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Studies of dense plasmas in laser generated shock wave experiments Parfeniuk, Dean Allister 1987

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STUDIES OF DENSE PLASMAS IN LASER GENERATED SHOCK WAVE EXPERIMENTS By DEAN A. PARFENIUK B.A.Sc, University of British Columbia, 1981 M.A.Sc , University of British Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF PHYSICS) We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA February 1987 © Dean A. Parfeniuk, 1987 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 The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DF-fifVft-n A B S T R A C T Shock waves generated by laser-driven ablation in solids have provided a great opportunity for the study of dense plasmas. The work presented in this thesis include measurements of Hugoniot curves and the reflectivity of shocked aluminum. In these experiments, planar aluminum targets were irradiated with a 0.53/xm, 2ns (FWHM) laser pulse at irradiances up to ~ 10UW/cm2. Temporally and spectrally resolved measurements of the target rear surface luminous emission have yielded the shock speed and temperature Hugoniot curve which showed good agreement with equation of state predictions. In addition, temporally resolved reflectivity measurements of the shocked target rear surface compared well with a theoretical model for the electrical conductivity of a dense plasma. For copper and molybdenum targets, both the luminescence and the reflectivity measurements indicated that the heating of the dense target material was dominated by radiation transport from the coronal plasma rather than shock heating. An analysis of the molybdenum results indicate that x-ray shine-through may be the dominant energy transport mechanism to the target rear surface, whereas for the copper targets the transport process appears to be much more complex. iii T A B L E O F C O N T E N T S ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES , vi LIST OF FIGURES vii LIST OF ACRONYMS x CHAPTER 1 INTRODUCTION 1 1.1 High pressure physics in laser-matter interactions 1 1.2 Present investigation . 5 1.3 Thesis outline 6 CHAPTER 2 THEORY 7 2.1 Introduction 7 2.2 Laser-driven ablation 7 2.3 Simulations of laser-target interactions 10 2.4 The SESAME equation of state for aluminum 15 2.5 The SESAME electrical conductivity model 20 2.6 Lee and More's electrical conductivity model 27 2.7 Hugoniot relations 35 2.8 The role of two dimensional effects 36 CHAPTER 3 EXPERIMENTAL FACILITY 39 3.1 Introduction 39 3.2 Neodymium-glass laser system and beam diagnostics 39 3.3 Irradiation conditions 41 CHAPTER 4 LUMINESCENCE MEASUREMENTS 44 4.1 Experimental setup 44 4.2 Measurement of shock induced luminous emission 46 4.2a Measurement of shock speed in aluminum 49 iv 4.2b Measurement of shock pressure in aluminum 53 4.2c Shock speed and pressure Hugoniot curve for aluminum 56 4.3 Measurement of shock temperature in aluminum 59 4.3a Spectral temperature measurement 60 4.3b Reliability of the spectral measurement 68 4.3c Brightness temperature measurement 72 4.3d Simulation of the temperature measurements 73 4.3e Shock temperature and speed Hugoniot curve for aluminum . . . 81 4.3f Discussion of the Hugoniot measurements for aluminum 81 4.4 Luminescence measurements in Mg, A l , Cu, and Mo 84 4.4a Thick foils at a moderate laser irradiance 84 4.4b Mechanisms for energy transport to the target rear surface . . 107 4.4c Models for radiation transport 112 4.4d Copper and molybdenum at lower intensities 121 4.4e Discussion of the Cu and Mo luminescence measurements . . . . 121 CHAPTER 5 REFLECTIVITY MEASUREMENTS 125 5.1 Experimental setup 125 5.2 Measurement of the reflectivity of shocked aluminum 127 5.3 Simulation of the reflectivity measurements 130 5.3a Sensitivity of the reflectivity diagnostic 138 5.3b Limitation on the detection of R$ 141 5.3c Comparison of the simulations with experiment 144 5.4 Reflectivity of high Z targets 152 5.4a Reflectivity measurements on Cu as a function of laser intensity 152 5.4b Two dimensional model for radiation transport 160 5.4c Reflectivity measurements on Mo as a function of laser intensity 166 5.4d Discussion of the reflectivity measurements on Cu and Mo . . . 