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The Index of Refraction of Lithium Fluoride at Pressures in Excess of 100 GPa by Dayne Eric Fratanduono Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Supervised by Doctor Thomas R. Boehly Professor David D. Meyerhofer Department of Mechanical Engineering Arts, Sciences and Engineering Edmund A. Hajim School of Engineering and Applied Sciences University of Rochester Rochester, New York 2010 ii CURRICULUM VITAE Dayne Eric Fratanduono was first interest in engineering when he attended Clarkson University from 2002 to 2006. In 2006 he graduated with great distinction from Clarkson University with a Bachelor of Science in mechanical engineering and physics (dual major). Interested in the Inertial Confined Fusion (ICF) campagain at the Laboratory for Laser Energetics at the University of Rochester, he pursued further research studies. At the University of Rochester he was awarded the prestigious Sproull Fellowship from the fall of 2006 until the fall of 2008. He received the Horton fellowship over the remainder of this time at the University. In 2008, he received a Master of Science degree in Mechanical Engineering and remained at the University of Rochester to fullfill the requires of the Doctoral degree in Mechanical Engineering. His thesis was performed at the Laboratory for Laser Energetics under the direction of Dr. D.D. Meyerhofer and Dr. T.R. Boehly which focused on material properties in the high energy density regime. Publications and selected professional presentations include: • D.E. Fratanduono, T.R. Boehly, M.A. Barrios, D.D. Meyerhofer, J.H. Eg- iii gert, R.F. Smith, D.G. Hicks, P.M. Celliers, D.G. Braun and G.W. Collins. “Refractive Index of Lithium Fluoride at Pressures up to 800 GPa.” Submitted to Phys. Rev. Letter in September, 2010. • M.A. Barrios, D.G. Hicks, T.R. Boehly, D.E. Fratanduono, J.H. Eggert, P.M. Celliers, G.W. Collins, and D.D. Meyerhofer. “High-precision measurements of the equation of state of hydrocarbons at 1-10 Mbar using laser-driven shock waves,” Physics of Plasmas, 17, 056307 (2010). • D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, J. H. Eggert, R. Smith, D. G. Hicks, P. M. Celliers, and G. W. Collins, “Measures of Strain-Induced Refractive-Index Changes in Ramp-Compressed Lithium Fluoride.” Contributed poster, OMEGA Laser Facility Users Workshop, Rochester, NY, 28-30 April 2010. • D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, R. Smith, J. H. Eggert, D. G. Hicks, P. M. Celliers, G. W. Collins, and R. Rygg, “Measurements of Strain-Induced Refractive Index Changes in LiF Using Direct-Drive Ramp Compression.“ Contributed talk, 51st Annual Meeting of the APS Division of Plasma Physics, Atlanta, GA, 2-6 November 2009. • D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, R. iv Smith, J. H. Eggert, D. G. Hicks, P. M. Celliers, and G. W. Collins, ”Measurements of Strain-Induced Refractive-Index Changes in Shocked LiF Using Laser-Driven Flyer Plates.“ Contributed talk, 16th APS Topical Conference in Shock Compression of Condensed Matter, Nashville, TN, 28 June-3 July 2009. • D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, J. Eggert, R. Smith, D. G. Hicks, and G. Collins, ”Measurements of StrainInduced Refractive Index Changes in Shocked LiF Using Laser-Driven Flyer Plates.“ Contributed talk, OMEGA Laser Facility Users Group Workshop, Rochester, NY, 29 April-1 May 2009. • D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, D. G. Hicks, P. M. Celliers, S. Wilks, and R. Smith, ”Optical Properties of Materials at High Pressure Using ’Sandwich’ Targets.“ Contributed talk, 50th Annual Meeting of the APS Division of Plasma Physics, Dallas, TX, 17-21 November 2008. • D. E. Fratanduono, M. A. Barrios, T. R. Boehly, D. D. Meyerhofer, D. G. Hicks, P. M. Celliers, S. Wilks, and J. E. Miller, ”Nonequilibrium Conditions in a Shock Front.“ Contributed talk, 49th Annual Meeting of the APS Division of Plasma Physics, Orlando, FL, 12-16 November 2007. v ACKNOWLEDGEMENTS I am deeply indebted to all the individuals who provided assistance throughout my academic career. Most importantly, I would like to thank mother, father and sister for their constant encouragement and support, without them this work would not have been possible. I am very appreciative for the guidance and supervision provided to me by Dr. Tom Boehly and Professor David Meyerhofer of the Laboratory for Laser Energetics (LLE) at the University of Rochester. The fruitful discussions, constant encouragement and difficult questions were greatly appreciated. LLE has been a wonderful institution to advance my scientific knowledge and I have greatly enjoyed my time there. The financial support that I received from the Laboratory of Laser Energetics and the Department of Mechanical Engineering was greatly appreciated. I would like to acknowledge the close professional contacts at Lawrence Livermore National Laboratory. The ability to collaborate with leading scientists in the field of high pressure science was inspiring. Most specifically, I would like to vi note the guidance I received from Dr. J. H. Eggert, Dr. D. G. Hicks, Dr. R. F. Smith, Dr. P. M. Celliers and Dr. R. E. Collins. Specially thanks to my colleagues at the Laboratory for Laser Energetics. Dr. M.A. Barrios, my office mate and group member, for the long discussions. The experimental technicians and systems scientist who were extremely helpful in the development and execution of this experiment. Most specifically, I would like to thank Andrew Sorce for his due diligence in ensuring that the ASBO diagnostics was in prime operating condition for my campaigns. Steve Stagnitto for time spent explaining the OMEGA laser system and ensuring that experiments were performed without error. Mark Bonino and target fabrication for their willingness to build targets with strict requirements. Lastly, Bob Boni for making time to discuss streak cameras and various diagnostics. I am very greatly for that time, and my understanding of those diagnostics is a direct result of his teachings. This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC52-08NA28302, the University of Rochester, and the New York State Energy Research and Development Authority. The support of the DOE does not constitute an endorsement by DOE of the views expressed in this work. vii ABSTRACT The compression of materials to high pressure can alter their optical properties in ways that provide insight into the resulting structural changes. Under strong shock compression, transparent insulators transform into conducting fluids as a result of pressure-induced reduction of the band gap and thermal promotion of electrons across that gap. A new ramp compression technique; direct-drive shaped ablation, is used to compress LiF to 800 GPa without generating shocks thereby producing high pressures at significantly lower temperatures than would be created by shock waves. In this study, ramp compressed lithium fluoride (LiF) is observed to remain transparent to 800 GPa, pressures seven times higher than previous shock compression experiments. The ramp compressed refractive index of LiF is measured at pressures up to 800 GPa and depends linearly on density over this range. This is the highest pressure refractive index measurement made to date. The linear dependence of the refractive index and density is examined using a single-oscillator model. This model indicates that the linear behavior is a result of monoatomic closure of the band gap. Extrapolation of these results indicates that viii the band gap closure (metallization) will be greater than 5,000 GPa, well above the Goldhammer-Herzfeld criterion for metallization (∼ 2,860 GPa). The high metallization pressure of LiF is attributed to its large band gap and isoelectronic counterparts that exhibit high metallization pressures. The high pressure transparency of LiF has technical utility as an optical window for materials studies since the transparency at high pressure allows in situ measurements of samples confined by that window. The observed transparency and measurement of LiF refractive index enables advancement of those experiments to higher pressure regimes. Contents ix CONTENTS Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 High Energy Density Physics . . . . . . . . . . . . . . . . . . . . . 7 1.2 Relevance of This Study . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2. Fundamentals of Fluid Dynamics . . . . . . . . . . . . . . . . . . . 15 2.1 Governing Equations of Fluid Dynamics . . . . . . . . . . . . . . 16 2.2 Remarks on the Conservation Equations . . . . . . . . . . . . . . 26 2.3 Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Planar Isentropic Flow . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Eulerian and Lagrangian Coordinates . . . . . . . . . . . . . . . . 40 2.6 Shock Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.7 Hugoniot Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Contents x 2.8 Isentropic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 51 3. Experimental Technique . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1 Interferometric Measurements Through Optical Windows . . . . . 53 3.2 Shock Refractive Index Experiments . . . . . . . . . . . . . . . . 60 3.3 Isentropic Refractive Index Experiments . . . . . . . . . . . . . . 65 3.4 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 93 4. Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.1 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Weighted Mean and Orthogonal Regression . . . . . . . . . . . . . 112 4.3 LiF Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.4 LASNEX Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.5 Temperature Measurements . . . . . . . . . . . . . . . . . . . . . 129 4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 133 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.1 VISAR Window Corrections . . . . . . . . . . . . . . . . . . . . . 135 5.2 Classical Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3 Single-Oscillator Model . . . . . . . . . . . . . . . . . . . . . . . . 143 Contents xi 5.4 Metallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 157 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Appendix 182 A. Direct Drive Laser Ablation Scaling . . . . . . . . . . . . . . . . . 183 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 A.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . . 185 A.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 A.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B. Weighted Mean Values . . . . . . . . . . . . . . . . . . . . . . . . . 193 C. LiF Shock Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 C.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . 200 C.2 Unsteady Shock Breakout of an Optical Window . . . . . . . . . . 203 C.3 Steady Shock Breakout of an Optical Window . . . . . . . . . . . 209 C.4 Analysis of Shot 58815 . . . . . . . . . . . . . . . . . . . . . . . . 211 C.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 List of Tables xii LIST OF TABLES 4.1.1 Shot Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.1.2 Fused Silica Etalon Parameters . . . . . . . . . . . . . . . . . . 98 4.2.1 Orthogonal Fitting Parameters . . . . . . . . . . . . . . . . . . 119 5.3.1 Dispersion Parameters for the Alkali-Halides with NaCl-Type Lattice Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.4.1 Metallization Pressure for Various Materials . . . . . . . . . . . 154 5.4.2 Band Gap Energy for Various Materials . . . . . . . . . . . . . 155 A.2.1 Shot Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 186 A.3.1 Laser Ablation Scaling . . . . . . . . . . . . . . . . . . . . . . . 190 B.1.1 Apparent and True Weighted Mean Values . . . . . . . . . . . . 193 C.4.1 Analysis of Shot 58815 . . . . . . . . . . . . . . . . . . . . . . . 213 List of Figures xiii LIST OF FIGURES 1.0.1 Fission Product Yields . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Sample EOS Target . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Mass Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.2 Momentum Element . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3 Energy Element . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Acoustic Perturbation for a System Initially at Rest . . . . . . 31 2.4.1 Receding Piston . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.2 Sample x-t Diagram for Characteristics Analysis. . . . . . . . . 38 2.4.3 Comparison of Analytic and Numeric Techniques . . . . . . . . 40 2.6.1 Wave Deformation . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.6.2 Shock Front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.7.1 Hugoniot and Isentrope of LiF . . . . . . . . . . . . . . . . . . 48 2.8.1 Hugoniot and Isentrope Temperature Dependence of LiF . . . . 51 3.1.1 Velocity Window Correction of an Optical Window . . . . . . . 54 List of Figures xiv 3.1.2 Velocity Window Correction of a Shocked Optical Window . . . 57 3.2.1 Gas-Gun Experimental Configuration . . . . . . . . . . . . . . . 60 3.2.2 Collision Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2.3 LiF Shocked Refractive Index Data . . . . . . . . . . . . . . . . 64 3.2.4 LiF Melt Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.1 Target Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3.2 Characteristics Numbering Scheme . . . . . . . . . . . . . . . . 69 3.3.3 Pulse Shape Design . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4.1 Mach-Zhender Interferometer . . . . . . . . . . . . . . . . . . . 83 3.4.2 Sample VISAR Data. . . . . . . . . . . . . . . . . . . . . . . . 86 3.4.3 VISAR Configuration . . . . . . . . . . . . . . . . . . . . . . . 87 3.4.4 Processing VISAR Data . . . . . . . . . . . . . . . . . . . . . . 88 4.1.1 Shot 57575: VISAR Data . . . . . . . . . . . . . . . . . . . . . 99 4.1.2 Shot 57575: Backwards Characteristics Diagram . . . . . . . . . 101 4.1.3 Shot 57575: Forwards Characteristics Diagram . . . . . . . . . 103 4.1.4 Monte-Carlo Error Analysis . . . . . . . . . . . . . . . . . . . . 107 4.1.5 Shot 57575: Apparent versus True Particle Velocity 4.1.6 Shot 57575: Refractive Index . . . . . . . . . . . . . . . . . . . 110 4.1.7 Apparent versus True Velocity of All Experiments . . . . . . . . 111 4.2.1 Weighted Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 . . . . . . 109 List of Figures xv 4.2.2 Ratio of the Apparent to True Velocity . . . . . . . . . . . . . . 118 4.2.3 Comparison of the Weighted Mean and Data Using Pulse Shape RM3702 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.2.4 Target Design with an Embedded Gold Layer . . . . . . . . . . 122 4.2.5 Comparison of Weighted Mean and Embedded Gold Layer Target Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3.1 Refractive Index of LiF to 800 GPa . . . . . . . . . . . . . . . . 125 4.4.1 Characteristics Analysis of Shot 56113 with Shock Formation in the LiF Window . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.4.2 Comparison of Hydrocode Simulations and the Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.5.1 Shot 57575: Temperature Measurement . . . . . . . . . . . . . 130 4.5.2 Shot 57577: Temperature Measurement . . . . . . . . . . . . . 132 5.2.1 Lorentz Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2.2 LiF Refractive Index at Various Frequencies . . . . . . . . . . . 142 5.3.1 LiF Refractive Index in the Optical Region . . . . . . . . . . . 146 5.3.2 LiF Single Oscillator Model . . . . . . . . . . . . . . . . . . . . 149 5.4.1 Xenon Band Broadening . . . . . . . . . . . . . . . . . . . . . . 152 A.3.1 Ablation Pressure Versus Laser Intensity . . . . . . . . . . . . . 189 List of Figures xvi A.3.2 Shot 54944: Ablation Pressure . . . . . . . . . . . . . . . . . . 190 C.1.1 Shock Release Target Design . . . . . . . . . . . . . . . . . . . 201 C.1.2 Shot 58815: VISAR Data . . . . . . . . . . . . . . . . . . . . . 202 C.2.1 Shock Front in an Optical Window . . . . . . . . . . . . . . . . 204 C.2.2 Shock Breakout of an Optical Window. . . . . . . . . . . . . . . 205 C.2.3 Density and Refractive Index Profiles at Various Stages of Shock Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 C.4.1 Shot 58815: Velocity Profile . . . . . . . . . . . . . . . . . . . . 213 C.4.2 Shock Refractive Index Measurements . . . . . . . . . . . . . . 214 C.4.3 Shot 58815: Velocity Profile Difference . . . . . . . . . . . . . . 215 Foreword 1 FOREWORD The author was the principal investigator (PI) for all experiments analyzed and discussed in the body of this thesis. Chapter 4 and 5 (Experimental Results and Discussion) is based on the publication submitted to Physical Review Letters: D.E. Fratanduono, T.R. Boehly, M.A. Barrios, D.D. Meyerhofer, J.H. Eggert, R.F. Smith, D.G. Hicks, P.M. Celliers, D.G. Braun and G.W. Collins, “Refractive Index of Lithium Fluoride at Pressures up to 800 GPa,” submitted on September 24th, 2010. The author of this thesis performed all the analysis described herein and in that publication. LASNEX simulations (Section 4.4) were performed by David Braun at Lawrence Livermore National Laboratory (LLNL) to verify the consistency of the analysis technique outlined here. The remaining analysis and text in this thesis was developed by the author and the guidance of colleagues at LLNL. The experiment discussed in appendix C was performed by Hye-Sook Park of LLNL. Analysis, derivation and discussion using that data was performed solely by the author. 1. Introduction 2 1. INTRODUCTION The first test of a nuclear weapon occurred on July 16th, 1945 at the Alamogordo Bombing range in a remote part of New Mexico,1 beginning the scientific exploration of the field of high energy density physics.2 This detonation increased scientific interest in atomic physics, fission, and fusion. Further testing ensued, and after the Second World War, the United States deemed the development of nuclear weapons essential to national security. For the next fifty years, the Unites States and the former Soviet Union carried out aggressive campaigns to increase their understanding of nuclear weapons. During this period, the United States and the former Soviet Union performed 1,030 and 715 nuclear tests, respectively,1, 3 accounting for over 80% of the world’s nuclear detonations. Tests were conducted to gather information on nuclear devices, nuclear phenomena, and material properties at extreme conditions. Eight countries have since developed and detonated nuclear weapons (United States-1945, Soviet Union-1949, Great Britain-1952, France-1960, China-1964, India-1974, Pakistan-1998 and North Korea-2006).4 During this period, numer- 1. Introduction 3 ous nuclear agreements were proposed among these countries and others with the goal of halting or minimizing detonations. In 1954, the first agreement to stop nuclear testing was proposed by the prime minister of India. Three years later (1957), President Eisenhower announced a moratorium on nuclear testing. At this time, the United States, Great Britain and the former Soviet Union (the only countries in possession of nuclear weapons) agreed to halt testing.1 However in 1960, France detonated its first nuclear device causing the Soviet Union to resume testing, with the United States and Britain following. In the ensuing years, the United States, Soviet Union and Great Britain along with other non-nuclear states agreed to the Limited Test Ban treaty outlawing the testing of nuclear weapons in the atmosphere, outer space and underwater.5 It was not until 1996 that an agreement banning all nuclear testing was adopted by a majority of nations and nuclear detonations nearly ended. The Unites States signed the Comprehensive Nuclear Test Ban Treaty in 1996,6, 7 with their last nuclear explosion occurring four years prior.4 The United States has not yet ratified the treaty, but is currently abiding by its provisions. Great Britain, China, France and Russia followed along with seventy other nonnuclear states. These nations agreed to cease nuclear explosions and refrain from causing or encouraging other nations to carry out nuclear testing. This ban on nuclear testing caused the United States to pursue other forms of scientific re- 1. Introduction 4 Fig. 1.0.1: Fission product yields for thermal nuclear fission of U235 92 . For each U235 92 atom that undergoes fission, the probability of by-products (in mass number) is shown.10 search to ensure the viability of their nuclear stockpile. The most recent is the Stockpile Stewardship and Management Program (SSMP)8 whose purpose is to maintain and enhance the safety, security and reliability of the United States nuclear weapons stockpile without conducting underground testing.9 The specific aims of the SSMP are to establish a critical understanding of the material science, hydrodynamic and hydro-nuclear phenomenon in the high energy density regime (defined as material pressures greater than 100 GPa). Coincident with these events, research in large-scale fission and fusion energy production began. In 1934 Oliphant et al.11 discovered the fusion reaction while findings by Hahn and Strassmann in 1939 demonstrated the first fission reaction.12 1. Introduction 5 These processes illustrate that large scale energy production is possible through nuclear means. Nuclear fission is a process in which nuclear decay is initiated by a neutron.13 A neutron colliding with U235 results in the nuclear reaction, 236 141 92 n + U235 92 → U92 → Ba56 + Kr36 + 3n(170 MeV), (1.1) 92 ∗ where fission fragments (Ba141 56 and Kr36 ) are created. Equation 1.1 shows that a single neutron can be used to initiate decay in an U235 . The advantage of fission is that the neutron (neutral particle) can easily penetrate the electron cloud of the Uranium atom. The reaction generates three neutrons that can then be used to initiate a chain reaction leading to a self sustaining energy source. These reactions are easily initiated and sustained explaining their usefulness for power generation. Nuclear fission has a number of difficulties and environmental concerns. As figure 1.0.1 shows, various fission fragments are created during Uranium fission. A majority of these fission fragments are highly unstable (radioactive) and their environmental impact is of concern.14 Proper treatment, storage and segregation from the environment is difficult. Significant engineering technologies are required to control the chain reactions that occur during the fission process. Over 400 nuclear reactors have been developed with two reactor meltdowns occurring.15 It ∗ For this reaction a variety of nuclear fragments form. The probability that nuclear fragments found in equation 1.1 form is less than 10 % (other fragments may results). The probability of fragment formation is shown in figure 1.0.1. 1. Introduction 6 is because of these concerns that the safety and viability of nuclear fission reactors are questioned. Nuclear fusion represents a much safer form of nuclear energy production. A typical sustainable fusion reaction uses deuterium and tritium, D + T → He4 (3.5 MeV) + n(14.1 MeV). (1.2) This reaction is significantly more difficult to produce than fission. For deuterium and tritium to undergo a nuclear reaction, their nuclei must come within approximately a nuclear diameter of one another overcoming the electro-static repulsive force. Significant energy is required to fuse the elements together. The benefits of fusion reactions, when compared with their fission counterpart, is that the nuclear by-products are an inert gas and a neutron. The neutron carries away the majority of the energy generated during the process and could be used for a power plant. Most importantly, fusion energy production exceeds that of fission. The fission reaction of equation 1.1 releases 207 MeV per reaction or 0.9 MeV per nucleon. In the fusion reaction, 17.2 MeV is generated or 3.5 MeV per nucleon. The energy density of D and T is higher than that of U235 . The economical and environmental benefits of nuclear fusion motivates research in this field.16 In the 1950’s various confinement techniques were developed to initiate fusion: 1. Introduction 7 Princeton University invented the magnetic confinement stellerator,17 Los Alamos National Laboratory created the magnetic pinch18 and the Soviet Union developed a toroidal model of a magnetic thermonuclear reactor that later evolved into the tokamak.19 These magnetic confinement devices aimed to create the high density and temperatures required in fusion reactions by compressing and heating plasmas confined by magnetic fields. Initial attempts were unsuccessful due to plasma instabilities.20 With the development of the laser in 1960,21 the ability to generate high intensities created an additional pathway to controlled fusion. The concept of inertial confinement fusion was developed in which a fuel capsule is compressed to high temperature and pressure conditions using laser energy. Programs, such as the Laboratory for Laser Energetics were developed to advance this field. Due to the high pressures and material densities required to initiate fusion, the inertial confinement fusion community began scientific research into high pressure, temperature and density matter.22 Consequently, inertial confinement fusion and SSMP become intertwined due to their common interests in high energy density physics. 