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Laser-driven shock waves in quartz Waterman, Alfred James 1990

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LASER-DRIVEN SHOCK WAVES IN QUARTZ By ALFRED JAMES WATERMAN B.A.Sc. (Hons.), University of British Columbia, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1990 © ALFRED JAMES WATERMAN, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of p ^WSKS  The University of British Columbia Vancouver, Canada Date S£PT XV f\<\0 DE-6 (2/88) Abstract The formation and propagation of laser-driven shock waves has been observed by op-tical shadowgraphy in fused quartz, a-quartz and sodium chloride. Target materials were irradiated with a 0.53 /xm , ~ 2.5 ns FWHM laser pulse at intensities ranging be-tween 0.2 — 2 x 1013 W/cm2, producing peak pressures varying from 0.3 — 3 Mbar at the shock front. Observations in both varieties of quartz reveal transient, high-speed shock propagation followed by deceleration towards a steady asymptotic shock speed. Similar high-speed transients were not seen in sodium chloride. The results in quartz were found to be in significant disagreement with both one-dimensional and two-dimensional hy-drodynamic calculations based on equilibrium equations of state. The non-steady shock propagation is interpreted as being due to a relaxation process in the phase transforma-tion of quartz into the high-pressure stishovite phase which occurs at the shock front. The effects of such a relaxation process on the shock dynamics and shock compression process are considered for the case of a direct relaxation from quartz into stishovite, as well as for an indirect relaxation process in which the -transformation of quartz into stishovite is preceded by shock-induced amorphization of the quartz. It is shown that either scenario would result in higher shock speeds and less compressible shock states than those obtained under equilibrium conditions. u T a b l e of C o n t e n t s A b s t r a c t i i L i s t o f T a b l e s v i L i s t of F i g u r e s vii A c k n o w l e d g e m e n t xiii 1 I n t r o d u c t i o n 1 1.1 Introduction to High-Pressure Research 1 1.2 Techniques for the Generation of High Pressures 2 1.2.1 Static Pressure-Loading Methods 3 1.2.2 Dynamic Pressure-Loading Methods 3 1.2.3 The Laser as an Instrument of High-Pressure Research 5 1.2.4 Thesis Motivation and Objectives 7 1.2.5 Thesis Outline 9 2 S h o c k W a v e T h e o r y 10 2.1 Introduction 10 2.2 Shock Compression By Laser-Driven Ablation 10 2.2.1 Coronal Plasma 12 2.2.2 Ablation Zone 13 2.2.3 Laser-Driven Shock Wave Formation 15 2.3 The Shocked State and the Hugoniot 16 iii 2.4 Complex Aspects of Shock Compression 20 2.4.1 Material Strength Effects and Elastic-Plastic Flow 22 2.4.2 Phase Transformations 25 2.5 The Equation of State 28 2.5.1 Sesame Equation of State Models 29 2.5.2 New EOS Calculation for a-quartz 34 2.6 Computer Simulations 36 2.6.1 LTC Hydrocode 38 2.6.2 SHYLAC2 Hydrocode 40 3 Experiment 41 3.1 Introduction 41 3.2 Laser Facility 41 3.3 Streak Camera 43 3.4 Irradiation Conditions 44 3.4.1 Laser Focal Spot Measurements 44 3.4.2 Effective Intensity 48 3.5 Experimental Details 49 3.5.1 Targets 49 3.5.2 Experimental Arrangement 50 3.5.3 Experimental Procedure and Shadowgram Measurements 54 4 Experimental Results and Interpretations 60 4.1 Introduction 60 4.2 Experimental Observations in Quartz 60 4.3 Experimental Observations in Sodium Chloride 68 iv 4.4 Comparison of Experimental Observations with Computer Simulations in Quartz 74 4.5 High-Speed Shock Transient 86 4.6 Shock-Induced Amorphization of Quartz 99 .5 Conclusion 103 5.1 Summary of Results 103 5.2 Suggestions for Future Work 105 Bibliography 107 v List of Tables 4.1 Experimental data for fused quartz 67 4.2 Experimental data for a-quartz 67 4.3 Experimental data for CH-coated NaCl 71 vi List of Figures 2.1 A schematic illustration of the laser-driven shock process 11 2.2 A sequence of pressure (or density) profiles illustrating how a shock wave evolves 17 2.3 An illustration of an ideal shock transition 18 2.4 Schematic diagram of the Hugoniot curve in the P — V plane. Also shown are the isentrope and the isotherm 21 2.5 Illustration of the effects of material strength on the shock compression process 23 2.6 An illustration of the Hugoniot for a phase transforming material. Also shown for reference is the isotherm 26 2.7 Fused quartz Hugoniot obtained using SESAME table 7380 31 2.8 The density-temperature ranges of the five models that were used in ob-taining the SESAME equation of state of a-quartz 32 2.9 The Hugoniot of a-quartz obtained from SESAME 33 2.10 Single-phase or "quasi-a-quartz" Hugoniot obtained from the new EOS calculation 37 3.11 Schematic of the Nd-glass laser facility used in the experiments 42 3.12 Time-integrated laser focal spot intensity distribution. 46 3.13 Time-integrated intensity profile averaged over the central 7 /xm of the focal spot 47 3.14 Schematic of the complete experimental arrangement 51 vii 3.15 Details of the experimental arrangement near the target 52 3.16 A digitized representation of a typical main laser pulse. Also shown is the oscilloscope image which was digitized (inset) 55 3.17 An example of a shadowgram obtained with only probe beam illumination of the target 58 3.18 Shadowgram obtained in fused quartz under main beam irradiation. . . . 59 4.19 Shadowgram obtained in fused quartz 62 4.20 Shadowgram obtained in a-quartz 63 4.21 Shock trajectory and shock velocity curve for fused quartz derived from the shadowgram in figure 4.19 64 4.22 Shock trajectory and shock velocity curve for a-quartz derived from the shadowgram in figure 4.20 65 4.23 Shadowgram obtained in sodium chloride 70 4.24 Shadowgram obtained in polystyrene-coated sodium chloride 72 4.25 Shock trajectory and shock velocity curve in sodium chloride obtained from the shadowgram in figure 4.24 73 4.26 Comparison of measured and calculated shock trajectories in fused quartz. 76 4.27 Comparison of measured and calculated shock velocity profiles in fused quartz 77 4.28 Comparison of measured and calculated shock trajectories in a-quartz. . 78 4.29 Comparison of measured and calculated shock velocity profiles in a-quartz. 79 4.30 A plot illustrating the temporal evolution of the ablation pressure and the shock pressure as calculated by LTC 81 4.31 A plot of shock velocity versus laser irradiance 84 Vlll 4.32 Schematic illustration of the equilibrium Hugoniot curve of a phase trans-forming material and the metastable Hugoniot for the low-pressure phase. 87 4.33 A schematic illustration of the evolution occuring at the shock front of a phase transforming material which has been loaded by a pressure step. . 90 4.34 A plot of the non-equilibrium Hugoniot points obtained from the experi-mental shock transients in fused quartz 93 4.35 Same data as in figure 4.34 shown in the P — V plane 94 4.36 A plot of the non-equilibrium Hugoniot points obtained from the experi-mental shock transients in a-quartz 95 4.37 Same data as in figure 4.36 shown in the P — V plane 96 4.38 A plot of the experimental relaxation time obtained from the data. . . . 98 4.39 A plot of the 300 K isotherm which fits Hemley's static compression mea-surements on a-quartz 101 i x Acknowledgement I would like to express my sincere gratitude to my research supervisor, Dr. Andrew Ng, for his support, understanding and encouragement throughout the course of this work. I am especially grateful to Dr. B.K. Godwal for the tremendous amount of time and energy he put forth in producing some truly original calculations for this thesis, as well as for happily sharing his considerable theoretical expertise on many different occassions. I am equally indebted to L. Da Silva who assisted me greatly with all phases of this project and thoughtfully provided me with countless helpful suggestions. Thanks are also due to G. Chiu for his help in carrying out the measurements and to A. Cheuck for his able technical support and his efforts in obtaining the necessary supplies. The financial assistance of the National Sciences and Engineering Research Council and of the UBC Physics Department are also gratefully acknowledged. x Chapter 1 Introduction 1.1 Introduction to High-Pressure Research For several decades, scientists and engineers have been conducting investigations into the behaviour of matter at extremes of pressure. These investigations have explored a wide range of different phenomena, resulting in many notable scientific discoveries and technological advances. In particular, a great deal of research has been concerned with obtaining accurate, high-pressure equation of state (EOS) data for a variety of different materials, including many materials of geological interest'^ . This information has also proved invaluable in the geophysical and astrophysical sciences, where it has been used to help piece together an accurate representation of the earth's internal structure and composition ^ , as well as to postulate plausible physical models for some of the other planetary interiors ^. In other disciplines, such as the material sciences, high-pressure research has also found important application. Here investigations have addressed such diverse topics as high-stress structural design®, material strengthening and hardening'^ , and the pressure-induced formation of various synthetic materials (eg. industrial grade diamonds at high pressure. High-pressure research is also an important aspect in inertial confinement fusion (ICF) 1 Chapter 1. Introduction 2 science. In ICF research, several intense laser beams (or charged particle beams) are used to compress a small fuel pellet (e.g. deuterium-tritium) to extreme conditions of density and temperature (and hence pressure) ^ \ in an attempt to ignite and sustain a ther-monuclear fusion reaction in the pellet core. Interest in fusion research is world-wide due to its potential significance as an energy source of the future. In fact in the United States alone, fusion research is presently being actively pursued at a number of different faculties, including Lawrence Livermore National Laboratory, Los Alamos National Lab-oratory, Sandia National Laboratories, and the Laboratory for Laser Energetics at the University of Rochester. Based on this brief discussion, one can see that interest in high-pressure research is certainly quite pervasive. But so far we have not considered how these extremes of pressure are actually achieved experimentally. In § 1.2 we address this point in some detail, outlining a variety of techniques which are used in generating high pressures, as well as indicating the maximum pressures we can expect to achieve with each of these different methods. 1.2 Techniques for the Generation of High Pressures In discussing the various methods for producing high pressures, it is customary to dis-tinguish between static and dynamic techniques. In fact, there are many significant differences between these two approaches, including the methods of pressure loading, the range of attainable pressures, pressure uniformity, maximum sample size, cost, and the nature of the required diagnostic equipment. In general, dynamic techniques achieve the highest pressures but are more resource intensive, while static methods achieve more uniform pressure but are restricted to smaller sample volumes. We now consider each of these approaches in more detail. Chapter 1. Introduction 3 1.2.1 Static Pressure-Loading Methods Much of the early progress in the development of static high-pressure apparatus is cred-ited to P.W. Bridgman. His involvement in this field spanned several decades and, by the late 1940s, he had devised piston-cylinder apparatus which routinely achieved pres-sures in excess of 0.1 Mbart nl Today, virtually all static high-pressure devices are of the piston-cylinder or multianvil var i e tyIn such devices, extreme pressures are generated mechanically by compressing a cell assembly containing the sample under study. Condi-tions of near hydrostatic pressure are achieved in the sample by filling the cell assembly with a pressure-transmitting medium (normally a fluid). These systems have been used to compress large sample volumes (several cm3) up to pressures as high as 0.10 Mbar. At the same time, the recent advent of the diamond anvil cell has permitted measurements to be made at even higher pressures, but at the expense of sample volume. Bellt13], for example, has obtained precise measurements (< ±5% pressure error) on minute sample volumes of aluminum (dimensions of fim ) at pressures of nearly 2 Mbar. Moreover, there are currently prospects for extending these measurements to pressures approaching 5 Mbart1 4l Yet with any of the static methods, the maximum pressure that one can expect to realize will be limited, ultimately, by the finite yield-strength of the load-bearing material. Dynamic techniques, on the other hand, do not suffer from this limitation and we discuss these methods next. 1.2.2 Dynamic Pressure-Loading Methods During the early 1940s, dynamic (or shock) techniques began to emerge as an alternative means of producing high pressures. Today, these methods are in wide spread use and, in fact, represent the only present means of attaining pressures in excess of a few Mbar. Chapter 1. Introduction 4 With dynamic methods, high pressures are realized by applying an intense, impul-sive load to the surface of the experimental sample. This rapid loading drives a strong compressional disturbance, or shock wave, into the sample interior, which can then be monitored with the aid of high-speed diagnostic equipment. A variety of dynamic loading methods have been used in order to drive the shock. Early studies employed high-explosive lens systems, which were capable of generating shock pressures of a few hundred kbar^' Later, the development of the explosive-driven flyer-plate method allowed precise measurements (< ±2% pressure uncertainty) to be made at pressures near 10 Mbarl17- 1 8 l Currently, the state-of-the-art in precision is obtained with the two-stage light-gas gun^' 2^]. m this device, a light gas (such as hydrogen or helium) compressed by a propellent-driven piston, is used to accelerate a small projectile or impactor (typically 12-25 mm in diameter) to high speeds (up to 8 km/s). The projectile then collides with the target specimen of interest, launching a strong shock into the target interior. High-speed instrumentation typically records both the impactor velocity and the shock velocity inside the target. In this way, pressures as high as ~ 5 Mbar have been generated, with < ±1% error In order to obtain pressures above 10 Mbar one must employ either nuclear explosive devices or high-power lasers. In fact, experiments using underground nuclear explo-sions have achieved the highest pressures .to date. Ragan, for example, has used this method to generate shock pressures of 20-70 Mbar in a number of different metals and compounds And more recently, Soviet researchers have performed nuclear-driven impedance-mismatch experiments which have produced record-setting shock pressures of 400-4000 Mbar in aluminum I22]. Yet despite these impressive achievements, cost and security aspects associated with the nuclear-driven shock technique render it inaccessible to the vast majority of high-pressure researchers. Consequently, its has primarily been used as a means of providing Chapter 1. Introduction 5 precise (< ±5% errors), EOS data points in the ultra-high pressure regime. On the other hand, high-power lasers can achieve comparable extremes of pressure, but are far more versatile and practicable than nuclear-driven shock methods. Since a study of laser-driven shocks represents the primary aim of this thesis, we shall devote § 1.2.3 to a brief discussion of this important tool of high-pressure research. Further elaboration will be given, as required, in subsequent chapters of this thesis. 1.2.3 The Laser as an Instrument of High-Pressure Research As we alluded to in § 1.1, much of the early and present interest in high-power laser sys-tems has been centered around their application in heating and compressing small fuel pellets in inertial confinement fusion schemes'^ ' These studies have greatly en-hanced the understanding of both laser-matter interaction processes and high-temperature plasma physics. Recently, however, high-power lasers have also come to be recognized as a valuable tool in high-pressure research In fact, currently, these systems provide the only means of performing laboratory-scale shock experiments in the 10-100 Mbar pressure range'^ ' Moreover, the pressure is easily controlled in such experiments by simply varying the incident laser intensity. Yet the laser-driven shock method differs, in many respects, from any of the previously described dynamic techniques. For one thing, the shock is not produced as a result of an impact or explosions as with the other dynamic methods, but is instead generated through the laser ablation p r o c e s s a s described in chapter 2. In addition, experiments involving laser-driven shocks are inherently short-lived (usually a few ns in duration) and spatially microscopic. This fact necessitates the use of many high-speed optical and electro-optical diagnostics which are not required elsewhere. In most laser-driven shock experiments, the high-speed diagnostics measure either Chapter 1. Introduction 6 shock propagation or shock transit time. In general, the type of measurement made depends on whether the target material is initially transparent or opaque. In the case of an opaque target, like a metal foil, one normally measures the arrival of the shock at the target rear surface ^ — i.e. the shock transit time. Shock transit time has been measured experimentally by observing an abrupt change in either rear-surface luminescence '^ ^ or rear-surface reflectivity, with the use of a high-speed streak camera. A shock trajectory (i.e. shock position vs. time) can then be mapped out by repeating similar transit-time measurements for several targets of different thicknesses. On the other hand, when the target material is initially transparent, one normally observes the shock directly using a technique known as optical shadowgraphy. The first studies to use this technique were made by van Kessel and his associates in the early 1970s, on targets made of solid hydrogen and P l e x i g l a s A s described in more detail in chapter 3, this method allows the shock propagation through the target to be continuously monitored on a high-speed streak camera. In this case, therefore, the entire shock trajectory can be obtained in a single observation. Relatively few measurements of this kind have been made as compared with measure-ments using opaque targets. The few investigations to-date include the aforementioned studies of van Kessel et. al. on solid hydrogen and Plexiglas targets, as well as studies of Plexiglas by Fedosejevs et al.