triggers for the streak camera.
For the main trigger, the four signals were the inputs of the tilt timer which measured the time between the arrival of the first and last signals. The output of the tilt
timer was delayed by a constant time I1 plus half of the time between the arrival first and last signals after the arrival of the first signal. This output pulse was a trigger synchronized to the arrival of the shock wave at the center of the sample. If a single pin failed to provide a signal, the output trigger pulse from the tilt timer would be late and the backup trigger triggered the streak camera. For the backup trigger, the last set of signals from the power divider were summed using a diode combiner, and triggered a pulse generator, which provided a clean trigger pulse. This pulse was delayed with an Ortec
DB463 cable delay box.
The outputs of the tilt timer (main trigger) and the cable delay box (backup trigger) were combined by a diode combiner before entering a second cable delay box and finally triggering the streak camera. The first cable delay box was set to delay the backup trigger pulse by 40 ns after the arrival of the main trigger pulse in the case of zero
tilt. The second cable delay box (D2) was set to synchronize the arrival of the shock wave
69 in diamond with the streak camera sweep.
During ambient testing, an unused output on the SRS generated a simulated PZT
pin signal which triggered the streak camera. This output was delayed (D1) to synchronize the firing of the dye laser with the sweeping of the streak camera.
3.3.6 Spectral and Temporal Calibration
Spectral and temporal calibrations of the detection system were needed to quantitatively measure the frequency shifts of the Raman spectra and the time in which they occurred. Small changes in the temperature, alignment and other variables caused the calibration to be slightly different for each experiment. Hence, the detection system was calibrated spectrally and temporally using the following methods.
Several gas lamps were used to achieve spectral calibration of the detection system. The spectra from HgCd, Cs and Kr lamps, shown in Figure 3.13, provided eight spectral peaks which were fit using Gaussian + Lorentzian functions. A fourth-order polynomial fit to the peak positions provided a relation for converting CCD pixel positions to wavelength. Since there was some distortion in the streak camera, the spectral calibration was repeated for each bin. Spectral calibrations were typically 0.72 cm-1/pixel, however this varied over the spectral range of the CCD and the time for the streak camera sweep.
The detection system was calibrated temporally with an electro-optical modulator
(EOM) that pulsed a continuous wave 543 nm HeNe laser on for 50 ns. An SRS delay generator changed the relative delay between the sweeping of the streak camera and the trigger sent to the (EOM). The pixel position of the beginning laser pulses and the
70
12000 (a) HgCd 10000 546.07 nm
8000
6000
4000
2000
0 (b) Cs ) 546.07 nm 550.26 nm
s 8000 t i n u
y 6000 r a r t
i 540.67 nm
b
r 4000 556.82 nm a ( 557.35 nm y t i
s 2000 n
e t n I 0 15000 (c) Kr 557.00 nm
12500
10000 556.19 nm 7500
5000
2500
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 Pixel
Figure 3.13: Calibration spectra from (a) HgCd, (b) Cs, and (c) Kr lamps.
71 relative delays were fit with a linear function. This function provided a relation for calculating relative times between events from the pixel positions on the output of the
CCD detector. Temporal calibrations were typically 200 ps/pixel at the 200 ns sweep range.
72 References
1. J. E. Field, The Properties of Natural and Synthetic Diamond (Academic, San
Diego, 1992).
2. R. Mykolajewycz, J. Kalnajs, and A. Smakula, “High‐precision density
determination of natural diamonds,” J. Appl. Phys. 35, 1773 (1964).
3. H. J. McSkimin and P. Andreatch, Jr., “Elastic moduli of diamond as a function of
pressure and temperature,” J. Appl. Phys. 43, 2944 (1972).
4. S. P. Marsh, LASL Shock Hugoniot Handbook (University of California Press,
Berkeley, 1980).
5. A. C. Mitchell and W. J. Nellis, “Shock compression of aluminum, copper, and
tantalum,” J. Appl. Phys. 52, 3363 (1981).
6. W. J. Carter, “Hugoniot equation of state of some alkali halides,” High Temp. -
High Press. 5, 313 (1973).
7. J. R. Drabble and R. E. B. Strathen, “The third-order elastic constants of
potassium chloride, sodium chloride and lithium fluoride,” Proc. Phys. Soc. 92,
1090 (1967).
8. L. M. Barker and R. E. Hollenbach, “Laser interferometer for measuring high
velocities of any reflecting surface,” J. Appl. Phys. 43, 4669 (1972).
9. L. M. Barker, K. W. Schuler, “Correction to the velocity‐per‐fringe relationship
for the VISAR interferometer,” J. Appl. Phys. 45, 3692 (1974).
10. D. H. Dolan, Foundations of VISAR analysis, Sandia National Laboratories
Report No. SAND2006‒1950, 2006.
11. J. M. Boteler, “Time resolved Raman spectroscopy in diamonds shock
73 compressed along [110] and [100] orientations,” Ph.D. thesis, Washington State
University, WA, 1993.
12. S. Root, “Physical and chemical changes in liquid benzene multiply shocked to 25
GPa,” Ph.D. thesis, Washington State University, WA, 2007.
74 Chapter 4
NONLINEAR ELASTIC RESPONSE: RESULTS AND ANALYSIS
This chapter describes the experimental results and analysis related to the elasticity of diamond single crystals. Shock wave compression experiments provided the nonlinear elastic response of diamond samples subjected to uniaxial strain along the
[100], [110], and [111] orientations. Sections 4.1 and 4.2 describe how the experimental data were obtained from the elastic wave measurements, and the analysis of that data. In
Section 4.3, the nonlinear elastic response of diamond is calculated using finite strain theory and the complete set of third-order elastic constants are determined. The elastic
Hugoniots for the three crystal orientations are presented in Section 4.4.
4.1 Elastic Wave Measurements
A series of experiments, using the configuration described in Section 3.2, was conducted to determine the nonlinear elastic response of diamond. In all cases, laser interferometry was used to obtain the particle velocity histories. Table 4.1 lists the relevant experimental parameters for these experiments. Samples were oriented along the
[100], [110], and [111] orientations. Some of the [100] oriented samples were synthetic crystals, while all others were natural crystals. As shown here, projectile velocities ranged from 2.1 mm/μs to 3.7 mm/μs. Impact tilts were measured using piezoelectric pins embedded in the target. The peak elastic stress values listed in Table 4.1 were calculated using the nonlinear elastic Hugoniots developed in Section 4.4; they
75 Table 4.1: Experimental parameters for the VISAR measurements of diamonds shocked along different crystal orientations. Copper impactors were used in all but one experiment.
Peak Sample Projectile Impact Experiment Sample Sample Elastic Thickness Velocity Tilt Number Orientation Type Stress (mm) (mm/μs) (mrad) (GPa) 1 (09-618) [100] Synthetic 0.444 2.144 8.3 ± 0.8 61.0 2 (10-2S03) [100] Synthetic 0.444 2.463 5.7 ± 0.5 71.8 3 (09-2S19) [100] Synthetic 0.477 2.982 1.3 ± 0.3 90.3 4 (09-2S25) [100] Synthetic 0.445 3.027 5.2 ± 0.6 92.0 5 (09-2S26) [100] Synthetic 0.447 3.638 9.6 ± 0.9 115.1 6 (09-2S23) [100] Synthetic 0.480 3.654 11.1 ± 1.9 115.7 7 (09-2S22) [100] Synthetic 0.480 3.676 10.1 ± 1.0 116.5 8* (09-2S21) [100] Synthetic 0.441 3.187 3.7 ± 0.4 118.0 9 (09-2S20) [100] Natural 0.492 2.981 2.7 ± 0.7 90.3 10 (09-2S28) [100] Natural 0.494 3.033 4.6 ± 1.0 92.2 11 (09-2S27) [100] Natural 0.507 3.045 4.2 ± 0.4 92.6 12 (09-2S24) [100] Natural 0.514 3.644 9.9 ± 0.9 115.3 13 (09-2S29) [100] Natural 0.510 3.722 13.6 ± 1.4 118.3 14 (10-608) [110] Natural 0.437 2.089 3.7 ± 0.4 61.9 15 (10-2S08) [110] Natural 0.620 2.786 6.7 ± 0.7 88.0 16 (10-2S16) [110] Natural 0.603 2.812 7.5 ± 0.8 89.1 17 (10-2S04) [110] Natural 0.631 3.584 4.7 ± 0.4 121.0 18 (10-611) [111] Natural 0.534 2.111 1.8 ± 0.2 62.9 19 (10-2S18) [111] Natural 0.549 2.785 7.7 ± 0.9 88.2 20 (10-2S10) [111] Natural 0.520 2.825 8.5 ± 0.8 89.8 21 (10-2S09) [111] Natural 0.531 3.591 8.0 ± 0.8 121.5
* A Ta impactor was used in this experiment.
76 correspond to the elastic wave amplitudes (~60-120 GPa) if the diamond samples had remained elastic.
4.1.1 Particle Velocity Histories
Representative particle velocity histories measured in the case of a purely elastic response are shown in Figure 4.1, and in the case of an elastic-inelastic response are shown in Figure 4.2. Following impact, the shock wave traversed the buffer material, and upon reaching the diamond sample, produced a jump in the particle velocity history measured at the buffer-sample interface by a VISAR probe. The transmitted shock wave traversed the diamond sample and was partially reflected and transmitted at the sample- window interface, where the particle velocity history was measured by a second VISAR probe. In all experiments, the velocity history at the buffer-sample interface showed that the shock wave incident on the diamond sample was a single step wave with a rise time of 200 ps or less, faster than the resolution of the VISAR. When the sample response was purely elastic, as in Figure 4.1, a single flat-top wave was observed at both the buffer- sample and sample-window interfaces. When the response was elastic-inelastic, as in
Figure 4.2, the particle velocity at the sample-window interface showed the arrival of a sharp elastic wave followed by a decay and the subsequent arrival of a second wave. The velocity histories measured at both interfaces were truncated following the arrival of release waves from the sample edge, or at the loss of interferometer contrast or beam intensity.
77
2.5
Truncated at loss of beam intensity 2.0 ) s / m
m 1.5 (
y t i c
o l e
v 1.0
t t n n e a r r a a p p p p A A 0.5 Measurement interfaces: Buffer-sample Sample-window 0.0 0 10 20 30 40 50 60 Time (ns)
Figure 4.1: Representative velocity histories showing a purely elastic response measured at each interface of the diamond from experiment 1. The history measured at the buffer- sample interface shows the arrival of the shock wave in the diamond samples. The history measured at the sample-window interface shows the wave after traversing the diamond sample. In this experiment, both histories show a single flat topped wave.
These histories have not been corrected for refractive index changes due to the shock compression of the diamond sample and LiF window.
78 Figure 4.2: Representative velocity histories showing an elastic-inelastic response measured at each interface of the diamond for experiment 3. The history measured at the buffer-sample interface shows the arrival of the shock wave in the diamond samples. The history measured at the sample-window interface shows the wave after traversing the diamond sample; it consists of a sharp elastic wave followed by a rapid decay and the arrival of a second wave. These histories have not been corrected for refractive index changes due to the shock compression of the diamond sample and LiF window.
79 4.1.2 Elastic Shock Wave Velocities
The elastic wave velocities, listed in Table 4.2, were calculated from the shock wave traversal time and the sample thickness in each experiment. The arrival time of the elastic shock wave at each interface of the diamond corresponds to rapid jump in the interface velocity history. For a non-zero impact tilt, calculation of the traversal time of the elastic wave was somewhat complicated by the fact that the interface velocity measurements were made at laterally separated points at those interfaces. Thus, the effect of impact tilt needs to be considered in determining the elastic wave velocity.
The traversal time of the wave was determined following the method of Knudson et al. [1]. The two velocity histories were scaled to have the same amplitude and then iteratively time-shifted until the residual between the velocity histories at the velocity jump was minimized. The time shift from this procedure corresponds to the traversal time of the wave, neglecting the effects of shock wave tilt. The tilt measurements provided both the orientation and magnitude of the tilt of the shock front. Using the tilt measurements, the time taken for the wave to travel laterally between the VISAR measurement points was calculated. This time was the correction to the traversal time for the lateral separation of the measurement points due to tilt. The correction was typically around 1 ns, about 4% of the typical traversal time of 25 ns. Using standard methods of error analysis [2], the total uncertainty in the elastic shock wave velocity was calculated from the uncertainties in traversal time (~100-200 ps), tilt (~50 ps) and sample thickness
(~2 μm). The uncertainties in the shock wave velocity for each experiment are listed in
Table 4.2.
80 Table 4.2: Elastic wave measurements in diamond shock compressed along different crystal orientations. The particle velocities have been corrected for the refractive index change of the LiF windows.
Experiment Sample Shock Velocity Particle Velocity Stress Density Number Orientation (mm/μs) (mm/μs) (GPa) (g/cm3) 1 (09-618) [100] 18.46 ± 0.11 0.941 ± 0.011 61.1 ± 1.1 3.704 2 (10-2S03) [100] 18.39 ± 0.11 1.098 ± 0.012 71.0 ± 1.2 3.738 3 (09-2S19) [100] 18.56 ± 0.12 1.338 ± 0.012 87.3 ± 1.3 3.788 4 (09-2S25) [100] 18.42 ± 0.10 1.449 ± 0.012 93.9 ± 1.3 3.815 5 (09-2S26) [100] 18.29 ± 0.11 0.917 ± 0.011 59.0 ± 1.1 3.701 6 (09-2S23) [100] 18.41 ± 0.19 0.847 ± 0.011 54.8 ± 1.3 3.684 7 (09-2S22) [100] 18.49 ± 0.17 0.939 ± 0.012 61.0 ± 1.3 3.703 8* (09-2S21) [100] 18.28 ± 0.17 0.917 ± 0.012 58.9 ± 1.3 3.701 9 (09-2S20) [100] 18.27 ± 0.09 1.368 ± 0.012 87.9 ± 1.2 3.800 10 (09-2S28) [100] 18.57 ± 0.12 1.388 ± 0.012 90.7 ± 1.4 3.799 11 (09-2S27) [100] 18.52 ± 0.11 1.313 ± 0.012 85.5 ± 1.3 3.783 12 (09-2S24) [100] 18.24 ± 0.15 0.849 ± 0.011 54.4 ± 1.2 3.687 13 (09-2S29) [100] 18.19 ± 0.14 0.829 ± 0.011 53.0 ± 1.1 3.683 14 (10-608) [110] 20.40 ± 0.19 0.867 ± 0.011 62.1 ± 1.4 3.671 15 (10-2S08) [110] 20.42 ± 0.14 1.217 ± 0.012 87.4 ± 1.4 3.738 16 (10-2S16) [110] 20.21 ± 0.15 1.223 ± 0.012 86.9 ± 1.5 3.741 17 (10-2S04) [110] 20.54 ± 0.11 1.128 ± 0.011 81.5 ± 1.3 3.719 18 (10-611) [111] 19.93 ± 0.10 0.885 ± 0.011 62.0 ± 1.1 3.678 19 (10-2S18) [111] 20.54 ± 0.22 1.207 ± 0.012 87.2 ± 1.8 3.734 20 (10-2S10) [111] 20.49 ± 0.12 1.167 ± 0.011 84.0 ± 1.3 3.727 21 (10-2S09) [111] 20.67 ± 0.17 0.951 ± 0.011 68.6 ± 1.4 3.685
* A Ta impactor was used in this experiment.
81 4.1.3 Elastic Wave Amplitudes
The in situ particle velocities, also listed in Table 4.2, were calculated from the velocity histories measured at the sample-window interface as follows. In experiments where a two-wave response was observed, only the initial interface velocity jump corresponding to the elastic wave was considered. A window correction was applied to the velocity histories to correct for changes in the refractive index of the LiF windows [3] due to shock wave compression.
In experiments with a two-wave response, the velocity histories showed an artificial smoothing or rounding of the interface velocity jump caused by a finite tilt, the measurement spot size, and the rapid rise and decay of the wave. Due to the measurement spot size (~150 μm), the velocity histories were measured over a finite area in the measurement plane. In impacts with finite tilt, the velocity of all points within this area did not change simultaneously. When the shock wave front sweeps across the measurement area, there is a distribution of velocities in time. The velocity histories recorded by the VISAR were a convolution of the actual velocity history and the movement of the shock front across the measurement spot.
A simple model was developed to correct for the effect of tilt on the elastic wave amplitude. Reverse sawtooth waves with different decay rates, simulating the rise and decay of the elastic wave in the actual velocity histories, were convoluted with the response of the VISAR due to tilt. From this model, the empirical relation for the actual interface velocity jump uLiF was found to be
m uLiF = uLiF−t λ , (4.1)
m where uLiF is the measured velocity jump, t is the time taken for the line of closure to
82 cross the measurement spot and λ is slope of the decay immediately following the
m velocity jump. In extreme cases, uLiF and uLiF can differ by up to 10%. The uncertainty in the sample-window interface velocity included contributions from the background noise in the velocity history (~5-10 m/s) and the variation in multiple reasonable fits to the VISAR fringe data (~5-10 m/s).