1.1 High Energy Density Physics High energy density2 has been defined as energy densities greater than 1011 J/m3 which typically corresponds to material pressure exceeding 100 GPa. Recent ad- 1. Introduction 8 vances in driver techniques (lasers, particle beams, Z-pinch generators, magnetic flyer plates) have produced greater heating and compression enabling scientists to explore and develop a fundamental understanding of matter at ever increasing energy density. These advances have initiated new discoveries that have benefited nuclear initiatives (Stockpile Stewardship and Management Program) and inertial confinement fusion while initiating new innovations and ideas.2 High energy density physics (HEDP) encompasses many scientific fields (not limited to Astrophysics, Laser Plasma Interactions, Fluid Dynamics, Condensed Matter, and Equation of State Physics). This work is most relevant to the field of Equation of State (EOS) Physics that uses compression experiments to study material properties at high pressure. EOS data are needed for hydrodynamic simulations of inertial confinement fusion, to confirm theoretically predicted states of matter, and to aid in the understanding of solid-state dynamics at high strain rates.23 In astrophysics, understanding the evolution of a giant planet requires analysis of the thermodynamic and transport properties of compressed hydrogen and other materials at pressures greater than 100 GPa. Geophysicists understanding of the earths interior requires knowledge of the phase diagram of iron at high pressure and determination of its melt-line (in the pressure/temperature plane).24 Hydrodynamic codes solve the three conservation equations (mass, momentum and energy) which requires one additional equation: EOS or constitutive relation 1. Introduction 9 for closure.25 Understanding material behavior in the high energy density regime assists these efforts. Many techniques are employed to compress materials (i.e. isothermal compression, isentropic compression, isenthalpic compression, and shock wave compression). HEDP experiments have primarily used shock wave compression and isentropic compression. Shock waves are generated in materials through laser ablation, flyer plates, high energy explosives and nuclear explosions.26, 27 Each technique transfers momentum to the surface of the target, inducing stress waves that coalesce into a shock wave. A shock wave is a discontinuity in density and pressure that carries energy as it propagates through a medium. The wave, propagating faster than the local sound speed, causes a “step-like” change in the properties of the material and the flow variables on each side of the discontinuity. These changes are related by the Rankine-Hugoniot conditions: the conservation equations for mass, momentum and energy across the shock. These relations include terms for the particle and shock velocities. In many experiments, measurements of these velocities is essential for material studies. Entropy increases across the shock front, indicating an irreversible, dissipative process that will cause an increase in temperature.27 Isentropic compression is a reversible process that can be achieved through the use of diamond anvil cells,28 magnetic confinement,29 and laser ablation.30 This 1. Introduction 10 technique requires that material be gradually compressed to maintain constant entropy, creating a continuous pressure and density profile within the material. Quasi-isentropic compression is an approximation to an isentrope that minimizes the entropy increase. These methods differ from shock wave compression in that many states can be reached at a nearly constant temperature during a single experiment.27 Shock and ramp compression are two techniques that are commonly used to explore the phase space of materials in the high energy density regime. Ramp compression to greater than 100 GPa is a relatively new technique that provides access to states of matter previously inaccessible to laboratory experiments. 1.2 Relevance of This Study EOS experiments are important to a variety of fields as they enable one to correctly determine the phase space of materials, close the fluid dynamic system of equations, and provide insight into other high pressure phenomena. A typical experiment involves compression of a material and the measurement of the resulting material or compression wave velocity. A sample target geometry for laser driven HEDP experiments is shown in figure 1.2.1a. Laser ablation drives compression waves that traverse the sample and reach the rear surface. Diagnostics measure the velocity in the target and or at the rear surface to determine the compressed material properties. 1. Introduction 11 Fig. 1.2.1: Sample EOS target geometry for high pressure experiments In some cases, optical windows are used to observe compressed materials that are confined by a “window” through which that material is observed. An example of this style of target is shown in figure 1.2.1b. Optical windows are used because they suppress complicated wave interactions when studying materials that undergo phase transformation,31 and confine fluids/melted materials enabling in situ measurements.32 The simplest example of the need for high pressure windows can be understood by imagining a target design in which the EOS of an ideal gas is to be studied. Due to experimental constraints, the gas must be confined in a compression cell. Velocity interferometry is used to determined the characteristics of the compressed gas. To view the gas, a transparent optical window is required. As the cell compresses, the material properties and the refractive index of that optical window change. If these changes are not accounted for, systematic errors in the velocity interferometer measurements will be introduced. Therefore, knowledge of the high pressure behavior of an optical window is required for precise interferometry measurements. 1. Introduction 12 Lithium Fluoride (LiF) is of interest for shock and ramp compression experiments because its transparency at high pressure allows in situ particle-velocity measurements at the sample/window interface and for spectroscopic optical measurements of samples.33 Recent experimental developments using ramp compression have created a need for a transparent window, whose refractive index is known at high pressures (≫ 100 GPa). Before this work, LiF had been characterized only to 115 GPa.34 Velocities are measured in dynamic compression experiments through optical windows using interferometry.33, 35 Knowledge of the optical properties of the compressed window is required to properly correct velocity measurements.36, 37 A new technique to determine the refractive index of materials at high pressures is presented. This study fully characterizes ramp compressed LiF windows to pressures seven times higher than previous studies (800 GPa). This will enable the scientific community to extend measurements of various materials to pressures previously inaccessible. The results suggest that this window will remain transparent at significantly higher pressures by linking the dependence of the refractive index on the density to an effective single-oscillator model. This model implies that the band gap closes with increasing density and predicts a high metallization pressure for LiF (∼ 4200 GPa), suggesting that LiF will be a vital optical window for extreme pressure experiments. 1. Introduction 1.3 13 Thesis Outline Chapter 1 described the history and development of high energy density physics as an emerging scientific field. It discussed the needs of this research area to various scientific communities. Most specifically, the area of EOS physics, the subject of this work. The relevance of this body of knowledge will prove useful to experimentalists and theorists throughout this field. Fluid dynamic and thermodynamic equations central to the body of this work are derived in Chapter 2. To familiarize the reader with these equations and their applications, solutions to various systems are presented. The fundamental differences between shock and ramp compression are discussed. Chapter 3 discusses previous techniques used to measure the high pressure refractive index of shocked LiF. Equations relating measured observables to isentropically compressed refractive index are discussed. This relationship is used to design an experiment to measure the ramp compressed refractive index of LiF. Pulse shapes, drive specifications and target constraints are identified. Diagnostics necessary for this study are then discussed. The LiF ramp compressed refractive index analysis and results are discussed in Chapter 4. Twenty-four experiments are examined in detail. Chapter 5 discusses the implications of these measurements on optical interferometry experiments and the dependence of the refractive index and density is examined using a single-oscillator model. The 1. Introduction 14 metallization pressure for LiF is predicted using this model and compared to the metallization pressure of other materials. Chapter 6 summarizes and concludes the findings of this study. Three appendices are provided. Appendix A discusses the laser intensity to ablation pressure scaling law for ramp compression of diamond targets using a 351 µm laser. Appendix B is a tabular appendix that contains the values of the apparent and true particle velocity determined from the weighted mean of all experiments. Appendix C discusses a new experimental technique to measure the shocked refractive index of an optical window. 2. Fundamentals of Fluid Dynamics 15 2. FUNDAMENTALS OF FLUID DYNAMICS Fluid mechanics is the study of fluids and the forces imparted upon them. It is often divided into two parts; the study of stationary fluids (fluid statics) and fluids in motion (fluid dynamics). As with all systems, there are conservation laws governing the dynamic and static processes. In principle, there are an infinite number of fluid dynamic conservation equations, where each equation depends on a higher order moment (i.e. the conservation of mass equation depends upon a momentum term). Due to the infinite equations governing fluid dynamics, assumptions are required to close the system of equations. These assumptions may deal directly with the characteristics of the fluid flow or with the thermodynamic properties. Fluid mechanics is most commonly defined by three laws of mechanics (conservation of mass, momentum and energy), a thermodynamic state relation, and the boundary conditions specific to that system.38 Using these relations, the state 2. Fundamentals of Fluid Dynamics 16 of a moving fluid with known thermodynamic properties (velocity, density, and pressure) can be defined as a function of position and time.27 In the following sections, the conservation equations for fluid dynamics are derived. Fluid derivations identify the need for a thermodynamic relation to complete the set of equations. Specific fluid systems (acoustic perturbations, isentropic flow, and shock compression) are discussed and analyzed using the derived equations. The method of characteristics, a solution technique central to the body of this work, is introduced and employed to solve the case of isentropic flow. 2.1 Governing Equations of Fluid Dynamics Derivation of the conservation equations for fluid dynamics are taken from the Hydrodynamic Instability lectures given by Professor Riccardo Betti.39 While discrete elements are referenced, they contain a large number of atoms or molecules. These derivations assume that discrete elements within the flow are macroscopic and the fluid can be regarded as a continuous medium.40 2.1.1 Conservation of Mass Consider a volume (V ) enclosed by a surface (S) as shown in figure 2.1.1. The mass (M) of the fluid element is defined as the integral of the density (ρ) over the volume, M= Z V ρdV. (2.1) 2. Fundamentals of Fluid Dynamics 17 Fig. 2.1.1: Mass Element The mass of the fluid element may also be defined as a function of the fluid traveling through the surface. Assume a surface element dS, as shown, with ~ · ~n) flowing through the surface element. U ~ is the velocity permeating velocity (U the surface and ~n represents the normal vector to that surface (positive direction ~ The discrete mass (dm) flowing through dS in time defined as outward on S). interval (dt) is ~ · ~n)dSdt. dm = ρ(U (2.2) Total mass (dM) leaving the volume becomes dM = − I S ~ · ~n)dSdt. ρ(U (2.3) 2. Fundamentals of Fluid Dynamics 18 Fig. 2.1.2: Momentum Element This defines the mass flux traveling through the surface. Combining equation (2.3) and the time derivative of equation (2.1) gives Z V ∂ρ dV = − ∂t I S ~ · ~n)dS, ρ(U (2.4) for a fixed volume. This represents the integral form of mass conservation. Using the divergence theorem, equation (2.4) is reduced to the differential form, ∂ρ ~ = 0. + ∇ · ρU ∂t (2.5) The mass conservation equation (2.5) relates the element density and the momentum. The single equation with two unknowns (ρ, U) is not a closed system. The ~ ) illustrates the need for a higher order moment conservation momentum term (ρU equation. 2. Fundamentals of Fluid Dynamics 2.1.2 19 Conservation of Momentum Consider a discrete fluid element in one dimension as shown in figure 2.1.2. The element is defined such that the mass within the volume remains constant (i.e. a Lagrangian fluid element). The element, with fixed cross section (A) and variable length (dx), is free to move along the x-axis in time. The discrete mass (dM) of the element is defined as the density (ρ) times the element volume, dM = ρ(t)dx(t)A. (2.6) P Applying Newton’s second law of motion ( F = ma) to the fluid element (dM), the force (F ) is expressed as X F = dU (x(t), t)ρ(t)dx(t)A dt The acceleration (a) is written in terms of the element velocity (a = (2.7) dU (x(t), t)). dt Expanding equation 2.7 the total derivative gives X F = ∂U ∂t +U ∂U ρ(t)dx(t)A. ∂x (2.8) If a pressure (P ) is applied to the two surfaces shown in figure 2.1.2, equation 2. Fundamentals of Fluid Dynamics 20 (2.8) maybe rewritten substituting the relation for the applied pressure, ∂U dx ∂U dx ρ(t)dx(t)A. +U P (x − ) − P (x + ) A = 2 2 ∂t ∂x (2.9) The fundamental theorem of calculus reduces equation (2.9) to the conservation of momentum equation for one dimension, ∂U ~ ∂P ∂U ρ =− +U . ∂t ∂x ∂x (2.10) This derivation assumes that the applied pressure is the only external force acting on the element. In three dimensions, equation (2.10) transforms to a three equation set with five unknowns, ∂U ~ ~ · ∇)U ~ = −∇P + ρ~g , ρ + (U ∂t (2.11) where the force of gravity (g) is included for clarity and in most applications can be neglected.41 Combining the conservation of mass and momentum equations ~ and P ). The gradient of gives a four equation set with five unknowns (ρ, U pressure term (∇P ) describes the work done on the element. As observed in the derivation of mass conservation equation, we see that conservation of momentum depends on a higher order energy term, P . 2. Fundamentals of Fluid Dynamics 21 Fig. 2.1.3: Energy Element 2.1.3 Conservation of Energy The first law of thermodynamics states that in an isolated system the total energy remains constant. The energy is free to change forms (e.g. transformation of kinetic to potential energy), while obeying this principle. Various mechanisms exist by which energy is transformed within a closed system and the conservation of the energy equation relates these mechanisms to the total energy of the system. The transfer of energy is a dynamic process requiring rate equations to describe energy conservation. The total energy of a system is described by the energy stored in the form of internal energy (e) and kinetic energy( 12 ρU 2 ). The rate of change of energy stored per unit volume is simply ∂ (ρe + 12 ρU 2 ). ∂t This represents the total energy within the discrete volume. The energy of the element can be changed by mass flow into and out of the volume, heat transfer across the surface, work performed on and energy generated within the volume. The net rate of energy flow across a unit volume is affected by the rate at 2. Fundamentals of Fluid Dynamics 22 which mass flows in and out of a fluid element. As the mass enters and leaves the element, energy of the form (e + 12 U 2 ) is deposited or removed. Thermal energy is transfered by the movement of particles from one region to another and is termed convection. For the fluid element shown in figure 2.1.3, the convection across the y + ∆y/2 face per unit time is (ρUy (e + U2 ))y=∆y/2 ∆x∆z. 2 The fundamental theorem of calculus reduces the rate of flow in the y-direction per unit volume to ∂ (ρUy (e ∂y + U2 )). 2 A similar derivation follows for the other directions leading ~ (e + to a rate of energy flow per unit volume: −∇ · (ρU U2 )). 2 The rate of heat transfer follows a similar derivation. The heat transfer is defined as the transfer of thermal energy through conduction between neighboring elements. The heat flux (q) describes the transfer of energy between element faces of figure 2.1.3 and the rate of heat transfer per unit volume is −(∇ · ~q). The two remaining energy terms to be derived are work performed on and energy generated within the volume. Work is performed on a element volume through gravitational effects or external forces. The rate of work performed per ~ . Work can be performed by the surface forces unit volume by gravity is ρ~g · U ~ . Heat generation occurs within an element if a acting on the body or −P ∇ · U heat source is present. This is defined as the power released per unit volume per unit time or Q̇. 2. Fundamentals of Fluid Dynamics 23 Combing all terms, the conservation of energy equation for a fluid element is 2 1 ∂ ~ (e + U )) − P ∇ · U ~ − ∇ · ~q + ρ~g · U ~ + Q̇. (2.12) (ρe + ρU 2 ) = −∇ · (ρU ∂t 2 2 To determine a relationship for the pressure, required by the conservation of momentum equation, two unknowns have been introduced (internal energy and heat flux). This assumes the internal heat generation is known. The pressure and internal energy can be related through thermodynamic state relations. However, the heat flux has created the need for an additional conservation equation. It is easy to see how a recursion relation exists for the equations governing fluid mechanics (“n” unknowns with “n-1” equations) where each equation depends on a higher order momentum. To close the system of equations, thermodynamic properties of the fluid and assumptions regarding the heat flux are required. In the simplest case, the heat flux may be neglected or approximated using Fourier’s Law of thermal conduction (q ≈ −k∇T ). The temperature (T ) and thermal conductivity (k) are related to the thermodynamic properties of the fluid that are required for a complete set of equations. 2.1.4 Thermodynamic State Relations Thermodynamics characterizes the relation among measurable independent quantities that describe the current state of the system. These parameters are 2. Fundamentals of Fluid Dynamics 24 path independent and do not characterize dynamic changes to the system. The basic laws of thermodynamics are used to determine the mathematical relations among these independent quantities. The thermodynamic state describes the set of values that must be specified to exactly reproduce the current state of the system.42 Thermodynamic properties are defined by state variables. The interrelation among the variables is termed the state relation or equation of state (EOS). The most commonly known EOS is the ideal gas law (P V ≈ T ), which defines the relationship among pressure (P ), volume (V ) and temperature (T ). In many systems, an analytic representation of the state variable interdependence is unknown as is the number of quantities required to specify the state. The number of state variables required to delineate a system is determined by experimental observations and fundamental thermodynamic laws governing the system.42 The relation between state variables is determined from the laws of thermodynamics that describe the transport of heat and work in a closed system. The first law states that energy is conserved within a system. A change in internal energy (U) is related to the work (w) done on the system (denoting the negative sign) and the addition of heat (q), dU = δq − δw. (2.13) 2. Fundamentals of Fluid Dynamics 25 Fundamentally, work and heat addition are path dependent processes and are not state variables. This is noted in the notation as δ where as changes state relations are expressed as ∆U = Uf − Ui due to their path independence. Equation (2.13) can be described in terms of state relations by assuming that the work done on a system is reversible and is equivalent to the pressure times a discrete change in volume (δw = P ∆V ). The second law of thermodynamics (entropy increases in time) states that entropy increase is related to the discrete heat transfer divided by the temperature (∆S = δq/T ).43 The relation between the state variables defined by the first and second law of thermodynamics is ∆U = T ∆S − P ∆V. (2.14) This equation may be rewritten such that state relation depends on two state variables U(S, V ), ∆U = dU dS V ∆S + dU dV ∆V. (2.15) S Further state relations are determined using similar thermodynamic principles and arguments. The defining relations for the enthalpy (H), free energy (F ) and 2. Fundamentals of Fluid Dynamics 26 Gibbs free energy (G) are ∆H = T ∆S + V ∆P, (2.16) ∆F = −S∆T − P ∆V, (2.17) ∆G = −S∆T + V ∆P. (2.18) For a thermodynamic system, there are eight defining parameters with four equations (2.15-2.18). To define a thermodynamic system, four parameters must be specified, allowing the calculation of all other thermodynamic quantities using the equations above. For instance, common numerical equations of state, such as Sesame44 or Quotidian Equation of State,45 define the internal energy and pressure as functions of density and temperature (U(ρ, T ), E(ρ, T )). Such a definition is sufficient in describing the thermodynamic properties of a system and the interdependence of the formulas derived can be used to determine all of the other parameters. 2.2 Remarks on the Conservation Equations Fluid mechanics is described by the three laws of fluid mechanics, the thermodynamic state relation and the appropriate boundary conditions. Using Fourier’s approximation to close the system of equations leaves five fluid equations and 2. Fundamentals of Fluid Dynamics 27 seven unknowns. The thermodynamic equation of state provides two additional equations closing the set and determining the fluid flow. The hydrodynamic equations can further be simplified using assumptions specific to a known system. For instance, many classifications of fluid flow exist, with each classification accompanied with an assumption that may simplify the = 0), inviscid flow (as was equation set. Such assumptions are steady flow ( dU dt ~ = 0). Steady assumed in the derivation of 2.11), and incompressible flow (∇ · U flow assumes that conditions may vary in spatial location but are constant in time, inviscid flow assumes that the viscosity is zero, and incompressible flow assumes that changes in pressure do not effect the volume of the fluid. Each assumption reduces the complexity of the hydrodynamic equation set. If further is known about the fluid flow, assumptions in terms of the state variables maybe made (e.g. isentropic, isobaric, adiabatic etc...). For instance, isentropic flow occurs when changes in the fluid system are small and gradual. For such a flow, entropy is assumed constant and the state relation, 2.14, is reduced to a simple form ∆U = −P ∆V . 2.3 Acoustic Waves To gain a general understanding of the governing equations of fluid dynamics, consider a small sound disturbance centered on the origin in a one-dimensional spatially uniform system. Changes in the density (ρ), velocity (U) and pressure 2. Fundamentals of Fluid Dynamics 28 (P ) are small and expressed as ρ = ρ0 + ρ̃, (2.19) U = U0 + Ũ, (2.20) P = P0 + P̃ , (2.21) where the subscript, 0, and tilde denote the initial state of the system and the infinitesimal perturbation, respectively. The mass and momentum conservation equations for the acoustic waves are determined by linearizing equations (2.5) and (2.11), ∂ ρ̃ ∂ Ũ + ρ0 ∂t ∂x ∂ Ũ ∂ P̃ ρ0 + ∂t ∂x = 0, (2.22) = 0, (2.23) assuming the system is initially at rest (U0 = 0). The derived equations relate the density, velocity and pressure of the system and this two equation set is not sufficient to close the system of equations; requiring a thermodynamic relation. It has been shown that a thermodynamic state variable can be described by two other parameters. The pressure may then be described by the density and 2. Fundamentals of Fluid Dynamics 29 entropy (P (ρ, S)): ∆P = ∂P ∂ρ ∆ρ + S ∂P ∂S ∆S. (2.24) ρ For acoustic disturbances, the changes in fluid properties are small and develop gradually over time. The entropy is assumed to be constant and the dependence on entropy in equation (2.24) can be removed, ∆P = ∂P ∂ρ ∆ρ. (2.25) S As previously stated, state relations are path independent and equation (2.25) can be simplified to P̃ = ∂P ∂ρ ρ̃. (2.26) S Combining this result, with the conservation of mass and momentum equations (2.22, 2.23) for small perturbations yields the one dimensional wave equation, 2 ∂ 2 Ũ 2 ∂ Ũ = C , E ∂t2 ∂x2 2 ∂ 2 ρ̃ 2 ∂ ρ̃ = C , E ∂t2 ∂x2 (2.27) (2.28) where CE2 = ∂P ∂ρ , S (2.29) 2. Fundamentals of Fluid Dynamics 30 and CE is assumed to be constant. General solutions to the wave equations are written as the superposition of two particular solutions with the waves propagating in opposite directions.46 The solutions are Ũ = f1 (x − CE t) − f2 (x + CE t), ρ̃ = ρo ρo f1 (x − CE t) + f2 (x + CE t), CE CE (2.30) (2.31) (2.32) where f1 and f2 are arbitrary functions determined from the initial conditions.27 The solutions illustrate that the disturbances travel at velocity of CE which is formally referred to as the sound speed. Consider a small perturbation for a system that is initial at rest (figure 2.3.1). The perturbation is shown in blue and at some time later (red), where the perturbation has split into two separate waves traveling in opposite directions. The dt trajectory ( dx ) that the waves trace in x-t space is referred to as the pathline or characteristic (shown in black) and is inversely related to the sound speed (CE ) of the material. In a more rigorous approach, the next section considers a system in which the disturbances are not small, but entropy remains constant. 2. Fundamentals of Fluid Dynamics 31 Time (t) Initial Perturbation Wave Splitting Characteristics Position (x) Fig. 2.3.1: Acoustic perturbation for a system initially at rest. 2.4 Planar Isentropic Flow The equations of conservation of mass (2.5) and momentum (2.11) are used to determine the wave equation for one-dimensional isentropic flow. The acoustic wave solution (Section 2.3) assumes that entropy is constant and that the perturbations are small. However, in the case of planar isentropic flow, no constraint is placed upon the size of the perturbation. The conservation equations cannot be linearized (as was performed in the acoustic wave analysis). The conservation equations are reduced to the simple form, ∂U 1 ∂P ∂U ∂P ± = 0, + (U ± CE ) + (U ± CE ) ∂t ∂x ρCE ∂t ∂x where the sound speed (CE ) need not be constant. (2.33) 2. Fundamentals of Fluid Dynamics 32 As was shown for acoustic waves, disturbances propagate along specific pathlines or characteristics. For one dimensional isentropic flow, there are two families of characteristics that propagate in opposite directions and the solutions to equations (2.33) can be found along these characteristics. The derivatives can be defined along the forward and backwards propagating characteristics dx dt ± = U ± CE or d dt = ± ∂ ∂ + (U ± CE ) . ∂t ∂x (2.34) where (±) indicates the forward and backwards traveling waves. Substituting this assumption into equation (2.33) gives 1 dP dU ± = 0 and dt ρCE dt dx dt ± = U ± CE along C± , (2.35) where C± are the positive and negative characteristics. The pressure, density and sound speed are uniquely related through the thermodynamic relation CE2 = (dP/dρ)S . A complete solution to equation (2.35) can described by two functions in the x,t plane; the velocity (U) and one of the three thermodynamic relations.27 Integrating equation (2.35) gives J± = U ± Z dP = U ± σ, ρCE (2.36) 2. Fundamentals of Fluid Dynamics 33 where J± are arbitrary constants commonly referred to as the Reimann invariants for isentropic flow. If we assume a perfect gas with constant specific heats, the velocity and sound speed of the gas can be expressed in terms of the Reimann invariants. Assume that P = ρ0 ργ and CE2 = γρ0 ργ−1 , (2.37) where γ is the ratio of specific heats or the isentropic exponent. The Reimann invariants are defined as J± = U ± 2 CE . γ−1 (2.38) The velocity and sound speed may then be expressed in terms of these invariants, U= J+ + J− 2 and CE = γ−1 (J+ − J− ) 4 (2.39) as can the characteristic equations C± : dx dt ± = γ+1 3−γ J± + J∓ 4 4 for J± = constant. (2.40) It is important to note that the Reimann invariants possess a very important property: since J+ is constant along the C+ characteristics, dx dt + depends solely on the J− invariant, and vise versa. Therefore, if J− is constant everywhere, dx dt + too is 2. Fundamentals of Fluid Dynamics 34 constant. This important property is fundamental to understand the propagation of characteristics. From this approach, we see that if the Reimann invariants are known, all other parameters can be determined. This general approach is applied to the simple case of a receding piston to demonstrate the power of this technique (taken from the work of Zel’dovich27 ). Two separate solutions (analytical and numeric) of this example are discussed in the following section. 