$5] and AmiranofF et alS^\ and of fused silica (fused quartz) by Ng et al. [ 3 7 ] . Yet such a technique obviously has some advantages. For example, a detailed record of this kind can reveal important information about the initial stages of shock formation as well as providing insight into the detailed nature of the shock dynamics. Additionally, because of the microscopic time and space scales inherent to laser shock experiments, such a technique is also ideally suited to studying short-lived or time-dependent phenomena which might occur behind the shock front. One example would be in the application of Chapter 1. Introduction 7 this technique to the study of shock-induced phase transformations. Indeed, many materials have been observed to undergo such phase transformations as a result of shock wave loading by other methods. A number of these observations have been compiled in an extensive review of phase transforming materials by Duvall and Graham®, who have also included in their work a discussion of the kinetics and thermodynamics of shock-induced phase transformations. As the authors point out, the majority of shock-induced phase transitions are detected by way of wave-profile measure-ments made inside the shock-loaded solid. These observations have been used to yield estimates of the transition pressure as well as to provide some information about the transformation kinetics. However, optical shadowgraphic measurements can provide valuable information about shock-induced phase transformations which cannot be obtained with these other meth-ods. For one thing, the sub-nanosecond time scales accessible to this technique permit the study of much faster transformation processes than can be detected with other meth-ods. Moreover, as we will see later, the possibility of being able to continuously monitor the shock propagation inside the material can be very useful in establishing information about the rate at which the transformation takes place behind the shock front. 1.2.4 Thesis Motivation and Objectives In a recent study conducted at U B C[37, 38] laser-driven shocks were observed, for the first time, in a known phase transforming material called fused quartz. Detailed ob-servations of the formation and propagation of the high-pressure (0.3-5 Mbar) shocks generated in this material were found to be in noticeable disagreement with the results of computer simulations based on standard equilibrium modelling of the laser-matter inter-action and shock propagation. Specifically it was observed that above a certain laser in-tensity, the measured shock trajectories exhibited a prominent two-wave structure which Chapter 1. Introduction 8 was only marginally observed in the simulations. Moreover, the second, high-pressure wave emerged anomolously late in the measurements and furthermore was observed to propagate in an initially nonsteady manner which was not predicted by the simulations. The non-steady shock propagation was interpreted in terms of relaxation processes oc-curing at the shock front due to the high-pressure transformation of fused quartz into stishovite'^ ' 41, 42] j [ o w e v e r this model could not account for the observed delay in the formation of the high-pressure shock wave. Moreover, because the experiments were performed with a small laser focal spot, there remained lingering concerns about the possible influence of two-dimensional effects on shock propagation, which were not accounted for in the one-dimensional computer analysis. It therefore became apparent that further study would be necessary to resolve these and other concerns. In this thesis we report on the results of a new study of laser-driven shock waves in quartz. The target materials to be investigated here include the previously studied fused quartz, as well as a crystalline form of silica known as a-quartz. Additionally, some measurements will also be made in sodium chloride which, in this study, will function as a high-pressure standard against which to assess the validity and origin of the previously observed anomalies in quartz. The measurement technique, namely optical shadowgraphy, is the same as that used by Celhers in the earlier study. In these experiments, however, a much larger laser focal spot (~ 3 times larger diameter) will be used in order to mitigate any possible two-dimensional effects in the measurements. Moreover, both time-resolved and time-integrated intensity distributions of the laser focal spot will be determined. In all three substances, shock trajectories will be obtained at laser intensities well above the threshold which had previously resulted in a prominent two-wave structure in fused quartz. However our investigation will show no evidence for such a two-wave structure in either of the quartz materials or in sodium chloride. On the other hand, our Chapter 1. Introduction 9 observations will show non-steady shock propagation in both quartz materials, but not in sodium chloride. We will show, via one-dimensional and two-dimensional computer simulations, that this non-steady shock propagation in quartz cannot be explained by two-dimensional motion at the shock front. We will then examine possible physical explanations for the observations in quartz and assess the plausibility of each model. 1.2.5 T h e s i s O u t l i n e In chapter 2 we review some basic shock wave theory, including specific material relevant to shock compression of quartz. Also discussed is the physical content of the computer codes used in this investigation. Chapter 3 describes the experimental arrangement and associated diagnostic equipment which were used to obtain the shock trajectory, measure-ments. Chapter 4 presents the experimental results and comparisons with computer sim-ulations. This is followed by a discussion of possible interpretations for the observations in quartz. Finally, in chapter 5, we will summarize the major findings and conclusions of this study and offer some suggestions for further work. Chapter 2 Shock Wave Theory 2.1 Introduction We begin this chapter with a discussion of the laser ablation process and show how it gives rise to a high-pressure shock wave. We then discuss the nature of the shock state and introduce the concept of the shock Hugoniot. We will consider complex aspects of dynamic compression, including elastic-plastic flow and shock-induced phase transforma-tions. Following this we consider high-pressure equation of state models for quartz. We then conclude the chapter with a brief description of the computer codes which were used to simulate shock dynamics in quartz. 2.2 Shock Compression By Laser-Driven Ablation Laser-matter interaction encompasses a great diversity of physical processes which occur over a wide range of temperature and density states. Many of these phenomena have been well researched both theoretically and experimentally in the literature'^ ' In this section, however, we shall limit the scope of our discussion to include only those physical processes which are specifically relevant to the formation of a laser-driven shock. A more comprehensive treatment can be found in any of the references above. We begin our discussion by illustrating the situation in the vicinity of a laser-irradiated solid target, as shown in figure 2.1. In the figure we have identified three distinct regions 10 Chapter 2. Shock Wave Theory 11 CORONAL P L A S M A SHOCK COMPRESSED SOLID SHOCK FRONT ABLATION FRONT ABLATION ZONE Figure 2.1: A schematic il lustration of the laser-driven shock process. Chapter 2. Shock Wave Theory 12 of interaction which include the coronal plasma region, the ablation zone, and the shock-compressed region. We now discuss each of these regions in more detail. 2.2.1 Coronal Plasma At intensities > 10 1 1 W/cm2, laser irradiation will heat the target surface sufficiently to cause vaporization of a thin surface layer of the material. This "boiled off" material will also be highly ionized due to the intense heating, resulting in the formation of a low-density ionized gas cloud in front of the target, which is known as the coronal plasma. As this coronal plasma expands outward into the vacuum, it is continually heated by absorption of the incident laser irradiation. This absorption can occur through a variety of different mechanisms, which depend on the wavelength and intensity of the incoming laser light t 4 3 l At sub-micron wavelengths and moderate intensities (< 10 1 4 W/cm2), the dominant absorption mechanism is inverse bremsstrahlung or free-free absorption . In this process, a photon is absorbed by an electron in Coulomb collision with a neighbouring ion. Countless similar interactions in the coronal plasma heat the plasma electrons to extremely high temperatures on the order of 200-2000 eV'^1. The ions in the coronal plasma are also heated, via electron-ion collisions. However, because of the relatively low frequency of such collisions in the plasma, the ion temperatures are much lower (by a factor of ~ 10). Laser light penetrates and heats the plasma in this manner up to the so-called critical density layer, where the laser frequency becomes resonant with the local electronic plasma frequency. Any light which reaches the critical density layer is either reflected back into the coronal plasma or resonantly absorbed ^  at the critical surface, but in any case does not propagate beyond this point. The critical density, n .^, refers to the density of electrons at the critical surface, and is defined (in MKS units) by the following relation!43] Chapter 2. Shock Wave Theory 13 n, (2.1) where u>£ is the (angular) frequency of the laser light, e0 is the permittivity of free space, and me and qe are, respectively, the electronic mass and charge. As an example, for 0.53 fim laser light, as used in our experiments (see chapter 3), nCT = 4.5 x 1027 m~3. This is much less than the solid density of electrons which is n5 ~ 1029 m - 3 . 2.2.2 Ablation Zone. Just beyond the critical density surface is a region of energy transport known as the ablation zone. The ablation zone forms the transition region between the hot, rarefied coronal plasma and the cold, dense target interior. Consequently, large temperature and density gradients exist in this region. These large temperature gradients result in a large inward heat flux past the critical density layer. The inward propagation of this heat front causes ablation (i.e. vaporization) of the target material, expelling it out into the coronal plasma at high speeds (107 cm/s). This process gives rise to a pressure at the ablation surface (ablation pressure) which, in turn, launches a compression wave into the cold target interior. (This will be described in more detail in § 2.2.3). In the ablation zone, heat transport is mainly carried out by electron thermal con-duction. (The contribution to heat conduction due to the much slower moving ions is usually negligible.) Under the assumption that the electron mean free path (Ae) is small in comparison with the scale length of temperature (~ (VT/T) - 1 ), we find that the heat flux due to electron thermal conduction is governed by the well known Fourier law!46!: where q is the electronic heat flux, Te is the electron temperature and ne is the elec-q = -/ceVTe (2.2) tron thermal conductivity. Spitzer^' 4 ^ performed the first detailed calculation of Chapter 2. Shock Wave Theory 14 electron thermal conductivity in a plasma, obtaining an expression of the form .90.095(Z + 0.24)Te6/2 "• = 1 9 5 5 X 1 0 (1+0WhA < 2 3 ) where Te is the electron temperature, Z is the atomic number, and In A is the Coulomb logarithm. The strong temperature dependence of ne shows that heat conduction in the ablation zone is highly nonhnear. Fourier's law breaks down at sufficiently high intensities for which Ae > (VT/T) - 1 . Experiments in this intensity regime have shown that the actual heat flux is much less than that which would be predicted from the classical transport model described above'4 '^ This reduction in transported heat flux is known as electron thermal flux inhibition, and while various mechanisms have been invoked to explain it^^, the phenomenon is presently not that well understood. In computer simulations one nor-mally circumvents this problem by artificially limiting the actual heat flux, qa, to some fraction of the Spitzer heat flux q : qa = fq. The "fiux-hmiter", / , is then determined either from experiment or estimated from some particular heat transport model. Although electronic thermal conduction is by far the most dominant energy transport process in the ablation zone, other transport mechanisms do exist. Among the most important of these processes is the transport of hot (or suprathermal) electrons. These highly energetic electrons are produced in the coronal plasma through the decay of various plasma waves, and can reach temperatures on the order of 2-200 Because of their high temperature, these hot electrons have extremely long mean free paths and can pass easily through the ablation zone and the shock compressed region of the target, before depositing their energy in the uncompressed region ahead of the shock front. In this way, hot electrons can "preheat" the target (i.e. heat up the target before the shock arrives) to high temperature. Chapter 2. Shock Wave Theory 15 In addition to hot electron transport, radiation transport represents another signifi-cant preheating mechanism in laser-irradiated targets, and we will consider this process in detail in § 4.3. Both hot electron transport and radiation transport are highly un-desirable in laser-driven shock experiments, since they result in reduced shock pressures (for a given laser intensity) as well as a poorly characterised initial state. We will"see in chapter 4, however, that both hot electron transport and electron ther-mal flux inhibition are negligible in our investigation. This is because our experimental irradiation conditions (see chapter 3) fall within the so-called "classical" laser-matter coupling regime'^ 3, ^ for which $£,A2 < 1014 pro. 2W/cm2 ($£ is the laser irradiance, A is the laser wavelength) and these effects can be ignored entirely. On the other hand, radiation transport will be shown to be an important process in laser-irradiated sodium chloride targets, as discussed in § 4.3. 2.2.3 Laser-Driven Shock Wave Formation The laser-driven shock wave arises essentially out of momentum conservation. In the process of ablation, a large outward momentum flux is created by the high-speed, outward expansion of the ablated material. This outward flux gives rise to a pressure at the ablation surface (i.e. the ablation pressure), which drives a compressional disturbance into the cold target interior. At early times, when the laser intensity is relatively weak, the ablation pressure will only be sufficient to launch an elastic wave (see § 2.4.1). However, as the laser intensity continues to build, the ablation pressure will also increase, leading to further compression of the material behind the wave front. For example, at a certain time the pressure (or density) profile across the wave front may resemble that shown in figure 2.2a. Here the transition from initial to final states across the wave front is seen to be very smooth and gradual. Such a smooth compression profile is not maintained for very long, however, because the sound speed is not the same for all points in the wave Chapter 2. Shock Wave Theory 16 front. In particular, for most normal materials, the sound speed, c, is known to increase with increasing density. (In the case of an ideal gas, for example, c ~ p3^ - where p is the density and 7 > 1 is the ratio of specific heats'55 .^) Thus increments of higher density in the wave front will propagate faster than increments of lower density, and the profile will rapidly steepen as shown in figure 2.2b. Eventually, as shown in figure 2.2c, the wave front will become so steep that it is almost discontinous. This abrupt, irreversible transition is known as a shock wave and we will discuss some of its peculiar properties in the next section. 