171 V CHAPTER 6 SUMMARY AND CONCLUSIONS 173 6.1 Summary 173 6.2 New contributions 175 6.3 Future work 176 REFERENCES 178 L I S T O F T A B L E S T A B L E I Laser paramaters vii L I S T O F F I G U R E S 1- 1 Laser-target interaction 2 2- 1 Laser-driven ablation 9 2-2 Profiles of hydrodynamic variables from a simulation 16 2-3 Temperature-Density regimes of EOS models in SESAME 18 2-4 Equation of state isotherms for aluminum 21 2-5 Electrical conductivity from the SESAME library 26 2-6 Temperature-Density regimes of the A m y p models in Lee and More's o . . . . 31 2-7 Electrical conductivity from Lee and More's conductivity model 34 2- 8 Principal Hugoniot curves for aluminum 37 3- 1 Experimental facility and beam diagnostics 40 3- 2 Focal spot distribution at tightest focus 42 4- 1 Experimental setup for the luminescence measurements 45 4-2 Spatially resolved measurements of the luminous emission from Al 46 4-3 Spatial profiles of the luminous emission from Al 48 4-4 Luminous intensity of a target rear surface as a function of time for Al . . . 50 4-5 Shock transit time as a function of foil thickness for <^>60 ~ 1 x 1014W/cm2 52 4-6 Laser absorption as a function of incident laser intensity for Al 54 4-7 Shock speed as a function of absorbed laser intensity for Al 55 4-8 Ablation pressure as a function of absorbed laser intensity for Al 57 4-9 Shock speed and pressure Hugoniot curve for aluminum 58 4-10 Dispersion of the optical system used for the spectral measurement 61 4-11 A streak record of a time resolved spectrum of the target rear side 64 4-12 Rear surface luminescent spectrum from Al integrated for 200ps 65 4-13 Rear surface temperature as a function of time 66 4-14 Average rear surface temperature as a function of time 67 4-15 Spectral resolution of the optical system 70 4-16 Measured tungsten spectrum for 2000K 71 4-17 Simulated profiles of density, temperature and luminous density 75 4-18 Calculated spectral and brightness temperatures for Al 78 4-19 Calculated time integrated spectral temperature and spectra for Al 80 4-20 Shock temperature versus shock velocity Hugoniot curves for Al 82 4-21 Time resolved spectra for Mg, A l , Cu and Mo for <^>60 ~ 1013W/cm2 85 4-22 Luminous intensity as a function of time for Mg, A l , Cu and Mo 88 4-23 Luminous intensity on a logarithmic scale as a function of time 90 4-24 Temperature as a function of time for Mg, A l , Cu and Mo 93 4-25 Time integrated spectra for Mg, Al , Cu and Mo 95 4-26 Density and temperature profiles for aluminum from a simulation 98 4-27 Density and temperature profiles for copper from a simulation 99 4-28 Density and temperature profiles for molybdenum from a simulation . . . 100 4-29 Trajectories of density layers from the simulation for aluminum 101 4-30 Trajectories of density layers from the simulation for copper 102 4-31 Trajectories of density layers from the simulation for molybdenum 103 4-32 Comparison of the copper simulation to the measurement 104 4-33 Comparison of the molybdenum simulation to the measurement 105 4-34 X-ray spectrum from aluminum targets at </>60 ~ 8 x 1013W/cm2 109 4-35 X-ray energy flux from A l , Cu and Mo I l l 4-36 Required x-ray conversion efficiency versus absorption coefficient 114 4-37 Comparison of Cu simulations including x-ray transport to measurements 116 4-38 X-ray opacity as a function of photon energy for Cu 117 4-39 Comparison of Mo simulations including x-ray transport to measurements 119 4-40 X-ray opacity as a function of photon energy for Mo 120 4-41 Comparison of Cu simulations to measurements at 4>QO ~ 7.5 x 1012W/cm2 122 4- 42 Comparison of Mo simulations to measurements at <f>so ~ 7.5 x 10l2W / cm2 123 5- 1 Experimental setup for the rear surface reflectivity measurements 126 5-2 A streak record of the rear surface reflectivity from aluminum 128 5-3 A normalized streak record of a rear surface reflectivity measurement . . . . 129 5-4 Reflectivity versus time for Us ~ 1.7 x 106cm/s in aluminum 131 5-5 Reflectivity versus time for Us ~ 2.2 x lOPcm/s in aluminum 132 5-6 Reflectivity versus time for Us ~ 2.5 x 106cm/s in aluminum 133 5-7 p, T and a as a function of shock speed in aluminum 135 5-8 Reflectivity for a shocked aluminum surface versus shock velocity 136 5-9 Simulated profiles of density, temperature and dR/dx 139 5-10 Reflectivity versus time for different probe laser wavelengths 140 5-11 Reflectivity versus time for different values of the electrical conductivity 142 5-12 Estimated reflectivity as a function of time 145 5-13 Comparison of simulated reflectivity to measurements; Us ~ 1.7 x 106cm/s 146 5-14 Comparison of simulated reflectivity to measurements; Us ~ 2.2 x 106cm/s 147 5-15 Comparison of simulated reflectivity to measurements; Us ~ 2.5 x 106cm/s 148 5-16 Density-temperature regimes probed in the reflectivity measurements . . . 