The in situ particle velocity up and the shock wave velocity D are related to the longitudinal stress P1 = ‒σ1 (positive in compression), density ρ, and energy per unit mass E through the Rankine-Hugoniot jump conditions [4]:
ρ D−u Mass conservation: 0 = p (4.2) ρ D
Momentum conservation: P1 = ρ0 D up (4.3)
P1 up 1 2 Energy conservation: E−E 0 = − up (4.4) ρ0 D 2
The in situ particle velocity and longitudinal stress corresponding to the elastic wave in the diamond samples were calculated using the Rankine-Hugoniot jump conditions by impedance matching [4] with the LiF window [5]. A linear shock velocity- particle velocity relationship for diamond was used in the impedance matching calculation since the Hugoniot of diamond was not known prior to this work. It was defined as
D=CL +B u p , (4.5) where D is the shock velocity, CL is the longitudinal sound speed at ambient conditions, up is the in situ particle velocity, and B is a constant. The particle velocities and longitudinal stresses in the diamond sample and LiF window must be continuous at the
interface. Using this fact, Eq. (4.3), B = (D−C L)/u p from Eq. (4.5), and the LiF
83 Hugoniot given in Eq. (3.3), the following expression for longitudinal stress equilibrium at the interface was solved numerically for the in situ particle velocity up in the diamond sample:
D−C L 2 LiF ρ0 C L (2up−uLiF )+ (2up−uLiF) = ρ0 uLiF (5.148+1.353uLiF ) . (4.6) ( ( up ) )
The longitudinal stress and density in the diamond sample were then determined from Eq.
(4.3) and Eq. (4.2), respectively. The uncertainties in the calculated in situ particle velocity and stress were estimated from repeating the impedance matching calculations using the maximum and minimum shock velocity and interface velocity values. Table 4.2 summarizes the results of these calculations for each experiment.
4.2 Elastic Wave Analysis
Figure 4.3 shows the shock velocity-particle velocity data measured for the [100] orientation. Both natural and synthetic diamond crystals were shock compressed along this orientation. The results for both crystal types were quite comparable, as shown in
Table 4.2, which suggests that the differences in the growth of these crystals did not affect their elastic response. For this reason, the elasticity of synthetic diamonds was not investigated for compression along [110] and [111]. There is some scatter in the data for the [100] oriented samples, but it is within the uncertainty of the measurements and does not correlate to crystal type. The measured shock velocities were all significantly larger than the ambient sound speed.
The shock velocity-particle velocity measurements for the [110] orientation are shown in Figure 4.4. Again the data show some scatter within the uncertainty. A greater
84 Figure 4.3: Shock velocity and particle velocity measurements for shock compression along [100]. The solid curve is the prediction from incorporating the effective third-order elastic constant C'111.
85 Figure 4.4: Shock velocity and particle velocity measurements for shock compression along [110]. The solid curve is the prediction from incorporating the effective third-order elastic constant C'111.
86 increase in shock velocity with particle velocity was observed, when compared to the
[100] orientation.
Figure 4.5 shows the results for compression along the [111] orientation. Similar to the [100] and [110] orientations, the shock velocity for the [111] orientation increased with particle velocity.
An elastic response calculated from using finite strain theory to the first-order in strain does not lead to the formation of a shock wave. Finite strain theory to the second- or higher-order in strain is needed to examine the formation of a compressive shock wave for all three orientations. The third-order fits shown in these figures, and discussed in the next section, are in good agreement with the data.
4.3 Nonlinear Elastic Response
4.3.1 Theoretical Background
Finite strain theory, presented in Chapter 2, provides the theoretical framework for determining the nonlinear elastic response of diamond from the results of the shock compression experiments. The longitudinal stress under uniaxial strain conditions is given by [6]
ρ 1 σ ' = 0 C ' η' + C ' η' η' , (4.7) 1 ρ ( 11 1 2 111 1 1) where the primes indicate that the variables are expressed in the laboratory coordinate system. The longitudinal stress σ'1 (negative in compression), longitudinal strain η'1, and
87 Figure 4.5: Shock velocity and particle velocity measurements for shock compression along [111]. The solid curve is the prediction from incorporating the effective third-order elastic constant C'111.
88 density ρ are all known from the elastic wave measurements, and the second-order elastic constants C11' are known from acoustic measurements in the literature [7]. Therefore, three effective third-order elastic constants C'111 (one for each orientation) can be determined from Eq. (4.7) using the shock compression measurements. These effective constants are expressed in the crystallographic coordinate system as:
[100]: C '111 = C111 (4.8)
1 [110]: C ' = ( C +3C +12C ) (4.9) 111 4 111 112 166
1 [111]: C ' = (C +6C +2 C +12C +24C +16C ) . (4.10) 111 9 111 112 123 144 166 456
For each orientation, Eq. (4.7) can be written as:
ρ 1 [100]: σ = 0 C η + C η η (4.11) 1 ρ ( 11 1 2 111 1 1)
ρ 1 1 [110]: σ ' = 0 (C +C +2C ) η' + (C +3C +12C ) η' η' (4.12) 1 ρ ( 2 11 12 44 1 8 111 112 166 1 1)
ρ 1 σ ' = 0 C +2C +4 C η' + 1 ρ ( ( 11 12 44) 1 [111]: 3 (4.13) 1 (C +6C +2C +12C +24 C +16C ) η' η' 18 111 122 123 144 166 456 1 1)
The derivation of these expressions are discussed in greater detail in Chapter 2.
4.3.2 Determination of Third-order Constants
Nonlinear, least-squares fits of Eqs. (4.11)-(4.13) to the experimental data for each orientation provided the effective third-order elastic constants listed in Table 4.3. The nonlinear fit was calculated using the 'trust-region-reflective' algorithm of the lsqnonlin() function in MATLAB [8]. Not enough experiments were performed, in
89 Table 4.3: Effective longitudinal third-order elastic constants C'111 of diamond determined for shock compression along different orientations.
Orientation Value (GPa) [100] ‒7603 ± 600 [110] ‒15146 ± 1067 [111] ‒14631 ± 1183 large part because of the destructive and costly nature of the experiments, to use statistical approaches to determine the uncertainty in these constants from the fitting procedure. Therefore, the fitting procedure was repeated, varying the experimental data within the measurement uncertainty, to provide the variation of the third-order constants due to the uncertainty in the measurements. The total uncertainty in the effective third- order constants was estimated conservatively as the quadrature sum of the variation in the constants due to the measurement uncertainty and two standard deviations determined from the fitting routine.
The experimental data and the resulting third-order fit for the [100] orientation are shown in Figure 4.6. The figure shows that the elastic response of natural and synthetic crystals was indistinguishable and a single curve provided a good fit to all the data. However, differences were observed in the time-dependent inelastic response of the natural and synthetic crystals. These differences will be discussed in Chapter 7. Also shown in Figure 4.6 are predictions of the nonlinear elastic response from different theoretical calculations [9-12]. These predictions are in reasonable agreement with the experimental data. For this orientation, the nonlinear predictions depend only on the third-order elastic constant C111, which showed smaller variation between the different
90 Figure 4.6: The elastic response of natural and synthetic diamond shocked along [100].
Both crystal types lie along the same longitudinal stress-density curve. Predictions made using second- and third-order elastic constants from the literature, and the third-order elastic constants determined in this work are shown.
91 studies.
Figure 4.7 shows the stress-density compression experimental data and third-order fit for the [110] orientation, and the predictions from different theoretical calculations
[9-12]. Although there is some scatter in the data, the fit clearly shows that the measured diamond response is stiffer than the response predicted by the theoretical models for this orientation. Figure 4.8 shows the experimental data and the third-order fit for the [111] orientation, again with the predictions from different theoretical calculations [9-12]. The difference between the fit to the data and the theoretical predictions is more pronounced for this orientation than the other two orientations.
In contrast to the [100] orientation, the theoretical predictions for compression along [110] and [111] show considerable differences from each other and from the measured data. For the [110] and [111] orientations, the predictions depend on three and six of the third-order elastic constants, respectively. The calculated constants, other than
C111, show larger variations between the different studies, resulting in predictions that vary as well. Beyond 1% compression, the second-order elastic constants showed significant deviations from the elastic response for all three orientation.
Six independent constants describe the third-order elastic response of diamond
[6]. The shock wave compression experiments presented here provide three linear combinations of the isentropic third-order elastic constants, given in Table 4.3. Pressure derivatives of the second-order elastic constants of diamond [7], listed in Table 2.2, used in conjunction with Eqs. (2.15)-(2.17) provide three additional linear combinations of the third-order elastic constants [13].
Eqs. (2.15)-(2.17) and Eqs. (4.8)-(4.10) are six, linearly independent equations
92 120 Elastic Response: Second-order only ) 100 a Grimsditch et al. P
G Anastassakis et al. ( 80 s Cousins s
e Nielsen r t
s Present
l 60
a Data: n i
d Present u t
i 40 g n o L 20
0 0.00 0.02 0.04 0.06 0.08 0.10 Density compression ( / - 1) 0
Figure 4.7: The elastic response of diamond shocked along [110]. Predictions made using second- and third-order elastic constants from the literature, and the third-order elastic constants determined in this work are shown.
93 120 Elastic Response:
) 100 Second-order only a Grimsditch et al. P
G Anastassakis et al. ( 80 s Cousins s e
r Nielsen t s
Present
l 60
a Data: n i
d Present u t
i 40 g n o L 20
0 0.00 0.02 0.04 0.06 0.08 0.10 Density compression ( / - 1) 0
Figure 4.8: The elastic response of diamond shocked along [111]. Predictions made using second- and third-order elastic constants from the literature, and the third-order elastic constants determined in this work are shown.
94 that relate the third-order elastic constants to experimental data from the three shocked orientations and from three hydrostatic measurements [7]. The complete set of third- order elastic constants was determined by solving this system of six equations. The uncertainties in these constants were determined from the uncertainties in the measured quantities using standard methods of error analysis [2]. The experimentally determined third-order elastic constants of diamond are listed in Table 4.4 along with published sets of constants obtained from different theoretical calculations [9-12].
With the exception of C144 and C456, the experimentally determined constants are in reasonable agreement with the constants calculated by Anastassakis et al. [10].
However, the theoretically calculated constants [9-12] show significant variations with each other and with the experimentally determined constants. The large discrepancies between the different sets of theoretically calculated constants confirmed the need for experimental measurements. The differences between the theoretically calculated constants and the experimentally determined constants presented here demonstrate the
Table 4.4: Third-order elastic constants of diamond in GPa determined from different theoretical calculations [9-12], and from this work.
Ref. 9 Ref. 10 Ref. 11 Ref. 12 Present
C111 ‒6260 ‒7367 ‒6475 ‒6300 ± 300 ‒7603 ± 600
C112 ‒2260 ‒2136 ‒1947 ‒800 ± 100 ‒1909 ± 554
C123 112 1040 982 0 ± 400 835 ± 1447
C144 ‒674 186 115 0 ± 300 1438 ± 853
C166 ‒2860 ‒3292 ‒2998 ‒2600 ± 100 ‒3938 ± 375
C456 ‒823 76 ‒135 ‒1300 ± 100 ‒2316 ± 743
95 need for improvements to current theoretical approaches.
4.4 Experimentally Determined Elastic Hugoniots for Different Orientations
The longitudinal stress-particle velocity relationships were fit to the nonlinear elastic response obtained using the third-order elastic constants for the three orientations.
For each orientation, the elastic response was fit using the quadratic relationship
2 P' 1 = ρ0 C L up+Bu p for 0 Using the longitudinal stress and particle velocity values given in Table 4.2, the longitudinal stress-particle velocity relationships were found to be: 2 [100]: P' 1 = 61.58u p+2.65u p , umax = 1.40mm /μ s (4.15) 2 [110]: P' 1 = 64.39u p+6.50u p , umax = 1.25mm/μ s (4.16) 2 [111]: P' 1 = 65.31up+6.05up , umax = 1.25 mm/μ s (4.17) Within the range of the fit, the stress values agree to better than 0.5 GPa with the elastic response calculated from the third-order elastic constants, as shown in Figures 4.6-4.8 as solid curves. Expressed in the shock velocity-particle velocity plane using Eq. (4.3) for momentum conservation, P1 = ρ0 D up , Eqs. (4.15)-(4.17) become: 96 [100]: D = 17.52+0.755up (4.18) [110]: D = 18.32+1.85u p (4.19) [111]: D = 18.58+1.72u p (4.20) It should be noted that these are linear expressions and are different from the third-order elasticity fits shown in Figures 4.3-4.5. Because the prediction from third-order elasticity expressed in the longitudinal stress-particle velocity plane is not quadratic, the fit expressed in the shock velocity-particle velocity plane is not linear. Expressed in terms of longitudinal stress and density compression, the analogous relations are: 2 3 [100]: P' 1 = 1079μ+984μ −6665 μ (4.21) 2 3 [110]: P' 1 = 1180μ+4356μ −16776μ (4.22) 2 3 [111]: P' 1 = 1213μ+4026μ −15940μ (4.23) where μ=ρ/ρ0−1 is the density compression. Eq. (4.21)-(4.23) are in good agreement (< 0.5 GPa difference) with the elastic response calculated from the third- order elastic constants, shown as solid curves in Figures 4.6-4.8. The Hugoniots presented above were used for the impedance matching calculations in the remainder of this work to determine the elastic stress states in diamond. It is important to emphasize that the above relations hold only when the diamond response is elastic. 4.5 Summary of Elasticity Results The experimental results presented in this chapter are summarized below: 1. The elastic response of diamond crystals was measured under shock compression 97 along the [100], [110] and [111] crystallographic orientations. These measurements provided the longitudinal stress-strain states at the elastic shock wave front. 2. Both natural and synthetic diamonds were shock compressed along [100]. Although differences were observed between the time-dependent inelastic response of these samples, the elastic behavior was indistinguishable within the uncertainty of the measurements. 3. The shock compression measurements along [100], [110], and [111] provided three linear combinations of third-order elastic constants. Together with the hydrostatic pressure derivatives of the second-order elastic constants, six linearly independent equations relating the third-order elastic constants were obtained. The complete set of third-order elastic constants of diamond was determined from these equations, representing the first set derived entirely from experimental data. Comparing these constants with those determined using theoretical models demonstrates the need for refinement to the theoretical models used to predict the nonlinear elastic response of diamond. 4. Analytic expressions for the experimentally determined elastic Hugoniot for the [100], [110], and [111] orientations of diamond were summarized. 98 References 1. M. D. Knudson, M. P. Desjarlais, and D. H. Dolan, “Shock-Wave Exploration of the High-Pressure Phases of Carbon,” Science 322, 1822 (2008) . 2. J. R. Taylor, An Introduction to Error Analysis, 2nd ed. (University Science Books, California, 1982) . 3. J. L. Wise and L. C. Chhabildas, in Shock Compression of Condensed Matter, edited by Y. M. Gupta (Plenum, New York, 1986), pp. 441‒454. 4. R. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz, and W. J. Carter, in High- Velocity Impact Phenomena, edited by R. Kinslow (Academic, New York, 1970), p. 293. 5. W. J. Carter, “Hugoniot equation of state of some alkali halides,” High Temp. - High Press. 5, 313 (1973). 6. J. M. Boteler, “Time resolved Raman spectroscopy in diamonds shock compressed along [110] and [100] orientations,” Ph.D. thesis, Washington State University, 1993. 7. H. J. McSkimin and P. Andreatch, Jr., “Elastic Moduli of Diamond as a Function of Pressure and Temperature,” J. Appl. Phys. 43, 2944 (1972). 8. MATLAB® 7.10.0 (R2010a, The MathWorksTM, Natick, MA, 2010). 9. M. H. Grimsditch, E. Anastassakis, and M. Cardona, “Effect of uniaxial stress on the zone-center optical phonon of diamond,” Phys. Rev. B 18, 901 (1978). 10. E. Anastassakis, A. Cantarero, and M. Cardona, “Piezo-Raman measurements and anharmonic parameters in silicon and diamond,” Phys. Rev. B 41, 7529 (1990). 11. C. S. G. Cousins, “Elasticity of carbon allotropes. I. Optimization, and subsequent 99 modification, of an anharmonic Keating model for cubic diamond,” Phys. Rev. B 67, 024107 (2003). 12. O. H. Nielsen, “Optical phonons and elasticity of diamond at megabar stresses,” Phys. Rev. B 34, 5808 (1986). 