2.3 The Shocked State and the Hugoniot Figure 2.3 schematically illustrates the case of a one-dimensional, uniform shock wave at an instant during its propagation through the initially undisturbed solid material. Although it is shown as a discontinuity, the shock front actually has a very small but finite thickness on the order of a few hundred atomic layers'56' 5 7 l Across the shock front the thermodynamical state of the matter changes abruptly; from ambient conditions in front of the shock to a state of increased pressure, density, and temperature behind the shock. For a steady shock, these states are related by the Rankine-Hugoniot (R-H) equations which express conservation of mass, momentum, and energy across the shock front'58]: E-E0 = ±(P + P0){V0-V). (2.6) Here V = 1/p is the specific volume, P is the pressure, E is the internal energy, D is the shock speed, and u is the particle speed of the shock compressed material (the particle Chapter 2. Shock Wave Theory 17 Figure 2.2: A sequence of pressure (or density) profiles il lustrating how a shock wave evolves. Chapter 2. Shock Wave Theory P.V.E u D ^0 ' V ) ' ^ 0 X Figure 2.3: An illustration of an ideal shock transition. Chapter 2. Shock Wave Theory 19 speed of the material ahead of the shock is zero). In this notation, subscripted quantities refer to the initial state, whereas unsubscripted quantities refer to the shocked state. The R-H equations are generally valid in describing steady shocks in solids, liquids or gases. However, these equations alone are insufficient to determine completely the thermodynamic state of the shock-compressed material. To see this, we first note that in most shock experiments, the initial state of the material (P0,E0, Vo) is typically either known or can be neglected because of the high shock intensity. We then have a situation in which the five unknown parameters of the shocked state (D, u, P, V, E) are linked by a system of only three equations (equations (2.4)-(2.6)). The system is therefore underdetermined, having two free parameters. If the equation of state (EOS) of the material is known (say in the form E = E(P, V)) then it may be combined with equation (2.6) to yield a specific functional relationship between P and V which depends on initial conditions. When plotted in the P — V plane this curve defines the locus of final states which are achievable through a single shock-compression from a given initial state (Po> Vo). This curve is variously referred to as the shock adiabat, the Rankine-Hugoniot curve, or simply the Hugoniot. By eliminating variables among equations (2.4)-(2.6), the Hugoniot may be obtained in a plane which relates any two of the five shock parameters (e.g D and u, P and D etc.). A schematic of the Hugoniot curve as it appears in the P — V plane is shown by the solid curve in figure 2.4. For reference, the isentrope and isotherm centered on the same initial state are also shown. The Hugoniot lies above the isentrope because the shocked material is irreversibly heated by the shock compression process, and is consequently in a state of higher entropy. The straight Hne (shown dotted in figure 2.4) connecting the initial unshocked state (P0, V0) to the final shocked state (P, V) is known as the Rayleigh Chapter 2. Shock Wave Theory 20 line. By combining equation (2.4) and equation (2.5) we obtain the relation (2.7) which shows that the slope of the Rayleigh line is proportional to D2. This provides us with a useful graphical relationship between shock speed, pressure, and compression, namely D 2 oc AP/AV. In any case, if the equation of state of the material is known, or can be calculated, then the thermodynamic state of the shocked region can be inferred by an experimental determination of any one of the five shock parameters. For many reasons, however, the EOS of a material may not be known from theoretical considerations alone, so it must be measured. In this case, two parameters (among P,V,E,u,D ) must be simultaneously measured to completely define the shocked state. In hypervelocity impact experiments, the most frequently measured quantities are the shock speed and the particle speed^. Shock pressure has also been measured over a hmited pressure range'59]. In laser-driven shock experiments, the shock speed is the quan-tity which is most often measure d[30, 31]. Shock temperature has also been measured in both impact' ] and laser'31] shock experiments, but it is only indirectly related to the parameters of the shocked state through the material EOS. Shock temperature measure-ments do, however, provide information which is useful in establishing the partitioning of internal energy, E. This information often can be used to considerable advantage in constructing reliable material equations of state at high temperature and density. We will have more to say about the equation of state and how it is calculated in § 2.5. 2.4 Complex Aspects of Shock Compression The simple treatment of § 2.3 adequately describes shock behaviour in gases and in ideal (inviscid) liquids. It also applies to solids in the strong shock hmit where material strength Chapter 2. Shock Wave Theory 21 Figure 2.4: Schematic diagram of the Hugoniot curve in the P — V plane (solid curve). Also shown are the isentrope (dashed) and the isotherm (dot-dashed) centered on the same initial state. The Rayleigh hne (dotted) is used to calculate shock speed. Chapter 2. Shock Wave Theory 22 effects are negligible and the shocked material behaves, for all intents and purposes, like a hydrostatically compressed fluid. At lower shock pressures, however, it is necessary to consider the properties of a solid which distinguish it from a fluid. In this regime, there are strength effects in which the material supports a stress which is not purely hydrostatic, but also contains an element of shear stress. This can lead to elastic wave propagation in the solid or, at shghtly higher stress levels, to elastic-plastic flow. Further complexity arises when one considers that many solid materials are known to undergo shock-induced phase transitions. Such a material can no longer be described by a simple single-phase Hugoniot like that shown in figure 2.4. This leads to modifications in the wave profiles which develop as a result of shock wave loading. Both elastic-plastic flow and shock-induced phase transitions are known to occur in quartz. Since these effects have been observed within a pressure range accessible to the present investigation, it is possible that they might be important in the interpretation of our results. For this reason we devote § 2.4.1 and § 2.4.2 to a brief review of these concepts. More detailed reviews of the effects of material strength and shock-induced phase transformations on wave-profile measurements can be found in the excellent works by Zel'dovich and Raizer^ and by Duvall and Graham' 2.4.1 Material Strength Effects and Elastic-Plastic Flow Material strength effects are manifested in the Hugoniot curve by a cusp, or region of negative curvature, as shown in figure 2.5. Here, the wave profiles which evolve un-der dynamic loading will depend sensitively on the magnitude of the driving stress, or longitudinal stress 07. In the case of low driving stresses, an elastic wave is transmitted which deforms the material uniaxially in the direction of motion. This uniaxial compression results in an anisotropic distribution of stress behind the wave front, in which the longitudinal Chapter 2. Shock Wave Theory 23 Figure 2.5: Illustration of the effects of material strength on the shock compression process. The Hugoniot curve of an elastic-plastic material is shown by the solid curve. For comparison, the hydrostat, or bulk compression Hugoniot is also indicated (dashed curve). The inset shows the pressure distribution in the compression process for: (a) a pressure P 0 which shghtly exceeds the HEL and results in a two-wave structure (elastic-plastic flow) ; and (b) a pressure Pb which substantially exceeds the HEL so that the elastic wave is overtaken by the plastic wave to form a strong shock. The Rayleigh lines (dotted lines) are used in calculating wave speeds. Chapter 2. Shock Wave Theory 24 component of stress, 07, differs from the transverse (or lateral) component of stress, ot. The difference between these two components gives rise to a shear stress, r, given by r = ~ <rt). (2-8) Elastic wave propagation is not possible in a hydrostatic fluid, for which o~\ — o~t — P, and r = 0. In this case, only bulk compression waves are possible, corresponding to the Hugoniot shown by the dashed curve in figure 2.5. In a solid, uniaxial elastic deformation occurs only up to the so-called Hugoniot elastic hmit (HEL) , at which point partial shear failure occurs because of the finite strength of the material. Shear failure is indicated by the kink in the Hugoniot curve which occurs at the pressure Phei. In an intermediate regime above the HEL, the compressed solid undergoes permanent plastic deformation, as well as supporting shear stress. In this regime, a single-step wave transition is unstable, and will separate into a two-step transition consisting of an elastic wave followed by a slower moving plastic wave. This two-wave structure is known as elastic-plastic flow, and it is illustrated in figure 2.5 (inset) for the pressure Pa. Here the elastic wave compresses the material up to the Hugoniot elastic hmit Phei, while further compression to the final state Pa takes place across the wave front of the plastic wave. Two-wave structures are stable so long as the elastic wave speed is greater than the plastic wave speed. This condition will be met at any stress level for which the Rayleigh line slope of the elastic wave is larger than that of the plastic wave, as is the case for the state Pa. At sufficiently high pressures (e.g. Pt), however, the plastic wave will be the faster of the two, and this two-wave structure will collapse into a single, strong shock as shown in figure 2.5 (inset). Elastic wave propagation has been observed in both fused quartz and a-quartz ^ \ In fact both of these materials are characterized by unusually high Hugoniot elastic limits. In fused quartz the Hugoniot elastic hmit is ~ 0.1 Mbar, while in a-quartz the HEL Chapter 2. Shock Wave Theory 25 varies between 0.05-0.15 Mbar depending on the orientation of the shock with respect to the crystal axes. For x-cut a-quartz, as used in our experiments, the HEL is ~ 0.08 Mbar. Experiments have shown that for stress levels just above the HEL, both fused quartz and a-quartz suffer a complete loss of inherent material rigidity Compression of quartz above the HEL may thus be regarded as exactly hydrostatic, as in an ideal fluid. 2.4.2 Phase Transformations Many solid materials can exist in different structural forms or polymorphs under differ-ent conditions. In a material of this kind, the stable phase at any given pressure and temperature will be the one with the lowest Gibbs free energy. If the pressure and/or temperature are altered, a phase transition from one structural form to another may oc-cur. Such a transition involves an internal rearrangement of the solid, and often results in a large change in density and entropy. Figure 2.6 illustrates the Hugoniot curve (solid curve) of a phase transforming material with initial volume Vo 1 • Also shown for comparison is the isothermal compression curve for such a material (dashed curve). As can be seen, the Hugoniot in this case bares some similarity to the Hugoniot of an elastic-plastic material. At low shock pressures, the shocked material remains in its original phase (phase 1), as shown by the concave shape of the Hugoniot in this region. At the transition pressure Pt, the transformation from phase 1 to phase 2 begins. This is manifested in the Hugoniot by a discontinuous change in slope and the initiation of a volume collapse from Vj towards V2. (On the isotherm, there is a discontinuous change in volume at the transition pressure.) For pressures which span this volume change, the material is in a two-phase state in which phase 1 and phase 2 1For the purposes of this discussion we assume that the compression is purely hydrostatic so that CT( = cr; = P, and there are no shear stresses. Chapter 2. Shock Wave Theory •26 Figure 2.6: An illustration of the Hugoniot for a phase transforming material (solid curve). Also shown for reference is the isotherm (dashed curve). The dotted lines are the Rayleigh lines. Chapter 2. Shock Wave Theory 27 coexist in appropriate fractions to make up the mixed phase volume. The transformation is completed at a compression corresponding to V2. At higher compressions (i.e. higher pressures) the Hugoniot curve assumes a shape which is characteristic of the structure of phase 2. Analysis of wave propagation in a phase transforming material is actually quite similar to that described in connection with elastic-plastic flow. When the pressure P is below the transition pressure Pt, no transformation takes place and an ordinary shock passes through the material. Similarly, for very large pressures (e.g. Pb in figure 2.6) only one shock will propagate. In this case, however, the material behind the shock is in phase 2, with the transition from phase 1 to phase 2 occuring across the shock front. Finally there is, as before, a region of intermediate pressure for which a two-wave structure will be stable (e.g. Pa in figure 2.6). The leading wave compresses the material in phase 1 up to the transition pressure Pt, while compression to the final state is accomplished in a slower second wave in which the actual phase transformation occurs. Wave speeds can be calculated using the same Rayleigh line constructions as for the elastic-plastic transition. Upon shock compression, both fused quartz'39, 4 0 ' 4 1 , 4 2 ] and a-quartz[39' 4 1 , 6 1 , 6 2 j are known to undergo a high-pressure phase transformation into a form of SiC>2 known as stishovite. Whereas both low pressure forms of quartz have the tetrahedral bonding in which each Si atom is linked to four 0 atoms, stishovite is known to have the rutile bonding structure in which each Si atom has six oxygen neighbours. This change in bonding structure results in a much higher density for the stishovite phase (~ 4.29 g/cm3 at standard conditions'63' as compared with 2.20 g/cm3 for fused quartz and 2.65 g/cm3 for a-quartz). Stishovite was first synthesised in static compression experiments conducted in the Soviet Union by Stishov and Popova'64], but was later found to occur naturally in meteorite impact craters'6 .^ Identification of stishovite as the high-pressure phase of quartz under shock loading conditions was made by McQueen'66 .^ For both fused Chapter 2. Shock Wave Theory 28 quartz and a-quartz, the stishovite transformation commences under shock conditions at pressures just above the Hugoniot elastic limit for quartz (~ 0.1 Mbar). It then reaches completion at pressures of ~ 0.3 Mbar in fused quartz, and at shghtly higher pressures of ~ 0.4 Mbar in a-quartz. Further discussion of shock compression in quartz and the shock-induced transformation into stishovite is given in § 2.5 and in chapter 4. 2.5 T h e E q u a t i o n of S ta te The equation of state (EOS) describes the relationship among the various thermodynamic quantities of a system in equilibrium. Its detailed understanding is of fundamental im-portance to many different fields in both pure and applied science. Specifically, equation of state data can provide important information about phenomena such as phase tran-sitions and material properties (resistivity, specific heat, etc.). In addition, equation of state data is used extensively in computer modelling of material behaviour under a va-riety of different conditions. Good review articles on the equation of state of condensed matter can be found in the works by Godwal et a l . ^ and by Rossi68!. In many cases, the equation of state is specified in the form E — E(V,T), in which the internal energy (E) is given as a function of temperature (T) and specific volume (V). For computational purposes, the internal energy is often partitioned into a cold (or elastic) contribution Ec, which depends only on density, and thermal contributions due to the ions Ei and the electrons Ee, which depend on density and temperature : E = E(V,T) = EC(V) + Ei(V,T) + Ee(V,T). (2.9) Each of these contributions is then calculated separately according to some physical model of the material under consideration. One EOS model is thus distinguished from another by the methods used in calculating the various terms. Chapter 2. Shock Wave Theory 29 In § 2.5.1 we outline the computational methods used in the SESAME equation of state calculations for both fused quartz and a-quartz, while in § 2.5.2 we describe a new EOS model for a-quartz which was developed at UBC. 2.5.1 Sesame Equation of State Models The SESAME data library'69], developed at Los Alamos National Laboratory, provides equation of state data and other material properties (average ionization,thermal conduc-tivity, opacity, etc.) for a large number of different materials. The data tables combine both theoretical and experimental data, in order to provide an accurate representation of material properties over a wide range of temperature and density. These tables have been widely used and quoted, and are perhaps the most extensive and reliable of any available. Computer simulations of experimental results obtained in this work (see chapter 4) were carried out using the SESAME data tables compiled for the silicon dioxide sys-tem. The SESAME library includes equation of state data for both fused quartz and a-quartz. We will now- briefly describe some of the theoretical methods which were used in calculating these equations of state. SESAME EOS for Fused Quartz The equation of state data for fused quartz, with initial density 2.2 g/cm3, is given in SESAME table 7380. Details of the computational procedure used to obtain this table have not been disclosed, however, it was most "probably" l7 ]^ generated using the Barnes-Cowan-Rood procedure. In this method the cold pressure contribution is described by a modified Morse potential!71] of the form Pc = arl2/3(T}el*v - eb°v) (2.10) Chapter 2. Shock Wave Theory 30 where v = p/po,v — 1 — r\ 1^3,ba = 3 + br — 3K0/a (K0 is the bulk modulus at 0 K), and the parameters a, br are obtained by matching the Morse cold curve to a Thomas-Fermi-Dirac (TFD)t72l cold curve at high pressures. In order to account for the stishovite phase transition in quartz, the calculation of the cold curve was actually made using two separate modified Morse potentials (one for each phase). These potentials were then joined at the known transition point as determined from Hugoniot data. The contribution of the ions to the internal energy was calculated with the Debye model t73] which requires that the Griineisen parameter be known as a function of density. Below a temperature of 1 eV, the thermal contribution of the electrons is negligible so that only the cold curve and ionic contributions are important in this region. At high temperatures and densities the EOS was modelled using the Thomas-Fermi-Dirac theory. And finally, in the region of expanded densities (less than solid) the EOS was expressed as a virial series'74!, whose coefficients were deduced from known material properties of quartz at standard conditions. Figure 2.7 shows Hugoniot curves for fused quartz in both the P — V and D — u planes, as calculated using the SESAME equation of state. SESAME EOS for a-quartz The equation of state for a-quartz, with initial density 2.65 g/cm3., is tabulated in SESAME table 7382. The calculation was made by Reel75! using a combination of five different theoretical models, each one being valid over a certain range of tempera-tures and densities. Figure 2.8 indicates the region of temperature-density space where each model is applied. As can be seen, the EOS data covers a very wide range of temper-atures (2.5 x 10 - 2 - 2.5 x 104 eV) and densities (1.5 x 10 -6 - 1.5 x 102 g/cm3). We now follow with a brief description of the individual models which were used in formulating the EOS. Chapter 2. Shock Wave Theory 31 U (Wcm/s ) Figure 2.7: Fused quartz Hugoniot obtained using SESAME table 7380. Upper plot shows the Hugoniot in P — V plane. Lower plot shows the Hugoniot in D — u plane. Chapter 2. Shock Wave Theory 32 10' L U i d 10 (2 10 • 6 O C C I P I T A L TIGER TFCMIX [ 5 IMASTER GRAY-. -2 10 10 1 10* DENSITY (g/cm3) Figure 2.8: The density-temperature ranges of the five models that were used in obtaining the SESAME equation of state of a-quartz. Figure 2.9: The Hugoniot of a-quartz obtained from SESAME. Upper figure shows the calculated Hugoniot in the P — V plane. The lower figure shows the calculated Hugoniot in the D — u plane . Chapter 2. Shock Wave Theory 34 1. TIGER'76] - This is the low temperature, low-density region in which ionization is negligible. Here the system is regarded as a mixture of dissociation products of SiC"2, as well as liquid and solid SiOV 2. GRAY' 7 7! - In this high density, low temperature region, both ionization and disso-ciation are negligible. Here the system is modelled as a mixture of liquid, a-quartz, and stishovite phases, within a self-consistent framework developed by Grover. 