150 5-17 Normalized streak records of probe beam reflection from Cu targets . . . . 154 5-18 Reflectivity versus time for the copper targets 157 5-19 Coordinates for the two dimensional direct heating model 162 5-20 Temperature contours from the two dimensional wave heating model . . . 163 5-21 Temperature contours for different values of the opacity 165 5-22 Normalized streak records of probe beam reflection from Mo targets . . . . 167 5-23 Reflectivity versus time for the molybdenum targets 169 L I S T O F A C R O N Y M S X A C T E X represents a computer program which generates equation of state data based on quantum statistical mechanical theory. The computer program was devel-oped by F.J. Rogers and H.E. DeWitt 5 9 . AIJ represents an equation of state model developed by D.A. Liberman et a l . 7 0 ' 7 1 called atom in jellium. APW Agmented plane wave method6 3. F W H M Full width half maximum. GRAY is an equation of state model developed by Grover, Royce and Young 6 2. LASNEX is a hydrodynamic code developed by American scientists for simulating iner-tial confinement fusion57. LTC is an extensively modified version of the hydrodynamic code MEDUSA. LTC (Laser Target Code) was developed at the University of British Columbia. MEDUSA is a hydrodynamic code that was developed at Rutherford Appleton Laboratory 5 0' 5 1. MFA represents the self-consistent Hartree-Fock-Slater mean-field approximation. This was used for equation of state calculations in the SESAME library 7 0. PEC is a hydrodynamic code which was used to calculate plasma profiles of an unloading free surface. PEC (Plasma Expansion Code) was developed at the University of British Columbia. SESAME is a equation of state and material properties library. SESAME data is avail-able on request from Los Alamos National Laboratory29. TFD represents the Thomas-Fermi-Dirac equation of state25. TFK represents the Thomas-Fermi-Kirzhnits equation of state26. TFNUC is a computer program which generates equation of state data 6 4 , 6 5 . This code was used to generate some of the SESAME data. CHAPTER 1: INTRODUCTION 1 C H A P T E R 1 INTRODUCTION 1.1 High pressure physics in laser-matter interactions Motivated primarily by the possibility of attaining the density, temperature and confinement conditions necessary for inertial confinement fusion 1 , 2, research on laser-matter interactions has been extremely active over the past decade. Such investigations also permit the study of many fundamental physical processes. Figure 1-1 shows a schematic diagram of a solid target irradiated by laser radiation at high intensities. Also identified in the figure are the various plasma regions: (i) the coronal plasma has densities below the critical density (the critical density is the density at which the plasma frequency is equal to the frequency of the incident laser light), (ii) the ablation zone is the plasma between the critical density layer and the solid density layer (the largest thermal fluxes exist in this region) and (iii) the shock compressed region is where the density exceeds solid density. The physics of nonlinear waves in plasmas have been studied extensively through laser-induced parametric instabilities3,4 in the coronal plasma. Absorp-tion of the intense laser light in the coronal plasma near the target surface results in electron thermal conduction at flux levels approaching the free streaming limit, the flux being highest in the ablation zone. This allows the possibility of resolving current uncertainties in the theoretical description of heat transport 5 - 8 . The ab-lation process has been studied extensively in the pas t 9 - 1 2 . For short wavelength CHAPTER 1: INTRODUCTION 3 (< 1/im) laser radiation, the ablation process produces pressures of the order of lOMbar even at moderate laser irradiances (~ 1014W/cm2). With these high pres-sures, strong shock waves are generated which can be used to study the properties of compressed solids13-16. Moreover, the coronal plasma is a rich source of x-rays17-20 which also provides a unique source for probing the compressed solid. Studies of shock compressed metals are addressed in this thesis. The equation of state, electron transport and x-ray transport in compressed solids are of great importance to the study of many phenomena in geophysics, planetary physics, high-density plasmas, nuclear explosions and inertial confinement fusion. A detailed review of equations of state for condensed