2.4.1 Receding Piston Imagine a one-dimensional semi-infinite system, in which a perfect gas with constant specific heats occupying infinite half space (x > 0) is constrained by a piston at (x = 0). Initially (t < 0), the system is at rest with constant density, pressure and sound speed. At t = 0, the piston is withdrawn with velocity (w = −U0 (1 − e−t/τ )) where U0 and τ are positive, real constants (figure 2.4.1). The piston path (X(t)) is described by the integral of the piston velocity, X(t) = Z t 0 t − τt w dt = −U0 τ − (1 − e ) . τ (2.41) Analytic Solution Characteristics propagating in the negative direction originate from the undisturbed region. Negative characteristics cannot emanate from the piston front because they are bounded by that surface. For this reason, the invariant J− is 2. Fundamentals of Fluid Dynamics 35 Fig. 2.4.1: Semi-infinite ideal gas, occupying half space (x > 0), is bound by a piston at x = 0. At t = 0 the piston is withdrawn from the gas and the corresponding fluid flow is determined. constant over all of the (x,t) space, greatly simplifying the problem. As shown previously, the slopes of the C+ characteristics are affected only by the J− invariants. In this case, the C+ characteristics will be straight lines due to the constant values of the J− invariants. The fluid velocity and sound speed are related using this invariant, 2 2 CE = constant = − CE0 , γ−1 γ −1 J− = U − −2 (CE0 − CE ), γ −1 γ−1 = CE0 + U. 2 U = CE (2.42) (2.43) (2.44) Substituting equation (2.44) into the relation for the slope of the positive characteristic (2.35) gives dx dt + = U + CE = CE0 + γ+1 U, 2 (2.45) 2. Fundamentals of Fluid Dynamics 36 and upon integration x = (CE0 + γ+1 U)t + f (U). 2 (2.46) f (U), a constant of integration, is determined by solving equation (2.46) at the piston interface. The arbitrary function defined at that location is f (w) = X(t) − [w + CE (w)]t, (2.47) where w denotes the velocity at the piston interface. The sound speed is determined from the negative Reimann invariant (J− ) or equation (2.44) which, expressed in terms of w, becomes CE (w) = CE0 + γ−1 w. 2 (2.48) Substituting this relation into equation (2.47) gives γ −1 f (w) = X(t) − w + CE0 + w t. 2 (2.49) The time dependence is removed using the expressions for the piston velocity 2. Fundamentals of Fluid Dynamics (t = −τ ln(1 + w )) U0 37 t and position (X(t) = −U0 τ [ τt − (1 − e− τ )]), w γ+1 . w + U0 ln 1 + f (w) = −wτ + τ CE0 + 2 U0 (2.50) The transcendental equation for velocity as a function of space (x) and time (t) becomes γ+1 γ+1 U x = CE0 + U0 + U t + −Uτ + τ ln 1 + U + CE0 , (2.51) 2 U0 2 which is only valid in the interval of X(t) < x < CE0 t. CE0t is the lead characteristic that represents the interface between the stationary gas and the gas set in motion by the receding piston. Since the velocity has been determined as a function of space and time, the corresponding relation for the sound speed may be calculated. All other parameters are determined from the assumed EOS. Numerical Solution In a similar approach, the characteristics are calculated and propagated in the (x,t) plane to determine the fluid flow. The Reimann invariants (J± ) are determined from the boundary conditions of the problem. As previously stated, the J− invariant is constant everywhere and J+ invariant is determined from the 2. Fundamentals of Fluid Dynamics 38 Fig. 2.4.2: Sample x-t diagram for characteristics analysis. Reimann invariants J+ and J− are show in blue and red, respectively. The piston trajectory is shown in black. The intersection of the solid blue and red lines is determined using the boundary conditions and the slope of the characteristics. 2. Fundamentals of Fluid Dynamics 39 piston velocity, 2 CE0 , γ−1 2 = CE0 + 2w. γ −1 J− = − (2.52) J+ (2.53) Using equation (2.39), the velocity and sound speed at the intersection of every Reimann invariant is determined. The difficulty in this technique arises in determining the location and time of each intersection. The trajectory of each characteristic is determined from dx dt ± = U ± CE0. (2.54) The values at each intersection must be determined in sequence since the trajectory of later characteristics depend on the interaction of previous ones. A sample (x-t) for characteristic analysis is shown in figure (2.4.2). Mapping all of the characteristics provides the location and time of each intersection. From this geometrical interpretation, similar results are produced when compared to those obtained in the analytic approach (to within round-off errors). Solutions using both methods are shown in figure 2.4.3. The numerical solution does not provide values at every location in (x-t) space and interpolation of the 2. Fundamentals of Fluid Dynamics 40 Fig. 2.4.3: Velocity (U), Sound Speed (CE ) and density (ρ) profiles arising from the motion of a receding piston in a semi-infinite system. Analytic (solid line) and numeric (points) solutions are shown. Both solutions produce identical results (to within round-off errors). results is required. This can be circumvented by increasing the density of characteristics to better approximate the analytic solution. 2.5 Eulerian and Lagrangian Coordinates Often it is useful to transform the conservation equations from the Eulerian coordinate system (of which they were derived) into the Lagrangian system. For one-dimensional systems, Lagrangian coordinate systems often enable simplified exact solutions to fluid flow equations.27 Lagrangian flow describes fluid properties along the pathline of the individual particles whereas Eulerian flow describes the state of a fluid at a specific location through which various particle pass.47 Examples in this chapter have focused on the use of the Eulerian coordinate sys- 2. Fundamentals of Fluid Dynamics 41 tems since readers are more familiar with such a coordinate system. However, as is shown in the following chapter, the Lagrangian coordinate system is used to describe the fluid flow specific to the body of this work. Eulerian and Lagrangian systems can be visualized by imagining two perspectives: a passenger in a boat versus observing the passage of that boat. Sitting in a boat as it travels downstream describes Lagrangian flow. A specific particle (boat) is tracked through the fluid along specific pathlines. Standing on the bank and observing the passage of the boat represented the Eulerian system. The passage of particles at a specific location in time is observed. Each of the coordinate systems possesses qualities that make them optimal in certain situations. For instance, if one desired to measure the flow rate through a channel it would be easier to measure the current at a specific location (Eulerian) than to determine the fluid path of each particle to calculate the flow rate (Lagrangian).47 In one-dimensional systems, the Lagrangian coordinates describes the location and trajectory of each particle allowing one to observe particle movement through the domain of the system. Transformations of one-dimensional fluid equations from Eulerian to Lagrangian coordinates is straight forward. In the Lagrangian coordinate system, the fluid trajectory of a particle is described by the fluid trajectory (h) and time (t). In Eulerian coordinates, this is represented as x(h, t). Conservation of mass requires 2. Fundamentals of Fluid Dynamics 42 that the fluid element mass is conserved in both systems or ρ(t)dx = ρ0 dh, (2.55) where ρ0 describes the initial density of the fluid element.40 Transforming between coordinate systems requires use of the material (convective) derivative that describe the derivative taken along a fluid path. In Eulerian and Lagrangian systems the material derivative is defined as27 D ∂ = + U · ∇(Eulerian), Dt ∂t D ∂ = (Lagrangian). Dt ∂t (2.56) (2.57) Using these conditions, the one-dimensional mass and momentum conservation equations are transformed from the Eulerian into the Lagrangian coordinate system, ∂ρ ρ2 ∂U =− , ∂t ρ0 ∂h 1 ∂P ∂U =− . ∂t ρ0 ∂h (2.58) (2.59) Conservation of mass (2.58) and momentum (2.59) equations are used to determine the characteristics in Lagrangian coordinate system. Through substitution 2. Fundamentals of Fluid Dynamics 43 one finds, 1 ∂P ∂U ∂P ∂U ± = 0, ± CL ± CL ∂t ∂h CL ρ0 ∂t ∂h (2.60) where h is the Lagrangian coordinate and CL is the Lagrangian sound speed. The Lagrangian sound speed is related to the Eulerian sound speed (CE ) by CL = ρ CE . ρ0 (2.61) Following the techniques outlined in Section 2.4, the characteristic equations become 1 dU ± dP = 0 and ρ0 CL dh dt Z dP . ρ0 CL = CL along C± . (2.62) ± The Riemann invariants become J± = U ± 2.6 (2.63) Shock Formation If characteristics with increasing sound speed are allowed to propagate for an infinite time they will intersect. After this point, these characteristics become multivalued which is physically unrealistic. Consider the case of a wave propagating to the right in an ideal gas with known thermodynamic properties. The 2. Fundamentals of Fluid Dynamics 44 initial velocity disturbance of the wave (U(x, t0 )) is known. The positive characteristics are defined as straight lines with slope dx/dt = (γ + 1)U/2 + CE0. Figure 2.6.1a shows an initial disturbance and its corresponding characteristics in Figure 2.6.1b and at later times (t1 and t2 ). As time progresses, the wave profiles steepen (figure 2.6.1c) and eventually “overshoots” becoming multivalued (figure 2.6.1d). When the characteristics cross, the method of characteristics is multivalued and a continuous solution does not exist. The wave profile develops into a discontinuity in velocity, pressure and density. The solution for systems in which wave profiles steepen into a discontinuity is the starting point for the development of shock wave theory. Realistically, “overshooting” does not occur. As the wave profiles deform they become extremely steep and discontinues are formed, preventing the system from becoming multi-valued. To understand this discontinuity the conservation equations are applied to the fluid flow. Consider a shock discontinuity propagating through an ideal gas with initial values (ρ0 , P0 , CE0 , U0 ) and unknown state following the discontinuity (ρ1 , P1 , CE1 , U1 ) as shown in Figure 2.6.2. Assume that the shock propagates with velocity D and ν1 is the post-shock fluid velocity with respect to the shock front (ν0 = D − U0 and ν1 = D − U1 ). Using the equation of mass conservation (2.5) and noting that ∂ρ/∂t = 0, we find that ∂(ρν)/∂t = 0. Integrating this relation gives ρ1 ν1 = ρ0 ν0 in Lagrangian space. In the Eulerian 2. Fundamentals of Fluid Dynamics 45 Fig. 2.6.1: Wave deformation. Diagram of a finite amplitude wave propagating to the right. Figure (a) depicts the initial disturbance as a sinusoidal function. As time progresses, the wave profile steepen (figure (c)) and eventually “overshoots” becoming multivalued as shown in figure (d). Characteristics are illustrate at the times relative figure (a), (c) and (d). Figure (b) shows that as time advances characteristics intersect for finite amplitude waves. Fig. 2.6.2: Thermodynamic properties before before and after a shock front in the shock reference frame. Density (ρ), pressure (P ), and particle velocity (U) before and after the shock front. Shock propagates with velocity (D) and the mass velocities (ν) through the shock front are shown. 2. Fundamentals of Fluid Dynamics 46 frame of reference, this is the conservation of mass flux equation, ρ1 (D − U1 ) = ρ0 (D − U0 ). (2.64) Using the same approach, the conservation equations for momentum and energy can be determined in Eulerian space, P1 + ρ1 (D − U1 )2 = P0 + ρ0 (D − U0 )2 , E1 + P1 (D − U1 )2 P0 (D − U0 )2 + = E0 + + . ρ1 2 ρ0 2 (2.65) (2.66) These equations relate the flow variables at each side of the discontinuity and are called the Rankine-Hugoniot relations. No assumptions have been made regarding the properties of the fluid and equations 2.64, 2.65, and 2.66 represent the general conservation equations across a discontinuity or shock front.27 2.7 Hugoniot Curves Shock Hugoniot curves represent possible states that are achievable due to a shock discontinuity. The principle Hugoniot is determined from the initial standard density and pressure conditions. For any material there is only one principle Hugoniot and an infinite number of reshock Hugoniots that may originate at any point along the principle Hugoniot.48 The first two Rankine-Hugoniot equations 2. Fundamentals of Fluid Dynamics 47 (equations 2.64 and 2.65) have seven unknowns. If the initial conditions are known (P0 , ρ0 , and U0 ) are known, that leaves two equations with four unknowns. Hugoniots are experimentally determined by measuring two parameters, typically the shock (D) and fluid velocity (U). In many materials, a linear relation is observed between the shock and fluid velocity,49 D = C0 + sU, (2.67) where C0 is often, but not always, the sound speed under standard conditions and s is typically the derivative of the bulk modulus at zero pressure.∗ The shock conservation equations (2.64, 2.65, and 2.66) can be used to determine all other parameters. Due to the arbitrary change in the entropy across the shock front, a fully-defined EOS requires a measurement of the thermal state of the shocked material. If the EOS for a material is known, the corresponding principle and nonprinciple Hugoniots can be determined.27 Through a simple manipulation of the Rankine-Hugoniot equations (2.64, 2.65, 2.66), one can show that 1 E1 (P1 , ρ1 ) − E0 (P0 , ρ0 ) = (P1 + P0 ) 2 ∗ ρ1 − ρ0 ρ1 ρ0 . (2.68) It is important to note, that this linear dependence is not observed in all materials and often a higher order fit is required to accurately represent the relation. 2. Fundamentals of Fluid Dynamics 48 Fig. 2.7.1: Hugoniot and Isentrope for LiF calculated using Sesame table 727150 at standard conditions. By specifying the initial conditions within a known EOS (P (T, ρ), E(T, ρ)), the Hugoniot can be found. For instance, using the Sesame table50 7271 for LiF the principal Hugoniot curve is shown in figure 2.7.1. The Hugoniot curve (red line) describes the possible end states for shocked LiF with standard initial conditions (T = 298 k, ρ = 2.64 g/cc) shown as the black point. For a single shock experiment, the LiF would compressed to a known end state (red point). The line connecting the initial and final state is referred to as the Rayleigh line. It is important to note that a shocked material only experiences the beginning and end state unlike the curve for isentropic compression. The blue isentrope in figure 2.7.1 is described in the following section. 2. Fundamentals of Fluid Dynamics 2.8 49 Isentropic Curves Similar to the Hugoniot curves, the isentrope can be determined using the known thermodynamic relations of a material. The thermodynamic relation (Section 2.14) for internal energy (E) can be described in terms of the temperature (T), entropy (S), pressure (P) and density (ρ), ∆E = T ∆S + P ∆ρ, ρ2 (2.69) where the specific volume (V) has been removed using the relation V = 1/ρ. For isentropic systems, the entropy remains constant, reducing the state relation to ∆E = P ∆ρ, ρ2 (2.70) for ∆ρ/ρ ≪ 1. Equation 2.70 determines the isentrope for a material with known thermodynamic properties. The isentrope for LiF is determined from standard conditions using Sesame table 727150 and is shown in figure 2.7.1 as the blue line. To reach the final state (blue point), the material follows the isentrope with each intermediate state experienced as long as the pressure rise is gradual enough. This is fundamentally different from the Hugoniot case in which the material only experiences the initial and final states. Isentropic compression experiments have 2. Fundamentals of Fluid Dynamics 50 a special desirability because in a single experiment a continuum of states can be achieved. Moreover, since entropy addition increases the temperature, isentropic compression is typically lower temperature than shock compression. In juxtaposing the Hugoniot and isentrope for LiF, figure 2.7.1 shows that for the same final pressure, the isentrope achieves a higher density. In isentropic compression, more energy is used in compression compared to shock compression because the entropy remains constant. For the shock case, higher temperatures are reached, as shown in figure 2.8.1, because the entropy increases. The rapid increase in temperature along the Hugoniot is due to the large shock-induced increase in entropy. For an isentropic compression wave propagating in a solid, the relationship among the temperature, density and entropy is T = T0 ρ ρ0 Γ0 exp ∆S , CV (2.71) where T is the temperature, ρ is the density, Γ0 is the Grüneisen parameter,27 S is the entropy of the system, Cv is the specific heat at constant volume, and the subscript 0 indicates the initial value. For the isentropic case (∆S = 0), the temperature increases by T = T0 ρ ρ0 Γ0 , (2.72) 2. Fundamentals of Fluid Dynamics 51 Fig. 2.8.1: Hugoniot and isentrope temperature dependence of LiF calculated using Sesame table 727150 at standard conditions. while for the shock case, the increase in entropy at the shock front will cause an exponential increase in temperature. 2.9 Concluding Remarks In the previous sections, the conservation of mass, momentum and energy equations were derived. From these derivations, it was evident that there are an infinite set of conservation equations. To close the system of equations, a thermodynamic relation and assumptions regarding the heat flux are required. In many systems, further understanding of the thermodynamic processes enables additional assumptions to be made, reducing the complexity of the system of 2. Fundamentals of Fluid Dynamics 52 equations. Fluid dynamic equations were applied to three separate cases (acoustic waves, planar isentropic flow, and shock formation). The solutions to each of these cases were provided. The evolution of isentropic compression waves into shock waves was discussed. Determination of isentropic and Hugoniot curves from EOS state data was shown. In terms of this work, understanding of the planar isentropic flow and its solutions are most important. Fundamental understanding of the method of characteristics and the limitations are pivotal to this research. 3. Experimental Technique 53 3. EXPERIMENTAL TECHNIQUE Interferometric measurements of flow velocities made through optical windows are discussed in this chapter. Equations relating the observed (apparent) particle velocity to the (true) particle velocity when measured through an optical window are derived for both shock and ramp compression. Previous experimental techniques that measured the shock-induced refractive index of an optical window are explored. These techniques are extended to an experimental design to measure the isentropically compressed refractive index of LiF. The target design and experimental constraints are identified. Diagnostics used to measure the required velocities and temperatures are discussed. 3.1 Interferometric Measurements Through Optical Windows Materials that remain transparent at extremely high pressures are useful in high energy density experiments. Shock and ramp compression experiments often employ optical windows that confine the samples but allow in situ particle-velocity 3. Experimental Technique 54 measurements at the sample/window interface. Velocities are measured using interferometry whose sensitivity depends on the refractive index of the material and the window. The optical window can be compressed during the experiment and changes in its refractive index must be accounted for in velocity measurements. If unaccounted for, systematic errors are introduced. The dependence of the refractive index on the compression of the window is required to provide accurate velocity measurements. The effects on interferometry measurements are considered in the following examples. Consider a system in which the velocity of a reflecting surface is observed through an optical window as shown in figure 3.1.1. The optical path length in the window (Zw ) is expressed as the integral of the refractive index (n(x, t)) over Fig. 3.1.1: Velocity of a reflecting surface observed through an optical window. 3. Experimental Technique 55 the window length (xf s (t) − x(t)), Zw (t) = Z x(t) n(x, t)dx, (3.1) xf s (t) where x(t) is the position of the reflecting surface and xf s (t) is the free surface position. For velocity interferometry measurements that occur at a reference plane in vacuum, the total optical path length (ZT ) from the observer (xV ) to the reflecting surface (x(t)) is expressed as ZT (t) = Z x(t) xf s (t) n(x, t)dx + Z xf s (t) dx. (3.2) xV Now consider that the reflecting surface (x(t)) is a piston that moves with the true particle velocity U(t), compressing the window material. Since the window material is constrained to move with the piston, its velocity is also the true particle velocity (U(t)). The optical thickness represents the apparent position of the reflective surface. This apparent position depends on the motion of the interface (U(t)) and the refractive index of the window. The time derivative of the total optical path length determines the apparent particle velocity. The difference between the apparent particle velocity and the true particle velocity is attributed solely to changes in the refractive index of the window. Taking the time derivative 3. Experimental Technique 56 of 3.2 determines the apparent particle velocity (Uapp ), d Uapp (t) = dt Uapp (t) = Uapp (t) = d dt d dt "Z "Z "Z x(t) n(x, t)dx + xf s (t) x(t) Z # x(t) # d [xf s (t) − xV ] , dt n(x, t)dx + Uf s (t), xf s (t) # dx , xV n(x, t)dx + xf s (t) xf s (t) (3.3) where Uf s is the window free surface velocity. To determine the true velocity (dx/dt = Utrue ) of the reflecting interface (i.e. the velocity that would be measured if the window were not present) knowledge of the refractive index as a function of time and space is required. Equation 3.3 is simplified for the case of shock and ramp compressed windows in the next section. 3.1.1 Shock Compressed Window Figure 3.1.2 shows an optical window that is compressed by a single steady transparent shock whose position is defined as xD (t). Prior to shock arrival at xf s , the window free surface is at rest (Uf s = 0). Let n0 and n represent the spatially uniform refractive indices ahead of and behind the shock, respectively. Equation 3. Experimental Technique 57 Fig. 3.1.2: Velocity of a reflecting surfaces observed through an optical window. 3.3 becomes d Uapp (t) = dt Uapp (t) = "Z x(t) ndx + xD (t) Z xD (t) # n0 dx , xf s (t) d [n(x(t) − xD (t)) + n0 (xD (t) − xf s )] , dt Uapp (t) = n(Utrue − D) + n0 D. (3.4) The true velocity of the reflecting surface as viewed through the shock compressed window is Utrue = Uapp + (n − n0 )D . n (3.5) Thus, the shock compressed refractive index (n) can be determined if the true and apparent interface velocities are measured and the Hugoniot (D(UT rue )) is known. 3. Experimental Technique 3.1.2 58 Ramp Compressed Window Hayes36 derived the relation for the dependence of the true and apparent velocities for isentropically or ramp compressed windows. The derivation assumes that waves are simple and that shocks do not occur in the window. Hayes showed that for simple waves the apparent and true particle velocities are directly related to refractive index and density gradients within the window dUapp dn =n−ρ . dUtrue dρ (3.6) Equation 3.6 can be expressed in integral form Z ρ n f (ρ′ ) ′ 0 , n(ρ) = ρ − ′ 2 dρ ρ0 ρ0 ρ (3.7) where f (ρ) represents the derivative of the apparent to true particle velocity and n0 , ρ0 are the required boundary conditions. Equation 3.7 illustrates that the influence of the refractive index in ramp compressed optical windows with density gradients is much more complicated than in single shock experiments. However, by measuring Uapp and Utrue over a range of ramp compressed pressures, the refractive index can be determined. 