3. OCCIPITAL'78- 791 - In this high temperature, low-density region, the Si02 molecule is assumed to be completely dissociated into a plasma of electrons, ions, and neu-tral atoms, with the individual species concentrations determined from the Saha equation. 4. TFCMIX' 8 0! - In this high density, high temperature region, the EOS calculation is based on a modified Thomas-Fermi theory which includes various correction terms. 5. MASTER' 8 1! - In this high density, moderate temperature region the material is as-sumed to be only in the stishovite phase. The EOS is modelled using Thomas-Fermi theory which includes a phenomenological correction that attempts to account for atomic shell structure and electronic correlation effects. In figure 2.9 we show the SESAME Hugoniot for a-quartz in both the P — V and D — u planes. 2.5.2 New EOS Calculation for a-quartz We now describe a new EOS model for a-quartz which was developed at UBC by Dr. B.K. Godwal (on leave from the Bhaba Atomic Research Centre, Bombay, India). In this model, the a-quartz remains in its original phase at all compression levels and does not Chapter 2. Shock Wave Theory 35 undergo a phase transition to stishovite, as in the SESAME EOS calculations. That is, the quartz to stishovite transition is explicitly suppressed in this new model. The new calculations divide the pressure P, and specific internal energy E, into the usual sum of the cold, ionic and electronic thermal excitation contributions: E(V, T) = EC(V) + Et(V, T) + Ee(V,T) (2.11) P(V,T) = PC(V) + Pi(V,T) + Pe(V,T) (2.12) In this model, the cold compression terms (EC,PC) are evaluated using two-parameter scaling relations due to Dodson'8^: Ec(u) = a{3(l-/3)+/32-(9/2l/)-f(9/3/I/2)-[3(4^-l)/(2I/3)]-/32[(31nl/+l)/I,3]} (2.13) Pc{u) = (3a/2)(i/2 - 4/3i/ + 2/32lni/ + 4/3 - 1). (2.14) where v = (p/po)1^3 and Q , / 3 are the parameters. These parameters are related to the zero-pressure bulk modulus B0, and to its pressure derivative B'0 = ^ |p_ 0 , through the relations a = B„(l - /?)"2 (2.15) * - 1 - 5 § J (2.16) For a-quartz these parameters have the values a = 14.5 Mbar and (3 = 0.83. (We could not calculate a corresponding "quasi-fused-quartz" Hugoniot because Dodson's formulation can not be used when the pressure derivative of the bulk modulus is negative, as is the case for fused quartz at pressures up to 20 kbar.'83^) Chapter 2. Shock Wave Theory 36 Ionic and electronic thermal contributions were calculated separately. The thermal contribution of the ions was calculated using the classical Debye model!84] in which Ei = SkgT per atom, where ks is Boltzmann's constant. The corresponding ionic pres-sure was then determined from P/ = ^Ei/V, with the Griineisen parameter, 7, being obtained from the SESAME data table for a-quartz (table 7382). The electronic ther-mal contribution, on the other hand, was evaluated using expressions obtained from Thomas-Fermi theory[85]. These new EOS calculations were used to obtain a single-phase Hugoniot for a-quartz (or quasi-a-quartz Hugoniot), as shown by the dotted curve in figure 2.10. Also shown for comparison is the SESAME Hugoniot for a-quartz (dashed curve) which includes the stishovite phase transition. Within the phase transition region (0.1-0.4 Mbar), the "quasi-a-quartz" Hugoniot yields much higher pressures (at a given volume) due to the suppressed transition. Further discussion of this "quasi-a-quartz" Hugoniot model will be given in chapter 4, when we discuss possible interpretations for our experimental observations in quartz. 2.6 Computer Simulations The laser-driven shock results from the complex interaction of many time-dependent and non-linear processes. In this situation, analytical calculations are of hmited utility, and one must rely almost exclusively on the use of computer simulations. In our studies of shock propagation in quartz we made use of two different codes: LTC and SHYLAC2. LTC models laser-matter interactions as well as shock propagation, and was used to provide a realistic model of our experimental conditions. SHYLAC2, on the other hand, models only shock dynamics, and was used to assess the effects of two-dimensional shock propagation. The basic physical processes included in each of these Chapter 2. Shock Wave Theory 37 Figure 2.10: Single-phase or "quasi-a-quartz" Hugoniot obtained from the new EOS calculation (dotted curve). Also shown is the SESAME Hugoniot for a-quartz (table 7382) which incorporates the quartz-stishovite phase transition (dot-dashed curve). Chapter 2. Shock Wave Theory 38 codes are discussed in § 2.6.1 and § 2.6.2, while the results of simulations made with these programs are presented in chapter 4. 2.6.1 LTC Hydrocode LTC (Laser Target Code) is a one-dimensional, two-temperature, Lagrangean2 hydrody-hydrodynamics in both single and multi-layered targets. It includes laser absorption by inverse bremsstrahlung, electronic and ionic heat conduction, and electron-ion energy exchange. Target hydrodynamics are considered in one-dimensional planar geometry. The target material is regarded as an inviscid and compressible fluid made up of interpenetrating electron and ion components which are each described by separate temperatures, in order to account for weak energy coupling between the two species in the hot, low-density region of the coronal plasma. Fluid motion is governed by a set of coupled differential equations which represent conservation of mass, momentum, and energy for electrons and ions in one dimension: namic code, which was developed at UBC by P. Celhersl38l This code was adapted from the laser fusion code Medusa'86!, a n c J is u s e d to model laser-matter interaction and shock Dp du ~Dt = ~Pdx~ (2.17) Du ~Dt ldPt p dx (2.18) Zl± - _ P _ Dt dt (2.19) 2 In a Lagrangean system one considers a division of the fluid into individual fluid elements or cells which are assigned coordinates which are constant in time. Moreover, each cell moves with its own local fluid velocity in such a way that the mass of each cell is conserved Chapter 2. Shock Wave Theory 39 Dt 1 dt + Qi (2.20) In the above equations D/Dt — d/dt +ud/dx represents the Lagrangean (or convective) time derivative, Pt = Pe -f Pi the total pressure, V = l/p the specific volume, e the specific internal energy, and Q the heat source terms which describe the various energy transport and absorption processes included in the LTC model. Specifically, contribu-tions to the electronic heat source term (Qe) include laser energy deposition by inverse bremsstrahlung, electronic thermal conduction, and electron-ion energy exchange. Con-tributions to the ionic heat source (Qi) are ionic thermal conduction and electron-ion energy exchange. LTC uses equation of state data obtained from the SESAME data library (see § 2.5.1). In addition, SESAME tables are used to provide other material dependent quantities such as average ionization state, electronic and ionic thermal conductivity, and opacity data (when radiation transport is desired). Alternatively, electronic thermal conductivity may be calculated directly in LTC according to the Spitzer formula (equation ( 2.3)). Solution of the governing equations is accomplished in two phases at each time step of the calculation. In the first phase of the calculation only adiabatic hydrodynamics are considered by solving equations (2.17)-(2.20) with the heat source terms set to zero (i.e. Qe = Qi = 0). Explicit solution is obtained using the PPM (piecewise parabolic method) developed by Collela and Woodward'87' 8 8^. This method was chosen for its recognized superiority in treating shock hydrodynamics. In the second phase of the calculation, the energy equations (2.19)-(2.20) are solved self-consistently by including all of the temperature and density dependent source terms (Qe,Qi)- Here an iterative solution is required because many of the EOS functions and transport coefficients depend on temperature and density in a highly nonlinear fashion. Chapter 2. Shock Wave Theory 40 2.6.2 SHYLAC2 Hydrocode In analysing our experimental results, it was necessary to assess the importance of two-dimensional motion on shock propagation inside the target. This was accomplished using a two-dimensional, Lagrangean hydrodynamic code known as SHYLAC2, which was developed at the Uniyersite de Poitiers in France, by Cottet and Marty'89!. This code is purely hydrodynamic in treatment and therefore does not incorporate any laser-matter interactions. Instead, the pressure resulting from the laser-driven ablation process (ablation pressure) is replaced by an equivalent pressure pulse applied at the target boundary. With this pressure pulse as a boundary condition, SHYLAC2 solves the ideal fluid equations as they apply in two-dimensional cylindrical geometry. Since neither viscosity nor thermal conductivity are included in this ideal treatment, dissipation is introduced artificially into the equations through an artificial or numerical viscosity'^ l. This tends to limit the maximum spatial gradients in the fluid flow variables across the shock front, thereby ensuring numerical stability in the solution of the fluid equations. In SHYLAC2, equation of state data may be specified in either analytical or tabular form. The former option uses a Mie-Griineisen equation of state which has been fitted to existing high-pressure data to make it more realistic. This option has been used successfully in previous work'91! to model the shock-breakout region at the rear-surface of laser-irradiated thin (~ 25 fim ) metal targets. Alternatively, equation of state data may be obtained in tabular form from the SESAME data library (see § 2.5.1). This latter option was incorporated into the SHYLAC2 code at UBC' 9 2!, allowing SHYLAC2 to be run on any target material for which SESAME equation of state data are available. This SESAME equation of state option was used for all the two-dimensional simulations in this thesis. Results of these calculations are discussed in chapter 4. Chapter 3 Experiment 3.1 Introduction This chapter begins with a description of the laser facility and the high-speed optical streak camera which were used in the experiment. Following this we describe measure-ments of the laser focal spot intensity distribution. Next, we overview the entire experi-mental set-up used in obtaining the shock trajectory measurements as well as outlining the measurement technique. Finally, we present examples of the raw experimental data. 3.2 Laser Facility The present experimental investigation was carried out using an upgraded version of the laser facility used in the earlier work of CehW 3 7 - 3 8). As shown schematically in figure 3.11, the laser system includes a Nd-YAG oscillator and preamplifier, four Nd-glass amplifiers, three vacuum spatial filters, two beam expanders, and mirrors to direct the beam through all of these components. The oscillator and all of the amplifiers were manufactured by Quantel International!93]. The oscillator, which was passively Q-switched using a dye cell, produced a single output pulse at A0 = 1.064 /xm , which was very nearly gaussian in time (nominal 2.5 ns FWHM) and in space (TEMoo spatial mode). The amplifiers, which were independently controlled, allowed the output energy of the system to be varied continuously up to a 15 J maximum (at A0 = 1.064 /xm ). However, in order to avoid problems associated with longer wavelength lasers (see § 2.2.2), the 41 Chapter 3. Experiment 42 C D SHG TARGET CHAMBER sf \ SF3 [] A4 A3 Ll BE2 ~7 SF2 A2 [ ] BE1 A l \ J osc PA sn Figure 3.11: Schematic of the Nd-glass laser facility used in the experiments. OSC = Nd-YAG oscillator, PA = Nd-YAG pre-ampliner, A1-A4 = Nd-glass amplifiers, SF1-SF3 = vacuum spatial filters, BE1-BE2 = beam expanders, and SHG = second harmonic generator (conversion crystal). Chapter 3. Experiment 43 1.064 /im fundamental wavelength was frequency-doubled to A = 0.532 fim with a KDP type II conversion crystal, yielding up to 7 J of energy for use in the experiments. The incorporation of the three vacuum spatial niters into the system prevented the build up of high intensity modulations in the spatial profile of the laser beam (see § 3.4.1). In these devices, the actual filtering is done by focusing the beam through a diamond pinhole inside an evacuated tube, and then recollimating it with a second lens. Very little beam energy is lost to the filtering process. The laser system was used in either of two modes. In the so-called low-power mode, only the Nd-YAG oscillator and preamplifier were activated. Low energy output pulses could then be obtained either individually or as a pulse train with a repetition frequency of ~ 0.2 Hz. This mode was convenient for aligning the various optical systems used in the experiment, as described in § 3.5.3. The actual experiments were performed with the laser operating in high-power mode, in which all of the laser system amplifiers were activated. In this mode, the laser produced a single, high-intensity pulse which was used to drive the shock. At least 10 minutes of coohng time was allowed between successive shots, however, in order for the system to restabilize. 3.3 Streak Camera The primary diagnostic device used in the experiments was an optical streak camera system which was manufactured by Hamamatsut94l The system is comprised of four components: temporal disperser, temporal analyser, SIT camera and TV monitor. Light enters the system through a small slit located at the front of the temporal disperser. This entrance slit is variable in both length (0-24 mm) and width (0-5 mm), allowing control over the amount of input light. The particular model we used here was sensitive to light Chapter 3. Experiment 44 signals in the optical to near-UV spectrum. The system can be operated in either streak mode or focus mode. In streak mode, one obtains a time-resolved record of signal intensity as a function of one spatial coordinate ( along the axis of the slit). Several different streak speeds are available in this mode. The experiments were done on the slowest streak speed setting, since this afforded the longest possible observation time (5.64 ns). The time resolution in this case was hmited to ~ 30 ps. Alternatively, in focus mode, the streak camera produces a time-integrated record of the incident signal intensity as a function of two spatial coordinates (along the axis of the slit and perpendicular to it). This mode was used to measure the spatial intensity distribution of the laser focal spot, as described in § 3.4.1. In either mode of operation, the streak camera data record is stored by the temporal analyser in a 256 x 256 x 16-bit pixel array, which can be displayed on the TV monitor screen. Additionally, the necessary interface is also in place to permit transfer of these data records to microcomputer or to the UBC mainframe computer, for storage and more detailed analysis. 