matter was recently given by Godwal et al.21. The equation of state on the zero degree isotherm has been calculated using band structure calculations22,23. This isotherm is expected to be reasonably accurate for the entire pressure range from atmospheric pres-sure to pressures of hundreds of megabar. For nonzero temperatures all statis-tical models (Thomas-Fermi24, Thomas-Fermi-Dirac25, Thomas-Fermi-Kirzhnitz26 and the quantum statistical model27) become accurate in the high pressure limit (P > 1000M&ar). For simplicity, equation of state calculations at non-zero temper-atures employ a mean field theory to calculate the electronic structure and include only two body ion-ion correlations. However, in the intermediate regime of pressure (P < lOOOMbar) and temperature (lOOeV > T > 0) the validity of these mod-els is uncertain; thus, experimental measurements are needed to test theoretical estimates. Calculation of the electron transport coefficients is even more uncertain in this regime since they depend on the equation of state as well as on some estimate of the electron-ion collision cross-section. In the high density (p ~ solid density), low temperature (T < lOOeV) regime, the calculation of the electron-ion collision cross-section is difficult. In one approach, Lee and More28 used the Coulomb cross-section with several prescriptions for the electron relaxation time. In the SESAME data CHAPTER 1: INTRODUCTION 4 library 2 9 the collision cross-section was calculated using the electronic structure of the material in a partial wave analysis30. Both of these models include only two body collisions in the calculation of the transport coefficients. These calculations as well as the equation of state calculations are only accurate in the limit of zero temperature or in the limit of high pressure (P > lOOOMbar) and should be verified experimentally. Since dynamic pressures exceeding many megabar can be readily attained in solids using shock waves, equation of state data are generally derived from Hugoniot measurements21'31. An Hugoniot curve is a plot of one shock parameter as a function of another shock parameter. With the use of shock waves generated by nuclear explosions, accurate benchmark data have been obtained from the measurement of an absolute Hugoniot point 3 2. The shock wave velocity as well as the particle speed behind the shock wave were measured. Impedance matching measurements have also been obtained using shock waves generated by nuclear explosions33 (sample consists of two materials bound together to give a step discontinuity in the density). In the impedance matching method, the relationships governing the reflection of a shock wave from a contact surface between the two media are used to relate the states of the two materials34. This method requires that one of the materials has a known equation of state, thus, it is used as a standard to determine the equation of state of the other material. The advantage of this method is that the particle velocities need not be measured. The particle velocity is more difficult to measure than the velocity of the shock front. Pressures in excess of lOOMbar were obtained in these measurements. The development of the two stage light gas gun has resulted in accurate equa-tion of state data up to pressures of nearly 5M6ar 3 5 . Some of the most accurate Hugoniot measurements (< 1% error in pressure) have been obtained by Mitchell and Nellis 3 6 using this technique. On the other hand, ultrahigh pressure shock CHAPTER 1: INTRODUCTION 5 waves can be effectively produced in laboratory experiments by laser-driven abla-tion. Impedance matching measurements using laser generated shock waves have been obtained37. This has led to strong interest in obtaining Hugoniot measure-ments in laser-target interactions. 1.2 Present investigation The primary objective of this work is an experimental investigation of equa-tions of state, electrical conductivity and x-ray transport through compressed solids. For the equation of state studies and the electrical conductivity studies, aluminum was compressed by laser-generated shock waves. The Hugoniot curve of shock pres-sure as a function of shock speed was measured. The temperature on the Hugoniot curve as a function of shock speed was also measured. The shock speed and shock temperature were obtained from measurements of the luminous emission from the shock heated material 1 6 ' 3 8 ' 3 9 . The shock pressure was obtained from ion ablation measurements12. These were compared with the