13. R. N. Thurston and K. Brugger, “Third-Order Elastic Constants and the Velocity of Small Amplitude Elastic Waves in Homogeneously Stressed Media,” Phys. Rev. 133, A1604 (1964). 100 Chapter 5 REFRACTIVE INDEX: RESULTS AND ANALYSIS This chapter presents the results and analysis related to the refractive index measurements of diamond single crystals shocked along different orientations. The present work provides measurements of the photo-elastic response of diamond crystals at large stresses. Section 5.1 presents the experimental data analysis and the determination of refractive indices for diamond crystals shocked along [100] and [111]. In Section 5.2, second-order elasto-optic constants were fit to the refractive index data using nonlinear photoelasticity theory. Optical window correction factors for diamond crystals shocked along [100] and [111] are determined in Section 5.3. The main findings are summarized in Section 5.4. 5.1 Refractive Index Measurements The refractive index of shocked diamond was determined from selected impact experiments presented in Chapter 4. These were the experiments in which the diamond samples remained transparent to the laser light after the shock wave entered the sample. This was evident from the relatively high VISAR beam intensity signals measured at the sample-buffer interface following the arrival of the shock wave. The parameters for these experiments are listed in Table 5.1. 101 Table 5.1: Experimental parameters for the laser interferometery measurements in diamonds shocked along different crystal orientations. Peak Sample Projectile Impact Experiment Sample Sample Elastic Thickness Velocity Tilt Number Orientation Type Stress (mm) (mm/μs) (mrad) (GPa) 01 (09-618) [100] Synthetic 0.444 2.144 8.3 ± 0.8 61.0 02 (10-2S03) [100] Synthetic 0.444 2.463 5.7 ± 0.5 71.8 03 (09-2S19) [100] Synthetic 0.477 2.982 1.3 ± 0.3 90.3 04 (09-2S25) [100] Synthetic 0.445 3.027 5.2 ± 0.6 92.0 09 (09-2S20) [100] Natural 0.492 2.981 2.7 ± 0.7 90.3 10 (09-2S28) [100] Natural 0.494 3.033 4.6 ± 1.0 92.2 18 (10-611) [111] Natural 0.534 2.111 1.8 ± 0.2 62.9 19 (10-2S18) [111] Natural 0.549 2.785 7.7 ± 0.9 88.2 20 (10-2S10) [111] Natural 0.520 2.825 8.5 ± 0.8 89.8 5.1.1 The Optical Path Length When looking through an optical window, the VISAR [1-3] data monitor the change in the optical path length from the reflector to a stationary reference plane [4]. The optical path length changes arise due to both mechanical and optical changes caused by shock compression. Since the elastic mechanical response of shocked diamond is known from the nonlinear elastic constants presented in Chapter 4, the optical response of shocked diamond could be determined from a VISAR measurement looking through the diamond crystal. Figure 5.1 shows typical particle velocity histories that were analyzed to determine the refractive index of the shocked diamond. A VISAR, looking through the diamond sample in the direction of the shock propagation, measured the apparent 102 3.0 Cu-DiamondDiamond-Cu: interface: Diamond-LiFDiamond-LiF: interface: 2.5 Apparent velocity Apparent velocity InterfaceActual velocity velocity InterfaceActual velocity velocity ) s µ 2.0 / m m ( 1.5 y t i c Impactor Diamond o l e 1.0 V V I S A 0.5 R Copper Buffer LiF Window 0.0 -5 0 5 10 15 20 25 30 35 40 Time (ns) Figure 5.1: Measured or apparent interface velocity histories (closed symbols) and actual interface velocity histories after correcting for the optical response of the materials (open symbols) at the Cu-diamond (squares) and diamond-LiF (circles) interfaces for shock compression along [100] in experiment 1. Velocity histories are truncated at the arrival of edge effects or the loss of VISAR beam intensity. The inset shows the experimental configuration. 103 velocity up of the Cu-diamond interface (closed squares); this corresponded to the change in the optical path length through the shocked diamond. The apparent velocity ua contained contributions from both the mechanical motion of the Cu-diamond interface and the refractive index change of the shocked diamond. A second VISAR measured the apparent velocity of the diamond-LiF interface (closed circles), which corresponded to the change in the optical path length through the shocked LiF crystal. The actual velocity of the diamond-LiF interface (open circles) was determined using the window correction for LiF [5]. The actual velocity of the Cu-diamond interface was calculated from the diamond-LiF interface velocity jump using shock wave impedance matching [6] along with the shock response of LiF [7] and the Hugoniot relations for diamond given in Section 4.4. The actual velocity for the Cu-diamond interface shown in Figure 5.1 is the apparent velocity rescaled so that the initial jump agrees with the actual interface velocity. The large difference between the apparent and actual interface velocities is a consequence of the large refractive index of diamond. The shock wave velocity in diamond was determined following the method presented in Section 4.1. The refractive index of shocked diamond n was calculated from the apparent velocity of the Cu- diamond interface ua, the actual particle velocity up, the shock velocity D in the diamond, and the refractive index at ambient conditions n0 = 2.426 [8] using the following equation [9]: D n −u n = 0 a . (5.1) D−up 104 5.1.2 Shock Compression Along [100] Shock compression or uniaxial strain along [100] reduces the diamond symmetry from cubic to tetragonal [10]. The deformed crystal becomes optically uniaxial with the optical axis along the direction of the shock propagation. Representative particle velocity histories for this orientation are shown in Figure 5.1. The intensity of the VISAR beam at the Cu-diamond interface remained high for a short time after the arrival of the shock wave in experiments with peak elastic stresses as large as ~90 GPa. At larger peak elastic stresses, the beam intensity dropped dramatically, preventing the measurement of the particle velocity history at that interface. This is likely due to scattering of the VISAR beam within the sample due to the onset of inelastic deformation. Table 5.2 lists the shock wave velocities, apparent velocities of the Cu-diamond interface, in situ particle velocities, longitudinal stresses, densities, and the refractive indices calculated using Eq. (5.1) for the [100] oriented diamond. The measured refractive indices for shock compression along the [100] orientation are plotted in Figure 5.2. The refractive index increased from the ambient value of 2.426 at 532 nm [8] ~2.46 at ~90 GPa. Measurements of the diamond refractive index under hydrostatic pressure [11] showed a decrease in the refractive index with increasing pressure. The opposite nature of the change in the optical response under uniaxial strain and hydrostatic pressure show that the diamond refractive index is not simply a function of density, but also depends on the components of the strain tensor. Linear photo-elasticity theory [12-14] predicts an increase in the refractive index of diamond strained uniaxially along [100], but the experiments show a much larger increase than was expected. The second-order fit shown 105 Table 5.2: Elastic wave and refractive index measurements in diamond shock compressed along the [100] orientation. The apparent velocity corresponds to the particle velocity history measured at the Cu-diamond interface without the correction for the refractive index change of diamond. Shock Apparent Actual Experiment Stress Density Refractive Velocity Velocity Velocity Number (GPa) (g/cm3) Index (mm/μs) (mm/μs) (mm/μs) 01 18.46 ± 1.902 ± 0.941 ± 61.1 ± 2.4478 ± 3.704 (09-618)* 0.11 0.019 0.011 1.1 0.0019 02 18.39 ± 2.189 ± 1.098 ± 71.0 ± 2.4534 ± 3.738 (10-2S03)* 0.11 0.040 0.012 1.2 0.0028 03 18.56 ± 2.722 ± 1.338 ± 87.3 ± 2.4565 ± 3.788 (09-2S19)* 0.12 0.043 0.012 1.3 0.0031 09 18.27 ± 2.735 ± 1.368 ± 87.9 ± 2.4607 ± 3.800 (09-2S20)** 0.09 0.071 0.012 1.2 0.0046 10 18.57 ± 2.787 ± 1.388 ± 90.7 ± 2.4599 ± 3.799 (09-2S28)** 0.12 0.034 0.012 1.4 0.0027 * Synthetic crystal ** Natural crystal in the figure is discussed later in Section 5.2. Both natural and synthetic diamond crystals were shock compressed to ~90 GPa along this orientation. The refractive indices of both sample types were comparable, indicating that the growth process does not measurably affect the optical response of elastically compressed diamond. Because of this similarity in the refractive index along the [100] orientation, the refractive indices of synthetic diamonds oriented along [110] and [111] were not investigated. 106 2.48 [100] strain 2.47 Ambient: n = 2.426 Data Linear (Grimsditch(Ref. 14) et al.) x 2.46 Second-order fit e d n i e 2.45 v i t c a r 2.44 f e R 2.43 2.42 0.00 0.02 0.04 0.06 0.08 0.10 Density compression Figure 5.2: Calculated and measured refractive index of diamond shock compressed along the [100] orientation. The uncertainties in density compression are within the size of the symbols. The linear refractive index response (dashed line) was calculated from the elasto-optic constants of Grimsditch and Ramdas [14]. The second-order curve was fit to the data, yielding the second-order elasto-optic constant, p221111 = ‒0.263. 107 5.1.3 Shock Compression Along [110] Shock compression or uniaxial strain along [110] reduces the symmetry of diamond from cubic to orthorhombic [10]. Crystals of this orientation are optically biaxial and the optical axes lie in the [110]-[001] plane. Therefore, light propagating along [110] experiences birefringence. The birefringence of this orientation introduced complications in the measurements preventing the determination of the refractive indices. For all VISAR measurements presented in this work, the VISAR was operated in the dual-VPF configuration described in Section 3.2.4. This means that the same particle velocity history was measured simultaneously using two different etalon lengths in the VISAR. However, the apparent velocity histories measured at the Cu-diamond interface using different etalon lengths did not agree with each other, as shown in Figure 5.3. This difference was not reproducible and varied from experiment to experiment; in one case it was as large as 11%. For this reason, a value could not be accurately assigned to the apparent velocity, preventing the determination of refractive indices for this orientation. The likely cause for the discrepancy in the apparent velocities is that the VISAR used in these experiments was designed to work with unpolarized light. Partially polarized light returning from the birefringent sample would be affected by the polarizing optical elements in the VISAR cavity in unintended ways, resulting in an unbalanced response, and therefore a different recorded velocity, depending on the path taken through the VISAR. Also, the polarization state of the light returning from the birefringent sample and entering the VISAR could have easily varied from experiment to experiment, affecting reproducibility. Although neither the refractive indices nor the degree of birefringence of diamond 108 3.0 Diamond-Cu:Cu-diamond: 2.5 Apparent velocity (0.8" etalon) Apparent velocity (0.4" etalon) Interface velocity 2.0 ) s µ / m m 1.5 ( y t i c o l e 1.0 V Diamond-LiF: Apparent velocity 0.5 Interface velocity 0.0 -5 0 5 10 15 20 25 30 35 40 45 Time (ns) Figure 5.3: Measured or apparent interface velocity histories (closed symbols) and interface velocity histories after correcting for the optical response of the materials (open symbols) at the Cu-diamond (squares and triangles) and diamond-LiF (circles) interfaces for shock compression along [110] in experiment 14. Shock-induced birefringence resulted in different, non-reproducible apparent velocities at the Cu-diamond interface for different VISAR etalon lengths, preventing the determination of refractive indices for this orientation. The velocity histories are truncated at the arrival of edge effects or the loss of VISAR beam intensity. 109 shocked along [110] were determined, the complications encountered with these measurements are consistent with a change from cubic to orthorhombic symmetry and the lack of an optical axis along [110]. 5.1.4 Shock Compression Along [111] Shock compression or uniaxial strain along the [111] direction reduces the symmetry of diamond from cubic to trigonal [10]. In this strain state, the deformed crystal becomes optically uniaxial with the optical axis remaining along the direction of strain, similar to the case of shock propagation along [100]. Representative velocity histories measured for this orientation are shown in Figure 5.4. The velocity history at the Cu-diamond interface shows a single elastic wave for a short time (~10 ns) before the VISAR beam intensity is lost. This loss of intensity is likely due to the onset of inelastic deformation in the shocked diamond samples, as evidenced by the the stress relaxation behind the elastic wave observed at the diamond-LiF interface. Table 5.3 lists the shock wave velocities, apparent velocities of the Cu-diamond interface, in situ particle velocities, longitudinal stresses, densities, and the refractive indices calculated using Eq. (5.1) for the [111] oriented diamond. The measured refractive indices for shock compression along the [111] orientation are plotted in Figure 5.5. Although there is some scatter in the measured refractive indices, there is a definite increase under shock compression compared to the refractive index under ambient conditions. Linear photo-elasticity theory [12-14] also predicts an increase in the refractive index with strain, and the prediction shows reasonably good agreement with the measured refractive index data. This agreement is quite different from the [100] 110 3.0 Cu-diamondDiamond-Cu:Cu-diamond: interface: Diamond-LiFDiamond-LiF: interface: 2.5 Apparent velocity Apparent velocity InterfaceActual velocity velocity InterfaceActual velocity velocity ) s µ 2.0 / m m ( 1.5 y t i c o l e 1.0 V 0.5 0.0 -5 0 5 10 15 20 25 30 35 40 45 50 Time (ns) Figure 5.4: Measured or apparent interface velocity histories (closed symbols) and actual interface velocity histories after correcting for the optical response of the materials (open symbols) at the Cu-diamond (squares) and diamond-LiF (circles) interfaces for shock compression along [111] in experiment 18. Velocity histories are truncated at the arrival of edge effects or the loss of VISAR beam intensity. 111 Table 5.3: Elastic wave and refractive index measurements in diamond shock compressed along the [111] orientation. The apparent velocity corresponds to the particle velocity history measured at the Cu-diamond interface and has not been corrected for the refractive index change of diamond. Shock Apparent Actual Experiment Stress Density Refractive Velocity Velocity Velocity Number (GPa) (g/cm3) Index (mm/μs) (mm/μs) (mm/μs) 18 19.93 ± 1.764 ± 0.885 ± 62.0 ± 2.4461 ± 3.678 (10-611) 0.10 0.022 0.011 1.1 0.0018 19 20.54 ± 2.393 ± 1.207 ± 87.2 ± 2.4537 ± 3.734 (10-2S18) 0.22 0.024 0.012 1.8 0.0020 20 20.49 ± 2.431 ± 1.167 ± 84.0 ± 2.4467 ± 3.727 (10-2S10) 0.12 0.023 0.011 1.3 0.0019 orientation, where the measured refractive index was significantly larger than the prediction. Although linear photoelasticity theory predicts a different optical response for the [100] and [111] directions, as shown in Figures 5.2 and 5.5, the refractive indices for these orientations were nearly the same at a given stress. The changes in crystal symmetry under uniaxial strain and the birefringence observed in the [110] experiments rule out the possibility of diamond remaining optically isotropic under uniaxial strain. However, for shock compression along [100] and [111], the refractive index is nearly the same for both orientations. Theoretical calculations of the polarizability of uniaxially strained diamond crystals are needed to provide insight into the reasons for this similar response. 112 2.48 [111] strain 2.47 Ambient: n = 2.426 Data Linear (Grimsditch(Ref. 14) et al.) x 2.46 e d n i e 2.45 v i t c a r 2.44 f e R 2.43 2.42 0.00 0.02 0.04 0.06 0.08 0.10 Density compression Figure 5.5: Calculated and measured refractive index of diamond shock compressed along [111]. The uncertainties in density compression are within the size of the symbols. The linear refractive index response (dashed line) was calculated from the elasto-optic constants of Grimsditch and Ramdas [14]. 