3.4 Irradiation Conditions 3.4.1 Laser Focal Spot Measurements In this work, the intensity distribution of the main Nd-YAG/Nd-glass laser was char-acterized through both time-integrated and time-resolved focal spot measurements. To make these measurements, it was first necessary to image the laser focal spot onto the streak camera. This was facilitated by placing a grid of known spacing (83 pm ) in the plane of the target front surface (with the target removed), and imaging the grid through a series of lenses, onto the entrance slit of the streak camera. The grid was then viewed Chapter 3. Experiment 45 on the monitor screen with the streak camera in focus mode and the entrance slit open fully to a width of 5 mm. In order to prevent high-intensity damage to the steak camera, this was done with neutral density (ND) filters placed in the path of the imaging optics. This image of the grid was then optimally focused on the streak camera by fine-adjusting the positions of the imaging lenses. Magnification of the imaging system was determined by comparing the apparent grid spacing when viewed on the monitor, with its actual spacing (of 83 ^ m ). In this way the magnification of the imaging optics and streak camera system was found to be 68 ± 2. A typical focal spot intensity distribution is shown in figure 3.12. The contours shown connect regions having the same time-integrated intensity (normalized to 1.0). The spatial resolution in this measurement is estimated to be ~ 5 fim . As can be seen, the focal spot appears to be confined to an area of roughly 120 fim by 150 fim . Figure 3.13 shows both horizontal (X) and vertical (Y) scans averaged through a central 7 fim portion of the focal spot. These scans show that the intensity profile is essentially a distorted gaussian with intensity modulations of less than 3 to 1. Similar laser focal spot measurements were repeated over the full range of amplifier pumping levels used in the experiment. This test was necessary because each amplifier pumping level results in different thermal gradients in the amplifier rods, and this can cause the laser to focus differently. In these measurements, however, we observed only small variations in the shape and size of the laser focal spot. These minor differences resulted in variations of < 10% in the effective focal spot area, A^jf, as defined § 3.4.2. We also performed a time-resolved measurement to check for spatial hot spots in the focused intensity distribution. This was done with the streak camera operating in streak mode and the entrance slit open to ~ 200 fim . This measurement yielded the temporal evolution of intensity (with 30 ps resolution), across the central horizontal axis of the focal spot. The laser pulse was observed to be temporally a single gaussian pulse, and Figure 3.12: Time-integrated laser focal spot intensity distribution. Chapter 3. Experiment 47 0 50 100 150 200 Y-COORD [um] Figure 3.13: Time-integrated intensity profile averaged over the central 7 pm of the focal spot in the a) X-direction and b) Y-direction. Chapter 3. Experiment 48 no evidence of temporal hot spots was observed. This result was confirmed at each of the pumping levels used in the experiment. 3.4.2 Effective Intensity When a laser irradiates a target under focused conditions, it does so in a manner which is spatially nonuniform. This fact might lead one to conclude that a laser-driven shock would never propagate in a one-dimensional fashion, and that any comparisons be-tween experiment and one-dimensional hydrodynamic simulations would, therefore, not be valid. Experiments designed to look at shock breakout from the rear surface of thin-foil targets have shown, however, that a region of planar shock breakout does exist Consequently, the shock propagation within this region is quasi-one-dimensional and one can use this information to specify an effective (or average) experimental laser irradiance for use in one-dimensional computer simulations. In previous shock breakout studies in aluminum foils of 9 — 50 pm thickness, which were made using the same laser system as in our work, it was observed that the shock emerging from the rear surface of the foil is essentially planar, within an area containing ~ 80% of the focused laser energy'9 .^ In our study, similar breakout measurements could not be repeated on quartz or sodium chloride targets, because these materials were not obtained in the thin foil form (10-20 pm ) which would be required. On the other hand, the hydrodynamic behaviour of quartz and sodium chloride should not be much different than that of aluminum, since ia the inverse bremsstrahlung absorption regime, laser-driven ablation exhibits a weak dependence on the atomic number of the target material^8]. Thus the resultant breakout region, which is governed entirely by shock hydrodynamics, should be similar in all cases. We therefore assume in this experiment that the diameter of the planar shock front, Chapter 3. Experiment 49 will also correspond to that of the 80% energy region. Thus we define the effective (or average) experimental irradiance as * i = 0.80 E b u e r (3.21)' where Eia,er is the laser energy, rla,„ is the temporal FWHM of the laser, Ae;j = \E\^ is the focal spot area containing ~ 80% of the laser energy, and Deff is the diameter of this region. This prescription has been used to specify effective irradiance in previous one-dimensional hydrodynamic simulations in aluminum targets'31' 9 5]. The calculated ablation pressures and shock trajectories were found to agree well with the experimental results, suggesting the validity of such an approximation. In our work, equation (3.21) will be used to specify the effective irradiance in all the one-dimensional LTC simulations performed in quartz (see chapter 4). 3.5 Experimental Details 3.5.1 Targets The target materials investigated in this study included fused quartz'96], x-cut alpha quartz'97] and sodium chloride'98]. Some measurements were also made on sodium chloride targets which had been coated with a thin layer (10-20 pm ) of polystyrene (CH). The targets were manufactured in the shape of rectangular slabs of dimensions 2mm x 25mm x 75mm. All of them arrived with their side surfaces optically polished to a A/10 finish, and were guaranteed to be of extremely high chemical purity and virtu-ally free of any inclusions. In the lab, a polishing jig and various polishing compounds were used to produce a slight, concave curvature (R = 1.6 mm) along the 2 mm edge of the target slab. This procedure was necessary to allow the target interior to be clearly imaged using optical shadowgraphy, as described in § 3.5.2. Chapter 3. Experiment 50 3.5.2 Experimental Arrangement The experimental observations discussed in this work consist of a set of shock trajec-tory measurements obtained using optical shadowgraphy '^ The complete experimental arrangement used in making these measurements is shown schematically in figure 3.14. Details of the experimental set-up in the vicinity of the target are shown in figure 3.15. High intensity irradiation of the target was provided by the Nd-YAG/Nd-glass laser, which was operated at a frequency-doubled wavelength of 0.53 fim (see § 3.2). A series of three dichroic mirrors were used to direct this beam towards the target chamber, where it was then focused with / / l l optics, onto the central region of the concave target edge. Side-illumination of the target was provided by a low-power dye laser (Rhodamine 6G) operating at a wavelength of 0.57 fim . This dye laser, which we subsequently refer to as the probe beam, was optically pumped with a small amount of energy extracted from the main Nd-YAG/Nd-glass laser beam. A pulse stacker consisting of a sequence of four quartz flats, each spaced equally by ~ 30 cm, was used to extend the probe laser pulse width to ~ 6 — 7 ns FWHM (cf. the main laser pulse width of ~ 2.5 ns). This was of sufficient duration to span the experimental observation time (streak camera limited) of 5.64 ns. The probe beam was focused at normal incidence onto the side surface of the target using a long focal length lens (/ ~ 1 m). This provided nearly uniform illumination of the target edge over a region of several hundred microns in diameter. Probe beam light which passed directly through the target was collected by a 10 x microscope objective which was used to image the target interior. The curvature in the target edge prevented scattered or diffracted light from the target corner from entering the microscope objective and disturbing the imaging of the target interior. Specifically, the curved surface caused Chapter 3. Experiment 51 PULSE d PHOTODIODE (PULSE SHAPE) Figure 3.14: Schematic of the complete experimental arrangement. Chapter 3. Experiment 52 M A I N L A S E R P R O B E L A S E R V ? T A R G E T OPAQUE I TRANSPARENT z 2 C O V E R O B J E C T I V E Figure 3.15: Details of the experimental arrangement near the target. Chapter 3. Experiment 53 all such light to be either reflected of refracted out of the field of view of the objective. Additionally, this arrangement also served to block out coronal plasma emissions. The image viewed by the microscope objective was relayed, via a series of achromatic lenses, onto the entrance slit of the streak camera. The optics were arranged to image the target interior along the central axis of shock propagation, in the region near the target edge. This was accomplished by placing a grid of known dimensions (83 fim ) in the median plane of the target (with the target removed) and adjusting the lens positions until the grid was in focus when viewed on the streak camera. (This was the same procedure as used in the focal spot measurements described in § 3.4.1.) In this way the magnification of the probe beam imaging system was found to be 190 ± 2 or, equivalently, 0.5 fim /pixel on the streak camera monitor. A timing reference, or fiducial, was also denned for the experiments. This reference was chosen such that t = 0 corresponded to the instant at which the main laser reached peak intensity at the target front surface. In the experiment, this was implemented by having a small fraction of the main beam (i.e. the fiducial signal) traverse an optical path length to the streak camera which was very nearly equal to the optical path length of the probe beam. The residual timing difference between the main beam and the fiducial was then absolutely calibrated by firing the main laser onto a scattering center located at the target plane, and comparing the time delay of the scattered signal with that of the fiducial signal as observed on the steak camera. In this way, the fiducial signal was found to lag the main beam by 440 ± 100 ps. This difference was included in the analysis of the experimental results presented in chapter 4. In the experiment, a Gentec energy meterwas used to measure the the energy of the main laser pulse. The error in the energy measurement was estimated to be < ±5%. The main laser pulse shape was also recorded using a fast (350 ps risetime) Hamamatsu photodiode'1^ and a high-speed oscilloscope. These oscilloscope images were recorded Chapter 3. Experiment 54 on polaroid 667 film and later digitized in a form which was suitable for input to computer simulations (see chapter 4). Figure 3.16 shows a typical example of such a digitized laser pulse as well as the oscilloscope image from which it was obtained. 3.5.3 Experimental Procedure and Shadowgram Measurements Before making any shadowgram measurements, it was necessary to align the various optical subsystems of the experiment. This was accomplished in several stages using the main laser in low-power mode (see § 3.2). First, the main beam was aligned to the center of the concave target edge by arranging a direct back reflection from the polished surface of the target. This was done with the main beam suitably attenuated by ND (neutral density) filters. This ensured that the main beam hit the target at normal incidence and at the bottom of the concave surface which, as described later, corresponds to the observed edge in the shadowgram measurements. Next, the probe beam imaging optics were aligned to view the region of the target interior directly underneath the main laser focal spot. This alignment was accomplished by producing a small damage mark in target with a low-power shot from the main laser. The image of this damage mark was then centered vertically on the entrance slit of the streak camera by adjustment of turning mirrors in the probe beam imaging optics. Error in alignment of the imaging system with respect to the central axis of the main laser beam was thus estimated to be < ±5 pm . Prior to each shadowgram measurement, a cover glass was inserted between the tar-get and the microscope objective. This cover glass served to protect the objective from possible damage from plasma debris and was so thin as to have no effect on the target imaging system. Additionally, because of the high laser intensities used in these exper-iments, the measurements were performed under a moderate vacuum (~ 60 mtorr) in Chapter 3. Experiment 55 Figure 3.16: A digitized representation of a typical main laser pulse. Also shown is the oscilloscope image which was digitized (inset). Chapter 3. Experiment 56 order to prevent air breakdown. In the experiment, the streak camera was operated on the slowest streak speed setting (5.64 ns/15 mm) and the entrance slit width was set to between 200-400 pm . Scattered light from the main beam (or from plasma emissions) was prevented from reaching the streak camera by two lOOA band-pass interference filters centered at 5700A and ap-propriate ND attenuators, which were inserted in the path of the probe beam imaging optics. Figure 3.17 shows an example of a streak shadowgram obtained with probe beam illumination of the target, but no main beam irradiation (i.e. the main beam is blocked in this measurement). The dark region in this image represents the target interior which, under conditions of no main beam irradiation, remains transparent to the probe light at all times. On the other hand, the probe beam is not transmitted through the concave front surface of the target and this causes the light vertical region seen in the leftmost portion of the streak image. The boundary between the light and dark regions constitutes the observed edge (or spatial origin) of the target in this measurement. Accuracy in determining this edge position was estimated to be ~ ±3 pm . Similar streaks of the probe beam illumination were repeated after every few shock trajectory measurements. This allowed the probe beam uniformity to be checked and the location of the target edge to be recorded for future reference. In figure 3.18 we show an actual shock trajectory shadowgram measurement obtained under simultaneous probe beam and main beam illumination. The time origin is located at the peak of the main laser pulse and is determined from the fiducial signal (also shown) as described in § 3.5.2. The spatial origin defines the position of the target edge and is obtained prior to the shock trajectory measurement as outlined above. In these shadowgram measurements, the shock front is normally assumed to coincide with the time-dependent shadow boundary as observed in figure 3.18. This interpretation Chapter 3. Experiment 57 wil l be valid in the case of a strong shock, where the shocked material is compressed and ionized sufficiently that it is rendered opaque to the probe beam light. However, it may not hold at very early times in the measurement when the shock is too weak to produce significant ionization. In any case once a strong shock is formed, measurement of the position of the shadow boundary (x) does constitute a measurement of shock position, while the slope of the shadow boundary (Ax/At) yields the shock speed. Further examples of shadowgram measurements obtained in our experiment are presented in chapter 4. Chapter 3. Experiment 58 0 20 AO 60 POSITION (pm) Figure 3.17: An example of a shadowgram obtained with only probe beam illumination of the target. Chapter 3. Experiment 59 POSITION (Lim) Figure 3.18: Shadowgram obtained in fused quartz under main beam irradiation. The shadow boundary marks the position of the shock front. Also indicated is the fiducial signal. Chapter 4 Experimental Results and Interpretations 4.1 Introduction We begin this chapter with a discussion of the experimental observations obtained in fused quartz, a-quartz and sodium chloride. This is followed by a detailed comparison of the experimental data in the quartz case with results of computer simulations. Possible interpretations for the quartz data are given and discussed in some detail. 