predictions of different equations of state models. The measured shock temperature versus shock speed curve pro-vided a much more sensitive means of discriminating between the predictions of the different equation of state models than the shock pressure versus shock speed measurement. This is because the calculated shock temperature is primarily depen-dent on the thermal part of the equation of state, E(p, T), which varies significantly between the different equation of state models. The work presented here repre-sents the first reported measurements of the shock temperature versus shock speed curve1 6. The electrical conductivity of the shock compressed aluminum was studied by measuring the reflectivity as a function of time of the rear surface of aluminum foils when a shock wave emerged from the target rear surface 4 0 - 4 2. These measurements were compared to simulations based on two different electron transport models. It was found that the accuracy of the measurement was sufficient to assess the CHA P TER 1: INTROD UC TION 6 validity of each model. In particular, the results provided the first measurement of the electrical conductivity of a strongly coupled degenerate plasma. A plasma is said to be strongly coupled if its potential energy is greater than its kinetic energy, and degenerate if Fermi-Dirac statistics are required to describe the electrons. Finally, both the luminescence and the reflectivity measurements indicated that x-ray transport from the coronal plasma was the dominant heating mechanism of the rear surfaces of copper and molybdenum targets 4 3 , 4 4. A simple radiation transport model based on x-ray shine-through was used to interpret the data. This simple model may explain the molybdenum results. However, for the copper targets, the process of x-ray emission and transport appear to be quite complex. 1.3 Thesis outl ine In chapter 2, a brief review of the physics in laser-target interactions is given as well as an outline of the physics in the equations of state and the electron trans-port models used in the simulations. This is followed by a brief discussion of the Hugoniot relations and the role of two dimensional effects in this work. Chapter 3 contains a description of the experimental facility. The luminescence measurements and their interpretation are given in chapter 4. This includes the shock pressure versus shock velocity and the shock temperature versus shock velocity curves of aluminum as well as a description of the calculation used to simulate the tem-perature measurements. The results of the luminescence measurements on copper and molybdenum targets are also given as well as a simple x-ray transport model used to interpret the results. Chapter 5 presents the reflectivity measurements on aluminum targets and a description of the calculation used to simulate these mea-surements. The reflectivity measurements on copper and molybdenum targets are also presented. A summary of the results and conclusions is given in chapter 6 as well as suggestions for further investigation. CHAPTER 2: THEORY 1 C H A P T E R 2 THEORY 2.1 Introduction In this chapter, we begin with a qualitative description of laser-target in-teractions and an outline of the physics included in the hydrodynamic codes used to simulate our experiments. This is followed by a description of the equation of state in the SESAME 2 9 library. The electrical conductivity models used in our simulations are then discussed. Two electrical conductivity models were used: a model developed by Lee and More 2 8 which is almost analytic and the model in the SESAME data library which is based on partial wave Ziman theory30. The Rankine-Hugoniot relations are then discussed. Finally, the role of two dimensional effects in this work are discussed. 2.2 Laser-driven ablation In this section we present a qualitative description of the ablation process of a solid target irradiated by a high intensity {(f>L > 10 1 1 W/cm 2), short wavelength {^L < lum) and long pulse laser (r^ > 0.1ns). For long laser pulses, the hydrody-namic variables adjust to the changing laser intensity on a time scale shorter than the time scale for the change in the laser intensity since the plasma flows through the laser heated region in a shorter time than 0.1ns. This leads to a steady state ablation process 9 - 1 1. CHAPTER 2: THEORY 8 Figure 2-1 shows a schematic diagram of the plasma regions in a steady state ablative process. The figure shows a cut-away view of the plasma down the laser axis. This