113 5.2 Nonlinear Photoelasticity The refractive index measurements along the [100] direction show significant differences from the prediction of the linear photoelasticity theory. The first-order (linear) elasto-optic constants collected by Grimsditch and Ramdas [14] and listed in Table 2.4 agree with the results from a variety of experiments conducted at much lower stresses and strains. Therefore, nonlinear photoelasticity [15] is needed to predict the changes in the refractive index under large uniaxial strains along [100]. Using Eq. (2.44), the refractive index for light propagating along [100] in diamond uniaxially strained along [100] is 1 − 1 2 2 n[100]= + p1122 η11+ p221111 η11 , (5.2) ( n2 ) 0 where n0 = 2.426 is the ambient refractive index of diamond at 532 nm [8], η11 is the longitudinal strain, and p1122 and p221111 are the relevant first- and second-order elasto-optic constants, respectively. A nonlinear, least-squares fit of Eq. (5.2) to the experimental data for the [100] orientation provided the second-order elasto-optic constant p221111 = ‒ 0.263. The nonlinear fit was calculated using the 'trust-region-reflective' algorithm of the lsqnonlin() function in MATLAB [16]. The result of the fit is plotted in Figure 5.2. The refractive index measurements along [111] are in good agreement with the predictions from linear photoelasticity theory [12-14], and therefore the nonlinear terms must not contribute to the refractive index change along this orientation. Using Eq. (2.44), the refractive index for light propagating along [111] in diamond uniaxially strained along [111] is 114 1 1 1 n[111] = + ( p1111+2 p1122−2 p1212) η' 11+ ( p111111 +4 p111122+2 p221111 + ( n2 3 9 0 , (5.3) 1 − 2 2 2 p112233+4 p112323 +8 p111212 – 4 p232311 – 8 p231312 –8 p121211 ) η'11 ) where n0 = 2.426 is the ambient refractive index of diamond at 532 nm [8], η'11 is the longitudinal strain, and pijkl and pijklmn are the relevant first- and second-order elasto-optic constants, respectively. Therefore, the condition for no contribution from strain to the second-order, as observed in the shock wave experiments, is p +4 p +2 p +2 p + 111111 111122 221111 112233 . (5.4) 4 p112323+8 p111212 – 4 p232311 – 8 p231312 – 8 p121211 ≈ 0 The difference between the linear response observed along [111] and the nonlinear response observed along [100] cannot be explained from the shock wave experiments alone. Theoretical calculations of the polarizability of the strained diamond lattice are needed to further understand the difference in the results. 5.3 VISAR Window Corrections In experiments where an optical window is used as part of a laser-interferometry measurement, the measured velocity history must be corrected to account for changes in the optical path length through the window [4]. This was discussed earlier in this chapter and is shown graphically in Figure 5.1. The apparent interface velocity ua was corrected for the refractive index of the window, giving the actual particle velocity up of the interface being measured. The difference in apparent velocity and the actual particle velocity, Δu = ua ‒ up, or equivalently [4] Δ ν ua 1+ ν = . (5.5) up 115 gives the window correction factor for the VISAR window. The window correction factors for diamond shock compressed along [100] and [111] were fit to the apparent and actual particle velocity data (up to ~90 GPa) and were found to be 1.992 and 2.020, respectively They do not appear to depend on the peak stress. These two values are comparable, further emphasizing that the refractive index changes for both orientations were similar. These window correction factors can aid in the analysis of shock wave experiments utilizing diamond optical windows. When large diamond crystals are readily available, they will fill the need for a high-impedance optical window that can be used at large stresses. 5.4 Summary of Refractive Index Results The main results of this chapter are summarized as follows: 1. The refractive indices of diamond crystals were quantitatively determined for shock compression along the [100] and [111] directions. The refractive indices for [110] compression were not determined, but the data support a change from cubic to orthorhombic symmetry and the lack of an optical axis along [110]. 2. The refractive indices for shock compression along [100] and [111] increase with increasing uniaxial strain in contrast to hydrostatic pressure measurements which showed a decrease in refractive index with pressure. This difference demonstrates that the diamond refractive index does not depend on density alone. Instead, it depends on the components of the strain tensor. The latter finding is in agreement with photoelasticity theory. 3. The refractive index measurements along [111] were in good agreement with 116 linear photoelasticity theory. However, the measurements along [100] were significantly larger than the predictions from the linear theory. Extending the linear theory to incorporate the second-order elasto-optic constant p221111 = ‒ 0.263 provided a good fit to the [100] data. 4. The window correction factors for the [100] and [111] orientations were determined to be 1.992 and 2.020, respectively, for shock compression up to ~90 GPa. These results are needed for the use of [100] and [111] oriented diamonds as optical windows in shock wave compression experiments. 117 References 1. L. M. Barker and R. E. Hollenbach, “Laser interferometer for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669 (1972). 2. L. M. Barker, K. W. Schuler, “Correction to the velocity‐per‐fringe relationship for the VISAR interferometer,” J. Appl. Phys. 45, 3692 (1974). 3. D. H. Dolan, Foundations of VISAR analysis, Sandia National Laboratories Report No. SAND2006‒1950, 2006. 4. D. Hayes, “Unsteady compression waves in interferometer windows,” J. Appl. Phys. 89, 6484 (2001). 5. J. L. Wise and L. C. Chhabildas, in Shock Compression of Condensed Matter, edited by Y. M. Gupta (Plenum, New York, 1986), pp. 441‒454. 6. R. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz, and W. J. Carter, in High- Velocity Impact Phenomena, edited by R. Kinslow (Academic, New York, 1970), p. 293. 7. W. J. Carter, “Hugoniot equation of state of some alkali halides,” High Temp. - High Press. 5, 313 (1973). 8. A. Zaitsev, Optical Properties of Diamond (Springer, Berlin, 2001). 9. S. C. Jones and Y. M. Gupta, “Refractive index and elastic properties of z-cut quartz shocked to 60 kbar,” J. Appl. Phys. 88, 5671 (2000). 10. H. S. Peiser, J. B. Wachtman, Jr., and R. W. Dickson, “Reduction of space groups to subgroups by homogenous strain,” J. Res. Nat. Bur. Stand. A 67A, 395 (1963). 11. N. M. Balzaretti and J. A. H. da Jornada, “Pressure dependence of the refractive index of diamond, cubic silicon carbide and cubic boron nitride,” Solid State 118 Comm. 99, 943 (1996). 12. J. F. Nye, Physical Properties of Crystals (Clarendon Press, Oxford, 1957). 13. A. Yariv and P. Yev, Optical Waves in Crystals (Wiley, New York, 1984). 14. M. H. Grimsditch and A. K. Ramdas, “Brillouin scattering in diamond,” Phys. Rev. B 11, 3139 (1975). 15. K. Vedam and R. Srinivasan, “Non-Linear Peizo-optics,” Acta. Cryst. 22, 630 (1967). 16. MATLAB® 7.10.0 (R2010a, The MathWorksTM, Natick, MA, 2010). 119 Chapter 6 RAMAN SPECTRA OF SHOCKED DIAMOND: RESULTS AND ANALYSIS This chapter presents the results and analysis of the Raman spectroscopy measurements of shock compressed diamond crystals. The present work provided the first measurements of the Raman frequency shifts in diamond shocked along [111], and extended the existing measurements along [100] and [110] [1-3] to larger peak stresses. Section 6.1 presents the data, the analysis methods, and the resulting frequency shifts for the different orientations. In Section 6.2 the anharmonic force constants are determined from the measurements, and are compared with previous determinations from uniaxial strain [1-3] and uniaxial stress [4] measurements. Corrections to the spectral calibration [3] and mechanical response for the previously published results are presented in Section 6.3. These corrected values were used for all comparisons with the results of the present work. The main findings are summarized in Section 6.4. 6.1 Raman Spectroscopy Measurements The time-resolved Raman spectra of shock compressed diamond single crystals were measured in a series of ten impact experiments. Table 6.1 lists the orientation, type and thickness of the diamond sample used in each experiment. The measured projectile velocity in each experiment is also shown. The stress and density in the shocked state were calculated using impedance matching [5] and the diamond Hugoniot relations given in Section 4.4. Projectile velocities were chosen such that the diamond samples were 120 Table 6.1: Experimental parameters for Raman spectra measurements of diamond crystals shock compressed along different orientations. Sample Projectile Experiment Sample Sample Stress Density Thickness Velocity Number Orientation Type (GPa) (g/cm3) (mm) (mm/μs) 1 (08-024) [100] Synthetic 0.484 0.800 20.0 3.579 2 (08-603) [100] Synthetic 0.484 1.554 42.0 3.649 3 (11-609) [100] Synthetic 0.453 1.657 45.2 3.659 4 (08-610) [100] Synthetic 0.484 2.105 59.9 3.704 5 (09-601) [100] Natural 0.484 2.120 60.6 3.706 6 (08-019) [110] Natural 0.474 0.784 19.9 3.572 7 (09-601) [110] Natural 0.500 2.061 60.7 3.675 8 (11-024) [111] Synthetic 0.775 0.602 15.1 3.557 9 (11-615) [111] Synthetic 0.754 1.132 30.4 3.598 10 (11-612) [111] Natural 0.754 1.641 46.6 3.639 always subjected to elastic compression. In each experiment, data were recorded in the form of two-dimensional intensity plots of the time-resolved Raman spectra. Figure 6.1 shows a typical plot of the time- resolved Raman spectra as the shock wave propagated through a [111] oriented diamond crystal in experiment 10. Before the shock wave entered the diamond, only the ambient Raman peak at 1332.5 cm-1 was present, as shown in track 0 of Figure 6.1. As the shock wave traversed the diamond, the shifted Raman peaks appeared and gained in intensity while the intensity of the ambient Raman peak decreased. The shock wave reached the free surface of the diamond ~20-40 ns after entering the diamond sample, at which time the entire sample was in the shocked state. The intensities of the shifted Raman peaks 121 25000 ) s t i 20000 n u . b r 15000 a ( y t i s 10000 n e t n I 5000 0 ) k c 5 ra /t s n 10 5 .0 5 ( 15 k c ra 20 T 1300 1325 1350 1375 1400 1425 1450 Raman Shift (cm-1) Figure 6.1: Time resolved Raman spectra of diamond shock compressed along [111] to ~45 GPa in experiment 10. The ambient peak at 1332.5 cm-1 on the back-left (track 0) decreases in intensity while the two shifted peaks on the right, corresponding to the shocked diamond, increase in intensity as the shock wave propagates through the diamond. The shifted peaks reach their maximum intensity when the shock wave arrives at the diamond free surface (track 10). After the shock wave reflects from the free surface, the ambient peak reappears and the intensity of the shifted peaks decrease until the diamond is fully unloaded (track 20). The total recording time shown is approximately 100 ns. 122 were greatest and the intensity of the ambient peak vanished, as shown in track 10 of Figure 6.1. As the shock wave was reflected from the free surface, the diamond crystal was fully released to a state of zero stress. The intensity of the ambient peak returned while the shifted peaks decreased in intensity as shown in track 20 of Figure 6.1. Edge effects typically arrived at the measurement location ~50-100 ns after the shock wave entered the diamond sample, during or after the release to zero stress. 6.1.1 Analysis of Raman Spectra The frequency shifts of the Raman peaks corresponding to the shock compressed diamond were determined after filtering, calibrating, and fitting the raw intensity data from the detection system. These procedures were based on those used by Boteler [1] and Root [6]. The spectral resolution of the detection system was typically 5.4 cm-1 or ~8 data points and was limited by the spectral width of the excitation pulse from the pulsed dye laser. The temporal resolution was typically 4-5 ns or 20 data points and was limited by the diameter of the optical fiber connected to the detection system. The resolution of the detection system was discussed in greater detail in Chapter 3. The data were filtered to reduce noise and features smaller than the experimental resolution. Point-to-point variations due to CCD noise were filtered using a two-dimensional Gaussian low-pass filter with a standard deviation of 2 data points implemented in MATLAB [7]. The data were divided on the temporal axis into sections 20 data points wide, matching the temporal resolution. The Raman intensity in each section was averaged over time. A fourth-order, zero-phase, low-pass Butterworth filter in MATLAB [7] was used to filter the spectra in each section. This procedure smoothed spectral 123 features narrower than 5 points, below the spectral resolution of the detection system. Finally, the background was subtracted, and the spectral calibration determined from known gas lamp lines, presented in Section 3.3.6, was applied to each temporal section. The positions of the Raman peaks were determined by fitting the Raman spectra to a Gaussian-Lorentzian function using the software PeakFit [8]. The fits provided the position, width, and amplitude of each of the Raman peaks, as well as the uncertainty in the fit to each of these parameters. By comparing the positions of the shifted Raman peaks to the position of the ambient Raman peak prior to the shock wave entering the diamond, the shock induced frequency shifts were determined. The total uncertainty in the shock wave induced frequency shifts was obtained by summing the following contributions in quadrature: the detector resolution (~1.2 cm-1), the uncertainty in the spectral calibration (~1.2 cm-1), and the uncertainty in the fits to the ambient and shocked spectra (~0.5 cm-1). 6.1.2 Shock Compression Along [111] For uniaxial strain along the [111] direction, the three-fold degeneracy of the ambient Raman line is partially lifted. The frequency shifts for this orientation had not been measured prior to this work, but can be predicted based on shock wave measurements along [100] and [110] [1-3]. Figure 6.2 shows the ambient and shocked Raman spectra recorded in experiment 10. The Raman spectra of diamond shocked along [111] showed two shifted peaks: a singlet ΔωS corresponding to the phonon mode in the direction of strain, and a doublet ΔωD corresponding to the degenerate phonon modes in the (111) plane. The observed splitting to a singlet and doublet is consistent with a strain 124 22500 = 1332.5 20000 0 Ambient Shocked 17500 = 1349.0 D ) s t i 15000 n u . b r 12500 a ( y t i 10000 s n e t n 7500 I = 1436.0 cm-1 5000 S 2500 0 1300 1325 1350 1375 1400 1425 1450 1475 Raman Shift (cm-1) Figure 6.2: The ambient and shocked Raman spectra of diamond shocked along [111] to ~45 GPa in experiment 10. The singlet and doublet are observed in the shocked spectrum, showing the strain induced symmetry change from cubic to trigonal. 125 induced symmetry change from cubic to trigonal [1]. The stress in the diamond sample and the frequency shifts of the shocked peaks relative to the ambient triplet peak for uniaxial strain along [111] are listed in Table 6.2. Diamond samples were shocked up to a stress of ~45 GPa for this orientation to stay below the elastic limit. Synthetic diamond crystals were used in two of the experiments; but based on the Raman measurements along [100], no difference is expected for the different crystal types. Table 6.2: Experimental results for uniaxial strain along the [111] direction. -1 Experiment Frequency Shift (cm ) Stress (GPa) Number ΔωS ΔωD 8 (11-024) 15.1 36.3 ± 1.7 5.8 ± 1.7 9 (11-615) 30.4 71.3 ± 1.6 16.5 ± 1.5 10 (11-612) 46.6 104 ± 1.6 11.0 ± 1.5 6.1.3 Shock Compression Along [100] Under uniaxial strain along the [100] direction, the three-fold degeneracy of the ambient Raman line is again partially lifted [1]. Figure 6.3 shows the ambient and shocked Raman spectra recorded in experiment 1. The Raman spectra of diamond shocked along [100] showed two shifted peaks: a singlet corresponding to the phonon mode in the direction of strain, and a doublet corresponding to the degenerate phonon modes in the (100) plane [1]. Table 6.3 lists the stresses in the diamond crystals and the corresponding 126 9000 =1332.5 8000 0 Ambient Shocked 7000 ) s t i =1416.8 n 6000 s u . b r 5000 a ( y t i 4000 s n e t 3000 n I 2000 d=1385.8 1000 0 1300 1325 1350 1375 1400 1425 1450 Raman shift (cm-1) Figure 6.3: The ambient and shocked Raman spectra of diamond shocked along [100] to ~60 GPa in experiment 4. Two peaks corresponding to the singlet and doublet were observed in the shocked spectrum. A remnant of the ambient peak can be seen in the shocked spectrum because of the finite time resolution of the measurement. 127 frequency shifts of the shocked singlet ΔωS and doublet ΔωD peaks relative to the ambient triplet peak for uniaxial strain along [100]. Both natural (experiment 4) and synthetic (experiment 5) crystals were shocked to ~60 GPa along this orientation. The frequency shifts for the different crystal types were comparable, indicating that natural and synthetic crystals behave in the same way with respect to uniaxial strain induced changes in the Raman spectrum. Table 6.3: Experimental results for uniaxial strain along the [100] direction. -1 Experiment Frequency Shift (cm ) Stress (GPa) Number ΔωS ΔωD 1 (08-024) 20.0 29.3 ± 1.6 17.7 ± 2.4 2 (08-603) 42.0 62.5 ± 1.8 39.2 ± 1.8 3 (11-609) 45.2 67.6 ± 1.6 43.2 ± 1.8 4 (08-610) 59.9 84.3 ± 1.6 53.3 ± 1.6 5 (09-601) 60.6 87.0 ± 1.8 53.5 ± 2.0 6.1.4 Shock Compression Along [110] The three-fold degeneracy of the ambient Raman line is completely lifted under uniaxial strain along the [110] direction [1]. Figure 6.4 shows the ambient and shocked Raman spectra recorded in experiment 7. The Raman spectra of diamond shocked along [110] showed three shifted peaks: a singlet Δω2 corresponding to the phonon mode in the direction of strain, and two singlets Δω1 and Δω3 corresponding to the non-degenerate phonon modes in the (110) plane [1]. Table 6.4 lists the stress in the diamond and the corresponding frequency shifts of 128 7000 =1332.5 Ambient 6000 0 Shocked 5000 ) s t i n u 4000 =1339.9 . 3 b r a ( 3000 y t i s n e =1382.8 t 2000 1 n I 2=1449.5 1000 0 1300 1325 1350 1375 1400 1425 1450 1475 Raman shift (cm-1) Figure 6.4: The ambient and shocked Raman spectra of diamond shocked along [110] to ~60 GPa in experiment 7. Three singlet peaks are observed in the shocked spectrum, showing the complete removal of the three-fold degeneracy of the ambient Raman spectrum. 