4.2 Experimental Observations in Quartz In this work, the primary experimental observations consisted of a set of shock trajectory measurements, obtained by the streak shadowgraphy technique described in chapter 3. Figures 4.19 and 4.20 show examples of typical streak shadowgrams obtained with this technique in fused quartz and in a-quartz, respectively. The spatial origin (x = 0) locates the position of the target edge in the measurements, and was determined by streaking the target with probe beam illumination in the absence of main beam irradiation, as described in chapter 3. The temporal origin (t = 0), on the other hand, is denned by the instant at which the main laser reaches peak intensity at the target edge. This origin was derived from the peak of the fiducial signal (also shown in figures 4.19 and 4.20) and the measured time delay between the fiducial and the main beam (see also chapter 3). Spatial and temporal uncertainties in the measurements were estimated to be ±3 fiva and ±100 ps, respectively. Magnification of the shadowgraphic imaging system was 190 ± 2. This 60 Chapter 4. Experimental Results and Interpretations 61 was sufficiently large to permit observation of the target interior to a depth of ~ 80 pm . In the shadowgram images, the main laser irradiation is incident on the target from the left, resulting in shock propagation from left to right in the figures. The shock front itself is assumed to coincide with the time-dependent shadow boundary evident in these images. As described in chapter 3, this identification requires that the shock is sufficiently heated and ionized that it is rendered opaque to the probe beam. In figures 4.19 and 4.20, the shock can be seen to penetrate the target to depths of 50-60 pm within the experimental observation period (~ 5.64 ns). Overall in the experiment, the shock propagation distance varied between 40-65 pm . In order to analyse these measurements in more detail, an interactive edge-detection program was used to extract the shadow boundary (i.e. the shock trajectory) from the raw shadowgram images obtained in the experiment W^-l. Figures 4.21 and 4.22 show the output produced by this program for the shadowgrams in figures 4.19 and 4.20, respectively. The shock trajectory is represented on these plots by the sequence of triangular symbols. The program locates the position of the shock at each instant in time through a process of differentiation along the spatial direction. Apparent roughness in the shape of the shock trajectory can be attributed to noise or ambiguity in the original shadowgram signal. Spatial and temporal origins are defined as above, and are obtained by supplying as input to the program, both the measured target edge position and the measured time delay of the fiducial signal. An estimate of instantaneous shock speed was also made by fitting a piecewise linear curve to each of the measured shock trajectories. The results of these computations, expressed in units of 108 cm/s, are indicated in figures 4.21 and 4.22 by a discrete sequence of hexagonal points shown with error bars. The horizontal error bars denote the time interval over which a linear segment was fitted to the trajectory, whereas the vertical error bars denote the standard error in the slope of the fit (i.e. the shock speed) for that Chapter 4. Experimental Results and Interpretations POSITION (Lim) Figure 4.19: Shadowgram obtained in fused quartz. The laser irradiance in this surement is $L = 1.4 x 1013 W/cm2. Chapter 4. Experimental Results and Interpretations 63 POSITION (jjm) Figure 4.20: Shadowgram obtained in a-quartz. The laser irradiance in this measurement is $ £ = 1.0 x 1013 W/cm2. Chapter 4. Experimental Results and Interpretations 64 Figure 4.21: Shock trajectory and shock velocity curve for fused quartz derived from the shadowgram in figure 4.19. Open triangles represent the shock trajectory, whereas the hexagons represent estimates of shock speed. Also indicated are the peak transient speed Dp, the steady asymptotic speed D/, and the relaxation time r r, which were extracted from the shock velocity curve. Figure 4.22: Shock trajectory and shock velocity curve for a-quartz derived from the shadowgram in figure 4.20. Open triangles represent the shock trajectory, whereas the hexagons represent estimates of the shock speed. Chapter 4. Experimental Results and Interpretations 66 time interval. In contrast to the actual shock trajectory, the velocity curve is not sensitive to errors in the location of the spatial origin. These errors could shift the shock trajectory by possibly as much as ±3 pm but would leave the velocity curve unaltered. Moreover, because the velocity curve is derived from the slope of the shock trajectory it provides more detailed information about the dynamics of shock propagation inside the target. For this reason, a good deal of the analysis to follow will be based on a comparison of experimentally derived velocity data with the results of theoretical calculations. For the purposes of our analysis we specifically consider two shock velocities, Dp and Df, which are extracted from the shock velocity curve as shown in figure 4.21. Here Dp corresponds to the maximum shock speed attained in the measurement, whereas Df refers to the steady, final shock speed attained towards the end of the observation period (at t ~ 2 ns). Also indicated in the figure is the relaxation time between these two velocity states which we define as r r. Table 4.1 compiles these parameters and other experimental data for all the shock trajectory measurements obtained in fused quartz. Table 4.2 summarizes the same data for a-quartz. The results are presented in order of decreasing laser irradiance so that comparisons between the two tables can be made easily. The laser irradiance was obtained from equation (3.21) which involves the laser energy, E L , the effective laser focal spot size, Deff, and the laser temporal pulse width (FWHM), TX (these values are also tabulated). In the experiment, varied by nearly an order of magnitude, from 2 x 1012 W/cm2 to 1.9 x 1013 W/cm2. Experimental uncertainties incurred in these measurements are not tabulated, but are significant. For example, these include ±5% uncertainties in both EL and TL, as well as a ±10% uncertainty in Deff. These uncertainties lead to a ±30% uncertainty in based on equation (3.21). In addition, the the shock speeds Dp and Df extracted from Chapter 4. Experimental Results and Interpretations Table 4.1: Experimental data for fused quartz 67 SHOT EL TL Deff Df (J) (ns) ( ) (W/cm2) (km/s) (km/s) (ns) 5/31/2 8.0 2.2 140 1.9 x 1013 15 11 2.0 4/27/6 7.3 2.6 140 1.4 16 10 2.0 4/27/3 6.2 2.6 140 1.3 17 11 1.6 4/27/2 5.7 2.6 140 1.1 17 10 1.7 4/26/7 5.4 2.9 140 1.0 16 8.5 2.6 5/03/2 5.3 2.4 140 1.1 17 11 1.8 4/26/10 4.8 2.8 140 0.90 15 13 1.5 4/27/5 1.8 2.5 150 0.60 14 8.9 1.9 5/31/7 1.4 2.0 150 0.31 9.1 7.0 2.6 5/31/4 1.3 2.1 150 0.28 10 7.8 2.1 5/31/6 1.2 2.1 150 0.26 9.6 7.4 1.8 4/27/9 1.1 2.2 150 0.22 9.8 6.4 2.4 Table 4.2: Experimental data for a-quartz SHOT EL Defi DP Df (J) (ns) ( /^ m ) (W/cm2) (km/s) (km/s) (ns) 5/31/10 8.0 2.4 140 1.7 x 1013 14 10 1.0 5/31/11 7.6 2.6 140 1.5 16 11 1.5 4/11/13 6.8 2.6 140 1.4 19 8.7 2.2 4/11/12 6.1 2.7 140 1.2 18 10 1.8 4/11/11 5.5 2.9 140 1.0 15 9.0 2.6 4/11/2 5.4 2.6 140 1.1 16 13 1.2 4/11/17 1.6 2.1 150 0.35 12 9.7 1.9 5/31/8 1.4 2.0 150 0.32 9.5 7.6 1.5 Chapter 4. Experimental Results and Interpretations 68 the trajectories also had associated uncertainties, typically of ±(1 — 2) km/s. There are several points regarding the experimental observations in quartz which are worth mentioning at this time. First of all, our findings suggest that, at least with respect to shock propagation behaviour, both of the quartz materials investigated actually behave quite similarly. This similarity is clearly demonstrated in the shock trajectories shown in figures 4.21 and 4.22, for example, which are representative of the larger set of trajectory data. Both trajectories are obtained at similar laser irradiances (1.4 x 1013 W/cm2 in fused quartz and 1.0 x 1013 W/cm2 in a-quartz) and both exhibit similar phases of shock acceleration and deceleration. Moreover, the peak and final shock velocity states attained in these observations appear to be almost identical in each case. This trend can be seen to be generally true of the data by comparing table 4.1 with table 4.2 at similar laser irradiances. For example, the experimental values for Dp are seen to vary between 9.1-17 km/s in fused quartz, while in a-quartz the corresponding values vary almost identically between 9.5-19 km/s. The range of Df values is also quite comparable; 6.4-13 km/s in fused quartz and 7.6-13 km/s in a-quartz. Furthermore, the relaxation from peak to final velocity states is also observed to occur on similar time scales in both quartz materials (Tp ~ 1 — 3 ns). These results will be discussed further in § 4.4-4.6. 4.3 Exper imental Observations in S o d i u m Chloride Figure 4.23 shows an example of a typical streak shadowgram obtained in sodium chloride at a laser irradiance of $x, = 1.0 x 1013 W/cm2. The spatial and temporal origins are defined in the same way as in the quartz measurements (see § 4.2). The spatial and temporal uncertainties are, similarly, ±3 pm and ±100 ps, respectively. Evident in the shadowgram is a very prominent "bulge" in the shadow boundary which persists from t ~ —1.5 ns to t ~ 1.5 ns. Furthermore, the shadow boundary is seen to actually Chapter 4. Experimental Results and Interpretations 69 reverse direction at t ~ 0 ns. This last point in particular suggests that the shadow boundary could not have been produced by a strong ionizing shock front (as described in § 3.5.3) since such an interpretation would require that the shock actually reverses direction as it propagates, and this is physically impossible. We therefore conclude that in sodium chloride, some phenomena must be causing ionization ahead of the shock front at sufficient levels to to produce the observed shadow boundary. One possible source for this ionization is hot electrons'1^2' As discussed in § 2.2.2, hot electron transport is a definite concern at very high irradiances ($x, > 1014 W/cm2) and/or long laser wavelengths (A^ > 1 pm ). Under such conditions, these hot or suprathermal electrons have significant mean free paths and can actually penetrate the target to depths exceeding the distance traversed by the shock front. The deposition of these hot electrons in the region ahead of the shock front can lead to significant preheating and ionization of the unshocked material. Such preheating effects should not be significant in our experiment, however, since the laser irradiance was moderate (< 2 x 1013 W/cm2) and the wavelength < 1 pm . In this case, the hot electron mean free path will be < C 1 pm and so negligible preheating can occur ahead of the shock. Another possible source for this ionization is x-rays produced in the hot coronal plasma'1^4]. Radiation in a plasma may occur through a variety of mechanisms in-cluding bremsstrahlung (or free-free) emission, free-bound emission or bound-bound emission'1^5, 1 9 6 ] . The first two processes result in continuous emission spectra, whereas the latter process leads to line radiation. Line radiation is the primary source of x-rays (photons of energy > 1 keV) in the coronal plasma. However, since line radiation is produced by bound-bound transitions in the coronal plasma, it is important only in high Z targets (Z > 10) where the plasma is incompletely ionized. Detailed experimental and theoretical studies in aluminum (Z ;^ = 13), for example, have shown that ~ 2% of the laser energy can be converted into line radiation in the energy range of 1.5-2 keV' 1 0 4). Chapter 4. Experimental Results and Interpretations 70 Chapter 4. Experimental Results and Interpretations Table 4.3: Experimental data for CH-coated NaCl 71 SHOT EL (J) TL (ns) Deff ( A*m ) (W/cm2) DP (km/s) D t (km/s) 4/20/7 5.9 2.4 140 1.3 x 1013 14 13 4/20/5 5.1 2.8 140 0.93 14 12 Moreover, these high-energy x-rays have long mean free paths and can penetrate the target to significant depths ahead of the shock front (~ 10 fim ), leading to significant preheating and ionization of the unshocked material. One might consider that similar ra-diation processes could be important in sodium chloride, since both sodium and chlorine have atomic numbers greater than 10 (Zjva = 11, Zc/ = 17). To test this hypothesis, some shadowgram measurements were obtained in sodium chloride targets which had been coated with a low Z layer of polystyrene (Zcff = 6). The results of these measurements are summarized in table 4.3. Figure 4.24 shows an example of such a shadowgram in which the thickness of the polystyrene layer is 10 — 20 /im . In this measurement, a; = 0 denotes the position of the polystyrene-sodium chloride interface, which was determined with an accuracy estimated at ±4 fim . The shock is produced by partial ablation of the polystyrene layer which, because of its low atomic number, is a poor source of coronal plasma x-rays. The shock then becomes observable in the shadowgram once it is transmitted across the interface and into the sodium chloride. Evidently, the shock front is clearly discernable in the shadowgram and is not obscured by any "bulge" in the shadow boundary as in the un coated sodium chloride targets. This observation appears to confirm our conjecture that the "bulge" in the pure sodium chloride shadowgrams is probably the result of x-ray ionization. In addition to this, and perhaps even more important, the shock propagation in the Figure 4.24: Shadowgram obtained in polystyrene-coated sodium chloride. The interface is located at x = 0. The laser irradiance is = 1.3 x 1013 W/cm2. Chapter 4. Experimental Results and Interpretations 73 Figure 4.25: Shock trajectory and shock velocity curve in sodium chloride obtained from the shadowgram in figure 4.24. Chapter 4. Experimental Results and Interpretations 74 sodium chloride portion of the target is observed to be very steady. This is illustrated in figure 4.25 which shows the shock trajectory and shock velocity curve obtained from the shadowgram in figure 4.24. This steady shock propagation is in marked contrast to the highly nonsteady shock propagation observed in both fused quartz and a-quartz at similar laser irradiances (compare figure 4.25 with figure 4.21 and figure 4.22). This finding is important because it suggests that the transient high-speed shock propagation observed in quartz is most likely an intrinsic property of this material under shock conditions, rather than simply an artifact of the experiment. We will discuss this point further in § 4.4. 4.4 Comparison of Experimental Observations with Computer Simulations in Quartz In this section we present a detailed comparison of measured and predicted shock dynam-ics as obtained in both fused quartz and a-quartz. The analysis will include a comparison of experimentally and theoretically obtained shock trajectories and shock velocity pro-files. Two separate computational codes, identified by the acronyms LTC and SHYLAC2, have been used here in order to provide the necessary theoretical support. The physics contained in each of these codes is discussed in detail in § 2.6. It is worth mentioning again, however, that whereas the LTC code is a complete laser-matter interaction code in one-dimensional planar geometry, the SHYLAC2 code is a purely fluid dynamic code in two-dimensional cylindrical geometry. In carrying out the computer analysis with these codes, a standard run procedure was adopted and consistently applied in all the simulations. For example, in the case of the LTC simulations, each run required the specification of the temporal profile of the laser pulse shape, the average (or effective) experimental laser irradiance, as well as the Chapter 4. Experimental Results and Interpretations 75 material equation of state. The first of these input requirements was provided by digi-tizing the laser photodiode signal corresponding to each shock trajectory measurement obtained in the experiment, as shown, for example, in figure 3.16. The effective laser irradiance was then determined according to the empirical relation expressed in equa-tion (3.21) which, as noted in § 3.4.2, has been successfully used to characterize intensity in previous studies utilizing the LTC code. Finally, all the EOS data required in the simulations were obtained from the SESAME data library. Table 7380 was used for the calculations in fused quartz, whereas table 7382 provided the corresponding data for the a-quartz calculations. The run parameters for the two-dimensional SHYLAC2 simulations were also stan-dardized, but in a shghtly different way. Since the SHYLAC2 code does not incorporate any laser-matter interaction, the temporal history of the ablation pressure calculated from the corresponding LTC simulation was adopted as the temporal pressure pulse in these simulations. The spatial profile of the pressure pulse was taken to be gaussian with an 80 /im FWHM. This spatial FWHM was consistent with 80 % of the pulse energy being confined to an area of ~ 140 fim diameter, as observed in the experiments (see Deff values in tables 4.1-4.2). Finally, all EOS data were obtained from the SESAME data library using the same tables as in the LTC calculations. In figures 4.26-4.27, the results of these calculations are compared with the experi-mental findings for a typical example in fused quartz in both the x — t (shock position versus time) and v — t (shock speed versus time) planes. A corresponding comparison is made for a-quartz in figures 4.28-4.29. In both cases the measured shock trajectories are denoted by the triangular symbols, whereas the experimental shock velocity data are represented by the sequence of open circles shown fitted by a smooth curve (dotted hne). On the trajectory plots, two curves are shown which correspond to the LTC simula-tions. The dotted curve traces the path of the so-called elastic wave (see § 2.4.1) which Chapter 4. Experimental Results and Interpretations 76 TIME (ns) Figure 4.26: Comparison of measured and calculated shock trajectories in fused quartz. The measured trajectory is represented by the sequence of open triangular symbols. The dotted and solid curves are obtained from the LTC simulations and correspond to the low-pressure elastic wave (p = 2.3 g/cm3) and the high-pressure compression front (p = 3.5 g/cm3), respectively. The SHYLAC2 simulation results (dot-dashed curve) are shown for the p = 3.0 g/cm3 density contour. The irradiance is 1.4 x 1013 W/cm2. Chapter 4. Experimental Results and Interpretations 77 2 Figure 4.27: Comparison of measured and calculated shock velocity profiles in fused quartz. The experimental data are represented by the sequence of open circles. The dotted curve represents a smooth fit to the velocity data. The solid and dot-dashed curves are obtained from the LTC and SHYLAC2 codes, respectively. Chapter 4. Experimental Results and Interpretations 78 Figure 4.28: Comparison of measured and calculated shock trajectories in a-quartz. The measured trajectory is represented by the sequence of open triangular symbols. The dotted and solid curves are obtained from the LTC simulations and correspond to the low-pressure elastic wave (p = 2.3 g/cm3) and the high-pressure compression front (p = 3.5 g/cm3), respectively. The SHYLAC2 simulation results (dot-dashed curve) are shown for the p = 3.0 g/cm3 density contour. The irradiance is 1.0 x 1013W/cm2. Chapter 4. Experimental Results and Interpretations 79 2 Figure 4.29: Comparison of measured and calculated shock velocity profiles in a-quartz. The experimental data are represented by the sequence of open circles. The dotted curve represents a smooth fit to the velocity data. The solid and dot-dashed curves are obtained from the LTC and SHYLAC2 codes, respectively. Chapter 4. Experimental Results and Interpretations 