figure represents a laser-irradiated target at a fixed time. The plasma density is very low far away from the target and increases closer to the target until it approaches that of a solid. Furthermore, the plasma is continually expanding into the vacuum. The laser light propagates from the vacuum up the density gradient. The plasma becomes more collisional as the laser light reaches the higher plasma densities since the electron-ion collision frequency increases approximately linearly with electron density10. The electrons oscillate in the electric field of the incident laser light, colliding with ions during their oscillatory motion. The laser energy is absorbed in the plasma due to these collisions. This absorption process is called inverse bremsstrahlung absorption10. For short wavelength lasers this absorption mechanism becomes dominant since the laser light penetrates nearly to the critical density layer, where the critical density scales as l/A2^. The critical density is the density which yields a plasma frequency equal to the frequency of the incident laser light. In fact, this absorption process is very efficient when the scale length of the plasma is much longer than the laser wavelength10 (scale length of the coronal plasma shown in Figure 2-1) so that essentially all of the laser light is absorbed before it reaches the critical density layer. In a steady state ablation process, the plasma scale length is always much longer than the laser wavelength. The energy absorbed in the plasma maintains a hot (~ IkeV) low density plasma. This region is called the coronal plasma as shown in Figure 2-1. Near this hot coronal plasma is the cold target material. Thus, there are strong temperature gradients in this region, consequently, there is a power flux towards the dense target material. This power flux continually vaporizes and heats the target materia] as it expands into the vacuum. The region between the coronal plasma and the solid is known as the ablation zone. This ablation mechanism develops high pressures in the plasma. These pressures are required to balance the momentum CORONAL PLASMA SHOCK COMPRESSED SOLID SHOCK FRONT ABLATION FRONT ABLATION ZONE gure 2-1 A s c h e m a t i c d i a g r a m of a so l id target i r r a d i a t e d by laser r a d i a t i o n at, h i g h intensi t ies . CHAPTER 2: THEORY flux in the blow-off plasma. This description of steady state ablation is only valid for short wavelength (short wavelengths, \L < lfim, are required to ensure collisional absorption), high intensity (the laser intensity must be high enough to create a plasma, 4>i > 1011 W / cm2) and long pulse lasers (the pulse length must be long enough to ensure a steady state ablation process, > 0.1ns)10. Details of the ablation process have been studied extensively 9 - 1 2. The pressures generated in the coronal plasma and in the ablation zone are of the order of lOMiar for a moderate laser intensity, ~ 10l4W/cm2. This will drive a strong shock wave into the target material. The measurements reported in this thesis are mainly concerned with the state of the compressed and heated target material. 2.3 S i m u l a t i o n s o f l a s e r - t a r g e t i n t e r a c t i o n s The main objective of this section is to describe, briefly, the physics incorpo-rated in the hydrodynamic codes used to model our experiments. The dynamics of laser-target interactions involves a variety of mechanisms. Although any single process can be described using simple analytical models when decoupled from the other processes, a complete description requires numerical cal-culations using large computer codes, or simulations, which contain all the relevant physics. The code we used to model laser-target interactions is an extensively mod-ified version of the hydrodynamic code M E D U S A 4 5 ' 4 6 . This has been named LTC (Laser Target Code). The original version of MEDUSA used a standard first order Lagrangian dif-ferencing scheme45 incorporating the von Neumann-Richtmyer artificial viscosity45 for the treatment of shock waves, to solve the conservation equations of mass and momentum. In this scheme, temperature and density were taken as the independent variables. This version of MEDUSA gave accurate results when the target material is far from the cold curve (T > Tp where TF is the Fermi temperature). However, CHAPTER S: THEORY 11 since the internal energy of matter near the cold curve (T <C Tp) is relatively in-sensitive to the temperature, the code was not appropriate for cold targets. The