129 the singlet peaks relative to the ambient peak for the Raman spectra measurements for shock compression along [110]. Table 6.4: Experimental results for uniaxial strain along the [110] direction. -1 Experiment Frequency Shift (cm ) Stress (GPa) Number Δω1 Δω2 Δω3 6 (08-019) 19.9 17.9 ± 1.5 41.7 ± 1.8 2.5 ± 1.4 7 (09-601) 60.7 50.3 ± 1.7 117 ± 2.0 7.4 ± 1.6 6.2 Anharmonic Force Constants of Diamond The anharmonic force constants p, q, and r are parameters in a phenomenological model for predicting strain-induced changes in the Raman spectrum of crystals having the diamond structure [9]. Raman spectroscopy measurements of diamond under uniaxial strain [1-3] and uniaxial stress [4] provided sufficient experimental data to determine the three constants. However, uniaxial strain measurements along [111] were previously unavailable. These new results of this work provide a further check of the consistency of the theory and the existing experimental data. 6.2.1 Determination From Uniaxial Strain Experiments Ganesan et al. [9] related the Raman frequency shifts under strain to anharmonic force constants. Solving Eq. (2.45) for uniaxial strain along the [111] direction, given by Eq. (2.37), provides the frequency shifts and phonon polarization directions for that orientation. The frequency shifts of the doublet and singlet modes are [1]: 130 ( p+2q−2r ) η' Δ ωD = ω0 1+ −1 (6.1) 3ω2 {√ 0 } ( p+2q+4 r ) η' Δ ωS = ω0 1+ −1 , (6.2) 3ω2 {√ 0 } where ω0 is the frequency of the ambient Raman mode and η' is the Lagrangian strain along the [111] direction. The degeneracy is only partially lifted for uniaxial strain along this direction; the phonon modes in the (111) plane remain degenerate. The 1 1 corresponding polarization vectors are [11̄0 ] and [ 112̄ ] for the doublet, and √2 √6 1 [111] for the singlet [1]. √3 The frequency shifts and polarization directions for the [100] and [110] orientations have already been determined in the literature [1-3]. For completeness, the frequency shifts are also presented here. For uniaxial strain along [100], the frequency shifts of the singlet and doublet peaks are related to the anharmonic force constants by [1] p η' Δ ω = ω 1+ −1 (6.3) S 0 ω2 {√ 0 } q η' Δ ω = ω 1+ −1 . (6.4) D 0 ω2 {√ 0 } For uniaxial strain along [110], the frequency shifts of the three singlet peaks are related to the anharmonic force constants by [1] q η' Δ ω = ω 1+ −1 (6.5) 1 0 ω2 {√ 0 } ( p+q+2r ) η' Δ ω2 = ω0 1+ −1 (6.6) 2ω2 {√ 0 } 131 ( p+2q−2r ) η' Δ ω3 = ω0 1+ −1 . (6.7) 2ω2 {√ 0 } The measurements of the Raman spectra in diamond shocked along [110] provide three independent frequency shifts, from which all three anharmonic force constants may be determined. Individually, uniaxial strain measurements along [100] and [111] do not provide enough information to determine all three constants. Taken together, the Raman frequency shift measurements of all three orientations provide an overdetermined system of equations depending on the three anharmonic force constants p, q, and r. To take all of the uniaxial strain data into account, a least-squares optimization was used to best fit the anharmonic constants to all of the available shock compression data. The published uniaxial strain data [1-3] was corrected, as described in Section 6.3, before fitting. The lsqnonlin() function using the 'trust-region-reflective' algorithm in MATLAB [7] performed the least-squares fit. Table 6.5 lists the anharmonic constants determined from the published uniaxial strain measurements [1-3], the published uniaxial stress measurements [4], and the present work in combination with the published uniaxial strain measurements [1-3]. The anharmonic force constants determined from the two sets of uniaxial strain measurements are in good agreement. Inclusion of the present data, including the [111] orientation, does not appreciably change the values of the constants. The constants determined from uniaxial stress measurements [4] show larger differences from the uniaxial strain constants, but these differences are within the uncertainty of the determinations [4]. 132 Table 6.5: Anharmonic force constants of diamond determined from measurements of the Raman spectra of diamond under uniaxial stress and uniaxial strain. p q r Sources Experiment Type 2 2 2 ω0 ω0 ω0 Ref. 4 Uniaxial stress ‒2.81 ± 0.19 ‒1.77 ± 0.16 ‒1.9 ± 0.2 Refs. 1-3 Uniaxial strain ‒2.60 ± 0.23 ‒1.74 ± 0.08 ‒2.00 ± 0.09 Present, with Refs. 1-3 Uniaxial strain ‒2.65 ± 0.11 ‒1.72 ± 0.09 ‒2.01 ± 0.09 6.2.2 Discussion The frequency shifts of the singlet and doublet peaks for uniaxial strain along [111] are plotted in Figure 6.5. Since no prior measurements of the frequency shifts for shock compression along this orientation have been reported, these measurements provide a check of the proposed anharmonic constants. The frequency shift of the singlet peak is very close to the predictions, which show good agreement with each other. Except for the largest compression datum, the frequency shift of the doublet peak is again very close to the uniaxial strain predictions. Unlike the singlet, the predicted shifts of the doublet differ between uniaxial strain and uniaxial stress predictions. The present measurements support the published anharmonic constants, determined from uniaxial strain measurement along [100] and [110]. The frequency shifts of the singlet and doublet peaks predicted by the three sets of anharmonic constants, and the measurements for uniaxial strain along [100] are plotted as a function of density compression in Figure 6.6. The predictions from the present work show somewhat better agreement with the data than the earlier uniaxial strain prediction 133 (a) 120 Predicted Shift: 100 Ref. 4 Ref. 3 Present and Ref. 3 ) 80 1 - Data: m c Present ( t f i 60 h s n a m 40 a R 20 0 0.00 0.01 0.02 0.03 0.04 Density Compression (b) 25 Predicted Shift: Ref. 4 20 Ref. 3 Present and Ref. 3 ) 1 Data: - m 15 Present c ( t f i h s n 10 a m a R 5 0 0.00 0.01 0.02 0.03 0.04 Density Compression Figure 6.5: Measured and predicted frequency shifts of the (a) singlet peak ΔωS and (b) doublet peak ΔωD versus density compression for uniaxial strain along [111]. 134 (a) 100 Predicted Shift: Ref. 4 80 Ref. 3 Present and Ref. 3 ) 1 Data: - m 60 Ref. 3 c ( t Present f i h S n 40 a m a R 20 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Density Compression (b) 70 Predicted Shift: 60 Ref. 4 Ref. 3 50 Present and Ref. 3 ) 1 Data: - m Ref. 3 c 40 ( t Present f i h S 30 n a m a 20 R 10 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Density Compression Figure 6.6: Measured and predicted frequency shifts of the (a) singlet peak ΔωS and (b) doublet peak ΔωD versus density compression for uniaxial strain along [100]. 135 [3], especially for the singlet peak. The predictions from uniaxial stress measurements [4], obtained at much lower stresses, are reasonable but show the largest differences with the data. The frequency shifts of the three singlet peaks for compression along [110] are plotted in Figures 6.7 and 6.8. For the first two singlet peaks, the predictions and the data show good agreement for all three sets of anharmonic constants. However, for the third singlet peak shown in Figure 6.8, the uniaxial strain predictions are considerably better than the uniaxial stress prediction. This peak also had the smallest frequency shift and intensity, resulting in larger uncertainties. The frequency shifts predicted by the two sets of anharmonic constants from shock compression measurements match the measured data, within the experimental uncertainty, equally well. However, the predictions from uniaxial stress show a larger deviation from the measured frequency shifts. This is likely due to the limited stress range (<1 GPa) over which theuniaxial stress measurements were made. The shock wave or uniaxial strain measurements achieved much greater stresses (up to ~60 GPa) that could not be reached under uniaxial stress. Therefore, it is recommended that the anharmonic constants determined from all the uniaxial strain data (third row in Table 6.5) be used for calculating the frequency shifts under arbitrary strain conditions. The same are listed below for convenience: p =− ± 2 2.65 0.11 , (6.8) ω0 q =− ± 2 1.72 0.09 , (6.9) ω0 136 60 (a) Predicted Shift: 50 Ref. 4 Ref. 3 Present and Ref. 3 ) 40 1 Data: - m Ref. 3 c ( t Present f i 30 h S n a 20 m a R 10 0 0.00 0.01 0.02 0.03 0.04 0.05 Density Compression 140 (b) Predicted Shift: 120 Ref. 4 Ref. 3 100 Present and Ref. 3 ) 1 - Data: m Ref. 3 c 80 ( t f Present i h S 60 n a m a 40 R 20 0 0.00 0.01 0.02 0.03 0.04 0.05 Density Compression Figure 6.7: Measured and predicted frequency shifts of the singlet peaks (a) Δω1 and (b) Δω2 versus density compression for uniaxial strain along [110]. 137 20 18 Predicted Shift: 16 Ref. 4 Ref. 3 14 Present and Ref. 3 ) 1 - 12 Data: m c Ref. 3 ( 10 t f i Present h 8 S n a 6 m a 4 R 2 0 -2 0.00 0.01 0.02 0.03 0.04 0.05 Density Compression Figure 6.8: Measured and predicted frequency shifts of the singlet peak Δω3 versus density compression for uniaxial strain along [110]. 138 r =− ± 2 2.01 0.09 . (6.10) ω0 6.3 Reanalysis of Published Shock Wave Measurements As discussed in Section 2.4.2, the Raman spectra of diamond crystals shock compressed along the [100] and [110] orientations were measured in earlier studies [1-2]. During the course of this work, a systematic error was discovered in the spectral calibration of the previous experiments [3]. Also, the present work provided a non-linear elastic model for diamond based on experimental data in Section 4.2.3. Therefore, the previous experimental data [1-2] were reanalyzed. The details of this reanalysis are presented in this section. When comparing the results of the present work with the previously published results, these corrected values were used in earlier sections of this chapter. 6.3.1 Spectral Calibration Several of the previous experiments [1-2] were repeated in the course of the present work However the measured frequency shifts were consistently lower than the previous values by ~20%. In working with Dr. Boteler, who had carried out the earlier experiments, it was determined that this discrepancy was the result of a systematic error in the spectral calibration procedure used in the earlier work [1-2]. In the previous work, a Taylor series expansion was used to convert a linear spectral calibration in wavelength into a linear spectral calibration in frequency. However, the Taylor series was expanded about the wavelength of the incident laser at 514.5 nm, rather than the wavelength of the 139 the ambient Raman line at 552.369 nm. An erratum was subsequently published showing that the reported frequency shifts may be corrected using [3] 7 7 10 −1 10 Δ ωC (Δ ωB) = − 1332.5cm − , (6.11) 514.5nm ( 514.5nm )2 Δ ωB + 552.369nm 107 -1 where ΔωC are the corrected frequency shifts in cm and ΔωB are the previously reported frequency shifts in cm-1. Tables 6.6 lists the previously reported and the corrected frequency shifts for the published work. The corrected values are significantly different than those originally published, and are in much better agreement with the present work. 6.3.2 Mechanical Response The non-linear elastic model for diamond used in the previous work [1-2] was based on theoretical calculations [10] rather than experimental data, which were unavailable at that time. The third-order elastic constants determined in Chapter 4 provide the non-linear elastic response of diamond entirely from experimental measurements. Hence, the stress and density states in the previous work [1-2] were recalculated using the diamond Hugoniot relations given in Section 4.4. Tables 6.6 lists the previously reported and recalculated stress values reached in the experiments. At the moderate stresses reached in these experiments the difference in the longitudinal and lateral stresses is relatively small, making no significant, difference in the results. 140 Table 6.6: Summary of published [1-3] experimental results for uniaxial strain along (a) [110] and (b) [100], and the results of the reanalysis of the spectral [3] and mechanical data. Experiment numbers correspond to those in Ref. 2. (a) [110]: Previous Values Corrected Values Expt. Frequency Shift (cm-1) Frequency Shift (cm-1) # Stress Stress (GPa) (GPa) Δω1 Δω2 Δω3 Δω1 Δω2 Δω3 1 12.2 12.1 ± 1.3 29.8 ± 1.6 1.6 ± 1.3 12.2 10.5 ± 1.0 25.8 ± 1.2 1.4 ± 1.0 2 15.2 17.2 ± 1.5 35.9 ± 1.8 1.3 ± 1.5 15.2 14.9 ± 1.1 31.1 ± 1.4 1.1 ± 1.1 3 16.7 16.3 ± 1.5 † 2.1 ± 1.5 16.7 14.1 ± 1.1 † 1.8 ± 1.1 4 20.3 20.6 ± 1.2 50.5 ± 1.6 2.0 ± 1.2 20.4 17.9 ± 0.9 43.7 ± 1.2 1.7 ± 0.9 5 26.8 24.8 ± 1.8 65.4 ± 1.9 2.8 ± 1.8 26.9 21.5 ± 1.4 56.6 ± 1.4 2.4 ± 1.4 6 31.8 32.7 ± 1.3 75.4 ± 1.6 3.7 ± 1.3 32.0 28.3 ± 1.0 65.2 ± 1.2 3.2 ± 1.0 7 37.2 36.6 ± 1.0 † 3.5 ± 1.0 37.1 31.7 ± 0.8 † 3.0 ± 0.8 8 42.85 44.7 ± 1.5 98.9 ± 1.9 2.9 ± 1.5 42.9 38.7 ± 1.1 85.4 ± 1.4 2.5 ± 1.1 9 45.2 43.3 ± 1.5 105.8 ± 1.8 5.1 ± 1.5 45.4 37.5 ± 1.1 91.3 ± 1.4 4.4 ± 1.1 (b) [100]: Previous Values Corrected Values Expt. Frequency Shift (cm-1) Frequency Shift (cm-1) # Stress Stress (GPa) (GPa) ΔωS ΔωD ΔωS ΔωD 10 27.8 48.6 ± 1.8 32.1 ± 2.1 27.7 42.1 ± 1.4 27.8 ± 1.6 11 33.85 58.8 ± 1.0 † 33.9 50.9 ± 0.8 † 12 41.75 70.1 ± 1.9 46.5 ± 2.2 41.7 60.6 ± 1.4 40.3 ± 1.7 † Not observed 141 6.4 Summary of Raman Spectroscopy Results The main results of this chapter are summarized as follows: 1. The Raman spectra of diamond shock compressed along [111] were measured for the first time and two shifted peaks, as expected, were observed. The frequency shifts for this orientation are in good agreement with the predictions from the anharmonic force constants based on earlier shock wave experiments along [100] and [110]. 2. The frequency shifts of the Raman spectra of shock compressed diamond were measured to ~60 GPa along [100] and [110]. The present data have extended previous shock compression measurements [1-3] to this higher value. Both natural and synthetic diamond crystals were shock compressed to ~60 GPa along the [100] direction. The frequency shifts at larger elastic stresses and with different crystal types are in good agreement with earlier predictions [1-3]. 3. A new set of anharmonic force constants of diamond were determined from the measured frequency shifts and the reanalyzed shock compression data [1-3]. These values are in good agreement with the values from previous shock work [1- 3], but are somewhat different from the values determined from uniaxial stress measurements [4]. 142 References 1. J. M. Boteler, “Time resolved Raman spectroscopy in diamonds shock compressed along [110] and [100] orientations,” Ph.D. thesis, Washington State University, 1993. 2. J. M. Boteler and Y. M. Gupta, “Raman spectra of shocked diamond crystals,” Phys. Rev. B 66, 014107 (2002). 3. J. M. Boteler and Y. M. Gupta, “Erratum: Raman spectra of shocked diamond crystals,” Phys. Rev. B 79, 179901 (2009). 4. M. H. Grimsditch, E. Anastassakis, and M. Cardona, “Effect of uniaxial stress on the zone-center optical phonon of diamond,” Phys. Rev. B 18, 901 (1978). 5. R. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz, and W. J. Carter, in High- Velocity Impact Phenomena, edited by R. Kinslow (Academic, New York, 1970), p. 293. 6. S. Root, “Physical and chemical changes in liquid benzene multiply shocked to 25 GPa,” Ph.D. thesis, Washington State University, 2007. 7. MATLAB® 7.10.0 (R2010a, The MathWorksTM, Natick, MA, 2010). 8. PeakFitTM 4.01 (Jandel Scientific, San Rafael, CA, 1995). 9. S. Ganesan, A. A. Maradudin, and J. Oitmaa, “A lattice theory of morphic effects in crystals of the diamond structure,” Ann. of Physics 56, 556 (1970). 10. E. Anastassakis, A. Cantarero, and M. Cardona, “Piezo-Raman measurements and anharmonic parameters in silicon and diamond,” Phys. Rev. B 41, 7529 (1990). 143 Chapter 7 ELASTIC LIMITS: RESULTS AND ANALYSIS This chapter presents the results and analysis related to the determination of elastic limits of diamond single crystals shocked along different orientations. Hugoniot elastic limit (HEL) measurements are used to determine the material strength and the yield response under rapid dynamic loading [1]. Section 7.1 describes the analysis of the experimental data and presents the Hugoniot elastic limits for diamond shocked along different orientations. Some of the material presented in this section was published in Ref. 2. In Section 7.2, the Hugoniot elastic limits are used to determine the shear strengths of shocked diamond crystals. The shear strengths for compression along different orientations are compared. In Section 7.3, the role of the stress component normal to the slip plane is investigated to explain the observed response. The main findings are summarized in Section 7.4. 7.1 Elastic Limit Measurements A total of 21 impact experiments at several different peak elastic stresses provided wave profiles in diamond shock compressed along the [100], [110], and [111] orientations. Representative profiles for the [100] orientation are shown in Figure 7.1. At lower peak stresses, a single elastic wave was typically observed, while at higher peaks stresses the wave profiles showed a two-wave structure. The experimentally measured wave profiles showing an elastic-inelastic response were analyzed to determine 144 3.0 Impactor Diamond V I 2.5 S A R ) s µ / Copper Buffer LiF Window m 2.0 m ( y t i c 1.5 o l e V e l 1.0 c i Expt. 1: 61.161.0 GPa t r a Expt. 2: 72.071.8 GPa P 0.5 Expt. 4: 92.292.0 GPa Expt. 8: 116.9118.0 GPa 0.0 0 5 10 15 20 25 30 Time (ns) Figure 7.1: Representative particle velocity histories measured at the diamond-LiF interface in synthetic diamonds shocked along the [100] direction to the peak elastic stresses listed. The higher stress experiments showed the arrival of an elastic wave followed by a slower, inelastic wave. At ~90 GPa peak elastic stress, the diamond sample remained elastic for ~5 ns before rapidly relaxing. The inset shows the experimental configuration. 