80 is launched inside both quartz materials at early times when the driving pressure (i.e. the laser ablation pressure) is weak (< 0.1 Mbar). The solid curve, on the other hand, represents a second, high-pressure wave which develops when the elastic hmit is exceeded and the quartz is driven into the mixed phase region where it begins to transform into stishovite. The transformation takes place across the wave front of this second wave and is completed at pressures of ~ 0.3 Mbar in fused quartz and ~ 0.4 Mbar in a-quartz. For pressures exceeding ~ 0.4 Mbar, the second wave travels faster than the elastic wave and the two disturbances eventually coalesce to form a single, strong shock. (Coalescence is indicated in the trajectory plots by the merging of the dotted and solid hnes.) Once formed, this strong shock will continue to build in intensity and accelerate so long as the ablation pressure is increasing, as shown in figure 4.30. However, due to the spatial separation between the shock front and the ablation front, the shock pressure and shock velocity predicted by the LTC simulations do not attain their peak values until 1-2 ns after the ablation pressure reaches its maximum at t=0 ns. Subsequently, as the ablation pressure begins to drop, a rarefaction or unloading wave develops at the front surface of the target. This wave propagates into the shocked medium at the velocity of sound in the shocked region, which is greater than the shock speed. Eventually it catches up to the shock front, where it leads to attenuation of the shock pressure and a reduction in shock speed. This phenomena, referred to as shock damping'1^, is manifested in figure 4.27 and figure 4.29 by the gradual decay in the predicted shock velocity towards the end of the recorded time interval. Also indicated in figures 4.26-4.29 are the results of the two-dimensional SHYLAC2 simulations in which the applied pressure pulse is of finite temporal and spatial extent. These calculations are shown by the dot-dashed curves in both the x — t and v — t planes. As can be seen, the inclusion of a spatially finite pressure pulse and the allowance for two-dimensional motion in these calculations has some effect on the resultant shock Chapter 4. Experimental Results and Interpretations 81 TIME (ns) Figure 4.30: A plot illustrating the temporal evolution of the ablation pressure (solid curve) and the shock pressure (dot-dashed curve) as calculated by LTC. The pressure scale is shown normalized to the peak ablation pressure. Also shown (in arbitrary units) is the laser pulse shape (dotted curve) which was supplied as input to the simulation. Chapter 4. Experimental Results and Interpretations 82 propagation. This is particularly noticeable in the shock velocity profiles, where several differences between the LTC and SHYLAC2 results are clearly discernable. First, the SHYLAC2 simulations yield lower peak (or maximum) shock velocities. Second, these simulations predict an earlier occurence of the maximum speed. And third, the SYHLAC2 simulations exhibit lower final shock speeds. These differences are manifested in the shock trajectories" by a slight reduction in late-time shock penetration depths. The physical basis for these discrepancies can be traced to the process of lateral (or edge) rarefaction of the shock which is included in the two-dimensional SHYLAC2 analysis, but not in the one-dimensional LTC analysis. Lateral rarefaction is an analogous process to the front-side rarefaction process described earlier. In this case, however, the rarefaction wave propagates from the edges of the shocked region towards the axis of the shock (i.e. laterally), due to the spatially nonuniform pressure profile. As it propagates inward (from opposite sides of the lateral pressure profile), it reduces the shock pressure and slows down the shock. Both front-side rarefaction and lateral rarefaction contribute to shock damping in the two-dimensional SHYLAC2 simulations, while only front-side rarefaction is present in the LTC simulations. Thus the gap between the LTC and SHYLAC2 shock velocity profiles gives a rough measure of the contribution of the lateral rarefaction process to shock damping. In comparison with these simulation results, the observed shock behaviour exhibits some similarities as well as some important differences. Referring again to figures 4.27 and 4.29, we observe that during the initial stages of shock development (say for t < — 1 ns), both the measured and calculated velocity profiles are in reasonable agreement. Moreover, both theory and experiment suggest that the shock undergoes almost con-stant acceleration during this time interval. Subsequently, however, the measured and predicted shock behaviour begins to diverge rapidly. On the one hand, the simulations indicate a gradual easing of shock acceleration and a "flattening" of the velocity profile, Chapter 4. Experimental Results and Interpretations 83 followed by a gentle decay in shock velocity towards the end of the observation period. While on the other hand, the measurements reveal a very prominent and short-lived surge in shock velocity, followed by a rapid and pronounced velocity decay. The peak shock speeds shown by this experimental transient greatly exceed the maximum shock speed predicted by either LTC or SHYLAC2. As alluded to earlier, this high-speed shock transient was observed to be a univer-sal property of both fused quartz and a-quartz over a wide range of shock conditions, corresponding to laser irradiances varying from 2 x 1012 to 1.9 x 1013 W/cm2. These findings are summarized in figure 4.31, where we have represented each shock trajectory measurement by its peak transient shock velocity, Dp, and its final or terminal shock velocity, Df, as defined in § 4.2. Also included in the figure are the results of LTC and SHYLAC2 simulations. The solid line corresponds to the peak shock velocity attained in the LTC simulations as a function of laser intensity. This line defines the upper bound on shock velocity since it was obtained under the assumption that the shock propagation is strictly one-dimensional. (Recall that the two-dimensional SHYLAC2 simulations yield lower peak velocities.) The dot-dashed and dashed lines correspond to the shock veloci-ties obtained in the two-dimensional SHYLAC2 simulations at t = 2 ns and at t = 3 ns, respectively. These two different times were chosen because they delimit the time interval in which the final shock speeds, Df, were extracted from the measured trajectories. Inspection of the plot clearly shows that, with the exception of a few data points at the highest irradiances, the peak transient shock velocities measured in the experiment are significantly larger than the maximum shock velocities predicted by the LTC code. In fact in some cases the predicted and measured shock velocities differed by as much as ~ 50%. These discrepancies appear to be too large to explain on the basis of experimental errors. In a typical observation in fused quartz, for example, a measured peak transient speed of Dp ~ 10 km/s was recorded at an irradiance of $ L ~ 2 x 1012 W/cm2, whereas an Chapter 4. Experimental Results and Interpretations 84 Figure 4.31: A plot of shock velocity versus laser irradiance. The solid and open circles respectively denote the peak and final velocities extracted from the measured shock trajectories in fused quartz. The solid and open squares denote the same quantities in a-quartz. The solid hne represents the maximum shock speed predicted by the LTC simulations. The dot-dashed and dashed curves show the velocity states predicted by SHYLAC2 at t = 2 ns and t = 3 ns, respectively. Chapter 4. Experimental Results and Interpretations 85 irradiance of ~ 5 x 1012 W/cm2 would be required to produce the same peak shock speed in the LTC simultions. In order to match the calculated and measured peak shock speeds in this case, we would have had to underestimate our effective irradiance by ~ 150%. But such enormous errors seem very unreasonable. In addition, we can rule out the possibility that these high-speed transients could be the result of some sort of laser hot spot effect, since similar velocity surges were not observed in sodium chloride. The simulations also fail to predict the enormous decay in shock velocity which is evident in the measurements immediately after the shock reaches its peak speed. In fused quartz this velocity reduction was found to range from 15-65%, or 45% on average, while in a-quartz the corresponding values ranged from 22-116%, or 50% on average. These levels of velocity attenuation are much larger than the 10-25% velocity reductions predicted by the two-dimensional code, due to the combined effects of front-side and lateral rarefaction attenuation. Thus two-dimensional effects alone can not adequately account for the observed decay in the high-speed shock transients. On the other hand, in most cases there appears to be reasonable agreement between the observed and calculated terminal shock velocities. This asymptotic agreement in shock speeds suggests that the pressures predicted by the simulations are actually at-tained in the measurements. (Recall that P oc D 2 , so that agreement in shock speed implies agreement in shock pressure.) Moreover, the steady nature of both the measured and predicted shock at late times (see figures 4.27 and 4.29) suggests that this phase of the observed shock propagation in quartz is adequately modelled by the equilibrium EOS description included in the simulation codes. However, as noted above, these equilibrium calculations can not adequately account for the high-speed shock transients observed in the quartz measurements. In our study, it was not possible to test the success of these codes in predicting shock dynamics in sodium chloride, since the necessary equation of state data was not available for this Chapter 4. Experimental Results and Interpretations 86 material. Nevertheless our observations of shock propagation in (CH-coated) sodium chloride targets did not indicate any high-speed shock transients as observed in quartz (see § 4.3). It therefore seems very likely that these high-speed transients are a real effect in quartz, which can not be explained by shock states on the equilibrium Hugoniot. 4.5 High-Speed Shock Transient In this section we elaborate on a possible explanation for the high-speed transient be-haviour observed in the quartz measurements. Specifically considered is the effect of a finite phase transformation rate on shock propagation dynamics in a phase transform-ing material. We will demonstrate how this process can result in nonequilibrium shock behaviour in which the shock propagates at a higher speed than that predicted from equilibrium considerations. It will then be shown how relaxation behind this nonequi-librium front leads to a decay in shock velocity and, ultimately, to the attainment of a steady, equilibrium shock state which lies on the equilibrium Hugoniot. Finally, we will discuss whether such a model can adequately account for the high-speed shock transients observed in our measurements in quartz. Before doing this, however, we begin with a brief review of equilibrium shock propa-gation in a phase-transforming material (see also § 2.4.2). The equilibrium Hugoniot for a phase-transforming material is schematically illustrated in figure 4.32. We recall that, under equilibrium conditions, the transformation from the low-pressure phase (phase 1) to the high-pressure phase (phase 2) is initiated at pressure PA and is completed at pres-sure PB- Pressures intermediate between PA and PB define the mixed phase region in which phase 1 and phase 2 coexist in appropriate fractions to make up the mixed phase volume. We found that the wave propagation possible in such a material could be divided Chapter 4. Experimental Results and Interpretations 87 Figure 4.32: Schematic illustration of the equilibrium Hugoniot curve of a phase trans-forming material (solid curve). Also shown is the extrapolated, or metastable Hugoniot for the low-pressure phase, corresponding to the absence of a phase transition (dashed curve). Labelled points refer to quantities discussed in the text. Chapter 4. Experimental Results and Interpretations 88 into three separate regimes, depending on the magnitude of the driving pressure. At driving pressures below PA, for example, a single shock is formed in which the material behind the wave front remains entirely in phase 1 (i.e. no phase change occurs). This disturbance propagates at a speed given by the Rayleigh Hne connecting initial and final states on the equiHbrium Hugoniot. The second regime occurs for pressures between PA and PE (eg. Pp), in which the wave disturbance is spht into two distinct waves moving at different speeds. The first and faster wave compresses the material from the initial state up to the pressure PA. This wave is foUowed by a slower, second wave which further compresses the material from pressure PA up to the final pressure (PD in this example). In this case, the phase tansition takes place across the front of the second wave which, under equiHbrium conditions, will have a thickness determined by the relaxation rate of the phase transition^. The third regime occurs for pressures above PE- Here a two wave structure is also formed, but in this case, because the second wave travels faster than the first, the two wave structure eventually coalesces, forming a single, strong shock. Here again, the equiHbrium shock front will be thickened due to the finite rate at which the transformation can occur there. The above analysis accurately describes the shock dynamics in the case that "sufficient time" has elapsed to estabHsh equiHbrium behind the shock front. However, it does not account for the shock behaviour prior to this when finite transformation effects can cause the structure of the shock front to deviate significantly from its equiHbrium form. In order to address this aspect in more detail, let us begin by supposing that associated with the transformation process is a characteristic transformation time, r, which describes the rate at which the material can transform from phase 1 to phase 2 under shock conditions. Let us further suppose that at some well defined instant in time, say t = 0, this material is rapidly loaded by a large pressure step of magnitude Pjf as shown in Chapter 4. Experimental Results and Interpretations 89 figure 4.32. The evolution of the shock front in such circumstances can be roughly divided into three separate time regimes: t <C v, t ~ r, and t >^ r, as illustrated in figure 4.33. In the first case, for which t <C r, the shocked material has not had enough time to undergo appreciable transformation from phase 1 to phase 2. Consequently, at these early times, the shocked material is compressed in its original phase only and no transformation has yet occured. Hence the initial shock state does not he on the equilibrium Hugoniot, but rather it hes on the nonequilibrium or metastable extrapolation of the Hugoniot for phase 1 (dashed curve in figure 4.32) at the point M*, corresponding to the absence of a phase transition. The Rayleigh line corresponding to this metastable state (line OM") is steeper than that corresponding to the equilibrium Hugoniot state at the same pressure (line ON). As a result, the shock speed attained during this nonequilibrium phase of propagation will exceed the equilibrium shock speed. The shock front in this case represents a steep transition between initial and final states as shown in figure 4.33a. At slightly longer times, when t ~ T , some of the shocked material has started to relax towards equilibrium. The initial layers of material which were shocked have had enough time to complete the transformation, while layers closer to the shock front have had less time to respond and are only partially transformed. This relaxation causes the shock to decelerate noticeably from its maximum speed, giving rise to nonsteady shock propagation. During this time the shock front also begins to thicken due to this relaxation process as shown in figure 4.33b. Eventually, for t >^ r, the material behind the shock front has relaxed sufficiently to attain a steady wave profile, as shown in figure 4.33c. In this steady condition, the end state of the shock hes on the equilibrium Hugoniot at the point N in figure 4.32. The shock speed, D, is also constant now, and can be calculated from the slope of the Rayleigh line ON. One may also note that the leading edge of the steady shock front hes Chapter 4. Experimental Results and Interpretations 90 (A) X ft (B) X Figure 4.33: A schematic illustration of the evolution occuring at the shock front of a phase transforming material which has been loaded by a pressure step of magnitude Ps. Profiles are shown at three different times: (a) t < r, (b) t ~ T , and (c) t >^ r, and could correspond to pressure or to density. See also figure 4.32. Chapter 4. Experimental Results and Interpretations 91 along the metastable Hugoniot for phase 1 at the state M. The phase transition is then accomplished behind this leading edge on a distance scale, d ~ Dr. Transformation rate effects in studies of shock-induced phase transformations are not new and, in fact, have been previously interpreted using similar relaxation models to that described above. Many of these studies have been compiled in the extensive review of phase transitions under shock wave loading by Duvall and Graham'9]. Iron'1^8] and antimony'1^9] are two well-known materials in which specific attention has been paid to transformation kinetics. In each of these materials, transformation rate effects have been signalled by a decay in the leading edge of the shock wave profiles with increasing sample thickness. Finite transformation rate effects have also been studied theoretically by Hayes'110' m l and by Andrews'112] using computer programs which incorporate models of the transformation kinetics. Their results display many features which are qualitatively consistent with the model of nonequilibrium shock propagation described above. The shock-induced transformation of quartz (both fused quartz and crystalline a-quartz) into stishovite has also been well researched (see § 2.4.2 for details and references). This transformation is known to be quite complex1 and some evidence exists which sug-gests that it may occur "sluggishly" under shock conditions. For example, Chhabildas et al.'113] found evidence for transformation rate effects in fused quartz through exam-ination of shock loading and release profile measurements obtained at pressures within the quartz-stishovite mixed phase region. In other studies, similar transformation kinetic effects have also been observed in x-cut a-quartz'114] and in novacubte'115] (a naturally occuring polycrystalhne a-quartz mineral). 