3. Experimental Technique 3.1.3 59 Optical Windows With Refractive Index That Varies Linearly With Density Velocity corrections for optical windows that possess a linear behavior of refractive index and density (n = a + bρ) are straightforward. Recall equation 3.3 for shocked windows and assuming the free surface is stationary (Uf s = 0), Uapp (t) = d dt "Z x(t) # n(x, t)dx . xf s (t) (3.8) Substituting the linear relations for refractive index and density " # Z x(t) d a(x(t) − xf s (t)) + b Uapp (t) = ρ(x, t)dx . dt xf s (t) (3.9) The second term of 3.9 represents the derivative of mass conservation of the window and is zero. This gives Utrue (t) = Uapp (t) . a (3.10) Substituting this linear relation into the equation 3.6, for ramp compressed windows, gives identical results indicating that corrections to the velocity measure- 3. Experimental Technique 60 Fig. 3.2.1: Experimental configuration shock-compressed gas gun experiment.34 ments have a constant value. Various materials have demonstrated this behavior such as quartz, sapphire and lithium fluoride (LiF).37 3.2 Shock Refractive Index Experiments Numerous studies have been performed to measure the refractive index of shocked windows34, 51–54 with the earliest performed by Kormer.51 These experiments used explosives and gas guns as drivers to compress the samples. The experimental approach was to collide a flyer plate into a buffer with a window attached to the rear surface (shown in figure 3.2.1). The flyer plate collision generates a shock that propagates across the buffer and into the optical window. Velocity interferometry measures the flyer plate velocity prior to impact and the apparent window interface velocity. Collisional analysis48 is used to determine the true particle velocity at the 3. Experimental Technique 61 window interface. This requires that the EOS of each material be known. A graphical representation of the collisional analysis is shown in figure 3.2.2. At the time of the collision, continuity requires that the pressure and particle velocity at the interface must be continuous (termed impedance matching). If Hf lyer and Hbuf f er represents the flyer plate and buffer Hugoniots, respectively, impedance matching requires that Hf lyer (Uf lyer − Up ) = Hbuf f er (Up ). (3.11) The value of Up that satisfies this condition determines the particle velocity at the contact interface (Ucollision ). The collision between the flyer plate and buffer, generates a shock that propagates across the buffer and into the window. The analysis assumes that the shock is steady and does not decay as it travels through the buffer. If Hwindow describes the window Hugoniot, then continuity requires Hbuf f er (2Ucollision − Up ) = Hwindow (Up ). (3.12) The particle velocity (Up ) that satisfies this equation determines the true particle velocity at the buffer/window interface. The collisional analysis is simplified when the flyer-plate and buffer are of the same material or when the flyer plate collides directly with the LiF window. In this case, the analysis reduces to the single 3. Experimental Technique 62 equation Hf lyer (Uf lyer − Up ) = Hwindow (Up ). (3.13) Using equation 3.5, the shocked refractive index is expressed as nS = UApp − n0 D . UT rue − D (3.14) By measuring the flyer plate velocity prior to collision, the true velocity is determined using the collisional analysis previously outlined. The Hugoniot of the optical window determines the shock velocity. Lastly, by measuring the apparent interface velocity post-collision, the refractive index is determined. Experiments conducted by Wise and Chhabildas34 to measure the refractive index of Lithium Fluoride (LiF) windows used various impactor and buffer materials (Al, Cu, Be, Ta, LiF, Al2 O3 ) with known Hugoniots. The authors performed both symmetric (identical flyer-plate and buffer material) and non-symmetric impacts. The flyer plate and apparent particle velocities were measured using velocity interferometry. The collision analysis outlined above determined the true particle velocity. Using equation 3.14, the shocked refractive index was determined. The results indicated that up to 115 GPa, the refractive index of shocked compressed LiF demonstrates a linear behavior with density as shown in figure 3.2.3. These are the highest published shocked refractive index measurements of LiF made to 3. Experimental Technique 63 Fig. 3.2.2: Geographical representation of collisional analysis to determine the true particle velocity. The flyer plate Hugoniot (blue line) is reflected through the measured flyer plate velocity (blue point). The intersection of this Hugoniot with the buffer Hugoniot (red line) determines the conditions of the flyer plate and buffer collision (red point). The buffer Hugoniot (dashed red line) is reflected through this point and the intersection with the window Hugoniot (black line) determines the true particle velocity. 3. Experimental Technique 64 Fig. 3.2.3: Shock refractive index measurements made by Wise and Chhabildas.34 The data shows a linear dependence over the measured density, with pressure up to 115 GPa. date. Observations and predictions suggest that shock-driven LiF becomes opaque between 130 and ∼ 280 GPa when shock compressed.51, 55 This is due, in part, to melting by the shock. As was shown in Section 2.8, ramp compression experiments achieve much lower temperatures than shock compression. Figure 3.2.4 illustrates the Hugoniots (red line) and isentrope (blue line) for LiF.50 Included in the figure are experimental and theoretical molecular dynamic predictions for the melt line of LiF.51, 55 The figure shows that at ∼160 GPa, the shock Hugoniot crosses the melt line in 3. Experimental Technique 65 Fig. 3.2.4: LiF Hugoniot (red) and isentrope (blue) calculated using sesame table 7271.50 Shock, diamond anvil cell (DAC) and molecular dynamic (simulations) of the LiF melt are included. Boehler’s estimated melt line (black) is included. A large discrepancy between the results of Kormer and Boehler are observed in determining the shock melting of LiF.51, 55 the region where LiF becomes opaque. Isentropic compression is able to achieve significantly higher pressures without crossing the melt line suggesting that LiF will remain transparent at pressures well above 160 GPa enabling refractive index measurements at substantially higher pressure than previously demonstrated. 3.3 Isentropic Refractive Index Experiments The derivation by Hayes (3.6) shows that the refractive index of a window is determined by measuring the true and apparent particle velocities with knowledge 3. Experimental Technique 66 of the corresponding EOS, dn dUapp =n−ρ . dUtrue dρ In this work, the refractive index of ramp compressed LiF is measured using a twosection target consisting of a piston with an optical window attached to half of its rear (undriven) surface. Planar compression waves are driven into the front surface of the target, traverse the piston, and reach the rear surface. These compression waves extend across the two sections (bare and window) on the rear of the target, producing distinct compression features in those two sections due to dissimilar boundary conditions (free surface vs. impedance matching). The free surface (Uf s ) and the apparent interface (Uapp ) velocities are measured simultaneously. The true interface velocity (Utrue ) is determined using the method of characteristics. Velocity measurements at the free surface and piston/window interface are made and the characteristics relations in Lagrangian form (section 2.4) are used. The transverse dimension of the target and drive are sufficient such that a onedimensional analysis is justified. The technique, outlined by Rothman56 and Maw,57 corrects for the wave interactions at the free surface to provide particle velocities within the sample. Adaptation of this method allows for a time-dependent calculation of the true interface velocity (Utrue ) that can be compared to the apparent velocity (Uapp ). The technique is to measure the free surface velocity (Uf s ) of the piston material and using a backwards characteristics scheme determine the 3. Experimental Technique 67 Fig. 3.3.1: Target design used to determine the refractive index of an optical window. The drive produces uniform compression regions across both sections of the target. The piston free surface and piston/window interface experience the same loading. The boundary conditions at corresponding interfaces are shown. applied ablation pressure (P(h=0,t)), where h=0 defines the location of ablation. Once the applied ablation pressure is known, a forward characteristics scheme is used to determine the true particle velocity at the piston/window interface (Utrue ). Each step requires different boundary conditions. For simplicity the forward (F) and backwards (B) characteristics∗ are defined as ∗ F = Up + σ along dh dt = CL , (3.15) B = Up − σ along dh dt = −CL , (3.16) The notation for Reimann invariants J+ and J− are replaced with F and B, respectively. 3. Experimental Technique 68 where Up is the particle velocity, σ is the Lagrangian strain, h is the Lagrangian depth and CL is the Lagrangian sound speed. The Lagrangian strain and sound speed are related to the thermodynamic properties (density (ρ) and pressure (P )) by dP , ρ0 CL 1/2 ρ ∂P . = ρ0 ∂ρ σ = CL Z (3.17) (3.18) Up and σ are determined from the intersection of the ith positive and j th negative characteristics where i and j represent the indexing of the characteristics as illustrated in figure 3.3.2. That value is given by 1 Upi,j = (F i + B j ), 2 1 σ i,j = (F i − B j ). 2 (3.19) (3.20) The negative characteristics that emerge from the undisturbed material (t ≤ 0) must have B = 0. Using these equations, two characteristics schemes are developed to propagate the characteristics that determine the fluid flow. 3. Experimental Technique 69 Fig. 3.3.2: Index of characteristics is shown. Blue and red lines represent the forward and backwards characteristics. The indexing of characteristics is defined and the intersection of the F3 and B2 characteristics is illustrated for clarity. 3. Experimental Technique 3.3.1 70 Backwards Characteristics Scheme The backwards characteristics scheme uses the free surface velocity measurement (Uf s ) to determine the ablation pressure (P (0, t)). When the compression wave reaches the free surface, it releases along the reflected isentrope to zero pressure. The free surface velocity is directly related to the particle velocity by the velocity doubling rule or Uf s = 2Up . (3.21) The free surface boundary condition requires that P (L, t) = σ(L, t) = 0, (3.22) where L is the Lagrangian thickness of the piston. These conditions together require that F i = B i = Uf s (tif s ), (3.23) which define the F and B characteristics. The velocity and Lagrangian stress are found at the intersection of every positive and negative characteristic (equation 3. Experimental Technique 71 3.20). The location and time of these intersections is i,j h i,j t CLi,j−1CLi−1,j (ti−1,j − ti,j−1 ) + CLi,j−1hi−1,j + CLi−1,j hi,j−1 = , CLi,j−1 + CLi−1,j (3.24) CLi,j−1ti−1,j + CLi,j−1 ti,j−1 + hi−1,j − hi,j−1 , = CLi,j−1 + CLi−1,j (3.25) where CLi,j = CL (σ i,j ). The initial conditions (U(0, ti ), σ(0, ti )) are determined when the ith characteristic reaches the loading surface (h = 0). The ablation pressure is found using the piston’s equation of state: P (0, ti) = P (σ(0, ti)). This scheme is valid until the backwards characteristics reach the loading surface (plane at which the ablation pressure is applied) because interaction of these characteristics with the loading surface are uncertain. 3.3.2 Forward Characteristics Scheme The forward characteristics (F ) are determined from the ablation pressure (P (0, ti)), i i F = 2U(0, t ) = 2σ(0, t ) = 2 Z 0 P (0,ti ) dP . ρ0 CL (3.26) At the intersection of the forward traveling characteristics with the window interface, the impedance matching boundary conditions are applied, P1 (σ1 ) = P2 (σ2 ), (3.27) Up1 = Up2 , (3.28) 3. Experimental Technique 72 where subscripts 1 and 2 indicate the piston and window, respectively. Using equations 3.19 and 3.20, one may write Up1 = F1 − σ1 = Up2 = B2 + σ2 . (3.29) Assuming no negative characteristics in the window (B2 = 0)† equation 3.27 is rewritten P1 (σ1 ) = P2 (F1 − σ1 ). (3.30) The value of σ1 that satisfies this equation is used to determined the particle velocity at the interface (Utrue = Up1 = Up2 ). 3.3.3 Target Design The technique described requires a two section target that consists of a piston with a window attached to half of the rear surface (figure 3.3.1). Characteristic schemes require that that reflected characteristics do not reach the loading surface prior to experimental termination.‡ Thus, the window must be sufficiently thick such that reflected characteristics from the window free surface do not perturb the flow at the piston/window interface. † The assumption that B2 = 0 requires that the window be sufficiently thick such that compression waves do not reach the window free surface and reflect back into the interaction region prior to the termination of the experiment. ‡ Experiment termination is defined as the time at which peak compression is observed at the rear of the piston sample. 3. Experimental Technique 73 The proper piston thickness requires balancing the applied strain rate, the material properties, and the limits of the driver to obtain the desired pressure profile. As a material is compressed, its sound speed increases such that subsequent compression waves can overtake predecessors. If the applied pressure rises too quickly, the compression waves will coalesce and form a shock (terminating the analysis) as shown in section 2.6. Stiffer materials can be ramped more rapidly while maintaining shockless compression. Materials with high Lagrangian sound speed and a large Bulk Modulus are ideal for piston materials.58 Diamond is used as a piston because its low compressibility allows it to be rapidly compressed to high pressures without shocking. Recently, the ramp wave response of chemical vapor deposited (CVD) diamond was experimentally determined to 800 GPa.59 The use of an experimentally determined quasi-isentrope increases the utility of diamond as a piston material such that errors in the diamond isentrope are greatly reduced when compared to theoretical predictions. The method of characteristics was used to specify the ablation pressure and determine the optimal diamond piston thickness. Characteristics were tracked in the diamond piston and LiF window to ensure that shock formation does not occur prior to the conclusion of the experiment. The duration of the applied ablation pressure is determined by the OMEGA laser system capabilities. Single laser pulse durations can be from 1 to 3.7 ns. The arrival time of each beam can 3. Experimental Technique 74 be adjusted using the path length adjustment system allowing a laser irradiation of significantly longer durations. In addition to these constraints, the diamond piston introduces further considerations with respect to the ablation pressure. When stressed to the elastic limit (EL), diamond generates a two-wave structure60 that consists of an elastic precursor propagating at the elastic sound speed and an inelastic wave traveling at a reduced velocity. If the two-wave structure turns on during the bulk of the compression, large uncertainties in the refractive index determination are introduced in the vicinity of the EL. Therefore, the ablation profiles are designed such that the initial pressure pulse stresses the diamond above its EL, initiating the two wave structure and this effect is easily accounted for in the analysis. When the elastic wave of diamond (∼ 80 GPa)59 is impedance matched to LiF, the minimum achievable pressure of LiF is ∼ 30 GPa. An initial estimate of the target thickness can be determined from the sound speed of the elastic wave (∼ 20µm/ns)59 and the pulse duration. The minimum required diamond thickness (T) to optimize the pulse duration is T = (Elastic Wave Sound Speed)(Pulse Duration) . 2 (3.31) From this we see that the pulse duration is the limiting factor in determining the optimum diamond piston thickness. For a 3.7 ns pulse, the minimum target 3. Experimental Technique 75 thickness is ∼ 40 µm. Characteristic analysis shows that the arrival time of the reflected elastic wave at the ablation surface is slowed due to interaction of forward traveling waves, slightly reducing the constraint on the diamond thickness. Furthermore, the determination of the refractive index requires the derivative of the apparent to true particle velocities (equation 3.6). Steep compression profiles introduce significant errors in this determination while gradual compression rates introduce fewer errors. The data quality and the peak compression achieved must therefore be balanced. For these experiments pulse shapes were designed to have durations of 3.7 and 7 ns. The applied ablation profiles were determined such that shock formation did not occur in the diamond piston or LiF window. Three ablation profiles were defined such that the LiF window achieved pressures ranging from 30 to 800 GPa. Once the ablation profile is known, the drive technique and ablation pressure to laser power scaling are required to determine the laser pulse profile. 3.3.4 Driver Experiments were performed on the OMEGA laser61 at the University of Rochester’s Laboratory for Laser Energetics. OMEGA is a 60 beam Nd:Glass laser where the laser light is frequency converted from 1054 to 351 nm which provides up to 30 kJ of energy on target in the UV (the maximum energy per beam is ∼500 J). The beams are arranged to achieve uniform ablation on spherical targets. 3. Experimental Technique 76 To produce planar compression up to 12 beams are used to irradiate the target. Six of these beams arrive at an angle of 23.2◦ to the normal with the other six beams arriving at angles of 47.8◦ . The lower angle beams (23.2◦ ) are used when all 12 beams are not required. Laser light is smoothed using distributed phase plates62 to produce spots that having a planar region of about 876 µm diameter. In laser-driven experiments, high pressures are produced by the ablation of target material. The ablation can be driven by either x-rays from laser-driven hohlraums (indirect drive)59 or by direct laser irradiation (direct drive).63 Highpower lasers typically produce drive pulses of a few nanoseconds that readily produce high-pressure (> 100 GPa) shock waves. These have been used in reservoircoupled experiments where a shocked reservoir material releases across a vacuum gap and the hydrodynamics of the releasing shocked material determines the target loading.64, 65 Recent advances in the ability to control the temporal shape of laser pulses have enabled shaped-ablation ramp compression. Laser-driven halfraums have been used to produce ramp compression in diamond targets by directly ablating the sample material59 (indirect-drive shaped-ablation ramp compression). The present experiments demonstrate ramp compression by direct-drive shaped ablation. The laser directly irradiates the sample; there is no intermediate energy conversion nor coupling and the ablation pressure is controlled by temporal shap- 3. Experimental Technique 77 ing of the laser pulse. The benefits of this technique are that relatively lower energies are required compared to the other techniques and many gigapascals are achieved in a few nanoseconds.66 The compression profile is directly controlled by adjusting the laser pulse profile using the appropriate ablation pressure to laser intensity scaling law. Long pulse shapes produce the highest data quality because gradual compression rate introduces fewer errors in determining the derivative of the apparent to true particle velocities. It was determined that achieving a smooth transition when stacking two beams was difficult and introduced significant deviations in the measured profiles. Thus, single 3.7 ns ramp profiles were predominantly used. 3.3.5 Laser Pulse Shape To develop isentropic compression in diamond samples, the direct-drive ablation scaling law is required. An initial scaling law was provided by Hicks for direct-drive shock compression of low-Z materials67 0.6 TW P [GPa] = 55.3 I , cm2 (3.32) 3. Experimental Technique 78 where P is the ablation pressure and I is the laser intensity. The scaling law for diamond was determined from the present experimental results to be 0.75 TW . Pablation [GPa] = 42 I cm2 (3.33) The ablation analysis technique is discussed in the Appendix A. The laser power is determined from the spot size of the laser pulse and total energy. Due to the requirement that the transverse dimension of the two compression regions be sufficiently large such that each undergo identical compression, distributed phase plates62 were used to produce uniform planar region of ∼876 µm in diameter.68 Using this cross sectional area, the laser power is determined. Three pulse shapes were designed for these experiments. As previously mentioned, the pulse duration is the primary factor in determining the target thickness. The pulse shapes are shown in figure 3.3.3a. Figure 3.3.3 contains plots of the laser pulse power, the corresponding ablation pressure, the predicted diamond free surface velocity and the predicted diamond/LiF interface velocity. 3.3.6 Target Specification Laser pulse durations were 3.7 and 7 ns correspond to required diamond thicknesses of ∼ 46 µm and 100 µm respectively. The thicknesses were chosen to be slightly larger to ensure that the backward characteristics do not reach the ab- 3. Experimental Technique Fig. 3.3.3: 79 Three pulse shapes designs for these experiments. Fig- ures (a) and (b) show the laser power and ablation pressure for the three designed pulse shapes. Using the method of characteristics, the free surface velocity and true particle velocity are calculated, figures (c) and (d) respectively. The diamond target thickness used in these simulations were 46 µm for pulse shapes RM3503 and RM1134 and 100 µm for RM3504. Figures (c) and (d) are normalized to the arrival time of the elastic wave. 3. Experimental Technique 80 lation surface prior to laser termination. The cross section of diamond samples ranged from 1.1 mm to 2.5 mm square. The larger cross section of the diamond samples was an engineering control to ensure that the laser spot did not miss the target. The cross section of the diamond is sufficient such that the wave interactions at the edges of the target do not influence the fluid flow. Chemical vapor deposition (CVD) and ultra-pure nano-crystalline diamond were used. A 500 µm thick LiF window was mounted over half of the rear surface of these diamond targets. The thickness ensures that reflected waves do not interfere with diamond/LiF interface. High-purity LiF was orientated with the [100] axis along the pressure loading direction. A 1000 Å coating of aluminum was applied to the ablation side of the diamond to prevent the low-intensity leading edge of the laser from penetrating the target before the ablation plasma is formed. The rear surface of the LiF had an anti-reflection coating to reduce ghost reflections in the optical measurements of the apparent interface velocity. A velocity interferometer system for any reflector (VISAR)33, 35, 69, 70 was used to measure the diamond free surface velocity and LiF interface velocity. The reflectivity of these layers must be taken into account to achieve sufficient signal to noise ratio in the VISAR measurements. The interface reflectivity of uncoated LiF mounted on diamond was inadequate to achieve significant signal to noise ratio. After arrival of the elastic wave, a substantial decrease in reflectivity was observed 3. Experimental Technique 81 (∼ 80%). A 1000 Å metallic coating was applied to the diamond/LiF interface to increase the reflectivity of that surface for interferometric measurements of the interface velocity. The material layer was sufficiently thin so that it did not affect the characteristics analysis. Three coating materials were used for this purpose: titanium, aluminum and gold. Titanium was predominantly used because the change in reflectivity over the pressure range was closest to that of the diamond free surface creating comparable reflectivity. During target fabrication, samples were glued only at the edges such that the compressed diamond released directly into the LiF where there was no glue. A finite gap between the diamond and LiF window was inevitable in the target fabrication process. The gap was measured to be less than > 3 µm in all samples and was verified by the delay in elastic wave arrival times between the free surface and interface measurements.§ 3.4 Diagnostics Two diagnostics were used to measure the velocity and experimental temperature. A velocity interferometer system for any reflector (VISAR)33 measures the free surface velocity (Uf s ) and the apparent particle velocity (Uapp ). A streaked optical pyrometer (SOP) provides optical emission measurements to determine the § Initial targets were faced glued (thin layer of epoxy between the diamond and LiF window). The impedance of the glue reduced the interface particle velocity, introducing significant errors. 3. Experimental Technique 82 gray body temperature of the target.71 These diagnostics and their data reduction techniques are discussed below. 3.4.1 Velocity Interferometer System for Any Reflector Theory VISAR measures the velocity history of a reflecting surface. A schematic of the Mach-Zhender interferometer used in these experiments is shown in figure 3.4.1. Coherent laser light (probe beam) with wavelength λ is reflected off a moving target and imaged through a Mach-Zhender interferometer. At the first beam splitter, the light is separated into two legs. The light traveling through leg two is delayed by a time τ using an etalon. The etalon, typically made of fused silica, increases the optical path length of leg 2, imposing a delay. The etalon delay is τ= 2h 1 n− , c n (3.34) where h and n are the thickness and refractive index of the etalon, respectively, and c is the speed of light. The etalon adjusts the focal plane of the corresponding leg and this is accounted for by applying a translation distance (d) to the location the etalon mirror defined as 1 d=h 1− . n (3.35) 3. Experimental Technique 83 Fig. 3.4.1: Schematic of Mach-Zhender interferometer This ensures that the recombined images at the output beam splitter are both in focus. A second beam splitter recombines the two beams such that the detector, or streak camera, observes a brightness that depends upon the phases of the recombined beams (interference). The recombined image contains phase data (brightness record) regarding the difference in velocities at times separated by τ . For constant velocities, the change in phase is zero and a constant light amplitude is observed. The relation between the target velocity and changes in the observed light amplitude (interference pattern) are understood through the following example.72 Assume that the light from leg 2 is delayed by Nλ where N is an integer number. 3. Experimental Technique 84 The length of this delay is written as Nλ = cτ. (3.36) Since N is an integer, both beams arrive in phase at the output beam splitter (constructive interference). If the length of leg 2 is held constant, then changes in the wavelength of the probe beam will directly affect the amplitude of light observed at the detector. If the wavelength decreases by 1/2, the combined beams will arrive out of phase and destructively interfere. The Doppler shifts in the probe beam are observed as changes in light amplitude. These changes are directly related to the velocity of the target since movement of the target induces a Doppler shift in the probe beam wavelength or ∆λ(t) = −2λ U(t), c (3.37) where U(t) is the time varying velocity of the reflecting surface. Note that the change in wavelength is related to the total change in the amplitude (∆N) up to time t by taking the derivative of equation 3.36, ∆N(t) = −cτ ∆λ(t), λ2 (3.38) 3. Experimental Technique 85 and the velocity is defined as U(t) = λ ∆N(t). 