145 the how the crystal orientation and peak stress affects the Hugoniot elastic limits (HEL) of shock compressed diamond crystals. It should be noted that some aspects of these experiments were previously analyzed in Chapter 4 to determine the nonlinear elastic response of diamond. The nonlinear elastic constants presented in Chapter 4 are used to analyze the results presented in this chapter. 7.1.1 Particle Velocity Histories Table 7.1 (same as Table 4.1) lists the relevant experimental parameters for the experiments used to determine the elastic limits. The impact velocity, projectile tilt, and velocity histories at both interfaces of the diamond sample were measured in each experiment. The peak elastic stresses presented in Table 7.1 were calculated using the Rankine-Hugoniot jump conditions [3], the impactor Hugoniot, and the elastic Hugoniots for diamond given by Eqs. (4.15)-(4.17). The peak elastic stresses correspond to the stress wave amplitudes in the diamond if the diamond samples remained elastic. These peak elastic stress values provide a convenient approach to quantifying the stress input to the crystals. Synthetic [100] ‒ Both synthetic and natural diamond crystals were shocked along the [100] orientation. Representative particle velocity histories measured at the diamond‒LiF interface of synthetic diamonds compressed along [100] to different peak elastic stresses are shown in Figure 7.1. A purely elastic response was observed in compression up to ~70 GPa, shown by the flat-topped waves in experiments 1 and 2. At a peak elastic stress of ~90 GPa (experiment 4), a sharp elastic wave at nearly the same amplitude (~90 GPa) was observed, followed by a decay and the subsequent arrival of the 146 Table 7.1: Experimental parameters for the particle velocity measurements in diamond crystals shocked along different crystal orientations. Copper impactors were used in all but one experiment. Peak Sample Projectile Impact Experiment Sample Sample Elastic Thickness Velocity Tilt Number Orientation Type Stress (mm) (mm/μs) (mrad) (GPa) 1 (09-618) [100] Synthetic 0.444 2.144 8.3 ± 0.8 61.0 2 (10-2S03) [100] Synthetic 0.444 2.463 5.7 ± 0.5 71.8 3 (09-2S19) [100] Synthetic 0.477 2.982 1.3 ± 0.3 90.3 4 (09-2S25) [100] Synthetic 0.445 3.027 5.2 ± 0.6 92.0 5 (09-2S26) [100] Synthetic 0.447 3.638 9.6 ± 0.9 115.1 6 (09-2S23) [100] Synthetic 0.480 3.654 11.1 ± 1.9 115.7 7 (09-2S22) [100] Synthetic 0.480 3.676 10.1 ± 1.0 116.5 8* (09-2S21) [100] Synthetic 0.441 3.187 3.7 ± 0.4 118.0 9 (09-2S20) [100] Natural 0.492 2.981 2.7 ± 0.7 90.3 10 (09-2S28) [100] Natural 0.494 3.033 4.6 ± 1.0 92.2 11 (09-2S27) [100] Natural 0.507 3.045 4.2 ± 0.4 92.6 12 (09-2S24) [100] Natural 0.514 3.644 9.9 ± 0.9 115.3 13 (09-2S29) [100] Natural 0.510 3.722 13.6 ± 1.4 118.3 14 (10-608) [110] Natural 0.437 2.089 3.7 ± 0.4 61.9 15 (10-2S08) [110] Natural 0.620 2.786 6.7 ± 0.7 88.0 16 (10-2S16) [110] Natural 0.603 2.812 7.5 ± 0.8 89.1 17 (10-2S04) [110] Natural 0.631 3.584 4.7 ± 0.4 121.0 18 (10-611) [111] Natural 0.534 2.111 1.8 ± 0.2 62.9 19 (10-2S18) [111] Natural 0.549 2.785 7.7 ± 0.9 88.2 20 (10-2S10) [111] Natural 0.520 2.825 8.5 ± 0.8 89.8 21 (10-2S09) [111] Natural 0.531 3.591 8.0 ± 0.8 121.5 * A Ta impactor was used in this experiment. 147 second wave, indicative of an elastic-inelastic response. At a higher peak elastic stress of ~120 GPa (experiment 8), a two-wave structure was again observed. However, the amplitude of the elastic wave measured in experiment 8 was lower than in experiment 4. This result was somewhat surprising and is discussed in a later section of this chapter. Natural [100] ‒ For natural diamonds shock compressed along [100], the representative particle velocity histories are shown in Figure 7.2. In contrast to the synthetic [100] diamond, the particle velocity histories at ~90 GPa (experiment 9) initially showed an elastic flat-top response, persisting for ~5 ns. This difference between the synthetic and natural crystals was reproducible and was observed in all experiments at this peak elastic stress. At the higher peak stress of ~120 GPa (experiment 12), the natural crystals exhibited an elastic-inelastic response similar to the synthetic crystals. Again, the HEL was lower in the experiments at higher peak elastic stress. Table 7.1 lists the relevant experimental parameters for the elastic limit measurements in [100] oriented synthetic and natural diamond crystals. The difference in the particle velocity histories of the natural and synthetic crystals at ~90 GPa peak stress suggests that there is a longer incubation time for the onset of inelastic deformation in natural crystals or the stress relaxation in natural crystals is slower compared to synthetic crystals. At larger peak stresses, this difference was likely overdriven, and natural and synthetic crystals behave similarly. Natural [110] ‒ Figure 7.3 shows representative particle velocity histories for natural diamonds compressed along [110]. Again, a purely elastic response was observed at ~60 GPa. At ~90 GPa (experiment 15), the decay following the elastic wave was very gradual before the arrival of the inelastic wave. At ~120 GPa peak elastic stress 148 3.0 2.5 ) s µ / m 2.0 m ( y t i c o l 1.5 e V e l c i 1.0 t r a P Expt. 9: 90.590.3 GPa 0.5 Expt. 12: 115.7115.3 GPa 0.0 0 5 10 15 20 25 30 Time (ns) Figure 7.2: Representative particle velocity histories measured at the diamond-LiF interface in natural diamonds shocked along [100] to the shown peak elastic stresses. Flat-top elastic waves were observed up to 72 GPa. While the elastic limit values for the natural and synthetic diamonds were comparable, the time-dependent response was very different at ~90 GPa peak stress, though similar at ~115 GPa peak stress. 149 3.0 2.5 ) s µ / m 2.0 m ( y t i c 1.5 o l e v e l 1.0 c i t r Expt. 14: 62.0 GPa a Expt. 14: 61.9 GPa P 0.5 Expt. 15: 88.288.0 GPa Expt. 17: 121.3121.0 GPa 0.0 0 5 10 15 20 25 30 Time (ns) Figure 7.3: Representative particle velocity histories measured at the diamond-LiF interface in diamonds shocked along [110] to the peak elastic stresses listed. A purely elastic response was observed at ~60 GPa peak stress, while at at ~90 and ~120 GPa peak stress an elastic-inelastic wave profile was observed. 150 (experiment 17), a sharp decay was again observed that was similar to, but not as pronounced as for the [100] orientation. In addition, the delay between the elastic and intelastic waves in the [110] orientated diamond is significantly larger that in the [100] oriented diamond. Natural [111] ‒ Representative particle velocity histories for compression along [111] are shown in Figure 7.4. At ~60 GPa peak elastic stress (experiment 18), an elastic- inelastic response was observed. A elastic response was observed at this peak elastic stress for shock compression along [100] and [110]. This observation of a two-wave, elastic-inelastic response at a much lower stress along the [111] orientation suggests that the diamond HEL is substantially lower when strained uniaxially along this orientation. At higher stresses, an elastic-inelastic response similar to that seen for the other orientations was observed. In the Raman spectroscopy experiments presented in Chapter 6, diamond crystals were shock compressed up to ~45 GPa along the [111] orientation. In those experiments, the ambient Raman spectrum returned during the release from the shocked state. The recovery of Raman spectrum implies that a purely elastic response would be observed to at least 45 GPa for diamond crystals shocked along [111]. The particle velocity histories measured along all three orientations showed varying degrees of stress relaxation behind the elastic wavefront. For compression along the [100] and [111] orientations pronounced stress relaxation occurred behind the elastic wave resulting in a ~50% decrease in the amplitude. In contrast, the stress relaxation was much smaller for compression along the [110] orientation. The time-dependent response observed in these experiments prevents the use of an impedance matching approach beyond the HEL. A quantitative analysis of the wave profiles beyond the HEL requires 151 3.0 2.5 ) s µ / m 2.0 m ( y t i c 1.5 o l e v e l c 1.0 i t r a P Expt.Expt. 18:18: 62.962.9 GPaGPa 0.5 Expt.Expt. 20:20: 88.388.3 GPaGPa Expt.Expt. 21:21: 121.6121.6 GPaGPa 0.0 0 5 10 15 20 25 30 Time (ns) Figure 7.4: Representative particle velocity histories measured at the diamond-LiF interface in diamonds shocked along [111] to the peak elastic stresses listed. All three profiles show a sharp elastic wave followed by a decay and a second wave. The elastic wave amplitude at ~120 GPa peak stress was nearly the same as at ~60 GPa peak stress, and significantly less than at ~90 GPa peak stress. 152 the use of numerical simulations that incorporate a time-dependent inelastic deformation description for diamond. At present, such a material model does not exist for diamond and the development of such a model was beyond the scope of the present study. The remainder of this chapter will focus on an analysis of only the elastic wave amplitude observed in the measured particle-velocity histories. 7.1.2 Elastic Wave Analysis The longitudinal and lateral stresses, and the densities corresponding to the measured Hugoniot elastic limits are listed in Table 7.2. The stresses in this table correspond to the laboratory or wave propagation coordinate systems, defined in Section 2.2.4, and are denoted by primed variables. The longitudinal stress and density at the Hugoniot elastic limit were determined from the measured particle velocity amplitude and shock velocity, as discussed in Section 4.1. However, the particle velocity and shock velocity measurements, and the Rankine-Hugoniot jump conditions [3] do not provide the lateral stresses at the HEL. Hence, the lateral stresses were calculated using finite strain theory [4], presented in Section 2.2.4, and the third-order elastic constants (Table 4.4) determined in this work. For uniaxial strain compression along the [100] and [111] orientations, the lateral stresses are degenerate. However, for the same longitudinal stress, the lateral stresses for the [100] orientation are 3 to 4 times larger than the lateral stresses for the [111] orientation. For uniaxial strain compression along the [110] orientation, the lateral stresses are non-degenerate; the large difference between the two sets of lateral stresses is noteworthy. Shock wave compression results in different, highly triaxial stress states for 153 different directions of wave propagation in diamond. 7.1.3 Hugoniot Elastic Limits The longitudinal elastic stresses and densities corresponding to the Hugoniot elastic limits, listed in Table 7.2, varied depending on the crystal orientation and the peak elastic stresses imparted to the diamonds. [100] ‒ For natural and synthetic diamonds shocked along [100] to a given peak elastic stress, the longitudinal stresses at the HEL were comparable. However, the the particle velocity histories showed significant differences beyond the elastic wave. This suggests that the source or growth method of high quality, type IIa diamond does not affect the yield strength of the crystal, but appears to play a role in the post-yield response. Because of the similarity in the HEL along [100], the response of synthetic [110] and [111] oriented diamonds was not investigated. At ~90 GPa peak elastic stress, the measured Hugoniot elastic limits for the [100] orientation were nearly ~90 GPa. This suggests that a peak elastic stress of ~90 GPa barely exceeded the HEL in the [100] experiments. At ~120 GPa peak elastic stress, the measured HEL values ranged between 54.4 and 61.0 GPa, the smallest elastic limits at this peak stress among the three orientations. Surprisingly, the elastic limits at ~120 GPa were even lower than the amplitudes of the elastic waves observed at ~60 GPa. The significant decrease in the elastic limit beyond ~90 GPa peak stress suggests that diamond cannot support an elastic stress as large as it could at lower peak stresses. [110] - For shock compression to ~90 GPa peak stress along the [110] orientation, the Hugoniot elastic limits were within few percent of the peak elastic stress. 154 Table 7.2: Experimental results corresponding to the Hugoniot elastic limit measurements in diamond shock compressed along different crystal orientations. Lateral stresses were calculated from the third-order elastic constants given in Chapter 4. For shock compression along the [100] and [111] directions, the lateral stresses are equal. Peak Hugoniot Elastic Limit Experiment Sample Sample Elastic Longitudinal Lateral Density Number Orientation Type Stress Stress (GPa) Stresses (GPa) (GPa) (g/cm3) σ1' σ2' σ3' 1* (09-618) [100] Synthetic 61.0 61.1 8.98 3.704 2* (10-2S03) [100] Synthetic 71.8 71.0 11.0 3.738 3 (09-2S19) [100] Synthetic 90.3 87.3 14.3 3.788 4 (09-2S25) [100] Synthetic 92.0 93.9 16.1 3.815 5 (09-2S26) [100] Synthetic 115.1 59.0 8.79 3.701 6 (09-2S23) [100] Synthetic 115.7 54.8 7.86 3.684 7 (09-2S22) [100] Synthetic 116.5 61.0 8.93 3.703 8 (09-2S21) [100] Synthetic 118.0 58.9 8.79 3.701 9 (09-2S20) [100] Natural 90.3 87.9 15.0 3.800 10 (09-2S28) [100] Natural 92.2 90.7 15.0 3.799 11 (09-2S27) [100] Natural 92.6 85.5 14.0 3.783 12 (09-2S24) [100] Natural 115.3 54.4 7.98 3.687 13 (09-2S29) [100] Natural 118.3 53.0 7.77 3.683 14* (10-608) [110] Natural 61.9 62.1 0.47 4.57 3.671 15 (10-2S08) [110] Natural 88.0 87.4 0.37 6.03 3.738 16 (10-2S16) [110] Natural 89.1 86.9 0.36 6.09 3.741 17 (10-2S04) [110] Natural 121.0 81.5 0.41 5.64 3.719 18 (10-611) [111] Natural 62.9 62.0 2.46 3.678 19 (10-2S18) [111] Natural 88.2 87.2 3.23 3.734 20 (10-2S10) [111] Natural 89.8 84.0 3.13 3.727 21 (10-2S09) [111] Natural 121.5 68.6 2.53 3.685 * A purely elastic response was observed in these experiments. 155 This behavior was similar to the measurements along the [100] orientation. The HEL decreased to 81.5 GPa at ~120 GPa peak elastic stress. This is somewhat smaller than the HEL measured as ~90 GPa peak elastic stress. However, at ~120 GPa peak elastic stress the decrease in HEL was much less pronounced than it was for the [100] orientation. [111] - For shock compression along the [111] orientation to ~60 GPa peak elastic stress, the Hugoniot elastic limit was nearly the same as the peak elastic stress. At ~90 GPa peak elastic stress, the elastic limits were again slightly less than the peak elastic stress, similar to the measurements along the [100] and [110] orientations. When the peak elastic stress was increased to ~120 GPa, a Hugoniot elastic limit of 68.6 GPa was measured, which was smaller than the HELs observed at ~90 GPa peak elastic stress. The [111] orientation showed an elastic-inelastic response (although it was nearly elastic) at ~60 GPa peak elastic stress while the other orientations first showed an elastic-inelastic response at a higher peak elastic stress of ~90 GPa. In contrast, the HEL measured at the highest peak stress for the [111] orientation was larger than the HELs observed at the same peak stress for the [100] orientation. At ~90 GPa peak elastic stress, the particle velocity histories along the three orientations showed significant differences but the Hugoniot elastic limits were remarkably similar for the different orientations. At ~120 GPa peak elastic stress, the HELs decreased along the three orientations, compared to the HELs at ~90 GPa peak elastic stress. The decrease in the measured HEL was less pronounced along the [110] and [111] orientations than it was along the [100] orientation. In the next section, the measured HEL values are analyzed to determine the shear stresses along relevant slip systems. 156 7.2 Shear Strength of Diamond Unlike isotropic solids, relating Hugoniot elastic limits measured along different orientations in single crystals is difficult [1,5]. Instead, shear stresses along relevant slip systems of a singe crystal provide a better measure of material strength [1,5-6]. In this section, the resolved shear stresses for the relevant slip systems are calculated from the longitudinal and lateral stresses at the HEL. These values are used to determine the shear strength of diamond under shock loading. 