1 Recall that the transformation involves a major change in chemical bonding structure (from tetra-hedral to rutile), and an almost two-fold increase in density of the material. Chapter 4. Experimental Results and Interpretations 92 One might naturally consider that similar transformation rate effects could be oc-curing in our measurements. Indeed, our observations of transient high-speed shock propagation in quartz suggest some similarities with the model of nonequilibrium shock propagation described above. Thus we might speculate that relaxation processes oc-curing in the quartz to stishovite transformation could provide the underlying physical explanation for the high-speed shock transients observed in the quartz measurements. Assuming this scenario, we obtained an estimate of the nonequilibrium shock states attained in the measurements. To do this, we extracted from each of the measured shock velocity profiles both the measured peak transient speed, Dp, and the time at which it occured in the measurement, tp. We then estimated the pressure at the nonequilibrium shock front, P", from the pressure calculated in the one-dimensional LTC simulations, Psim(t), using the prescription P" = Ptim{tp)- The nonequilibrium Hugoniot points were then taken as the shock speed - shock pressure pairs (DP,P~). The corresponding compression levels (V/V0) for these nonequilibrium states were then inferred using the Rayleigh hne relation between pressure, volume and shock speed (equation (2.7)). A plot of the nonequilibrium Hugoniot points obtained in this way are compiled for fused quartz in figures 4.34 and 4.35. Also shown for comparison is the SESAME equi-Hbrium Hugoniot for fused quartz (sohd curve). The nonequiHbrium points for a-quartz are compiled in figures 4.36 and 4.37. Also shown for comparison are the SESAME equi-Hbrium Hugoniot for a-quartz (dot-dashed curve), and the "quasi-a-quartz" Hugoniot calculation for a-quartz (dotted curve), in which the quartz to stishovite transition is suppressed (see § 2.5.2). (This "quasi-a-quartz" Hugoniot corresponds to the metastable or nonequiHbrium Hugoniot discussed in figure 4.32.) Error bars are indicated on aU plots for a representative coUection of data points. In almost every case, the nonequiHbrium Hugoniot points derived from the experi-mental data can be seen to differ markedly from the equiHbrium Hugoniots. In a-quartz, Chapter 4. Experimental Results and Interpretations 93 2-| P (Mbar) Figure 4.34: A plot of the non-equilibrium Hugoniot points obtained from the exper-imental shock transients in fused quartz (solid circles). Also shown for comparison is the corresponding equilibrium Hugoniot for fused quartz (solid curve) obtained from SESAME table 7380. Figure 4.35: Same data as in figure 4.34 shown in the P — V plane. Chapter 4. Experimental Results and Interpretations 95 Figure 4.36: A plot of the non-equilibrium Hugoniot points obtained from the experi-mental shock transients in a-quartz (solid squares). Also shown for comparison is the corresponding equiHbrium Hugoniot for a-quartz obtained from SESAME table 7382 (dot-dashed curve), and the "quasi-a-quartz" Hugoniot calculation for a-quartz, in which the stishovite transition is suppressed (dotted curve). Chapter 4. Experimental Results and Interpretations 96 Figure 4.37: Same data as in figure 4.36 shown in the P — V plane. Chapter 4. Experimental Results and Interpretations 97 the "quasi-a-quartz" Hugoniot yields higher pressures and shock velocities than the equi-Hbrium Hugoniot. Yet the shock states predicted by these calculations stiU deviate sig-nificantly from the nonequiHbrium Hugoniot states inferred from the experiment. In fact, when viewed in the P — V plane, the experimental Hugoniot data 6eem to suggest that the shocked quartz is attaining transient states in which the quartz is highly incompress-ible. This incompressibiHty appears to be particularly evident for pressures less than ~ 2 Mbar. These states suggest densities at the nonequiHbrium shock front which are significantly less than stishovite density or, for that matter, significantly less than the density of metastably compressed quartz (i.e. less than the densities predicted by the "quasi-a-quartz" Hugoniot calculations). On the other hand, the asymptotic shock states attained in the measurements are consistent with shock states on the equiHbrium Hugoniot. This is indicated by the agree-ment between the measured final shock speeds, Df, and the final shock speeds predicted by computer simulations, as discussed in § 4.4. When taken together, these findings imply that the relaxation from nonequiHbrium towards equiHbrium must involve a significant amount of compression. This relaxation appears to occur on nanosecond time-scales. This is iUustrated in figure 4.38, where we have plotted the measured relaxation times (TT) in both fused quartz (soHd circles) and a-quartz (soHd squares), as a function of laser intensity. Evidently, the values of r r vary between 1-3 ns, with the average being ~ 2 ns. The data also indicate a weak negative scaling with laser intensity: rr ~ $2°' 1 5. This observed decrease in r r with increasing laser intensity suggests that the relaxation process might be occuring more quickly at the higher shock pressures and temperatures which are characteristic of these higher intensities. Moreover, combining the average value of rP with the average shock speed in the non-steady propagation regime of \{DT + Df) ~ 12 km/s, impHes effective relaxation distances of dr ~ 25 fim . Chapter 4. Experimental Results and Interpretations 98 [sui u i Figure 4.38: A plot of the experimental relaxation time obtained from the data. Fused quartz data are shown by the solid circles, whereas a-quartz data are denoted by the solid squares. The dashed line shows a linear fit to the data, resulting in a T , ~ $ dependence. -0.15 Chapter 4. Experimental Results and Interpretations 99 In summary, the data presented in figures 4.34-4.37 suggest nonequilibrium states which deviate significantly from equilibrium. The relaxation from these transient nonequi-librium states towards equilibrium apparently involves a significant amount of compres-sion at the shock front. Moreover, based on the experimental values for rr it appears that this relaxation process is accomplished on time-scales of the order of nanoseconds. Though these estimates of relaxation times are instructive, it should be kept in mind that they are only order of magnitude estimates of the true relaxation time for the quartz-stishovite transition. This is because the applied pressure pulse in our experiments is gaussian with a finite rise-time of ~ 2 ns and is not an idealized pressure step like that assumed in the model decribed above. Consequently, the time-scale on which the applied pressure pulse varies and the time-scale on which the relaxation processes are occuring are inextricably linked in our experiments, and we can not consider these two different processes independently. 4.6 ShoCk-Induced Amorphization of Quartz In an attempt to find a plausible explanation for the dynamic incompressibility inferred from the quartz measurements, we refer to the findings of recent static compression experiments involving members of the Si02 system. Hemley et al.' 1 1 6!, for example, have carried out such static compression experiments on a-quartz and on another crys-talline form of Si02 known as coesite. Their studies show that on static compression at 300 K, both a-quartz and coesite lose "long-range translational order" and actually become amorphous at pressures between 0.25-0.35 Mbar, rather than transforming to stishovite, the thermodynamically stable phase in this pressure range'117'. This amor-phization transition was observed to occur at a density of pa ~ 3.3 — 3.5 g/cm3, which Chapter 4. Experimental Results and Interpretations 100 is significantly less than the density of stishovite (p, ~ 4.6 g/cm3) in the same pres-sure range'1 1 8 ! . Moreover, when fitted to a third-order Birch-Murnaghan equation of state'119!, the Hemley data yield a room-temperature isotherm which actually lies above the SESAME Hugoniot curve for a-quartz, as shown in figure 4.39. In another static compression experiment, Grimsditch'120! found evidence to suggest that fused quartz also might undergo a pressure-induced amorphization transition. Ac-cording to the author, the transition occurs at pressures between 0.10-0.17 Mbar, and is signalled by a "non-trivial change in microscopic structure of fused quartz" as deter-mined by Raman spectra measurements. The new form of amorphous SiC>2 appeared to evolve continously from ordinary fused quartz, and was observed to be stable at ambient conditions. However, the density of this new phase was not determined in the study. The possibility that a similar amorphous phase could be produced directly by shock-loading quartz to comparable pressures has been pointed out by Hemley'116!. Indeed, if such amorphization of quartz were to actually occur at the shock front, it could signifi-cantly alter the process of shock compression as well as the shock propagation dynamics. For example, amorphization could introduce significant disorder into the shocked quartz through a combination of cooperative (or mutual) rotations of the linked SiC*4 tetrahedra, Si-O-Si bond bending, and distortions (or even disruptions) of the Si-0 bonds. These microscopic alterations could give rise to new degrees of freedom in quartz and associated entropy increases. Moreover, such an amorphous or disordered state would also be characterized by its own set of other thermodynamical and physical properties, including equation of state, bulk modulus, etc. Consequently, if such amorphization were to occur directly under dynamic loading, we would expect the compression process to be governed by the prop-erties of the amorphous phase, rather than by the properties of ordinary quartz. As noted earlier, Hemley's static compression measurements suggest that this amorphous phase of Chapter 4. Experimental Results and Interpretations 101 Figure 4.39: A plot of the 300 K isotherm which fits Hemley's static compression mea-surements on a-quartz (dot-dash curve). Also shown for comparison is the SESAME Hugoniot for a-quartz (dash-dash curve). Chapter 4. Experimental Results and Interpretations 102 quartz could be considerably less compressible than ordinary quartz (recall figure 4.39). Thus, if this incompressibility were to manifest itself under shock conditions, it could result in higher peak shock velocities than those obtained for shock states on the equiHb-rium quartz Hugoniot or the "quasi-quartz" Hugoniot (see § 4.5). Subsequent relaxation of the shocked material from this incompressible amorphous state into stishovite might then give rise to a comparable decay in shock speed to that observed in the measurements. Still, the exact nature of the shock dynamics which would arise under this scenario can not be predicted from this simple qualitative discussion. Such information could only be obtained with more detailed knowledge of the shock-induced disordered state, including its equation of state, and of the kinetics of the relaxation process into stishovite. Chapter 5 Conclusion 5.1 Summary of Results In this work we have used the laser-driven shock method to produce high-pressure shock waves in fused quartz, a-quartz, and sodium chloride. Experimental observations con-sisted of a set of shock trajectory measurements which were obtained by the optical shadowgraphy method. This arrangement permitted continuous observation of the tra-jectory within the initial 5-6 ns of shock formation and 40-65 /xm of shock propagation. Profiles of shock velocity as a function of time were then extracted from these trajectories using an analysis program. The major experimental findings in quartz include: • Smooth build-up of the shock at early times in both quartz materials, in keeping with the steadily rising ablation pressure apphed to the target surface. • Observation of transient, high-speed shock propagation in both fused quartz and a-quartz. • Subsequent rapid deceleration (or relaxation) of the shock from these peak velocity states towards a steady final shock speed occuring on a ~ 2 ns time-scale. • Similar peak and final velocities in both fused quartz and a-quartz at a given laser irradiance. 103 Chapter 5. Conclusion 104 • No evidence for a two-wave structure in quartz as had been previously observed in fused quartz at similar laser irradiances. In this study, sodium chloride was supposed to function as a high-pressure control substance whose shock dynamics were to be compared with those observed in quartz. It turned out, however, that direct observation of the shock was obscured in these measure-ments by ionization processes occuring ahead of the shock front. Though not considered in detail, this ionization was thought to be due to high energy x-rays (> 1 keV) produced in the high-Z sodium chloride plasma. This radiation effect was not evident in sodium chloride targets which had been coated with a thin layer of low-Z polystyrene. More-over, observations in these double layer targets indicated steady (or quasi-steady) shock propagation inside the sodium chloride portion of the target, and no high-speed shock transients as observed in quartz. The observations in quartz were found to be in noticeable disagreement with the predictions of both one-dimensional and two-dimensional hydrodynamic simulations in-corporating equilibrium equations of state. In particular, the peak shock speeds observed in the quartz measurements were found to significantly exceed the maximum shock speeds predicted by the simulations. In some cases the discrepancy in shock speeds was as large as 50%. Moreover, the prominent decay in the observed shock velocity could not be adequately explained by shock attenuation due to two-dimensional motion at the shock front. On the other hand, the steady final shock speeds attained in the measurements were found to agree quite well with the late-time shock speeds predicted from the two-dimensional simulations. Two nonequilibrium relaxation models were proposed for quartz, in an attempt to explain the origin of the high-speed shock transient. The first model considered the effect of a finite quartz to stishovite transformation rate on the resultant shock propagation Chapter 5. Conclusion 105 dynamics. A nonequilibrium or "quasi-a-quartz" Hugoniot was calculated assuming that shock compression of quartz occurs in the pure quartz phase, without transformation into stishovite. This analysis suggested higher shock speeds than those calculated from equilibrium considerations. However, the maximum velocity states predicted by this model were still well below the peak velocity states observed in the experiments. A modified version of this model was then considered, based on observations of pressure-induced amorphization in quartz in recent static compression experiments. In particular, it was hypothesized that at the shock front, the transformation of quartz into stishovite could be preceded by shock-induced amorphization (or disordering) of the quartz. It was argued that such disordering of the quartz could result in a signifi-cant increase in entropy at the shock front. Moreover, the shock compression process in such circumstances would be initially governed by the properties of this amorphous or disordered phase. Shock loading of the quartz into a sufficiently incompressible amor-phous state followed by relaxation into stishovite could explain the high-speed transient phenomena observed in the measurements. 5.2 Suggestions for Future Work As a natural follow-up to this study, one might consider a more systematic investigation of relaxation effects in quartz. For example, this could include a determination of relaxation time as a function of laser irradiance over a larger range of intensities than examined here. In particular, it may be interesting to explore these effects at pressures in and around the phase transition region for quartz, where transformation rate effects may be more dominant. The driving pressure pulse could also be optimized for this study in order to more closely approximate an idealized pressure step like that assumed in § 4.5. This would allow the relaxation processes to be studied independently, without Chapter 5. Conclusion 106 the complications arising from 6hock compression with a gaussian laser pulse. Two alternative schemes could be considered to provide the desired rapid pressure loading of the target. One possibility would be to use a flat-top laser pulse with a very rapid rise-time (picoseconds or less). Another possibility would be to use a double layer target in which some other material (eg. Al or CH) is bonded on to the quartz surface, with its thickness chosen to ensure complete shock build-up by the time the shock reaches the interface. In this case the rise-time of the pressure pulse applied to the quartz would be hmited by the properties of the shock front in the bonded material. 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