2τ (3.39) In this configuration, the light amplitude depends on the integrated field of view at the output beam splitter. A spatial dimension is imposed at the image plane by tilting the output beam splitter. This changes the relative optical path of light across the output field, creating a series of fringes that establish “zero” phase. Tilting the output beamsplitter produces a shear in the phases of the beams creating a linear fringe pattern across the field of view. The spatial dimension of the image enables simultaneous measurements at different target regions. A sample pattern is shown in figure 3.4.2a, where the x-axis is time and the y axis is space. At t1 a shift in fringes is observed that corresponds to a change in target velocity. The velocity profile determined from those fringes is shown in figure 3.4.2b. Application A two dimensional image is relayed to a streak camera slit producing a one dimensional spatial image that is “streaked” in time. The image is recorded on with time on x-axis and a one-dimensional image of the target on the other. VISAR consists of a probe beam, an interferometer, and an imaging system. The 3. Experimental Technique 86 Fig. 3.4.2: Sample VISAR data. (a) The “fringe comb” observed at the image plane. (b) The corresponding velocity profile. configuration of this system on OMEGA is shown in figure 3.4.3. The probe beam, Nd-YAG laser light (532 nm), is reflected off the target. The reflected light is collimated, imaged at the output beam splitter of a Mach-Zehnder interferometer, and this image is relayed to the slit of the streak camera.70 Analysis of this image to determine the target velocity is discussed below. The target velocity is related to the fringe phase (φ(x, t)) of figure 3.4.4a. The fringe phase is determined by fitting the fringe intensity, S(x, t), to the sinusoidal function S(x, t) = A(x, t) + B(x, t)cos[φ(x, t) + 2πf0 x + δ0 ], (3.40) where A(x, t) and B(x, t) are functions representing the varying background in- 3. Experimental Technique 87 Fig. 3.4.3: Configuration of the VISAR and SOP on OMEGA.73 tensity and the amplitude variation respectively, and f0 is the spatial frequency.70 The desired phase can be obtained using a Fourier-transform method.35, 74 In this technique, equation 3.40 is rewritten in terms of complex conjugates S(x, t) = A(x, t) + C(x, t) exp(2πif0 x) + C ∗ (x, t) exp(−2πif0 x), (3.41) where C(x, t) = B(x, t) exp(iφ(x, t)). 2 (3.42) Taking the complex logarithm of equation 3.42 yields log(C(x, t)) = log(B(x, t)/2) + iφ(x, t), (3.43) where the desired phase information is separated from the amplitude information. 3. Experimental Technique 88 Fig. 3.4.4: Phase extraction procedure using the Fourier method to determine the velocity. (a) VISAR Raw data (b) Fourier Spectrum (c) unwrapped phase (d) velocity field (e) final velocity (line out over the velocity field). Figure taken from Celliers et al.35 3. Experimental Technique 89 C(x, t) is determined by taking the spatial Fourier transform of equation 3.41, s(f, t) = b(f, t) + c(f − f0 , t) + c∗ (f + f0 , t), (3.44) where the Fourier transformation of functions are represented in lowercase, s(f, t) = Z ∞ S(x, t) exp(−2πixf )dx. (3.45) −∞ The background information (b(f, t)) is separated from the phase information by the constant fringe frequency (f0 ). s(f, t) is filtered such that c(f −f0 , t) is selected (figure 3.4.4b), d(f, t) = c(f − f0 , t). (3.46) Taking the inverse Fourier transformation of d(f, t) produces D(x, t) = C(x, t) exp((2πf0 x + δ0 )i), (3.47) where D(x, t) is a complex function. The phase is defined as φ(x, t) = −2πf0 x − δ0 + arctan Im[D] Re[D] , (3.48) 3. Experimental Technique 90 and the amplitude variation (B(x, t)) is given by B(x, t) = Re[D] . cos(φ(x, t) + 2πf0 x + δ0 ) (3.49) In equation 3.48, φ(x, t) is not uniquely determined; an integer number of 2π can be added or subtracted. The relation between the fringe phase and the total change in fringe amplitude up to time t is ∆N(t) = φ(t) − φ(t0 ) ± n, 2π (3.50) where the n is an integer that represents the 2π ambiguities. The velocity is determined from equation 3.39 as shown in figure 3.4.4e. Difficulties arise in this technique when Doppler shifts occur at rates faster than c/τ due to constructive interface occurring at integer multiples (referred to as 2π ambiguities). This is resolved by using two interferometers with different VPF’s. The velocity per fringe (VPF) is defined as VPF = λ , 2τ (1 + δ) (3.51) where δ depends on the dispersion of the etalon, dn −n0 , λ0 δ= 2 n0 − 1 dλ λ=λ0 (3.52) 3. Experimental Technique 91 and n0 is the refractive index of the etalon at the probe wavelength. Etalons are chosen such that the sensitivities of the two interferometers are not integer multiples. The Doppler shift is found by comparing the predicted velocities of different VPF integer fringe shifts. Since both interferometers observed the same velocity (where subscripts 1 and 2 represent the different interferometers), 1 U(t − τ1 ) = VPF1 (∆φ(t)1 ± n), 2 1 U(t − τ2 ) = VPF2 (∆φ(t)2 ± m), 2 (3.53) (3.54) and if the VPF’s of these interferometers are not integer multiples of one another, the integers n and m can be determined uniquely and the velocity profile is determined. 3.4.2 Streaked Optical Pyrometer The temperature of the target is determined from the optical self emission brightness temperature. The streaked optical pyrometer (SOP)71 uses the VISAR telescope to relay the target self emission to a streak camera having a subnanosecond temporal resolution. The camera output intensity, I, can be related to the source irradiance through the following equations,71 B∆xWs Ωlens I= ηM 2 Z dλTx (λ)SR(λ)Ls (λ, T ), (3.55) 3. Experimental Technique 92 where the source radiance (Ls (λ, T )) is given by Planck’s law, L(λ, T ) = 1 2hc2 . 5 hc/λT λ e −1 (3.56) B is the binning of the CCD, ∆x is the length of one pixel, Ws is the slit width, Ω lens is the solid angle viewed by the lens, η is the sweep rate, M is the magnification of the system, λ is the wavelength, Tx is the product of the transmission/reflection spectrum of the optical elements in the system, SR is the wavelength-dependent sensitivity of the streak camera system, Ls is source radiance given by Planck’s law, h is Planck’s constant, c is the speed of light, and T is the temperature.71 The SOP has two narrow wavelength bands: a red-channel is created by using a long-pass filter with a cutoff wavelength of 590-nm and the streak cameras insensitivity to wavelengths greater than ∼ 850 nm. Similarly, a blue-channel is defined by a short pass filter with a cutoff of 500 nm and a long pass filter with a cutoff of 390 nm.71 When used with a relatively narrow wavelength band, the wavelength is approximated as a delta function and equation 3.55 becomes T = T0 . ln(1 + AI ) (3.57) hc λ0 (3.58) T0 is photon energy detected, T0 = 3. Experimental Technique 93 and A is defined as A= 2B∆xΩlens < Tx SR > hc2 G , ηM 2 λ50 (3.59) where G is the gain and λ0 is the central wavelength of the band. In these experiments, the reflectivity of the target is determined from the VISAR measurements. Consequently, the optical emission measurements must be corrected for the emissivity of the target to determine the temperature accurately.71 The reflectivity (R) of the target is determined from equation 3.49. The gray-body temperature is defined as T = 3.5 T0 ln(1 + A(1−R) ) I . (3.60) Concluding Remarks The effect of shocked and ramp compressed optical windows on interferometry measurements was discussed. The single shock experimental technique to determine the compressed refractive index of an optical window was examined in detail. It was shown that a similar approach can be used to determine the ramp compressed refractive index of an optical window. Experimental constraints re- 3. Experimental Technique 94 garding this technique were illustrated and diagnostics required to determine the compressed refractive index were described 4. Analysis and Results 95 4. ANALYSIS AND RESULTS The LiF ramp compressed refractive index analysis and results are discussed in this Chapter. Twenty-four experiments are analyzed in detail. Seven experiments contained a thin layer of glue and the technique used to correct for the impedance of the glue is discussed. In all experiments, a strong linear dependence is observed between the apparent and true velocity resulting in a linear refractive index as a function of density. Concerns regarding x-ray preheat and the large oscillations observed on specific shots when determining the apparent and true velocity are discussed in detail. Hydrocode simulations are performed to verify the accuracy of the method of characteristics. Temperature measurements of LiF are provided. 4.1 Data Analysis Twenty-four experiments were conducted to determine the ramp compressed refractive index of LiF. The analysis method is discussed in detail and illustrated using data acquired from shot 57575. The target for this shot consisted of a 46 µm diamond piston with a 500 µm LiF window attached to half of its rear surface. The LiF window was edge glued to the diamond piston to ensure that 4. Analysis and Results 96 the impedance of the glue did not affect the velocity measurements. A 1000 Å aluminum coating was applied to the ablation side of the diamond to prevent the low-intensity leading edge of the laser from penetrating the target before the ablation plasma was formed. It had an aluminum shine through barrier and a 1000 Å titanium coating on the LiF. Four beams, using pulse shape RM3502 (shown in figure 3.3.3a), irradiated the target with a total energy of 270 joules. The sweep speeds for camera 1 and 2 were 9 ns and 5 ns, respectively. This corresponds to a temporal resolution of 11 ps and 6 ps.∗ Target specifications for all shots used in this study are found in table 4.1.1. Table 4.1.2 contains the relative etalon parameters determined from equations 3.34 and 3.51. The VISAR data for shot 57575 camera 1 is shown in figure 4.1.1a. Before t = 0 the fringes are horizontal because neither the free nor embedded surfaces are moving. The fringes in the top half result from probe light reflected off the embedded diamond/LiF interface. On the bottom, they are from the reflection off the diamond free surface. At laser initiation (t = 0) a sudden change in the free surface reflectivity is observed. This is because the diamond is transparent and the signal from that section is a combination of reflection from the free surface and the aluminum coating on the front surface. The drop in signal is attributed to the ablation of the aluminum coating on the front surface and formation of ∗ Cameras that used a 15 ns sweep speed had a temporal resolution of 16 ps. 4. Analysis and Results 97 Table 4.1.1: Shot Specifications Shot 54939† 54940† 54941† 54944† 54945† 54946† 54948† 55857‡ Pulse Shape RM3503 RM3503 RM3503 RM3503 RM3503 RM3503 RM3503 RM3503 J J J J J J J J Sweep Speed 9ns/15ns 9ns/15ns 9ns/15ns 9ns/15ns 9ns/15ns 9ns/15ns 9ns/15ns 9ns/15ns 55859 55860‡ RM3503 370 J RM3503 378 J 9ns/15ns 9ns/15ns 15a/7a 15a/7a 56109‡ RM3503 508 J 9ns/5ns 18a/7a 56112 56113 57569 57570 57571 57572 57574 57575 57576 57577⋆ 57579⋆ 57581⋆ 57583 RM3503 RM3503 RM3504 RM3504 RM3504 RM3504 RM3504 RM3702 RM3702 RM3702 RM3702 RM3702 RM3702 9ns/15ns 9ns/15ns 9ns/15ns 9ns/15ns 9ns/15ns 9ns/15ns 9ns/15ns 9ns/5ns 9ns/5ns 9ns/5ns 9ns/5ns 9ns/5ns 9ns/5ns 18a/7a 18a/7a 18a/7a 18a/7a 18a/7a 18a/7a 18a/7a 7a/18a 7a/18a 7a/18a 7a/18a 7a/18a 7a/18a ∗ † ‡ ⋆ Energy 362 480 601 422 558 421 532 442 384 J 387 J 711 J 767 J 774 J 767 J 623 J 270 J 345 J 272 J 362 J 362 J N/A Etalon parameters are given in table 4.1.2. Targets contained a thin glue layer. Targets contained a gold preheat shield. Nano-crystalline diamond was used. Etalon∗ Thickness Coating 18a/7a 18a/7a 15a/7a 15a/7a 15a/7a 15a/7a 15a/7a 15a/7a Ti Ti Ti Ti Ti Ti Ti Al 43 µm 46 µm 46 µm 46 µm 46 µm 46 µm 43 µm 15-[3 Au]40µm 46 µm 10-[3 Au]25 µm 10-[1 Au]35 µm 45 µm 45 µm 97 µm 99 µm 95 µm 97 µm 103 µm 46 µm 45 µm 44 µm 45.5 µm 45.5 µm 46 µm Al N/A N/A Au Ti Ti Ti Ti Ti Ti Ti Ti N/A Ti Ti Ti 4. Analysis and Results 98 Table 4.1.2: Etalon Parameters Etalon Thickness (µm) 7a 15a 18a 7.2095 15.1318 18.2268 Delay (ps) 37.32 78.35 94.37 VPF ( µm ns fringe 6.0965 3.2906 2.7318 ) the opaque plastic wave in the diamond. This abrupt drop in reflectivity triggers an immediate reduction of current within the streak tube. The space charge of the electron beam is affected, causing a rapid change in the magnification of the streak tube and an artificial shift in the positions of the fringes. Note that this shift is symmetric about the center of the data record with fringes shifted in the vertical direction away from center. Since the current is constant after that abrupt change, the streak camera equilibrates and the fringe position stabilizes. At 2.2 ns, the elastic precursor reaches the rear diamond surface and the fringe position abruptly changes in response to the velocity of that surface. At 3.5 ns, the LiF window undergoes compression. The delay in arrival of the elastic wave is attributed to a gap between the diamond piston and LiF window. The diamond free surface velocity propagates at a velocity of ∼ 2.2 µm/ns at breakout. This velocity and the delay time of the LiF window compression is indicative of the gap thickness (a gap 3 µm or less is inevitable in the target fabrication process). After this time, fringes move continuously to higher displacement (velocity) as the pressure increases. The resulting diamond free-surface velocity profile (blue) and 4. Analysis and Results 99 Fig. 4.1.1: (a) Shot 57575 VISAR data corresponding to camera 1. Time is shown on the x-axis with the spatial dimension on the y-axis. VISAR measurements at the diamond free surface and diamond/LiF interface are made simultaneously. (b) The corresponding velocity profiles determined from the VISAR measurements. The blue and red lines corresponds to the diamond free surface velocity and the diamond/LiF interface velocity, respectively. 4. Analysis and Results 100 the apparent interface velocity (red) are shown in figure 4.1.1b . Zero velocity for both portions of the target is chosen as the fringe position after the space-charge induced shift at t=0. The diamond free surface velocity is backwards propagated to determine the applied ablation pressure using the backwards characteristics scheme discussed in Section 3.3.1. Figure 4.1.2a shows a graphical representation of the characteristic calculations. The 0 µm depth corresponds to the ablation surface and 46 µm corresponds to the free surface (diamond thickness for shot s57575). Time increases vertically, and depth (in Lagrangian coordinates) increases to the right. The slope of the characteristics is the inverse Lagrangian sound speed (CL ). Each characteristic line is color coded by the pressure colorbar shown at the right (the temporal profile of the inferred ablation pressure shown in figure 4.1.2b). Characteristics propagate to the right, at the pressures they were initiated, until they reach the rear surface. A zero-pressure boundary condition is imposed for the free surface, producing a reflected wave. The analysis is invalid once the free surface characteristics reach the loading surface at t ∼ 4.8 ns because interaction of characteristics with the ablation region is unknown. The applied ablation pressure (shown in figure 4.1.2b) is used to determine the true particle velocity at the diamond-LiF interface using the forward characteristics scheme discussed in Section 3.3.2. The graphical representation of the 4. Analysis and Results 101 Fig. 4.1.2: (a) Shot 57575 backwards characteristics. The diamond free surface velocity measurement is used as the boundary condition at the Lagrangian depth of 46 µm. The characteristics are color coded by the pressure colorbar shown on the right. (b) The ablation pressure profile in time calculated for the Lagrangian depth of 0 µm. 4. Analysis and Results 102 forward characteristics is shown in figure 4.1.3a. To account for the finite gap between the diamond and LiF window, a time constraint is place on the boundary condition at the diamond/LiF window interface. The apparent interface velocity is used to determine when LiF compression occurs (t ∼ 3.5 ns). Prior to this time, the free surface boundary condition is imposed on the diamond piston. At later times (t > 3.5 ns) the impedance matching condition is used. This adequately accounts for the finite vacuum gap between the diamond and LiF. The thickness of this gap is determined by integrating the diamond free surface velocity profile until gap closure at 3.5 ns. The estimated gap thickness for this shot is ∼ 2.6 µm. Shock formation in the LiF window is predicted at ∼5.4 ns when the characteristics cross. This occurs at just over 59 µm downstream of the interface. At this time a release fan would be generated that propagates backwards and impedes the interface.75 The characteristics analysis does not account for shock formation or the generation of the release fan. It is estimated that a release fan reaches the interface at ∼5.9 ns as shown in figure 4.1.3. In this case, shock formation is predicted after the conclusion of the experiment (∼5.5 ns),† and the estimated arrival of the release fan is ∼0.4 ns after peak compression. This indicates that shock formation in the LiF window does not influence the experimental results. The characteristic corresponding to the laser termination (black line) is in close † The experiment conclusion is defined as the time at which peak compression is observed at the diamond/LiF interface. For shot 57575 this corresponds to 5.4 ns. 4. Analysis and Results 103 Fig. 4.1.3: (a) Shot 57575 forward characteristics. Prior to ∼ 3.5 ns the free surface boundary condition is imposed to account for the finite gap between the diamond and LiF window. The points corresponding to the predicted shock formation, laser termination and peak compression are shown. The characteristics are color coded by the pressure colorbar shown on the right. (b) The true velocity (black line) is determined from the forward characteristics at the Lagrangian depth of 46 µm. The free surface velocity (blue line) and apparent velocity (red line) are also shown. 4. Analysis and Results 104 agreement with experimental termination. The effect of shock formation on interface measurements and validation of the method of characteristics is discussed in Section 4.4. Figure 4.1.3b shows the measured free-surface velocity (blue curve) and measured interface velocity (red curve). The calculated true interface velocity is shown in black. A noticeable feature is that the apparent interface velocity (red curve) exhibits a deceleration (at ∼5.5 ns) that is not evident in the free surface measurement (blue curve). Deceleration or pull-back results when two decompression waves, traveling in opposite directions, intersect in the bulk material to produce a region of tension. The decompression waves in this study correspond a relaxation wave that originates at the loading surface after laser termination and decompression waves originating at the free surface or diamond/LiF interface. Since the diamond/LiF interface undergoes impedance matching (i.e. higher pressure), an elevated state of stress exists in that section. This supports a decompression wave (tension) and deceleration is observed. The deceleration is not observed at the diamond free surface because the diamond free surface has released to zero pressure (by definition of a free surface). One reason that deceleration is not observed at the diamond free surface is that the compressed diamond fractured and the resulting structure has no tensile strength, a requirement for deceleration. 4. Analysis and Results 105 Refractive index measurements cannot be made at the onset of pull-back because information about the drive pressure is lost. A Monte-Carlo procedure was performed to determine the errors associated with the calculated true particle velocity. This procedure randomly samples variables from the density distributions.76 Random numbers (z) are chosen from a normal distribution with mean 0 and standard deviation of 1, h −z 2 i 1 P (z)dz = √ exp dz. 2 2π (4.1) The Monte-Carlo variable (y) is expressed as the sum of the known value (x) and the uncertainty (σ) times the random number (z) chosen from the normal distribution, y = σz + x. (4.2) 1,000 simulations were performed for each camera (2,000 simulations per experiment). Four Monte-Carlo variables were defined for the simulations. These correspond to the precision of fringe shift measurements (2.5 % of a fringe), the uncertainties in the diamond and the LiF isentropes, and in the gap closure time. The uncertainty in the diamond isentrope provided by Bradley et al. is used.59 A conservative 10 % error in pressure is assumed for the LiF isentrope. Experiments conducted by Ao et al.77 to measured the LiF isentrope show that the discrep- 4. Analysis and Results 106 ancies between the experimentally determined isentrope and Sesame 727150 is ∼ 3 % of the Lagrangian sound speed over a pressure range of 0 to 114 GPa. This corresponds to a 6 % error in the pressure, which justifies the conservative estimate of 10 % in pressure for the LiF isentrope over the pressure range of 30 to 800 GPa. The uncertainties in the diamond and LiF isentropes are chosen to best preserve the shape of the isentrope. The uncertainty in the gap closure time was estimated to be ±0.1 ns. It was later determined that errors associated with the gap closure time are significantly less than all other sources. The results of the Monte-Carlo simulations for shot 57575 (camera 1) are shown in figure 4.1.4. The individual black points represent the Monte-Carlo simulation and the red line is the mean of these points with 1-sigma error bars in velocity and time. The inset shows the estimation of the 1-sigma error bars from the Monte-Carlo calculation. Recall that the apparent particle velocity as a function of true particle velocity determines the refractive index, dUapp dn =n−ρ . dUtrue dρ (4.3) A plot of the apparent versus true particle velocity is shown in figure 4.1.5. Errors associated with the apparent particle velocity correspond to the precision of fringe shift measurements (2.5% of a fringe). Errors in the calculated true particle 4. Analysis and Results 107 Fig. 4.1.4: Monte-Carlo simulation of shot 57575. The black points represent true velocity values determined using the Monte-Carlo routine. The red line is the mean of the Monte-Carlo simulations with 1-sigma error bars in time and velocity shown. The inset is an enlargement of the Monte-Carlo simulation illustrating the standard deviations of the data. 4. Analysis and Results 108 velocity correspond to the 1-sigma uncertainties in velocity and time added in quadrature. This includes the errors associated with the camera resolution and etalon delay. The total error in the true particle velocity is determined from otal δUTTrue = h 2 dU 2 T rue δUT rue + δtT rue dt dU 2 dU 2 i1/2 T rue T rue + τetalon + τsweep , dt dt (4.4) (4.5) where δUT rue and δtT rue are the uncertainties in the timing and velocity determined from the Monte-Carlo simulations, τetalon is the timing uncertainty corresponding to the etalon delay and τsweep is the timing uncertainty corresponding to the streak camera temporal resolution. Solving equation 4.3, the refractive index as a function of density for shot 57575 is determined (figure 4.1.6). This requires the derivative of the apparent to true particle velocity. Small deviations in their ratio produces large deviations in the refractive index. The boundary condition required to solve equation 3.6 is discussed in Section 4.3. The propagation of uncertainties through equation 3.6 are not straightforward. To remedy these issues, multiple experiments are conducted in which a weighted mean and orthogonal fit76 are performed to the determine the relation (Uapp (Utrue )) and uncertainties associated with the orthogonal fit. The 4. Analysis and Results 109 Fig. 4.1.5: The apparent versus true particle velocity for shot 57575 camera 1. uncertainties in the orthogonal fitting parameters are easily propagated though equation 4.3. A total of seventeen target shots were analyzed in the manner described above. These are shown as an ensemble of blue-green like-colored points in figure 4.1.7a. The pressures corresponding to these velocities are determined from the LiF isentrope50 as shown on the top axis. Seven additional that targets employed ∼2 µm of glue to fill the gap between the diamond and the LiF window are shown as the ensemble of red-yellow points. At low pressures, the compressibility of the glue and reverberations within it cause the data to deviate from the general trend of the vacuum-gap data. Once the glue “rang up” to higher pressure, the glue 4. Analysis and Results 110 Fig. 4.1.6: LiF refractive index determined using shot 57575 (black line). Shock measurements made by Wise,34 Lalone,54 and Jensen53, 78 are included as yellow, brown and red squares respectively. Extrapolation of the linear fit proposed by Lalone et al.54 is shown as the red line. The pressure scaling along the top axis corresponds to the LiF isentrope. 4. Analysis and Results 111 Fig. 4.1.7: Apparent versus true particle velocity of all experiments conducted in this study. The like colored points (blue-green) result from the seventeen targets that did not contain a glue layer. The peak compression of those targets is 500 GPa. The remaining seven points (red-yellow) are the results of those shots that contained glue which reach 800 GPa. The pressure scaling along the top axis corresponds to the LiF isentrope. 4. Analysis and Results 112 data follows the trend in the vacuum-gap data. To account for delay in arrival times of the compression waves at the LiF interface, the diamond thickness in the characteristics analysis is increased to account for the relative thickness of the glue. Although this modification is not without error, it is a better approximation than to neglect that thickness. The glue thickness is estimated from the delay in the arrival time of the elastic wave and the estimated Lagrangian sound speed of 5 µm/ns of the glue. Errors associated with the glue thickness are incorporated in the Monte-Carlo routine with an estimated error of ± 1 µm. The large error bars shown in figure 4.1.7 are due to the steep rate of rise of the pressure profile and uncertainties associated with the glue thickness. The weighted mean and orthogonal fit of all twenty-four experiments are discussed in the following section. 