7.2.1 Resolved Shear Stresses In the diamond lattice, inelastic deformation occurs due to slip along the {111} 〈〉110 slip system, the lattice plane with the widest separation in the direction of the shortest Burgers vector [7-8]. It has been observed that slip also occurs along the {111} 〈〉110 system in silicon [9-10], another crystal having the diamond structure. Hence, this slip system is the one considered in the remainder of this work. Resolved shear stresses for shock wave compression may be calculated using the method of Johnson et al. [5], summarized in Section 2.5.1. The resolved shear stresses for the {111}〈〉 110 slip systems of a cubic crystal undergoing uniaxial strain compression along the [100], [110], and [111] directions are: 1 [100]: τ13 ' ' = ∣σ 1'−σ 2'∣ , (7.1) √6 1 1 [110]: (a) τ13 ' ' = ∣σ 1'−σ 3'∣ , (b) τ13 ' ' = ∣σ 2'−σ 3'∣ , (7.2) √6 √6 √2 [111]: τ13 ' ' = ∣σ1 '−σ2 '∣ . (7.3) 3√3 157 The shear stresses correspond to the coordinate system of the slip system, defined in Section 2.5.1, and are denoted by double-primed variables. For the [110] orientation, two different, but equivalent {111}〈〉 110 slip systems are active. The shear stress for the slip planes corresponding to Eq. (7.2b) is related to the difference in the two lateral stresses. It is very small compared to the shear stress for the slip planes corresponding to Eq. (7.2a). The smaller shear stress was not expected to cause inelastic deformation. Hence, only the larger shear stress from Eq. (7.2a) was considered in the analysis. The remaining resolved shear stresses are related to the difference in longitudinal and lateral stresses. Since lateral stresses were not experimentally determined, they were calculated using the third-order elastic constants determined in Chapter 4. The calculated resolved shear stresses at the HEL are shown in Table 7.3. Figure 7.5 shows the resolved shear stresses for the {111}〈〉 110 slip system as a function of the peak elastic longitudinal stress for the three orientations. Similar to the Hugoniot elastic limits, the resolved shear stress values vary with both crystal orientation and peak stress. For the three orientations, resolved shear stresses increased monotonically up to a peak elastic stress of ~90 GPa, as expected. The largest resolved shear stresses observed were ~33 GPa for samples shocked along [110], followed closely by ~30 GPa for sample shocked along [100]. The largest resolved shear stress for samples shocked along [111] was ~22 GPa, the smallest of the three orientations. When the peak elastic stress was increased to ~120 GPa, the resolved shear stress values corresponding to the HEL dropped for all three orientations. The drop in resolved shear stress suggests a strength loss beyond a peak elastic stress of ~90 GPa. The 158 Table 7.3: Calculated resolved shear and normal stresses corresponding to the Hugoniot elastic limit for the {111}〈〉 110 slip system in shock compressed diamond. Peak Hugoniot Elastic Limit Experiment Sample Sample Elastic Resolved Resolved Number Orientation Type Stress Shear Stress Normal Stress (GPa) (GPa) (GPa) 1* (09-618) [100] Synthetic 61.0 21.3 26.3 2* (10-2S03) [100] Synthetic 71.8 24.5 31.0 3 (09-2S19) [100] Synthetic 90.3 29.8 38.6 4 (09-2S25) [100] Synthetic 92.0 31.8 42.0 5 (09-2S26) [100] Synthetic 115.1 20.5 25.5 6 (09-2S23) [100] Synthetic 115.7 19.2 23.5 7 (09-2S22) [100] Synthetic 116.5 21.3 26.3 8 (09-2S21) [100] Synthetic 118.0 20.5 25.5 9 (09-2S20) [100] Natural 90.3 29.8 39.3 10 (09-2S28) [100] Natural 92.2 30.9 40.2 11 (09-2S27) [100] Natural 92.6 29.2 37.8 12 (09-2S24) [100] Natural 115.3 19.0 23.5 13 (09-2S29) [100] Natural 118.3 18.5 22.9 14* (10-608) [110] Natural 61.9 23.5 42.9 15 (10-2S08) [110] Natural 88.0 33.2 60.3 16 (10-2S16) [110] Natural 89.1 33.0 60.0 17 (10-2S04) [110] Natural 121.0 31.0 56.2 18 (10-611) [111] Natural 62.9 16.2 9.1 19 (10-2S18) [111] Natural 88.2 22.9 12.6 20 (10-2S10) [111] Natural 89.8 22.0 12.1 21 (10-2S09) [111] Natural 121.5 18.0 9.9 * A purely elastic response was observed in these experiments. 159 40 ) 35 a P G 30 ( s s e 25 r t s r a 20 e h s 15 d e v l 10 Elastic response: o s [100] [110] [111] e R 5 Data: [100] [110] [111] 0 0 20 40 60 80 100 120 140 Peak elastic stress (GPa) Figure 7.5: Calculated resolved shear stress for the {111}〈〉 110 slip system as a function of the calculated peak elastic stress. The resolved shear stresses correspond to the HEL. The peak elastic stresses were obtained by assuming a nonlinear elastic response for the diamond. Experimental results are shown as symbols and expected elastic response calculated from the third-order elastic constants are shown as lines. The resolved shear stress decreased for all three orientations at the highest peak stresses, as indicated by the arrows. 160 magnitude of the drop in the resolved shear stress varied with crystal orientation. Of the three orientations, the smallest drop in resolved shear stresses was for the [110] orientation. The [100] orientation showed the largest decrease in the resolved shear stress. The resolved shear stresses for the [100] orientation were slightly larger than those determined for the [111] orientation at ~120 GPa peak elastic stress. The [110] orientation was most resistant to the onset of inelastic deformation, while the [111] orientation was least resistant. At ~90 GPa peak elastic stress, the [100] orientation showed a stronger response, similar to the [110] orientation. But at ~120 GPa peak elastic stress, the [100] orientation showed a weaker response, similar to the [111] orientation. A decrease in resolved shear stress with increasing peak stress was not observed in the laser shock experiments in Refs. [11-12]. This may, in part, be due to the larger peak stresses and shorter shock wave durations in those experiments. Also, the peak stresses were not accurately known in these experiments. 7.2.2 Shear Strengths Under Shock Compression The shear strength of single crystal diamond subjected to uniaxial strain compression along a particular orientation is expressed as the maximum resolved shear stress determined for that orientation. Table 7.4 lists the shear strengths under compression along each orientation and the ratio of the shear strength to the shear modulus for the {111} plane. For a cubic crystal, the shear modulus G for the {111} plane is [13] 1 G= (C −C +C ) , (7.4) 3 11 12 44 161 Table 7.4: Shear strength, ratio of strength to shear modulus, and the normal stress for the {111}〈〉 110 slip system of diamond for uniaxial strain along different orientations. Resolved Shear Strength: τ ' ' τ13 ' ' 13 Normal Stress: Orientation τ ' ' (GPa) N 13 G N σ33 ' ' σ33 ' ' (GPa) [100] 31.8 0.063 42.0 0.76 [110] 33.2 0.065 60.3 0.55 [111] 22.9 0.045 12.6 1.82 where Cij are the second-order elastic constants. Using the second-order elastic constants (Table 2.1) [14], the shear modulus of diamond for the {111} plane is 511 GPa. The shear strengths are comparable for uniaxial strain compression along [100] and [110], but the shear strength is significantly smaller for compression along [111]. The measured shear strengths in this work are significantly different from the theoretical calculations (~90-120 GPa) for perfect diamond crystals [15-17]. As is well known, defects in real crystals can cause slip to occur at much lower shear stresses, causing a reduction in strength from that of the perfect crystal [18-19]. McWilliams et al. [12] reported on the strength of diamonds that were laser- shocked to peak elastic stresses between ~0.1 and ~1 TPa. Except for the lowest stresses, the peak stresses were much greater than those reached in this work. The analysis of the resolved shear stresses corresponding to the HEL in Ref. 12 is similar to the analysis in this work. But because of the significant uncertainties in their measurements, and the experimental and conceptual difficulties with this work (discussed in Section 2.5.2), it is difficult to compare the results of this study with the results of Ref. 12. However, it can 162 be said that the resolved shear stresses reported in that work [12] are somewhat smaller, but comparable, to those presented in this work. The different shear strengths along the {111}〈〉 110 slip system for different loading directions show that wave propagation direction relative to the slip system is an important factor for determining diamond strength. In the following section, the shear strength dependence on orientation is re-examined by taking into account the stress magnitude normal to the slip planes. 7.3 Role of Normal Stress For some materials, it is only the resolved shear stress for a given slip system that determines the onset of inelastic deformation [6]. However, the present work in shock compressed diamond crystals has demonstrated that the critical resolved shear stress, corresponding to the HEL, depends on the direction of shock wave propagation. This finding suggests that there may be another factor contributing to the onset of inelastic deformation. Furthermore, the theoretical calculations of diamond strength for different loading conditions have shown that the stress magnitude normal to the slip plane can result in a larger critical resolved shear stress, resulting in a larger apparent shear strength [20]. 7.3.1 Resolved Normal Stress N The resolved normal stress on the slip plane is the element σ33 ' ' of the transformed stress tensor and the same was calculated using the method of Johnson et al. [10]. The resolved normal stresses for uniaxial strain along the [100], [110], and [111] 163 orientations are: N 1 [100]: σ33 ' ' = ∣σ 1' +2σ2'∣ , (7.5) 3 N 1 [110]: σ33 ' ' = ∣2σ 1'+σ3 '∣ , (7.6) 3 N 1 [111]: σ33 ' ' = ∣σ1' +8σ 2 '∣ . (7.7) 9 The last column of Table 7.3 lists the resolved normal stress corresponding to the HEL for each of the orientations. Figure 7.6 shows plots of the resolved shear stress as a function of the resolved normal stress for the {111}〈〉 110 slip system for shock compression along the three orientations examined in this work. The measured data for each orientation lie on the calculated elastic response curves; this is a consequence of calculating the lateral stresses using third-order elastic constants which constrained the data to the calculated elastic response curves. The slopes (in the linear approximation) of the three lines give the ratios of the resolved shear stress to the resolved normal stress for each orientation and are listed in Table 7.4. This ratio is largest for the [111] orientation, which is the orientation that showed the smallest critical resolved shear stress. Conversely, the ratio of the resolved shear stress to the resolved normal stress was smallest for the [110] orientation, but that is the orientation that showed the largest critical resolved shear stress. These findings suggest that the stress normal to the slip system plays a key role in determining the onset of inelastic deformation in shocked diamond. 164 40 ) 35 a P G 30 ( s s e 25 r t s r a 20 e h s 15 d e v l 10 Elastic response: o s [100] [110] [111] e R 5 Data: [100] [110] [111] 0 0 10 20 30 40 50 60 70 Resolved normal stress (GPa) Figure 7.6. Calculated resolved shear stress as a function of resolved normal stress for the {111}〈〉 110 slip system. The resolved shear and normal stress data correspond to the HEL. The ratio of resolved shear stress to resolved normal stress was largest (~1.8) for compression along [111] which has the smallest critical resolved shear stress, while the ratio was smallest (~0.55) for compression along [110] which has the largest critical resolved shear stress. 165 7.3.2 Comparison with Silicon and Germanium The elastic limits of shocked silicon and germanium crystals, both having the diamond structure, have been measured for the [100], [110] and [111] orientations [21-23]. In this section the shear strengths of shocked silicon and germanium single crystals are compared to the shear strengths of diamond measured in this work. The Hugoniot elastic limits of [100], [110], and [111] oriented silicon were first reported using the inclined prism technique [21]. More recently, particle velocity measurements in silicon shocked along the [100] and [111] orientations have provided significantly more accurate determinations of the elastic limits [22]. Using the second- and third-order elastic constants for silicon [24], the resolved shear and normal stresses for the {111}〈〉 110 slip system were calculated using Eqs. (7.1)-(7.3) and Eq. (7.5)-(7.7); the same are plotted in Figure 7.7. For silicon shocked along [100], the shear strengths from Gust and Royce [21] and Turneaure and Gupta [20] are comparable. In contrast, for silicon shocked along [111] the shear strength determined in the later study [22] is significantly larger. Considering only the [100] and [111] results from the more recent work [22], the stress normal to the slip plane appears to play a role in the shear strength of shocked silicon. However, the measured shear strength for silicon shocked along [110] is significantly [21] less than the expected shear strength, if the normal stress played a role. The HELs of [100], [110], and [111] oriented germanium were also measured using the less accurate inclined prism technique [23]. Since wave profiles measurements in shocked germanium are not available, only the HEL values from Ref. 23 were analyzed. Using the second- and third-order elastic constants for germanium [24], the 166 2.5 Elastic response: 2.0 [100] ) [110] a P [111] G ( s s e r 1.5 t s r a e h s d e 1.0 v l o s e Data: R Turneaure and Gupta (2007): 0.5 [100] [111] Gust and Royce (1971): [100] [110] [111] 0.0 0 1 2 3 4 5 6 7 Resolved normal stress (GPa) Figure 7.7. Calculated resolved shear stress versus resolved normal stress for the {111} <110> slip system in shock compressed silicon crystals. The lines show the elastic response calculated from the second- and third-order elastic constants of silicon [24], and the points are calculated from the elastic limits for each orientation [21-22]. 167 resolved shear and normal stresses in the {111}〈〉 110 slip system were calculated using Eqs. (7.1)-(7.3) and Eqs. (7.4)-(7.7); the same are plotted in Figure 7.8. The shear and normal stresses for the [100] and [110] were very similar, while the shear and normal stress values for the [111] orientation were smaller. For the three orientations in silicon and germanium crystals, there may be some correlation between the stress normal to the slip plane and the shear strength, but more data are needed to draw firm conclusions. 7.3.3 Comparison with Theoretical Calculations Theoretical simulations of shear deformation in diamond, silicon and germanium show significantly different responses [17,25]. After the shear strength was exceeded, diamond showed a dramatic 50% increase in volume, while silicon and germanium showed a gradual decrease in volume [17,25]. Ab initio simulations also showed that both the application of normal stress on the {111} plane and hydrostatic pressure increase the calculated critical shear stress in diamond, but decreased the calculated critical shear stress in silicon and germanium [20,25]. Roundy and Cohen suggested that this difference may be caused by carbon forming a graphitic structure of strong π bonds following the shear instability [17]. This does not mean that diamond transforms into graphite following inelastic deformation. Instead, when bonds in diamond break, carbon may form a three-fold coordinated graphite-like structure which results in a volume increase [17]. A similar structure was expected for the π-bonded chain reconstruction of {111} diamond surfaces [26]. Umeno et al. suggested that different response of diamond was caused by the 168 2.0 1.8 ) a P 1.6 G ( 1.4 s s e r 1.2 t s r a 1.0 e h s 0.8 d e 0.6 v l Elastic response: o s 0.4 [100] [110] [111] e R Data: 0.2 [100] [110] [111] 0.0 0 1 2 3 4 5 Resolved normal stress (GPa) Figure 7.8. Calculated resolved shear stress versus resolved normal stress for the {111} <110> slip system in shock compressed germanium crystals. The lines show the elastic response calculated from the second- and third-order elastic constants of germanium [24], and the points are calculated from the elastic limits for each orientation [23]. 169 strong covalency in its interatomic bondings [25]. Using a Tersoff interatomic potential model [27], they found that the dependence of the short-range attraction on the bond angle (bond-order) played a dominant role in shear deformation behavior of diamond and silicon. They associated the bond-order with the covalency of the bonding, which is stronger for carbon and weaker for silicon and germanium [25]. The larger bonding covalency in diamond resulted in an increase in the shear strength with normal stress, and an increase in volume during shear [25]. The shear strengths and normal stresses in diamond determined from the experimental measurements in this work and the theoretical calculations [17,20,25] are consistent with each other. The measurements and the calculations show that when the stress normal to the slip plane is larger, the shear strength is also larger. The theoretical calculations also show that shear deformation in diamond is accompanied by an increase in volume [17,20,25]. The correlation between the shear strength and the normal stress can be interpreted as the normal stress opposing the increase in volume. The theoretical calculations associated the volume increase with the formation of a graphitic structure along the slip plane [17,20], or as a decrease in the short-range interatomic attraction caused by bond-bending [25]. The normal stress may inhibit the bending and breaking of bonds, and the formation of new π bonds. This work provides the first experimental observation of this effect in diamond. In contrast to diamond, silicon and germanium have p electrons in their cores, which prevent them from forming the strong π bonds of a graphitic structure [17]. Silicon and germanium also have a weaker bonding covalency than diamond [25]. While the calculations show a definite decrease in the shear strength with normal stress for 170 silicon and germanium [20,25], the experimental results are less clear, but appear to show an increase in shear strength with normal stress [21-23]. This suggests that, while the normal stress may play a role in the shear strength, there may be another factor contributing to the shear strength of shocked silicon and germanium single crystals. 7.4 Summary of Elastic Limit Results The main results of this chapter are summarized as follows: 1. The Hugoniot elastic limits of diamond crystals shocked along different orientations have been measured in extremely well characterized loading conditions. 