4.2 Weighted Mean and Orthogonal Regression A weighted mean of the twenty-four experiments determines the relation between the apparent and true particle velocities.76 The weighted mean (ȳ) is defined as and the variance of the mean as P (yi/σi2 ) P ȳ = , 1/σi2 σµ = P 1 , (1/σi2) (4.6) (4.7) 4. Analysis and Results 113 where yi and σi are the measured value and uncertainties of the points to be averaged and the sums are over the twenty-four (i) experiments. The apparent velocity is considered the dependent variable and the values are equally spaced by ∆Uapp = 0.01 µm/ns. Prior to performing the weighted mean on all twentyfour measurements, the values obtained from camera one and camera two for a single measurement are combined using the same technique. To determine the weighted mean, error measurements in both the apparent and true velocity are combined. The equivalent error in the true velocity due to uncertainties in the apparent velocity is described as σtrue (equiv) = where dUtrue dUapp dUtrue σapp , dUapp (4.8) represents the inverse slope of the apparent to true velocity and is estimated to be 1/1.28 for these measurements. The total error in the true particle velocity is σtrue (total) = σtrue (total) = q q 2 2 σtrue + σtrue (equiv), (4.9) 2 2 /1.28)2 . σtrue + (σapp (4.10) 4. Analysis and Results 114 Once the errors in the weighted mean are determined (equation 4.7), they are converted to errors in the apparent velocity through the same technique, σapp (total) = 1.28σtrue (total). (4.11) Therefore, errors associated with the apparent and true velocity have been combined into a single term. Since the weighted mean is performed on measurements using cameras 1 and 2, the errors bars in figure 4.1.7 are shown only for the apparent velocity measurements. The twenty-four measurements are combined using the weighted mean. A reduced chi squared test is performed to ensure that errors are not underestimated. χ2ν is defined as χ2ν = χ2 , ν (4.12) where ν corresponds to the number of degrees of freedom and χ2 represents the measure of agreement between the observed and expected values. χ2 is defined as N X yi − ȳ 2 ), χ = ( σi i=1 2 (4.13) where N is the number of measurements, yi are the experimental values, ȳ is the 4. Analysis and Results 115 Fig. 4.2.1: The weighted mean performed using all twenty-four experiments is shown in black with uncertainties. The orthogonal fit performed using existing data is shown as the dashed line. Shock measurements made by Wise,34 Lalone,54 and Jensen53, 78 are included as yellow, brown and red squares respectively. The pressure scaling along the top axis corresponds to the LiF isentrope. weighted mean and σi is the uncertainties in yi . For χ2ν values less than one, the errors are underestimated and increased such that χ2ν becomes unity. Figure 4.2.1 shows the weighted mean (black points) of the data from figure 4.1.7 using the associated errors discussed above. As a record, the values of the weighted mean are included in Appendix B. The large errors between 700-800 GPa occur because only a single experiment reached those pressures. Included 4. Analysis and Results 116 in the figure are previous shock measurement data from Wise,34 Lalone,54 and Jensen.53, 78 A second-order orthogonal polynomial regression is performed to determine the relation between the true and apparent particle velocities.76, 79 In the orthogonal fit, the value of each coefficient is independent of higher-order terms, diagonalizing the covariance. The form of the fit is Uapp (Utrue ) = a0 + a1 (Utrue − β) + a2 (Utrue − γ1 )(Utrue − γ2 ). (4.14) The coefficients a0 , a1 and a2 (the centroid, average slope, and average curvature, respectively) are found by minimizing χ2 , the goodness-of-fit parameter. The fit requires additional parameters, β, γ1 and γ2 . Errors are not assigned to these parameters because they only depend on the independent variable. The orthogonal fit is performed using various combinations of the data sets to best determine the relation between the true and apparent particle velocity. The orthogonal fit is determined for the seventeen experiments that did not include the glue layer as well as for all twenty-four experiments. Those results are shown in table 4.2.1. Furthermore, orthogonal fitting was performed on the shock measurement data from Wise,34 Lalone,54 and Jensen.53, 78 Due to the observed commonality between 4. Analysis and Results 117 the shock and ramp compression measurements, orthogonal fitting is performed in which results of all experiments are combined. The errors determined in the orthogonal fit are not a true representation of deviations within the data. This is observed when shots are removed at random and orthogonal regression performed. To remedy this issue, uncertainties in the orthogonal coefficients are determined by randomly removing one to four shots and performing orthogonal fitting. The coefficients determined from 100 such groupings are averaged and the standard deviations provided (table 4.2.1). The orthogonal fit (grey dashed line) corresponding to the shock and ramp compression experiments is shown in figure 4.2.1. A plot of the ratio of the apparent to true velocity is shown in figure 4.2.2 illustrating that the values of the weighted mean deviate about the orthogonal fit. It was found that 66 % of the weighted mean error bars encompass the orthogonal fit using the ramp, glue and shock data. This percentage is in good agreement with one standard deviation (68.2 %), indicating a strong correlation between the data and orthogonal fit. Two concerns, one regarding the large deviations of single shots from linearity observed in figure 4.1.7 and secondly the effect of x-ray preheat on LiF refractive index, are addressed. Steep gradients of the velocity profiles produce large errors in the apparent versus true particle velocity plot for individual shots, since the correlation of events is limited by the etalon delay and temporal resolution of 4. Analysis and Results 118 Fig. 4.2.2: Ratio of the apparent to true velocity is shown. Black points correspond to the weighted mean off all twenty-four experiments. The first order orthogonal fit using all existing data is shown as the grey dashed line. Shock measurements made by Wise,34 Lalone,54 and Jensen53, 78 are included as yellow, brown and red squares respectively. The pressure scaling along the top axis corresponds to the LiF isentrope. 4. Analysis and Results 119 Table 4.2.1: Parameters Resulting from the Orthogonal Fit (Eq. 4.14) a0 [km/s] a1 a2 [s/km] β[km/s] γ1 [km/s] γ2 [km/s] Ramp 5.22 1.28 0.001 4.12 2.84 7.46 ± 0.01 ±0.004 Ramp 8.79 1.28 0.000 6.91 3.10 9.96 & Glue ± 0.01 ±0.009 34 Wise 1.39 1.289 0.000 1.09 0.87 2.89 ± 0.003 ± 0.002 Lalone54 0.48 1.276 0.01 0.38 0.25 0.61 ± 0.005 ±0.02 Jensen53, 78 0.56 1.26 -0.2 0.46 0.33 0.56 ± 0.04 ±0.6 Ramp 1.49 1.273 0.001 1.17 0.61 6.23 & Shock ±0.008 ±0.003 Ramp, Glue 3.06 1.275 0.001 2.41 0.71 9.53 & Shock ± 0.008 ± 0.002 the camera. Recall that this uncertainty contributes to the total uncertainty through the time derivative of the velocity profile and errors are proportional to the gradient of that profile. This accounts for the large error bars observed in many shots and observed deviations of shots from linear behavior. To reduce these deviations pulse shape RM3702 was designed to produce a more gradual velocity profile. Shot 57575 is a demonstration of this technique, the shot described in the previous section. Comparison of this shot with the weighted mean, shows that there is considerably less deviation from linearity (figure 4.2.3) in these points as compared to the experiments shown in figure 4.1.7. The data all fall within the error bars of the weighted mean fit indicating that the observed deviations are a results of the experimental technique and not changes in the refractive index. 4. Analysis and Results 120 Fig. 4.2.3: Comparison of the weighted mean and data acquired using a gradual applied pressure (RM3702). The pressure scaling along the top axis corresponds to the LiF isentrope. 4. Analysis and Results 121 Concerns regarding the effects of x-ray preheat on the refractive index of the LiF window were addressed using specially designed targets to prevent x-ray preheat from affecting the window. These targets consisted of a 10 to 15 µm diamond ablator, a 1 or 3 µm gold layer to prevent x-ray preheat backed with 25 to 40 µm diamond sample. The target design is shown in figure 4.2.4. When the ablation plasma is formed, it acts as a thermal source that typically produces photons of 1 to 2 keV in energy. A 3 µm gold layer transmits less than 0.15 % of those energies. The tail of that thermal distribution is of the order of 5 keV photon energy that corresponds to a transmission of 1.9 % through the gold barrier layer.80 It is reasonable to expect that this thermal shield prevents x-ray penetration of the LiF window. The results of shot 55857 which contained a 3 µm gold shield is shown in figure 4.2.5 (blue data). Excellent agreement between this measurement and the weighted mean suggests that the effects are x-ray preheat on the LiF window are inconsequential. This is expected since the direct ablation method is quite efficient and requires far less laser intensity than hohlraum drive or reservoir coupled experiments. 4.3 LiF Refractive Index The refractive index of LiF is inferred from the orthogonal fitting performed in the previous section. Recall from Section 3.1.2 that the relation of the refractive index (n), density (ρ) and the measured apparent (Uapp ) and true (Utrue ) particle 4. Analysis and Results 122 Fig. 4.2.4: Target design with embedded gold layer preheat shield Fig. 4.2.5: Comparison of the weighted mean with an embedded gold layer target (shot 55857). The pressure scaling along the top axis corresponds to the LiF isentrope. 4. Analysis and Results 123 velocities for ramp compression experiments dUapp dn = f (ρ), = dρ dUtrue Z ρ n f (ρ′ ) ′ 0 n(ρ) = ρ . − ′ 2 dρ ρ0 ρ0 ρ n−ρ (4.15) (4.16) This integral equation requires a boundary condition; the highest pressure refractive index measurements made by Wise and Chhabildas is used.34 Uncertainties are estimated from the deviation of measured values from linearity. The boundary condition is ρ = 4.23 ± 0.06 g/cc, n = 1.461 ± 0.003 and P = 108 ± 8 GPa. Both first and second order fits corresponding to the ramp, glue and shock data are used to determine the refractive index. The weighted mean (black line), first order (blue line) and second (red line) order results are shown in figure 4.3.1. Results from Wise,34 Lalone,54 and Jensen53, 78 are shown as the yellow, orange and red squares respectively. The corresponding pressure is shown in the top axis. The error bars corresponding to the first and second order orthogonal fit encompass the weighted mean and previous shock measurements. This indicates that there is no discernable difference in the refractive index determined using shock or ramp compression. Using the first order orthogonal fit, the solution to equation 4.15 is n = 1.275[±0.008] + 0.044[±0.002]ρ. (4.17) 4. Analysis and Results 124 Recall from Section 3.1.3 that the correction factor to VISAR measurements for materials that possess a linear refractive index is simply the zero density refractive index or a = 1.275[±0.008]. The second order orthogonal fit possesses a slight nonlinear behavior that is attributed to fitting the second order terms to the residuals of the first order. It should be noted that one previous shock-release refractive index study found a strong non-linear behavior that was not observed in this nor other studies.81 That experiment consisted of direct laser ablation of an iron target with a LiF window attached to the rear surface. Interface velocity measurements made through a LiF window were compared to hydrodynamic simulations performed using the applied laser intensity. The authors claimed that the only explanation for the discrepancy in the observed interface velocity when compared to the hydrodynamic simulations was due to changes in the refractive index. They found the refractive index to be non-linear from 100 to 250 GPa. In that study, a thin glue layer, < 1µm, was applied to the Fe/LiF interface. As observed in this study, the impedance of the glue layer causes a significant reduction in the apparent velocity. If unaccounted for, this significantly alters the refractive index calculation. This issue was not addressed in that study and may explain the perceived non-linear behavior. 4. Analysis and Results 125 Fig. 4.3.1: The refractive index of LiF determined using the weighted mean (black line), first order (blue line) and second (red line) order orthogonal fit to the ramp and shock data. Results from Wise,34 Lalone,54 and Jensen53, 78 are shown yellow, orange and red squares, respectively. Notice that there is little difference between the first and second order plots due to the small second order parameter (a2 ) and that the error bars of both fits encompass the weighted mean and shock data. The pressure scaling along the top axis corresponds to the LiF isentrope. 4. Analysis and Results 4.4 126 LASNEX Simulations One-dimensional hydrodynamic LASNEX82 simulations are performed to validate the method of characteristics. Hydrocodes model the complex behavior of continuous media and are not limited to special cases of fluid flow, as is the method of characteristics. LASNEX simulations were performed to understand the effects of shock formation in LiF windows downstream from the diamond/LiF interface. In several early experiments, shock formation occurred prior to termination of the experiment, generating a release fan that may impede the interface velocity. Recall, that the characteristics analysis is valid only for isentropic compression. Radiation hydrodynamic simulations were performed for shot 56113 to determine the effects of LiF shock formation on the true interface velocity. The target consisted of a 45 µm CVD diamond piston with a titanium coated LiF window attached to half of the rear surface. The diamond/LiF interface reached ∼ 570 GPa. Figure 4.4.1 shows that the arrival of the release fan due at the diamond/LiF interface due to shock formation in the LiF window occurs prior to peak compression. The LASNEX simulations use a diamond EOS with a Steinberg-Guinan strength model83 to recover the diamond elastic limit and the LiF sesame table 7271.50 The pressure drive is applied 10 µm inside the target-front surface to account for the 4. Analysis and Results 127 Fig. 4.4.1: Characteristics Analysis of shot 56113 with early shock formation in LiF window. Prior to ∼ 2.6 ns the free surface boundary condition is imposed to account for the finite gap between the diamond and LiF window. Points corresponding to predicted shock formation, laser termination and peak compression are shown. Characteristics are color coded by the pressure colorbar shown on the right. 4. Analysis and Results 128 Fig. 4.4.2: Comparison of hydrocode simulations and method of characteristics for shot s56113. (a) LASNEX ablation pressure is determined iteratively by matching the LASNEX free surface velocity (black) with the VISAR measurement (blue). (b) The ablation pressure is then used to determine the true interface velocity. Comparison of the LASNEX results (black) and method of characteristics (red) is shown. material that is ablated by the laser. This applied pressure is iterated until the simulated free surface velocity best agrees with the measured free surface velocity (figure 4.4.2a). The characteristic analysis for shot 56113 is shown in figure 4.4.1. Shock formation is predicted at 4.8 ns. The arrival of the release fan is predict at 4.9 ns, 0.3 ns before peak compression is achieved. The calculated interface velocity determined using the method of characteristics and LASNEX simulations is shown in figure 4.4.2b. Figure 4.4.2b shows that the hydro-code simulations and characteristics analysis infer nearly identical 4. Analysis and Results 129 true interface velocities. The black line is the LASNEX predicted true interface velocity and the red line is the true interface velocity calculated using the method of characteristics. The excellent agreement between the two techniques confirms the accuracy of the characteristic model and indicates that shock formation in the LiF window does not significantly perturb the interface velocity. The excellent agreement between the two techniques validates the accuracy of the experimental analysis. It should be noted that the step like discontinues that are observed in LASNEX calculated free surface and true interface velocity are attributed to the limitations of the Steinberg-Guinan strength model and knowledge of the diamond elastic limit. 4.5 Temperature Measurements Grey-body temperature measurements were made for both the diamond free surface and diamond/LiF interface. The temperature of the diamond free surface remains below 6000 K for all experiments and are in agreement with published results.59 Measurements of the diamond/LiF section were dominated by emission of the diamond piston. This precluded direct measurement of the LiF self-emission. The absorption (α), transmission (τ ), and reflectivity (ρ) are interrelated by α + τ + ρ = 1. (4.18) 4. Analysis and Results 130 Fig. 4.5.1: Temperature measurements corresponding to shot 57575. Blue and red lines correspond to the CVD diamond free surface and the titanium coated LiF interface, respectively. The black line corresponds to the LiF sesame 727150 predicted temperature. The large increase in temperature at the diamond free surface corresponds to the formation of the ablation plasma when the laser is initiated. The sudden reduction is due to the formation of the opaque elastic limit. This is not observed at the LiF interface due to the 1000 Å coating of titanium. Compression of the LiF window does not occur until 3.5 ns. If the transmission dominates, and the absorption is minimal, then Kirchhoff’s law indicates that the self-emission will be negligible.84 Such a low self-emission cannot be observed in the presence of strong emission from the diamond anvil SOP measurements for shot 57575 are shown in figure 4.5.1. The target consisted of a 46 µm CVD diamond piston with a titanium coated LiF window attached to half of the rear surface. Peak compression (∼ 280 GPa) occurs at 5.5 4. Analysis and Results 131 ns. The large increase in temperature that occurs at ∼ 0.1 ns is due to the formation of the ablation plasma at laser initiation and the partial transparency of the diamond piston. The sudden reduction in that emission is caused by the formation of the opaque elastic limit in diamond. Emission is not observed at the LiF interface due to the opaque 1000 Å titanium coating. The diamond/LiF interface temperature is lower than that of the diamond free surface. This is due to the thermal conductivity of the titanium layer between the diamond and LiF window. The LiF interface observed temperatures higher than predicted by LiF equation of state50 (black line of figure 4.5.1) which is attributed to the self-emission of the diamond sample. Experiments were conducted using nano-crystalline diamond to reduce the selfemission since the higher opacity of the nano-crystalline diamond compared to the CVD diamond shields the SOP from the laser plasma emission. Results of one such shot (57577) is shown in figure 4.5.2. The target consisted of a nano-crystalline diamond piston and uncoated LiF window mounted on half of the rear surface. The higher reflectivity of the nano-crystalline diamond compared to the CVD diamond enabled the use of uncoated LiF windows. Peak compression occurs at 5.4 ns corresponding to a pressure of ∼260 GPa. The diamond free surface (blue) and LiF interface (red) were observed to have nearly identical temperatures. The agreement between these measurements indicates that the self-emission of the LiF 4. Analysis and Results 132 Fig. 4.5.2: Temperature measurements corresponding to shot 57577. Blue and red lines correspond to the nano-crystalline diamond free surface and uncoated LiF interface, respectively. The black line corresponds to the LiF sesame 727150 predicted temperature. Compression of the LiF window does not occur until 2.3 ns. The agreement between these measurements indicates that the self-emission of the LiF is small and of the order of the error bars. is small and of the order of the error bars. Figure 4.5.2 shows that the SOP can resolve temperatures of ∼ 1000 K. This indicates that the LiF window remains below ∼1000 K for these experiments. This is in agreement with Sesame table 7271 for LiF50 that predicts that the temperature at 800 GPa to be 800 K. 4. Analysis and Results 4.6 133 Concluding Remarks Analysis and data reduction to determine the LiF ramp compressed refractive index was discussed in detail. A linear relation between the apparent and true particle velocity was observed up to 800 GPa. This was used to determine the refractive index as function of density for LiF. It was found that the refractive index depends linearly on the density up to 800 GPa. Hydrocode simulations were performed which verified the accuracy of the method of characteristics. Concerns regarding x-ray preheat and the large oscillations the apparent to true particle velocity measurements on single shot basis were addressed. Temperature measurements of the diamond free surface are consistent with published results and the LiF self-emission was compromised by the diamond piston but its temperature appears to remain below 1000 K for these experiments. 5. Discussion 134 5. DISCUSSION Materials whose refractive index depends linearly on density are commonly used as optical windows for high pressure experiments that use velocity interferometry system for any reflector (VISAR).33 The observed transparency and measurement of LiF refractive index to 800 GPa is important for advancing those experiments to higher pressure regimes. The linear behavior of the refractive index and density of LiF and the implications on VISAR experiments is examined in Section 5.1. The theory of classical propagation of light through a medium and the Lorentz oscillator model is introduced in Section 5.2 to explain the dependence of the refractive index on the dielectric function. A single-oscillator model is derived and applied to the LiF refractive index as a function of density (Section 5.3). The model indicates that the linear behavior of the refractive index and density is related to the band gap energy. In Section 5.4, the LiF metallization pressure is predicted by extrapolation of those results. The metallization pressure is compared to the Goldhammer-Herzfeld metallization85, 86 of LiF and other large band 5. Discussion 135 gap insulators. It is postulated that the high metallization pressure of LiF is due to the large band gap and that it is isoelectronic with Helium and Neon. 5.1 VISAR Window Corrections Optical windows that are used in VISAR experiments require corrections due to changes in their refractive index.37 For windows that exhibit a linear refractive index (n) as a function of density (ρ), n = a + bρ, (5.1) the true particle velocity (Utrue ) is related to the apparent particle velocity (Uapp ) by Utrue (t) = Uapp (t) , a (5.2) for both ramp and shock compression (the corrections were discussed in detail in Section 3.1).36, 37 Shocked quartz, sapphire and LiF exhibit a linear refractive index as a function of density.37 This study demonstrated that ramp compressed LiF is transparent and its refractive index depends linearly on density up to 800 GPa and is in agreement with studies along the Hugoniot.34, 53, 54 This extends the achievable pressure range of ramp compression experiments that require optical windows to 800 GPa. The linear behavior of shock and ramp compressed refractive 5. Discussion 136 index suggests that changes in the refractive index are dominated by pressure effects. Therefore, the linear behavior of the refractive index and density will also be observed for multi-shock experiments in which thermal excitation is insufficient to produce appreciable conduction electrons. 5.2 Classical Propagation The response of large band gap insulators (dielectrics) to electromagnetic fields is of particular concern to many areas of study (i.e. optics, physics, engineering). Transparent dielectric materials are used as lenses, prisms, films and windows. For these applications an understanding of the index of refraction (n) is critical in determining the propagation of light. The index of refraction is a measure of the dielectric function (ǫ), which describes the response and behavior of materials to electric fields that vary with space and time. The relation between the refractive index (n) and the dielectric response (ǫ) of a material can be derived using Maxwell’s equations in vacuum by replacing the free space variable (ǫo , µo ) with the materials reduced dielectric equivalents (ǫ, µ).87 Maxwell’s equations88 in free 5. Discussion 137 space become ~ = ρf , ∇·E ǫo ~ ~ = − ∂B , ∇×E ∂t ~ = 0, ∇·B ~ ~ = µo J~ + ǫo µo ∂ E , ∇×B ∂t (5.3) (5.4) (5.5) (5.6) ~ is the electric field, ρf is the free charge density, B ~ is the magnetic field, where E ~ = ǫE, ~ and c is the speed of light in vacuum. Assuming a linear medium (D ~ = B/µ) ~ H that is free of sources (ρf = 0) with zero conductivity (σ = 0), the solutions to Maxwell’s equations are ~ r , t) = Eo ei(~k·~r−ωt) ê, E(~ (5.7) ~ r , t) = Ho ei(~k·~r−ωt) ĥ, H(~ (5.8) ~ is the electric displacement field, H ~ is the magnetizing field, E0 and H0 where D are constants, ê and ĥ are polarization unit vectors, ω is the angular frequency and k is the wavenumber. The angular frequency and wavenumber are related by √ k = ω ǫµ, (5.9) 5. Discussion 138 and the refractive index (n) is defined as n= r ǫµ . ǫo µo (5.10) For materials in which the dielectric response is frequency dependent, the refractive index varies with frequency and this is termed wavelength dispersion. Equation (5.10) is a simple model that ignores the material’s conductivity (i.e. the conduction electrons) and accounts only for the core electrons.88 With the inclusion of conductivity, a more rigorous equation for the complex index of refraction can be derived that includes the response of all electrons within the material, ñ = s ǫµ iσ 1+ . ǫo µo ω (5.11) The propagation of light through a medium can be described in terms of the complex refractive index,88 ñ = n + iκ, (5.12) where the real part of ñ is the normal refractive index (n) and the imaginary part (κ) is the extinction coefficient. The real part of the refractive index refers to the reduction of light velocity within a medium due to phase lag caused by atomic oscillations.88 The extinction coefficient (κ) is related to the absorption 5. Discussion 139 in the medium. These two quantities are directly related to the electronic band structure. The dielectric function of a crystal lattice can be described using a classical model that includes multiple oscillators each with their own resonant frequency. These oscillators influence the propagation of light through that medium. Electronic resonances describe the interband absorption (band gap) required to excite an electron from the valence to conduction band. These oscillations are in the range from 1014 to 1015 Hz. In ionic materials (compounds in which the crystal lattice is held together through ionic bonds) dipole oscillations of charged atoms from their equilibrium positions gives rise to vibrational oscillations. These molecular vibrations are associated with strong absorption in the infrared or the frequency range of 1012 to 1013 Hz. Various other resonance modes exist within a crystal lattice (such as free electron oscillators), but are not important in this work. The Lorentz oscillator89 model suggests that the dielectric function (ǫ) of a crystal is related to the multiple resonances that occur, ǫ(ω) = 1 + ωp2 X j ωj2 fj , − ω 2 − iγj ω (5.13) where ωp is the plasma frequency, ω is the light frequency, ωj is the frequency of a 5. Discussion 140 particular resonance, fj is the strength of that resonance, and γj is the damping of that resonance. The plasma frequency is defined as ωp = s 4πNe2 , me (5.14) where N is the number of atoms per unit volume, e is the charge of an electron, and me is the mass of an electron. In this model, the nucleus is assumed to be immobile due to its large mass compared to that of the electron. The dielectric function is related to the complex refractive index by equation 5.10. For materials in the optical region the relative permeability (µ) is assumed to be unity and the dielectric function is approximated by ǫ ≈ ñ2 . (5.15) Using this model, the refractive index and absorption are estimated for LiF (shown in figure 5.2.1). Two resonances with frequencies equal to 3.2 × 1015 Hz and 9.3 × 1012 Hz are chosen to correspond with the electronic and vibrational modes of LiF,90 respectively. The damping has been set to 5% of the central frequency. The strength of each resonance is chosen such that the features are discernible at that frequency. As the frequency approaches resonance, a discontinuity in the refractive index is observed. The extinction coefficient (κ) is zero everywhere 5. Discussion 141 Fig. 5.2.1: The real part of the refractive index (n) and extinction coefficient (κ) for a hypothetical solid using the electronic and vibration modes of LiF determined using the Lorentz oscillator model.90 The damping has been set to 5% of the central frequency and the strength of transitions chosen to make the resonance features visible. except in regions near the resonance. Passing through a resonance with increasing frequency causes a reduction in the real part of the refractive index. The LiF90 refractive index and extinction coefficient that have been experimentally determined are shown in figure 5.2.2. Electronic and vibrational modes are observed at 3.2 × 1015 Hz and 9.3 × 1012 Hz, respectively. The refractive index measurements of this study were made at a wavelength of 532 nm or 5.64 × 1014 Hz. This is illustrated as the circle shown in figure 5.2.2. For these measurements, the wavelength was bounded by the electronic and vibrational resonance frequencies. Comparison of this data with the simple calculation performed using 5. Discussion 142 Fig. 5.2.2: Experimentally determined LiF index of refraction data.90 The extinction coefficient (κ) is of the order 10−8 in the transparent region. The probe frequency used in this thesis (5.64 ×1014 Hz) is shown as the white circle. the Lorentz oscillator model shows that a model of this form can adequately describe the measured data. This requires knowledge of the resonance modes, the strength of those resonances, and the damping associated with those modes. At wavelengths near that of the probe beam, a linear relation between the refractive index and frequency is observed. The single-oscillator model, proposed by Wemple and DiDomenico,91 suggests that for large band-gap insulators, the optical properties in the transparent region are dominated by the electronic resonance at higher frequency. Application of this model to the LiF refractive index measurements as a function of density indicates that the linear behavior is due to a reduction in the band gap energy.91 5. Discussion 5.3 143 Single-Oscillator Model The single-oscillator model91 has been successfully used to explain the optical properties of various composite and amorphous materials at photon energies below the electronic resonance (interband absorption edge).92–95 This model addresses the frequency dependence of the dielectric function in the transparency region (i.e. the region between the electronic and vibrational modes). It assumes that the effects of vibrational modes on the optical properties are small compared to the electronic modes and the former’s effect can be neglected. Due to the high transparency in this region, the damping is assumed to be zero and the Lorentz oscillator model (equation 5.13) reduces to the Kramers-Heisenberg,96 ǫ(ω) = 1 + ωp2 X j ωj2 fj , − ω2 (5.16) where ǫ is the dielectric response, ω is the frequency, ωp is the plasma frequency, ωj is the frequency of a particular resonance and fj is the strength of that resonance. The Kramers-Heisenberg dispersion relation represents the interband transitions as individual oscillators where each electron contributes one mode to the dielectric function. Wemple and DiDomenico91 showed that the summation of all resonances near the absorption edge can be approximated assuming ω < ωj . The first oscillator 5. Discussion 144 (f1 /(ω12 − ω 2 )) is retained and the higher frequency oscillators are assumed to occur at frequencies significantly greater than the probe frequency (ω ≪ ωj and j > 1). The remaining oscillators in equation 5.16 are Taylor expanded, X fj ω2 1 + , 2 2 ω ω j j j6=1 (5.17) and equation 5.16 takes the form ǫ(ω) = 1 + ωp2 X fj ω2 f1 2 1 + . + ω p 2 2 ω12 − ω 2 ω ω j j j6=1 (5.18) Equation 5.18 reduces to the single-oscillator model by retaining terms to order ω2, ǫ(ω) − 1 ≈ E02 F , − (~ω)2 (5.19) where F and E0 are related to the combination of all oscillator strengths (fn ) and frequencies (ωn ). Wemple and DiDomenico91 showed, using experimental data, that F is related to the single-oscillator energy (E0 ) and a measure of the strength of interband optical transitions (Ed ) by F = E0 Ed , (5.20) 5. Discussion 145 leading to the relation n2 − 1 = Ed E0 . − ~2 ω 2 E02 (5.21) Ed and E0 are determined from dispersion data in the transparent regime for ionic and covalent materials. If y = 1/(n2 − 1) and x = 1/(~ω)2, equation 5.21 can be written in the simple form y= E0 x − . Ed E0 Ed (5.22) By plotting the dispersion in the form of 1/(n2 − 1) vs. (~ω)2, the single-oscillator energy and the strength of optical transitions are determined using linear regression. The dispersion data for LiF in the optical range (black points) is shown in figure 5.3.1 with the linear regression (red dashed line) used to determine the single-oscillator strength and energy.97 The ambient values of Ed and E0 were determined by fitting the refractive index to measured values97 in the range 332 nm < λ < 732 nm (the region near the probe laser). These values corresponded to 16.66 eV and 15.38 eV for E0 and Ed respectively. In a survey of over 100 solid and liquid insulators, the single-oscillator model has been shown to fit the energy-dependent refractive index well. Wemple and DiDomenico91 empirically found that the oscillator energy (E0 ) was approximately 5. Discussion 146 Fig. 5.3.1: LiF refractive index in the optical region used to determine the single-oscillator energy (E0 ) and the strength of interband optical transitions (Ed ). The black points indicate the dispersion data97 and the red dashed line is the linear regression to that data. 5. Discussion 147 related to the lowest direct band gap energy (Et ) by E0 ≈ 1.5Et , (5.23) suggesting that the refractive index is directly related to the optical band gap energy. Equation 5.23 was re-examined specifically for the alkali-halides with the NaCltype crystal lattice.91 The oscillator energy (E0 ), the strength of interband transition (Ed ), and the exciton energy (Ex ) are shown in table 5.3.1. Et is compared to the exciton energy. In ionic materials, the energy required to promote an electron from the valence to conduction band is less than the band gap energy due to the Coulomb attraction between the electron-hole pair. This attraction reduces the required energy to promote an electron to the valence band and is termed the exciton energy. It was found that the oscillator energy (E0 ) is best related to the exciton energy (Ex ) or the direct band gap (Et ) for alkali-halides with NaCl-type lattice structure by E0 ≈ 1.36Ex . (5.24) The single-oscillator model was applied to the pressure induced closing of the H2 band gap over a density range exceeding an order of magnitude.92–94 Ten years later, it was found that the simple model successfully predicted the emergence of 5. Discussion 148 Table 5.3.1: Dispersion parameters for the Alkali-Halides with NaCltype lattice structure. Crystal LiF NaF KF NaCl KCl RbCl CsCl KBr RbBr KI RbI E0 (eV)91 16.7 15 14.8 10.3 10.5 10.4 10.6 9.2 9.1 7.7 7.7 Ed (eV)91 15.4 11.3 12.3 13.6 12.3 12.2 14 12.4 12.1 12.8 12.1 Ex (eV)98 12.9 10.66 9.88 7.96 7.79 7.51 7.85 6.71 6.64 5.88 5.73 E0 /Ex 1.29 1.41 1.5 1.29 1.35 1.38 1.35 1.37 1.37 1.31 1.34 excitonic absorption into the visible.99 Taken together, these studies show that the H2 exciton energy shifts from 14.5 eV to 2 eV with a slightly sublinear dependence on density over nearly 15-fold compression. The single-oscillator model has also been applied to compressed H2 O ice, demonstrating a linear reduction of the band gap over 2.3-fold compression.95 In that experiment, the refractive index was determined over the wavelength range of 560 nm to 740 nm at various pressures. Using this technique, Ed and E0 were determined over the pressure range of that study. It was found that Ed was insensitive to changes in pressure.100 For both H2 and H2 O, the data supports the assumptions that Ed is independent of density, E0 is proportional to the minimum optical band gap energy, and the gap closes nearly linearly with density. 5. Discussion 149 Fig. 5.3.2: Density dependence of the single-oscillator model (Et ). Weighted mean (black) and orthogonal fit (blue) with estimated error bars. The exciton energy98 (green point) and Goldhammer-Herzfeld metallization85, 86 (red point) are shown. Extrapolation suggests that LiF remains transparent well above the Goldhammer-Herzfeld metallization. 5. Discussion 150 In this work, the single-oscillator model is applied to the LiF refractive index data at 532 nm ((~ω)2 = 5.43 eV) shown in figure 5.3.1. Fixing Ed to its ambient value, E0 is calculated as a function of density using equation 5.21. The lowest direct optical transition, Et , (proportional to the band gap) is determined using the proportionality proposed in equation 5.24. The results are shown in figure 5.3.2. The refractive index data determined from the weighted mean is shown in black, and the linear orthogonal fit in blue. At ambient pressure, Et corresponds to the intense exciton (green circle) observed in ultra-violet absorption measurements.101 The model suggests that the linear behavior of the refractive index as a function of density is the result of a monoatomic decrease in the band gap energy. The metallization pressure predicted by the Goldhammer-Herzfeld criterion85, 86 is shown as a red circle in the figure 5.3.2. Metallization is discussed in the following section. 5.4 Metallization Metallization is defined as zero band gap at absolute zero (0 k). This describes the point at which the valence and conduction bands overlap and electrons are free to travel throughout the crystal lattice. At sufficiently high pressures, all insulators become metallic due to band gap closure. This is understood by examining the electronic band structure of a crystal lattice. As atoms are packed closely together to form the crystal lattice, the orbitals of those atoms overlap and 5. Discussion 151 the discrete eigenstates broaden into degenerate energy bands. Electronic bands of different energy arise due to the interaction among the electronic states of the nearest neighbors. An example of the band broadening of Xenon under compression is shown in figure 5.4.1.102 The conduction (5s and 5p) and valence bands (6s, 5d and 6p) are shown. Electrons become delocalized and mobile within the conduction band. As the molar volume decreases (density increases) the 5s and 5p bands of Xenon broaden and the band gap decreases. At high compression, the valence and conduction bands cross (intersection of 5p and 5d bands) and compressed Xenon becomes metallic. The earliest predictions of metallization were made almost simultaneously by Goldhammer85 and Herzfeld.86 Both began with the Lorentz-Lorenz equation,88 n2 − 1 4π = Na ρα 2 n +2 3 (5.25) that relates the refractive index (n) to the density (ρ), the polarizability (α) and Avogadro’s number (Na ). The authors noted that a unity reflectivity (R) requires an infinite refractive index. This is shown from the definition of reflectivity, R= n − 1 2 n+1 . (5.26) 5. Discussion 152 Fig. 5.4.1: Band structure energy calculation for Xenon with atomic labels.102 Electron degeneracies are shown as the shaded region. Valence bands (5s and 5p) broaden under compression. The band gap closes between the valence and conduction band (5d) at high compression (∼ 11 cm3 /mol). 5. Discussion 153 For a infinite refractive index, equation 5.25 reduces to 1= 4π Na ρα, 3 (5.27) defining the condition for metallization. For constant polarizablities, this simple model has been shown to be in good agreement with experimentally determined metallization pressures for a variety of materials.103–105 However, discrepancies have been observed when comparing Goldhammer-Herzfeld metallization pressure with experimental results and band structure calculations.97, 106, 107 These discrepancies are attributed to the simplicity of the Goldhammer-Herzfeld model as well as the assumption of constant polarizability. Table 5.4.1 compares the predicted Goldhammer-Herzfeld metallization85, 86 pressure with experimental results and band-structure calculations for various materials. Experimental results indicate that change in the refractive index are dominated by compression and insensitive to temperature changes. This is demonstrated by similar results for shock and ramp compression where the changes in temperature were significantly greater for shock compression, while refractive index results are identical. Therefore linear extrapolation of the data in figure 5.3.2 to zero band gap energy is a good estimate of the metallization pressure of LiF. The first five compounds in the table correspond to alkali-halides and the 5. Discussion 154 remaining are noble gases. For potassium iodide (KI) the Goldhammer-Herzfeld criterion is a good approximation to the metallization pressure determined using optical absorption. However, for rubidium iodide (RbI) the results indicate that this model can differ by up to 50% from the experimentally determined optical absorption value. Therefore, the discrepancy between the Goldhammer-Herzfeld criterion for LiF and the calculated metallization pressure of this study is not unreasonable. Table 5.4.1: Metallization Pressure for Various Materials 97 LiF NaF97, 104 RbI97, 106, 107 KI97, 106, 107 CsI97, 108, 109 He110 Ne111 Ar112, 113 Kr114, 115 Xe114, 116 † GoldhammerHerzfeld85, 86 Predictions ρ (g/cc) P (GPa) 11.2 2,860 8.8 300 10.4 127 8.85 125 10.7 87 8 5,000 20 6,000 9.6 – 13.3 – 12.9 150 Optical Absorption Measurements ρ (g/cc) – – 9.62 8.7 – – – – – – P (GPa) – – 85 115 111 – – – – 150 Band Structure Predictions ρ (g/cc) >14.2† 9.9 10.4 9.51 9.02 21.3 78.8 < 7.1 12.8 12.9 P (GPa) >5,000† 455 122 155 100 25,700 158,000 <230 316 132 Extrapolation of this study. The anion and cation that comprise the alkali-halides are isoelectronic with the noble gases. Isoelectronicity is defined as two elements or ions which posses the same number of electrons or the same electronic configuration. The simi- 5. Discussion 155 Table 5.4.2: Band Gap Energy for Various Materials Alkali-Halides LiF98 NaF98 KF98 RbF98 CsI117 EG (eV) 14.2 11.5 10.8 10.3 6.1 Noble Gas EG He110 Ne117 Ar118 Kr118 Xe117 (eV) 19.8 21.5 14.3 11.7 9.32 lar metallization pressures of CsI and Xe can be attributed to their comparable electronic configuration and band gap energies (table 5.4.2). Of the all alkalihalides, LiF has the highest predicted metallization pressure (table 5.4.1). Li+ and F− are isoelectronic with helium (He) and neon (Ne), respectively. He and Ne have the largest band gap and the highest metallization pressures predicted of all monatomic materials due to the predicted intershell band overlap.119 Band structure calculations performed by Boettger119 found that metallization of He and Ne is due to (n+1,l+1) conduction band overlapping with the (n,l) valence band. The metallization of the heavy noble gases (Argon, Krypton, Xenon) occurs at significantly lower pressures due to intrashell band overlap of the (n,l) valance band with the (n,l+1) conduction band.119 Augmented plane-wave band structure calculations were performed on LiF.120 The calculations determine the electronic band structure by approximating the electron energy states within the crystal lattice using spherical potentials centered at each atom with constant potentials in the interstitial region.88 Calculations in- 5. Discussion 156 dicate that the valence band consists of the 2p state of F− and the conduction band is comprised of the 2s state of Li+ , 2p state of Li+ and the 3d state of F− .120 An extension of Boettger’s119 results suggests that metallization of LiF occurs when the 2p state of F− crosses the 3d state of F− . Band structure calculations of LiF would provide insight into the band overlapping processes that cause metallization. All alkali-halides, except LiF and sodium fluoride (NaF), contain at least one species that is isoelectronic with a heavy noble gases. The heavy noble gases exhibit significantly lower metallization pressures than the light noble gases, as shown in table 5.4.1. The cation and anion of NaF are isoelectronic with Ne; the monoatomic element with the highest metallization pressure. However, band structure calculations indicate that the metallization pressure is 455 GPa,104 well below that of Ne and LiF. Comparing the band gap energies for those materials (Ne, LiF, and NaF), NaF has the smallest band gap (table 5.4.2), suggesting that the electronic band structure plays a crucial role in the metallization pressure. The low metallization pressure of NaF may also be attributed to the additional energy bands contributed by the Na− cation (2p) when compared to the Li− anion in LiF. Materials being isoelectronic with the light noble gases is not sufficient in producing high metallization pressure. The high metallization pressure of LiF is 5. Discussion 157 attributed to the large band gap energy (largest of all the alkali-halides) and its isoelectronicity with He and Ne. 5.5 Concluding Remarks The linear dependence of the refractive index and density of LiF was examined using the single-oscillator model. This model predicted that the linear dependence of the refractive index on density is the result of monoatomic closure of the band gap. Extrapolation of these results suggests that metallization will occur at 4200 GPa, well above the Goldhammer-Herzfeld criterion.85, 86 Comparison of LiF with its isoelectronic counterparts (He and Ne) suggests that the high metallization pressure is attributed to the intershell band overlap. The high metallization pressure indicates that LiF will remain transparent to pressures at least six times higher than observed in this study. If true, LiF will prove to be a valuable window material for extremely high pressure ramp compression experiments. 6. Conclusion 158 6. CONCLUSION The optical properties of LiF under extreme pressure were examined in this study. Using direct-drive shaped laser ablation, LiF was ramp compressed to 800 GPa at the OMEGA laser facility. A specially designed two section target was used to determine the refractive index of ramp compressed LiF. The target consisted of a diamond pusher with a LiF window mounted on half of the rear surface. Laser pulse profiles were designed to prevent shock formation in both sections of the target. VISAR measurements were made simultaneously in both sections of that target determining the free surface and apparent particle velocity. The method of characteristics and the free surface velocity was used to determine the true particle velocity. The accuracy of that technique was verified through LASNEX simulations.82 The relation between the apparent and true particle velocity determined the refractive index as a function of density. LiF remained transparent up to 800 GPa, pressure seven times higher than previous shock experiments. Under strong shock compression, transparent insulators transform into conducting fluids as a result of pressure-induced reduction 6. Conclusion 159 of the band gap and thermal promotion of electrons across that gap. The reduced temperature of ramp compression enable significantly higher pressure to achieved in LiF while remaining transparent. The refractive index was measured from 30 to 800 GPa; pressure seven times higher than previous shock experiments. As was found with low pressure shock experiments, the refractive index depends linearly with density up to 800 GPa. These are the highest pressure refractive index measurements made to date. Measurements indicate that LiF temperature remained below 1,000 K in these experiments which is consistent with equation of state predictions. A single-oscillator model was used to infer the pressure-induced band gap closure of ramp compressed LiF. Results indicate that the linear behavior of the refractive index on the density is a direct result of pressure-induced closure of the band gap. Extrapolation of these results indicates that LiF will remain transparent to at least 5,000 GPa, well above the Goldhammer-Herzfeld criterion of ∼ 2,860 GPa. The high metallization pressure of LiF is attributed to its large band gap and a structure that is isoelectronic with helium and neon. Helium and neon have the highest metallization pressure of all monatomic materials due to the predicted intershell band overlap. 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The ablation pressure (P ) is found to scale with the laser intensity (I) as P [GPa] = 42[±3](I[TW/cm2 ])0.75[±0.01] . A.1 Introduction High intensity lasers are increasingly used in the study of matter under extreme conditions by creating loading through laser ablation. Accurate knowledge of the scaling of ablation pressure with laser intensity is important for the design A. Direct Drive Laser Ablation Scaling 184 of those experiments. A variety of techniques have been used to determine the ablation parameters of materials such as x-ray spectroscopy,121 time-resolved xray radiography,122 layered-target burn-through measurements,123 time-resolved streak record of visible emission124 and velocity measurements.125 Those studies were often plagued with problems resulting from the presence of laser hot spots and edge effects associated with small laser spots required to reach high intensities. With the development of laser smoothing techniques and more powerful lasers,61, 62, 126 the presence of hot spots and edge effects was diminished allowing a more accurate determination of the scaling laws.127 Dynamic-loading experiments driven by lasers contain two classes: shock compression and ramp compression. Until recently, shock compression experiments were more predominant than ramp compression due to the pulse-shaping requirements for the former.58 Recent advances in the ability to control the temporal shape of laser pulses have enabled shaped-ablation ramp compression with high precision. The experiments described in this thesis demonstrate ramp compression by direct-drive shaped ablation. Here the laser directly irradiates the sample; there is no intermediate energy conversion nor coupling. The ablation pressure is controlled by temporal shaping of the laser intensity in time. Ramp compression allows for the pressure scaling to be observed over a range of pressures and not for a single point that is customary for shock-wave studies. A. Direct Drive Laser Ablation Scaling 185 Recently, direct-drive shaped ablation was developed to ramp compress targets.66 Free surface velocity measurements and knowledge of the target equation of state enable the determination of the pressure profile within the sample.56, 57 Design of such experiments requires accurate knowledge not only of the targets thermodynamic properties, but also the laser-intensity to ablation pressure scaling. Accurate laser-intensity ablation pressure scaling is required to achieve a desired pressure profile over a prescribed distance without generating shocks.58 The laser-intensity ablation scaling is determined for diamond in the next section. The diamond free surface velocity is backwards integrated to determine the ablation pressure. That calculated drive pressure is compared to the intensity on target, determined by the optical streak camera power measurements. An appropriate scaling law, using the functional form proposed by Lindl,128 is determined and compared with previous results. A.2 Experimental Technique The experiments were performed on the OMEGA laser61 at the University of Rochester’s Laboratory for Laser Energetics. Pressure profiles that compressed diamond from 100 to 970 GPa in 3.7 ns were produced using laser pulses that ramped from to 3.0 × 1012 to 7.1 × 1013 W/cm2 with a ∼ t3 shape. The target specifications for the twelve experiments conducted were shown in table A.2.1. A. Direct Drive Laser Ablation Scaling 186 The 351 nm wavelength laser light is smoothed using distributed phase plates62 to produce spots that had a planar region of about 876 µm in diameter.68 Table A.2.1: Shot Specifications Shot Pulse Shape 54939 54940 54944 54945 54946 54948 56112 56113 57575 57576 57577 57579 RM3503 RM3503 RM3503 RM3503 RM3503 RM3503 RM3503 RM3503 RM3702 RM3702 RM3702 RM3702 † Peak Intensity (TW/cm2 ) 44 57 57 71 56 57 53 52 19 28 24 31 Peak Ablation† Thickness Pressure (GPa) (µm) 660 43 850 46 810 46 970 46 710 46 770 43 710 45 720 45 370 46 440 45 420 44 540 45.5 Determined using the method of characteristics.56, 57 The diamond thicknesses for 3.7 ns ramp-compression experiments are included in table A.2.1. The cross section of diamond samples ranged from 1.1 mm to 2.5 mm square. Chemical vapor deposition (CVD) and ultra-pure nanocrystalline diamond were compressed with no discernable difference observed in their ablation pressure scaling. A 1000 Å coating of aluminum was applied to the ablation side of the diamond to prevent the low-intensity leading edge of the laser from penetrating the target before the ablation plasma is formed. The diamond free surface velocities were measured using a 532-nm probe beam A. Direct Drive Laser Ablation Scaling 187 and a line-imaging velocity interferometer for any reflector (VISAR) discussed in detail in Section 3.4.1.33, 35 This device detects the Doppler shifts of the probe light reflected off of the moving portions of the target. That light is imaged through a Mach-Zehnder interferometer onto the slit of an optical streak camera producing a series of fringes streaked in time. The position (phase) of those fringes is proportional to the velocity of the reflecting surface. The fringe position is measured within an accuracy of 2.5% of a fringe using Fourier analysis of the streak record. This is confirmed by the excellent agreement between the velocities derived from the two (redundant) VISAR channels. The free surface measurements are backwards integrated using the method of characteristics56, 57 to determine the ablation pressure as discussed in Section 3.3.1. The diamond isentrope measured by Bradley59 up to 800 GPa and extrapolated to higher pressures was used. The minimum achievable pressure in these experiments is 100 GPa due to the elastic limit of diamond.59 At pressures below that, the intermediate two-wave structure of the elastic limit at lower pressures introduces large uncertainties in the method of characteristics. The pulse shapes are designed such that the elastic limit is reached, followed by ramp compression. To determine the ablation scaling law, the free surface measurements are backwards integrated to the ablation front.129 LASNEX82 simulations estimate the ablation front to be 3 µm (± 1µm) from the diamond loading surface A. Direct Drive Laser Ablation Scaling 188 The laser intensity is determined by the optical streak camera power measurements and the fully characterized smooth laser spots.68 Laser power waveforms are obtained using P510 streak cameras,130 where the P510 indicates the Phillips streak tube used within the camera. The camera directly measures the power of each beam on target, and is calibrated to within 50 ps of the laser arrival time at target. The intensity on target is calculated using focal spot produced by the distributed phase plates and accounting for the beam angle relative to the normal of the diamond sample. Up to twelve beams irradiate the target. The beams are grouped into two clusters with angles of 23.2◦ (±0.1◦ ) and 47.8◦ (±0.1◦ ). The beam intensity on target is modeled using a Super-Gaussian functional dependence (e−r A.3 2 /r 2 0 ) with r0 = 438µm. Results The calculated ablation pressure versus laser intensity is shown in figure A.3.1 with estimated error bars. Linear regression is performed to determine the weighted mean of all twelve shots (as performed in Section 4.2) and is shown in black. According to Lindl,128 the ablation pressure (P ), scales with laser intensity (I) as P = P0 I C0 , (A.1) A. Direct Drive Laser Ablation Scaling 189 Fig. A.3.1: Ablation Pressure vs. Laser Intensity. Results of all twelve shots are shown as the shaded blue lines with error bars. The regression of that data using the function form proposed by Lindl128 is shown in black. where P0 is the ablation scaling constant and C0 is less than 1. These constants are determined by “linearizing” equation A.1, ln P = ln P0 + C0 ln I, (A.2) and performing linear regression of the weighted mean. The functional form for direct drive laser ablation of diamond at the wavelength of 351 nm is 0.75[±0.01] P [GPa] = 42[±3] I[TW/cm2 ] , and is shown as the black line of figure A.3.1. (A.3) A. Direct Drive Laser Ablation Scaling 190 Fig. A.3.2: Comparison of the ablation pressure determined from the method of characteristics (red line) and the ablation pressure determined using the scaling law (equation A.3) from the calculated laser intensity (blue line) for shot 54944. Table A.3.1: Laser Ablation Scaling Ref. Fabbro131 λ (nm) 260 Pulse (ns) 0.5 Intensity (W/cm2 ) 1013 − 1015 Goldsack121 530 1.0 1050 1.0 Dahmani132 1060 0.7 7 × 1012 − 6 ×1013 7 × 1012 − 6 ×1013 1013 − 1015 This Work 351 3.8 † ‡ I is in units TW/cm2 . Z is the atomic number. 