2. The Hugoniot elastic limits of natural and synthetic diamonds shocked along [100] are the same even though the wave profiles beyond the HEL show a very different inelastic response. This suggests that the source or growth method of high quality (type IIa) diamond crystals does not affect the yield strength of the crystal. However, the crystal type likely plays a role in the post-yield behavior. 3. The Hugoniot elastic limits for the three orientations increased up to ~90 GPa peak elastic stress. Beyond ~90 GPa peak elastic stress, the Hugoniot elastic limits decreased. This suggests that at larger peak stresses there was a loss of strength on the slip plane which reduced the measured HEL. The drop in HEL was relatively small along the [110] direction and very large along the [100] direction, suggesting that this strength loss was anisotropic. 4. Shear strengths corresponding to the {111}〈〉 110 slip system were determined for the three orientations. Shocked diamond was strongest along the [110] direction 171 and weakest along the [111] direction. For all orientations, the measured shear strengths were significantly less than the shear strengths predicted for a perfect diamond crystal. 5. The dependence of the shear strength on the shock propagation direction correlated with the stress normal to the slip plane. A larger shear strength was observed when the normal stress was larger. A smaller shear strength was observed when the normal stress was smaller. The stress normal to the slip plane appeared to inhibit the onset of inelastic deformation in diamond. 6. The shear strengths of diamond were compared the shear strengths of shocked silicon and germanium. The experimental measurements showed that there may be a correlation between the normal stress and the shear strength for silicon and germanium. Theoretical calculations predicted an increase in the shear strength with normal stress for diamond, which agree with the observations of this work. The normal stress may inhibit the bending and breaking of bonds, and the formation of new π bonds, preventing the onset of inelastic deformation in diamond. 172 References 1. Y. M. Gupta, “Effect of crystal orientation on dynamic strength of LiF,” J. Appl. Phys. 48, 5067 (1977). 2. J. M. Lang and Y. M. Gupta, “Strength and elastic deformation of natural and synthetic diamond crystals shock compressed along [100],” J. Appl. Phys. 107, 113538 (2010). 3. R. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz, and W. J. Carter, in High- Velocity Impact Phenomena, edited by R. Kinslow (Academic, New York, 1970), p. 293. 4. R. N. Thurston, “Wave propogation in fluids and normal solids” in Physical Acoustics, Vol. 1A, edited by W. P. Mason (Academic, New York, 1964), p. 1. 5. J. N. Johnson, O. E. Jones, and T. E. Michaels, “Dislocation dynamics and single- crystal constitutive relations: shock-wave propagation and precursor decay,” J. Appl. Phys. 41, 2330 (1970). 6. A. Kelly, W. R. Tyson, and A. H. Cottrell, “Ductile and brittle crystals,” Philos. Mag. 15, 567 (1967). 7. A. Kelly, Strong Solids (Clarendon, Oxford, 1966). 8. N. S. Brar and W. R. Tyson, “Elastic anisotropy and slip systems in diamond structure elements”, Scripta Metallurgica 6, 587 (1972). 9. J. Rabier, P. Cordier, J. L. Demenet and H. Garem, “Plastic deformation of Si at low temperature under high confining pressure”, Materials Science and Engineering A 309, 74 (2001). 10. J. Tan, S. X. Li, Y. Wan, F. Li and K. Lu, “Crystallographic cracking behavior in 173 silicon single crystal wafer”, Materials Science and Engineering B 103, 49 (2003). 11. R. S. McWilliams, “Elastic and inelastic shock compression of diamond and other minerals,” Ph.D. thesis, University of California, Berkeley, 2008. 12. R. S. McWilliams, J. H. Eggert, D. G. Hicks, D. K. Bradley, P. M. Celliers, D. K. Spaulding, T. R. Boehly, G. W. Collins, and R. Jeanloz, “Strength effects in diamond under shock compression from 0.1 to 1 TPa,” Phys. Rev B 81, 014111 (2010). 13. J. F. Nye, Physical Properties of Crystals (Clarendon Press, Oxford, 1957). 14. H. J. McSkimin and P. Andreatch, Jr., “Elastic moduli of diamond as a function of pressure and temperature,” J. Appl. Phys. 43, 2944 (1972). 15. W. R. Tyson, “Theoretical strength of perfect crystals,” Phil. Mag. 14, 925 (1966). 16. A. Kelly, W. R. Tyson, and A. H. Cottrell, “Ductile and brittle crystals,” Phil. Mag. 15, 567 (1967). 17. D. Roundy and M. L. Cohen, “Ideal strength of diamond, Si, and Ge,” Phys. Rev. B 64, 212103 (2001). 18. A. H. Cottrell, Dislocations and Plastic Flow in Crystals (Clarendon, Oxford, 1953). 19. A. Kelly, Strong Solids (Clarendon, Oxford, 1966). 20. Y. Umeno and M. Černý, “Effect of normal stress on the ideal shear strength in covalent crystals,” Phys. Rev. B 77, 100101 (2008). 21. W. H. Gust and E. B. Royce, “Axial yield strengths and two successive phase transisiton stresses for crystalline silicon,” J. Appl. Phys. 42, 1897 (1971). 174 22. S. J. Turneaure and Y. M. Gupta, “Inelastic deformation and phase transformation of shock compressed silicon single crystals,” Appl. Phys. Lett. 91, 201913 (2007). 23. W. H. Gust and E. B. Royce, “Axial yield strengths and phase-transisiton stresses for 〈〉〈〉〈〉 100 , 110 , and 111 germanium,” J. Appl. Phys. 43, 4437 (1972). 24. H. J. McSkimin and P. Andreatch, Jr., “Measurement of third-order moduli of silicon and germanium,” J. Appl. Phys. 35, 3312 (1964). 25. Y. Umeno, Y. Shiihara and N. Yoshikawa, “Ideal shear strength under compression and tension in C, Si, Ge, and cubic SiC: an ab initio density functional theory study,” J. Phys.: Condens. Matter 23, 385401 (2011). 26. D. Vanderbilt and S. G. Louie, “Total energy minimization for diamond (111) surfaces: Support for an undimerized π-bonded chain reconstruction,” Phys. Rev. B 29, 7099 (1984). 27. J. Tersoff, “Modeling solid-state chemistry: Interatomic potentials for multicomponent systems,” Phys. Rev. B 39, 5566 (1989). 175 Chapter 8 SUMMARY AND CONCLUSIONS 8.1 Summary The overall goal of this work was to examine the mechanical and optical response of shock-compressed diamond crystals, and to evaluate the effects of crystalline anisotropy on this response. Well characterized, plate impact experiments, resulting in uniaxial strain compression, were used to generate peak elastic stresses ranging from ~15 GPa to ~120 GPa along the [100], [110], and [111] orientations in natural and synthetic, Type IIa diamonds. The mechanical response and refractive index of shocked diamond were determined using laser interferometry measurements (VISAR) [1], which provided shock wave profiles at both interfaces of the diamond samples. Raman spectra were measured using a time-resolved spectroscopy system used in previous shock compression measurements [2]. The measured elastic wave profiles were used to examine the nonlinear elastic response of diamond shocked along [100], [110], and [111]. Natural and synthetic crystals shocked along [100] showed no difference in their elastic response. The profiles measured for the three crystal orientations, in combination with published acoustic data [3], were used to determine the complete set of third- order elastic constants for diamond, the first determination entirely from experimental data. Several of these constants differed significantly from those determined using theoretical models [4-7]. Laser interferometry [1] was used to measure changes in the optical path length 176 through diamond crystals shocked along [100] and [111], and the refractive indices were determined from these changes. The experimentally determined refractive indices were compared to predictions from linear photoelasticity theory [8] and good agreement was observed for shock compression along [111]. However, the linear theory underpredicted the measurements for shock compression along [100]. The linear theory was extended by fitting the nonlinear elasto-optic constant p221111 to achieve good agreement with the [100] data. Particle velocity correction factors were determined to permit the use of diamond crystals, oriented along [100] and [111], as optical windows in shock compression experiments. Diamond crystals shocked along [110] showed evidence of birefringence, but the refractive indices for this orientation could not be determined from the present measurements. Time-resolved Raman spectroscopy was used to observe the splitting of the triply degenerate 1332.5 cm-1 Raman line for diamond crystals shocked along [111]. Previous measurements [2] along [100] and [110] were extended to larger peak stresses. As expected, two shifted peaks were observed for shock compression along [111], and the frequency shifts were in good agreement with the predictions from previous work [2,9- 10]. The frequency shifts for [100] (natural and synthetic) and [110] (natural only) oriented crystals shocked to ~60 GPa also agree with the earlier predictions based on measurements at lower stresses. The anharmonic force constants for diamond determined from all available shock compression data were nearly the same as the published values determined from shock compression measurements along [110] [2,9-10]. The constants determined from shock compression measurements (to ~60 GPa) were somewhat different from the published constants determined from uniaxial stress measurements [11] 177 at lower stresses (< 1 GPa). The Hugoniot elastic limits of diamond crystals shocked to peak elastic stresses of ~120 GPa along the [100], [110] and [111] orientations were determined from the measured particle velocity histories. Natural and synthetic crystals shocked along the [100] orientation had comparable elastic limits at all peak elastic stress. This indicates that the yield strength of shocked diamond does not depend on the source or growth method of the crystal. However, the post-elastic responses of the two crystal types for shock compression along [100] were different at ~90 GPa peak elastic stress. This difference suggests that the crystal type may play a role in the post-elastic behavior. The largest elastic limits for the three orientations were observed at ~90 GPa peak elastic stress. The elastic limits decreased at ~120 GPa peak elastic stress by different amounts for each orientation. The HEL decrease was smallest for shock compression along [110] and largest for shock compression along [100]. This suggests an anisotropic loss of strength in diamond at the higher peak stress. Critical resolved shear stresses for the {111}〈〉 110 slip system were calculated from the present data: the [110] orientation showed the greatest shear strength and the [111] showed the smallest shear strength. The measured shear strengths were significantly less than the shear strengths predicted for a perfect diamond crystal [12-14]. The dependence of the shear strength on the shock propagation direction was correlated with the stress normal to the slip plane. Stress normal to the slip plane appeared to inhibit the onset of inelastic deformation in diamond. Theoretical calculations of shear in diamond also showed an increase in the shear strength of diamond when there was a compressive stress normal to the slip plane [14- 178 15]. They also showed that the volume dramatically increased after exceeding the shear strength. The volume increase and elastic limit were associated with the bending, breaking, and formation of new carbon-carbon bonds [14-15]. A stress normal to the slip plane may prevent the bonds from changing, resulting in a larger shear strength. 8.2 Conclusions The main findings of this study are compiled below: ● Nonlinear elastic response 1. The complete set of third-order elastic constants for diamond was determined solely from experimental data. Second-order elastic constants alone do not predict an elastic response that allows the formation of a compressive shock wave. Third-order constants are needed to accurately represent the elastic response of diamond beyond 1% compression. Several of the third-order elastic constants differed significantly from those determined from theoretical calculations. The elastic response of natural and synthetic diamonds was indistinguishable. 2. Experimentally determined elastic Hugoniot relations for diamond crystals shocked along [100], [110], and [111] were provided. ● Refractive index 3. The refractive indices of diamond crystals shock compressed along [100] and [111] increased with uniaxial strain, in contrast to the reported decrease in refractive index with pressure under hydrostatic compression. The diamond refractive index depends on components of the strain tensor, rather than on 179 density alone. 4. For shock compression along [110], the measured wave profiles suggested that the diamond crystal became birefringent. This result was consistent with the expected change from cubic to orthorhombic symmetry for uniaxial strain along this direction. 5. Linear photoelasticity theory predicted the refractive indices measured for shock compression along [111]. Nonlinear photoelasticity was needed to predict the refractive indices measured for compression along [100]. The second-order elasto-optic constant p221111 = ‒ 0.263 provided a good fit to the [100] data. 6. The particle velocity correction factors for diamond shocked along [100] and [111] were found to be 1.992 and 2.020, respectively, permitting the use of these diamond orientations as optical windows in shock wave experiments. ● Raman spectra 7. As expected, the Raman spectra of diamond shock compressed along [111] showed two shifted peaks. The frequency shifts were consistent with predictions based on previous measurements under shock compressions along [100] and [110]. 8. The frequency shifts measured for natural and synthetic diamonds along [100] and for natural diamonds along [110] to ~60 GPa were consistent with the previous shock compression measurements at lower stresses (to ~45 GPa). 9. The anharmonic force constants determined using all of the available shock compression data are similar to the values determined from the previous shock compression measurements. These constants predict the frequency shifts for the 180 three orientations and both crystals types for large elastic strains. ● Elastic limits and shear strength 10. Shock compressed diamond crystals showed an elastic-inelastic response at peak elastic stresses, ~90 GPa and larger, for compression along [100] and [110]; and at ~60 GPa and larger for compression along [111]. The Hugoniot elastic limits were highest for each orientation at ~90 GPa peak elastic stress but decreased at ~120 GPa peak elastic stress, indicating a loss of strength at larger stresses. 11. While natural and synthetic crystals had comparable elastic limits, the wave profiles showed a very different inelastic response. Because of this, the crystal type appears to play a role in the post-yield behavior. 12. Shock compression along different crystal orientations gave rise to different critical resolved shear stresses for the {111}〈〉 110 slip system. The dependence of the shear strength on the shock propagation direction correlates with the stress normal to the slip plane. Stress normal to the slip plane appeared to delay the onset of inelastic deformation in diamond. 13. Theoretical calculations predicted that a stress normal to the slip plane would increase the shear strength of diamond. These predictions agree with the results of this study. The stress normal to the slip plane may inhibit the bending and breaking of bonds, and the formation of new π bonds, preventing the onset of inelastic deformation in diamond. 181 References 1. L. M. Barker and R. E. Hollenbach, “Laser interferometer for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669 (1972). 2. J. M. Boteler, “Time resolved Raman spectroscopy in diamonds shock compressed along [110] and [100] orientations,” Ph.D. thesis, Washington State University, 1993. 3. H. J. McSkimin and P. Andreatch, Jr., “Elastic moduli of diamond as a function of pressure and temperature,” J. Appl. Phys. 43, 2944 (1972). 4. M. H. Grimsditch, E. Anastassakis, and M. Cardona, “Effect of uniaxial stress on the zone-center optical phonon of diamond,” Phys. Rev. B 18, 901 (1978). 5. E. Anastassakis, A. Cantarero, and M. Cardona, “Piezo-Raman measurements and anharmonic parameters in silicon and diamond,” Phys. Rev. B 41, 7529 (1990). 6. C. S. G. Cousins, “Elasticity of carbon allotropes. I. Optimization, and subsequent modification, of an anharmonic Keating model for cubic diamond,” Phys. Rev. B 67, 024107 (2003). 7. O. H. Nielsen, “Optical phonons and elasticity of diamond at megabar stresses,” Phys. Rev. B 34, 5808 (1986). 8. M. H. Grimsditch and A. K. Ramdas, “Brillouin scattering in diamond,” Phys. Rev. B 11, 3139 (1975). 9. J. M. Boteler and Y. M. Gupta, “Raman spectra of shocked diamond crystals,” Phys. Rev. B 66, 014107 (2002). 10. J. M. Boteler and Y. M. Gupta, “Erratum: Raman spectra of shocked diamond crystals,” Phys. Rev. B 79, 179901 (2009). 182 11. M. H. Grimsditch, E. Anastassakis, and M. Cardona, “Effect of uniaxial stress on the zone-center optical phonon of diamond,” Phys. Rev. B 18, 901 (1978). 12. W. R. Tyson, “Theoretical strength of perfect crystals,” Phil. Mag. 14, 925 (1966). 13. A. Kelly, W. R. Tyson, and A. H. Cottrell, “Ductile and brittle crystals,” Phil. Mag. 15, 567 (1967). 14. D. Roundy and M. L. Cohen, “Ideal strength of diamond, Si, and Ge,” Phys. Rev. B 64, 212103 (2001). 15. Y. Umeno, Y. Shiihara and N. Yoshikawa, “Ideal shear strength under compression and tension in C, Si, Ge, and cubic SiC: an ab initio density functional theory study,” J. Phys.: Condens. Matter 23, 385401 (2011). 183