5 × 1012 − 7 ×1013 Type Mat. P (GPa)†‡ Planar Shock Spherical Shock Spherical Shock Planar Shock Planar Ramp Al 56.9 × I0.75 Al 18.3 × I0.95 Al 13.1 × I1.14 C & Si 30 × I2/3 ×Z3/16 Dia. 42 × I0.75 A. Direct Drive Laser Ablation Scaling 191 Figure A.3.2 compares the ablation pressure calculated from the method of characteristics (red) and the ablation pressure determine using equation A.3 (blue) for shot 54944. Excellent agreement is observed between these two measurements. Various studies have been performed to determine the laser ablation scaling law for planar and spherical shock experiments. The results of those experiments are summarized in table A.3.1. Aluminum planar shock results of Fabbro et al.131 are in close agreement with the diamond ramp compression scaling results of this thesis. The difference in the wavelength of these studies is accounted for using the functional form proposed by Lindl.128 Lindl finds that for direct drive the ablation pressure scales as P ∼ I 2/3 λ , (A.4) where P is the ablation pressure, I is the laser intensity and λ is laser wavelength. Using equation A.4 and replacing the exponential with the experimental result, the functional form proposed by Fabbro for aluminum, adjusted for the wavelength dependence, becomes P [GPa] = 20.7 I[TW/cm2 ] 0.75 λ[µm] , (A.5) A. Direct Drive Laser Ablation Scaling 192 and ablation pressure scaling for ramp compressed diamond is I[TW/cm2 ] 0.75[±0.01] . P [GPa] = 19[±1] λ[µm] (A.6) Excellent agreement is observed for the ablation scaling of aluminum and diamond. A.4 Conclusion An ablation pressure scaling law has been determined for direct-drive ramp compression using diamond targets over the range of 100 to 970 GPa. Ablation pressure is determined by backwards integrating free surface diamond velocity measurements using VISAR to the ablation front determined from LASNEX simulations. The laser intensity is calculated from the power-on-target measurements and the characterized laser spot. The ablation pressure (P ) and laser intensity (I) scales as P [GPa] = 42[±3](I[TW/cm2 ])0.75[±0.01] for the laser wavelength of 351 nm. The scaling law is in agreement with the functional form proposed by Lindl and corresponds well with previous direct drive planar experiments of aluminum at a similar laser wavelength and intensity. B. Weighted Mean Values 193 B. WEIGHTED MEAN VALUES As a matter of record, the values determined from the weighted mean of the apparent and true particle velocity are tabulated in this Appendix (table B.1.1). The first fifty values correspond to shock refractive index measurements made by Wise,34 Lalone54 and Jensen.53 The remaining values correspond to the weighted mean determined in Chapter 4. The number of data points in the weighted mean has been reduced by quoting every tenth value. Table B.1.1: Apparent and True Weighted Mean Values True Velocity (µm/ns) 0.48 0.31 0.58 0.38 0.19 0.10 0.25 0.74 0.98 0.34 0.51 0.45 Apparent Velocity (µm/ns) 0.59 ± 0.008 0.38 ± 0.008 0.73 ± 0.008 0.48 ± 0.003 0.25 ± 0.003 0.13 ± 0.003 0.32 ± 0.003 0.95 ± 0.018 1.25 ± 0.008 0.43 ± 0.002 0.64 ± 0.004 0.58 ± 0.002 True Velocity (µm/ns) 5.57 5.59 5.62 5.65 5.68 5.71 5.74 5.78 5.82 5.86 5.90 5.96 Apparent Velocity (µm/ns) 7.09 ± 0.075 7.14 ± 0.081 7.19 ± 0.082 7.24 ± 0.086 7.29 ± 0.094 7.34 ± 0.095 7.39 ± 0.102 7.44 ± 0.108 7.49 ± 0.107 7.54 ± 0.111 7.59 ± 0.127 7.64 ± 0.134 B. Weighted Mean Values 194 Table B.1.1: Apparent and True Weighted Mean Values (cont.) True Velocity 0.67 0.69 1.01 0.11 0.30 0.65 0.66 0.66 0.70 0.70 0.70 0.81 0.84 0.84 0.89 0.98 0.98 0.99 1.00 1.01 1.01 1.12 1.13 1.13 1.14 1.36 1.36 1.37 1.37 1.38 Apparent Velocity 0.85 ± 0.003 0.88 ± 0.007 1.28 ± 0.014 0.14 ± 0.010 0.38 ± 0.010 0.82 ± 0.010 0.85 ± 0.010 0.84 ± 0.010 0.90 ± 0.010 0.89 ± 0.010 0.89 ± 0.010 1.03 ± 0.010 1.07 ± 0.010 1.06 ± 0.010 1.14 ± 0.010 1.25 ± 0.010 1.25 ± 0.010 1.26 ± 0.010 1.30 ± 0.016 1.29 ± 0.010 1.31 ± 0.013 1.43 ± 0.012 1.45 ± 0.010 1.44 ± 0.010 1.47 ± 0.010 1.74 ± 0.010 1.74 ± 0.010 1.76 ± 0.010 1.76 ± 0.010 1.77 ± 0.010 True Velocity 6.01 6.07 6.05 6.10 6.15 6.22 6.28 6.33 6.37 6.42 6.47 6.51 6.55 6.60 6.65 6.69 6.71 6.76 6.82 6.87 6.91 6.94 6.95 7.03 7.06 7.08 7.09 7.11 7.11 7.12 Apparent Velocity 7.69 ± 0.134 7.74 ± 0.152 7.79 ± 0.138 7.84 ± 0.13 7.89 ± 0.137 7.94 ± 0.131 7.99 ± 0.125 8.04 ± 0.135 8.09 ± 0.136 8.14 ± 0.129 8.19 ± 0.135 8.24 ± 0.124 8.29 ± 0.116 8.34 ± 0.122 8.39 ± 0.111 8.44 ± 0.101 8.49 ± 0.11 8.54 ± 0.102 8.59 ± 0.094 8.64 ± 0.102 8.69 ± 0.096 8.74 ± 0.097 8.79 ± 0.110 8.84 ± 0.119 8.89 ± 0.128 8.94 ± 0.150 8.99 ± 0.135 9.04 ± 0.132 9.09 ± 0.146 9.14 ± 0.131 B. Weighted Mean Values 195 Table B.1.1: Apparent and True Weighted Mean Values (cont.) True Velocity 1.39 1.39 1.41 2.64 3.78 3.78 4.09 4.09 1.52 1.58 1.60 1.62 1.64 1.65 1.67 1.70 1.75 1.84 1.87 1.90 1.96 1.99 2.02 2.05 2.09 2.10 2.14 2.18 2.23 2.24 Apparent Velocity 1.8 ± 0.024 1.79 ± 0.010 1.82 ± 0.010 3.4 ± 0.010 4.82 ± 0.026 4.81 ± 0.045 5.27 ± 0.017 5.28 ± 0.033 1.84 ± 0.103 1.89 ± 0.105 1.94 ± 0.111 1.99 ± 0.114 2.04 ± 0.119 2.09 ± 0.127 2.14 ± 0.137 2.19 ± 0.145 2.24 ± 0.044 2.29 ± 0.043 2.34 ± 0.042 2.39 ± 0.044 2.44 ± 0.042 2.49 ± 0.049 2.54 ± 0.050 2.59 ± 0.055 2.64 ± 0.055 2.69 ± 0.056 2.74 ± 0.055 2.79 ± 0.052 2.84 ± 0.053 2.89 ± 0.069 True Velocity 7.16 7.22 7.23 7.26 7.27 7.30 7.34 7.36 7.40 7.45 7.48 7.51 7.56 7.58 7.61 7.71 7.73 7.76 7.79 7.81 7.86 7.91 7.92 7.97 7.99 8.11 8.15 8.20 8.23 8.26 Apparent Velocity 9.19 ± 0.130 9.24 ± 0.173 9.29 ± 0.147 9.34 ± 0.139 9.39 ± 0.157 9.44 ± 0.138 9.49 ± 0.137 9.54 ± 0.159 9.59 ± 0.140 9.64 ± 0.136 9.69 ± 0.155 9.74 ± 0.135 9.79 ± 0.129 9.84 ± 0.147 9.89 ± 0.128 9.94 ± 0.155 9.99 ± 0.183 10.04 ± 0.158 10.09 ± 0.148 10.14 ± 0.175 10.19 ± 0.149 10.24 ± 0.137 10.29 ± 0.169 10.34 ± 0.141 10.39 ± 0.133 10.44 ± 0.188 10.49 ± 0.160 10.54 ± 0.146 10.59 ± 0.207 10.64 ± 0.170 B. Weighted Mean Values 196 Table B.1.1: Apparent and True Weighted Mean Values (cont.) True Velocity 2.28 2.31 2.36 2.42 2.47 2.51 2.56 2.64 2.68 2.71 2.75 2.78 2.81 2.84 2.88 2.91 2.95 2.98 3.04 3.07 3.11 3.16 3.25 3.28 3.31 3.34 3.38 3.42 3.45 3.48 Apparent Velocity 2.94 ± 0.062 2.99 ± 0.057 3.04 ± 0.058 3.09 ± 0.044 3.14 ± 0.046 3.19 ± 0.049 3.24 ± 0.049 3.29 ± 0.047 3.34 ± 0.049 3.39 ± 0.051 3.44 ± 0.053 3.49 ± 0.050 3.54 ± 0.050 3.59 ± 0.050 3.64 ± 0.045 3.69 ± 0.048 3.74 ± 0.054 3.79 ± 0.049 3.84 ± 0.057 3.89 ± 0.055 3.94 ± 0.051 3.99 ± 0.054 4.04 ± 0.058 4.09 ± 0.053 4.14 ± 0.056 4.19 ± 0.052 4.24 ± 0.047 4.29 ± 0.048 4.34 ± 0.049 4.39 ± 0.047 True Velocity 8.30 8.44 8.52 8.54 8.55 8.61 8.66 8.80 8.81 8.82 8.83 8.84 8.88 8.89 8.90 8.87 8.88 8.93 8.94 8.94 8.99 9.21 9.18 9.25 9.28 9.26 9.31 9.35 9.33 9.38 Apparent Velocity 10.69 ± 0.154 10.74 ± 0.210 10.79 ± 0.142 10.84 ± 0.123 10.89 ± 0.19 10.94 ± 0.163 10.99 ± 0.118 11.04 ± 0.071 11.09 ± 0.059 11.14 ± 0.055 11.19 ± 0.058 11.24 ± 0.056 11.29 ± 0.057 11.34 ± 0.059 11.39 ± 0.058 11.44 ± 0.111 11.49 ± 0.137 11.54 ± 0.108 11.59 ± 0.128 11.64 ± 0.148 11.69 ± 0.111 11.74 ± 0.209 11.79 ± 0.872 11.84 ± 0.128 11.89 ± 0.215 11.94 ± 0.075 11.99 ± 0.133 12.04 ± 0.216 12.09 ± 0.089 12.14 ± 0.138 B. Weighted Mean Values 197 Table B.1.1: Apparent and True Weighted Mean Values (cont.) True Velocity 3.51 3.54 3.57 3.60 3.62 3.65 3.68 3.70 3.73 3.77 3.81 3.86 3.92 3.97 4.10 4.15 4.18 4.22 4.27 4.30 4.35 4.39 4.43 4.45 4.49 4.53 4.57 4.62 4.64 4.69 Apparent Velocity 4.44 ± 0.052 4.49 ± 0.057 4.54 ± 0.057 4.59 ± 0.059 4.64 ± 0.062 4.69 ± 0.062 4.74 ± 0.065 4.79 ± 0.064 4.84 ± 0.064 4.89 ± 0.064 4.94 ± 0.059 4.99 ± 0.061 5.04 ± 0.061 5.09 ± 0.066 5.14 ± 0.087 5.19 ± 0.092 5.24 ± 0.095 5.29 ± 0.096 5.34 ± 0.099 5.39 ± 0.097 5.44 ± 0.103 5.49 ± 0.104 5.54 ± 0.106 5.59 ± 0.104 5.64 ± 0.105 5.69 ± 0.106 5.74 ± 0.106 5.79 ± 0.107 5.84 ± 0.105 5.89 ± 0.110 True Velocity 9.42 9.41 9.46 9.48 9.49 9.53 9.56 9.59 9.62 9.62 9.69 9.70 9.70 9.83 9.82 9.87 10.00 10.05 10.40 10.55 10.33 10.39 10.45 10.52 10.58 10.64 10.70 10.75 10.81 10.86 Apparent Velocity 12.19 ± 0.211 12.24 ± 0.105 12.29 ± 0.137 12.34 ± 0.205 12.39 ± 0.126 12.44 ± 0.136 12.49 ± 0.214 12.54 ± 0.146 12.59 ± 0.131 12.64 ± 0.214 12.69 ± 0.160 12.74 ± 0.115 12.79 ± 0.22 12.84 ± 0.173 12.89 ± 0.101 12.94 ± 0.199 12.99 ± 0.142 13.04 ± 0.062 13.09 ± 0.124 13.14 ± 0.394 13.21 ± 0.486 13.29 ± 0.467 13.36 ± 0.449 13.44 ± 0.427 13.51 ± 0.406 13.59 ± 0.379 13.66 ± 0.352 13.74 ± 0.325 13.81 ± 0.296 13.88 ± 0.265 B. Weighted Mean Values 198 Table B.1.1: Apparent and True Weighted Mean Values (cont.) True Velocity 4.82 4.86 4.90 4.91 4.90 4.94 4.97 4.98 4.99 5.01 5.03 5.07 5.10 5.15 5.19 5.27 5.32 5.37 5.40 5.43 5.47 5.50 5.53 Apparent Velocity 5.94 ± 0.101 5.99 ± 0.093 6.04 ± 0.085 6.09 ± 0.083 6.14 ± 0.080 6.19 ± 0.079 6.24 ± 0.081 6.29 ± 0.076 6.34 ± 0.073 6.39 ± 0.073 6.44 ± 0.068 6.49 ± 0.069 6.54 ± 0.072 6.59 ± 0.069 6.64 ± 0.069 6.69 ± 0.067 6.74 ± 0.062 6.79 ± 0.060 6.84 ± 0.061 6.89 ± 0.061 6.94 ± 0.064 6.99 ± 0.072 7.04 ± 0.073 True Velocity 10.91 10.95 10.99 11.03 11.07 11.11 11.16 11.08 11.12 11.18 11.24 11.38 11.50 11.65 11.85 11.30 11.50 11.64 11.84 12.04 12.21 12.37 12.61 Apparent Velocity 13.96 ± 0.250 14.04 ± 0.242 14.11 ± 0.239 14.19 ± 0.240 14.26 ± 0.240 14.34 ± 0.232 14.41 ± 0.216 14.49 ± 0.244 14.56 ± 0.215 14.63 ± 0.197 14.71 ± 0.176 14.84 ± 0.355 15.00 ± 0.295 15.15 ± 0.219 15.30 ± 0.197 15.45 ± 3.272 15.60 ± 2.780 15.75 ± 2.376 15.90 ± 2.212 16.05 ± 1.653 16.20 ± 1.188 16.35 ± 0.879 16.50 ± 0.473 C. LiF Shock Release 199 C. LIF SHOCK RELEASE This appendix describes a new technique for determining the shocked index of refraction of an insulator. Experiments were conducted by Hye-Sook Park of Lawrence Livermore National Laboratory to determine the effects of shockreleased LiF windows on optical interferometry measurements. A planar shock was driven through a tantalum target with two LiF windows of different thickness attached to rear surface. Simultaneous measurements were made through a thin and thick LiF window. Shock breakout was observed in the thin window and compared to measurements made through the thick window, where the shock is still within the solid LiF. The effects of that shock breakout on optical interferometry measurements were examined by comparing the measured velocities in both sections. At shock breakout, a discontinuity in the VISAR record is observed. The derivation shown below indicates that this discontinuity is directly related to the shocked refractive index of the window. A new measurement technique to determine the refractive index in that released material is proposed. C. LiF Shock Release C.1 200 Experimental Design Indirect-drive reservoir-coupled compression, shown in figure C.1.1, was used to compress two LiF samples of differing thickness. In this design, a gold hohlraum is driven using 40 OMEGA61 beams. The pulse shape is a 1 ns square pulse and the total energy on target is 20,000 J. The laser drive generates x-rays that drive the ablation of a 25 µm beryllium anvil. A shock is generated in the beryllium that is impedance matched into a 12.5% BrCH 200 µm thick reservoir. The shocked reservoir material eventually rarefies as it propagates across a 400 µm gap and stagnates at a 10 µm tantalum pusher. This stagnation produces ramp compression in the tantalum that in turn compresses the LiF windows. Two LiF windows, of differing thicknesses, are attached to the rear surface of the tantalum. One window is sufficiently thick such that compression waves do not reach the rear surface prior to the conclusion of the experiment. The second window is 10 times thinner than the thick window. The compression waves in that window reach the rear surface and release to zero pressure prior to the conclusion of the experiment. VISAR measurements are made simultaneously through both windows to determine the effects of the released window on interferometry measurements. The tantalum/LiF interface motion is measured using a 532-nm probe beam and a C. LiF Shock Release 201 Fig. C.1.1: Shock Release Target Design. line-imaging velocity interferometer for any reflector (VISAR) discussed in detail in Section 3.4.1.33, 35 This device detects the Doppler shifts of the probe light reflected off of the moving portions of the target. That light is imaged through a Mach-Zehnder interferometer onto the slit of an optical streak camera producing a series of fringes streaked in time. The position (phase) of those fringes is proportional to the velocity of the reflecting surface. The fringe position is measured within an accuracy of 2.5% of a fringe using Fourier analysis of the streak record. A single experiment∗ (shot 58815) was conducted. In that experiment, the compression profile of the tantalum was not properly controlled and a shock was generated in the tantalum and LiF windows. The VISAR data for shot 58815 is shown in figure C.1.2. At shock breakout of the thin LiF window, a discontinu∗ This experiment was designed and performed by Hye-Sook Park of Lawrence Livermore National Laboratory. C. LiF Shock Release Fig. C.1.2: VISAR data for shot s58815. 202 C. LiF Shock Release 203 ity in fringe position is observed. The fringe discontinuity can be understood by deriving the correction to the measured apparent particle velocity when observed through an optical window at times prior to and coincident with shock breakout. The difference between these observed apparent particle velocities defines the discontinuity at breakout. C.2 Unsteady Shock Breakout of an Optical Window The discontinuity in fringe position is explained by examining the optical path length of the LiF window prior to and coincident with shock breakout. The change in optical path length determines the observed discontinuity. In this derivation, no assumption is made regarding the shock steadiness. C.2.1 Prior to Shock Breakout Prior to the shock breakout (t < tB.O. ), the shock position in the window is shown in figure C.2.1. Recall from Section 3.1 that the apparent particle velocity (Uapp (t)) is determined by the time derivative of the integral of the optical path length (equation 3.2) or dh UApp (t) = dt Z xint (t) xf s (t) i n(x, t)dx + Uf s (t), (C.1) C. LiF Shock Release 204 Fig. C.2.1: Shock propagating through an optical window. where xint (t) is the interface position, xf s (t) is the free surface position, nS (x, t) is the compressed refractive index of the optical window, and Uf s (t) is window the free surface velocity. Prior to shock breakout, the window free surface is stationary and equation C.1 reduces to dh UApp (t < tB.O. ) = dt Z xint (t) xf s i n(x, t)dx . (C.2) The optical path length of the shocked (nS (x, t)) and unshocked (n0 ) regions is expressed as dh UApp (t < tB.O. ) = dt Z xint (t) xD (t) nS (x, t)dx + Z xD (t) xf s i n0 dx , (C.3) where xD (t) defines the shock front position and n0 is the initial refractive index of the window. Leibniz integral rule relates the differentiation of an integral whose C. LiF Shock Release 205 Fig. C.2.2: Shock Breakout of an Optical Window. limits are functions of the differential variable,133 d dt Z V (t) U (t) ′ ′ f (x, t)dx = V (t)f (V (t), t) − U (t)f (U(t), t) + Z V (t) U (t) ∂f (x, t) dx. (C.4) ∂t Applying equation C.4 to equation C.3 gives the relation for the observed apparent particle velocity or Uapp (t < tB.O. ) = Uint (t)nS (xint , t) − D(t)nS (xD , t) Z xint (t) ∂nS (x, t) + dx + n0 D(t), ∂t xD (t) (C.5) (C.6) where D(t) is the shock velocity and Uint (t) is the true interface velocity. The integral term in equation C.6 is directly related to the shock steadiness. C.2.2 Shock Breakout When the shock reaches the free surface (t = tB.O. ), as shown in figure C.2.2, a change in VISAR fringe position is observed. Rederiving the apparent particle C. LiF Shock Release 206 velocity, the free surface (xf s ) is no longer stationary (Uf s 6= 0). Recall equation C.1, dh UApp (t = tB.O. ) = dt Z xint (t) xf s (t) i nS (x, t)dx + Uf s (t). At shock breakout, the window is completely compressed by the shock and the integral cannot be separated into parts. Applying Leibniz’s rule (equation C.4), Uapp (t = tB.O. ) = Uint (t)nS (xint , t) − Uf s (t)nS (xf s , t) Z xint (t) ∂nS [x, t] + dx + Uf s (t), ∂t xf s (t) (C.7) (C.8) where the integral term is related to the shock steadiness. At shock breakout, there is a sudden change in the observed apparent particle velocity. Define this change as ∆ or ∆ = Uapp (tB.O. ) − Uapp (tB.O. − ǫ), (C.9) where 0 < ǫ ≪ tB.O. . Substituting the equations C.6 and C.8 and taking the limit as ǫ → 0 gives ∆ = Uf s (tB.O. )(1 − n(xf s , tB.O. )) + D(tB.O. )(n(xf s , tB.O. ) − n0 ) (C.10) Z ∂n(x, t) + δ(xf s )dx. (C.11) tB.O. ∂t C. LiF Shock Release 207 Notice that the dependence upon the shock steadiness and interface conditions have been removed. The integral in equation C.11 can be simplified by examining the density and refractive index profiles of the window (figure C.2.3) at various times. Figure C.2.3 shows that the free surface refractive index is n0 at all times except at shock breakout. The rear surface refractive index is described by the function Z t nS (x = xf s , t) = n0 + (nS (xf s , tB.O. ) − n0 )δ xf s − D(t′ )dt′ . (C.12) 0 Substituting this into the integral of equation C.11 gives Z xf s ∂n(x, t) dx = (n(xf s , tB.O. ) − n0 )D(tB.O. ), ∂t tB.O. (C.13) and ∆ is redefined as ∆ = Uf s (tB.O. )(1 − nS ) + 2D(tB.O. )(nS − n0 ), (C.14) where shocked refractive index of the window at breakout nS (xf s , tB.O. ) has been replaced with the simpler notation nS . Recall that no assumptions were made regarding the shock steadiness within the window. The discontinuity at shock C. LiF Shock Release Fig. C.2.3: 208 Density and refractive index profiles at various stages of shock propagation. The profiles indicate that the refractive index at the free surface is n0 at all times except at breakout. After shock breakout, as shown in figure d, the window releases to standard conditions assuming the shock pressure is below the melt and the window is in the solid state. C. LiF Shock Release 209 breakout depends on parameters at the free surface. By measuring ∆, D and Uf s at breakout, the refractive index of the compressed window is determined. If the EOS of the window material is known, then the number of measurable parameters reduces to two. C.3 Steady Shock Breakout of an Optical Window Equation C.14 is rederived assuming a steady shock with known shocked refractive index (nS ) to check the accuracy of equation C.14. Since shock steadiness does not influence the discontinuity at shock breakout, a derivation assuming a steady shock should arrive at the same result. Recall equation C.1 or dh UApp (t) = dt Z x(t) xf s (t) i n(x, t)dx + Uf s (t). Prior to shock breakout, the free surface is stationary (Uf s = 0) and the refractive index following the shock (nS ) is constant. For steady shocks equation C.3 C. LiF Shock Release 210 transforms to Z Z xD i d h xint nS dx + UApp (t < tB.O. ) = n0 dx , dt xD xf s i dh nS (xint − xD ) + n0 (xD − xf s ) , UApp (t < tB.O. ) = dt UApp (t < tB.O. ) = nS (Uint − D) + Dn0 . (C.15) At shock-breakout (t = tB.O. ), the window refractive index is uniform (nS ) and Uf s 6= 0, dh UApp (t = tB.O. ) = dt Z xint nS dx + xf s Z xf s xV i dx . (C.16) For steady shocks, equation C.16 transforms to UApp (t = tB.O. ) = nS (Uint − Uf s ) + Uf s + dnS (x − xf s ), dt (C.17) where the temporal derivative (dnS /dt) of the refractive index is required. From the steady shock assumption, the derivative of the refractive index can be expressed as dn nS − n0 = , dt L/D (C.18) UApp (t = tB.O. ) = nS (Uint − Uf s ) + Uf s + D(ns − n0 ). (C.19) leading to the general equation C. LiF Shock Release 211 Equation C.11 defined ∆ as ∆ = UApp (tB.O. ) − UApp (tB.O. − ǫ). The change in the apparent particle velocity at shock breakout for steady-shock is ∆ = Uf s (1 − nS ) + 2D(nS − n0 ), which is identical to the unsteady shock case (equation C.14). C.4 Analysis of Shot 58815 The velocity profile for shot 58815 is shown in figure C.4.1. Measurements made through the thick window (blue) are compared with measurements made through the rarefied window (red). At ∼42 ns, a discontinuity is observed corresponding to shock breakout in the thin LiF window. At ∼47 ns the rarefaction wave reaches the tantalum pusher and the interface accelerates. The observed apparent particle velocity prior to shock-breakout and the magnitude of the change in fringe position are measured. Due to the limitations of this experiment, the free surface velocity (Uf s ) and shock velocity (D) are not measured. To determine the shocked refractive index, the steady shock assumption is made such that the measured apparent particle velocity can be related to the free surface velocity. C. LiF Shock Release 212 Proper design of future experiments can circumvent this issue. Recall the steady shock assumption or Uf s (t) = 2Uint (t). (C.20) The true interface velocity is related to the measured apparent particle velocity through equation C.15, UApp (t < tB.O. ) = nS (Uint − D) + Dn0 , (C.21) and the change in fringe position (∆) is defined as ∆ = 2Uint (1 − nS ) + 2D(nS − n0 ). (C.22) A two equation set with three unknowns is determined by measuring the apparent particle velocity and change in fringe position at breakout. The system of equations is closed with the corresponding LiF equation of state.50 Values obtained in this experiment can be found in table C.4.1. A plot of the refractive index versus density determined in this experiment (yellow point) is shown in figure C.4.2 with previous measurements made by Wise34 (blue square) and Lalone54 (red diamond). Figure C.4.3 is a plot of the difference between the measurements made in the C. LiF Shock Release 213 Table C.4.1: Analysis of Shot 58815 Uapp (µm/ns) ∆ (µm/ns) D (µm/ns) Uint (µm/ns) ρS (g/cc) nS Asbo 1 4.382 ± 0.031 -1.911 ± 0.118 9.755 ± 0.089 3.427 ± 0.066 4.067 ± 0.023 1.456 ± 0.016 Asbo 2 4.331 ± 0.023 -1.926 ± 0.044 9.675 ±0.042 3.368 ±0.032 4.047 ± 0.011 1.451 ± 0.008 Weighted Mean 4.349 ± 0.0183 -1.925 ± 0.044 9.700 ± 0.038 3.386 ± 0.029 4.053 ± 0.010 1.452 ± 0.007 Fig. C.4.1: Shot 58815 velocity profiles. C. LiF Shock Release 214 Fig. C.4.2: Shocked refractive index measurements from various studies. Value determined in this experiment is shown in yellow. Experiments conducted by Wise34 and Lalone54 are shown as blue square and red diamonds, respectively. Red dashed line corresponds to the fit proposed by Lalone. C. LiF Shock Release Fig. C.4.3: 215 Plot of the difference between the apparent interface velocities observed through the thick window and the released apparent interface velocities. After shock breakout of the thin window, we see that prior to the reverberation the difference is constant. compressed and released material. Between shock breakout and the arrival of the rarefaction wave at the tantalum/LiF interface, a constant ratio is observed. Thin LiF measurements are made through a rarified material whose density varies from 2.63 to 4.28 g/cc. These measurements are consistent with those made through the thick LiF window (taking the ∆ offset into account). The only functional form that satisfies these conditions is for the refractive index to depend linearly with density over this range. C. LiF Shock Release C.5 216 Conclusion The shock breakout of a transparent optical window generates a discontinuity in the velocity observed through that window. Comparison of measurements made through released and compressed LiF windows indicates that the discontinuity is directly related to the shocked refractive index at breakout. The derivation of this phenomenon shows that the discontinuity at breakout is independent of the shock steadiness and from that derivation, a new shock refractive index measurement technique is presented.