original differencing scheme for the mass and momentum equations have been re-placed in LTC with a second order Godunov type scheme47 which used the density and internal energy as the independent variables. The resulting code operated well for all laser and target conditions. LTC is primarily designed to model the hydrodynamic and thermodynamic variables in one spatial dimension for a single fluid, two temperature plasma irra-diated by an intense laser beam. In this model, electrons and ions are assumed to have the same fluid velocity, implying no charge separation. However, each species maintains its own characteristic temperature due to weak coupling between the two in the low density and high temperature regimes of the plasma. Essentially, LTC integrates the conservation equations of mass, momentum and energy. The motion of the plasma is governed by the inviscid Navier-Stokes equation which represents the conservation of momentum. Here u is the velocity of the plasma which defines the motion of the Lagrangian coordinate according to dr dt [2-2] The hydrodynamic pressure, P, is defined by P = Pi + Pe [2-3] where the ion pressure, P{(p,E), and the electron pressure, Pe(p,E), were obtained from the SESAME data library for most of the simulations given in this thesis. For some of the simulations, an analytic approximation to the Thomas-Fermi-Kirzhnits CHAPTER 2: THEORY 12 (TFK) model 4 5 ' 4 6 was used for the electron pressure and an ideal gas law for the 4^ 46 ion pressure ' . An energy equation of the form dT dp s = £ V * + * i | 2 - 4 1 is used for both the ions and the electrons where S represents a power flux in the material and the terms which form S are called 'source' terms. In terms of the internal energy, E = E(p,T), the coefficient Cy and BT are: These were obtained from the SESAME data library. The source term, 5, for the electrons and ions is given by Si = H{-K + Qt Se = He + K + Q e + X [2-6] where H represents the heat flow due to thermal conduction, K is the rate of energy transfer between the ions and electrons, X is the rate of absorption of laser light and Q is the energy flux due to hydrodynamic motion (—PdV/dt, where V is the specific volume of the material). The heat source term is given by ff = - - v * v r [2-7] p where K is the thermal conductivity. The electron thermal conductivity was ob-tained from the SESAME data library whereas for the ion thermal conductivity, a Spitzer48 model was used. The energy transported by thermal conduction was CHAPTER 2: THEORY 13 not allowed to exceed some fraction of the free streaming limit 9. This is known as flux limited heat transport9 where the free streaming heat flux is that which results from a step discontinuity in the temperature porfile9. However, for our laser wavelength (\L < 1/xm) and laser intensities ((pr, < 2 x 1014W/cm2), this process is never invoked. The energy exchange term is given by 4 5 K = 0.59 x lQ-8ne(Ti - Te)T^^m~lZ2 log A [2 - 8] where m is the electron mass, Z, is the average charge state of the ions and log A is the Coulomb logarithm. The important point to note here is the density scaling which inevitably leads to Ti = Te in the high density regions of the target. Absorption of the laser light is assumed to occur via inverse bremsstrahlung up to the critical density. The source term X is given by A , ( r i ( ) = i ^ o ( 2 _ 9 | where 4>L, the local laser intensity, is related to the laser intensity at the plasma boundary, r = R0, by 4>L[r,t) = 4>L{Ro,t)exv[-a(Ro - r ) ] |2 - 10] The absorption coefficient, a, is taken from J.R. Stallcop and K.W. Billman 4 9 . LTC models anomalous absorption by dumping a fraction of the laser power 4 5 ' 4 6 reaching the critical density into the first overdense zone. This rather crude approach is adequate in the regime of laser intensities of interest here since the anomalous absorption is less than a fraction of a percent even at the highest laser intensities43. The calculation of the hydrodynamic motion is performed with an explicit Godunov type scheme47. This automatically takes into account shock heating, which is included in the term Qt. The energy equation is solved implicitly and uses CHAPTER 2: THEORY 14 the flux Q calculated in the hydrodynamic step as one of the source terms in S. A Crank-Nicolson45 scheme is used for the energy equation. This is essentially all the physics included in the simulations used to model laser-target interactions. The ponderomotive radiation pressure has been neglected. For our laser wavelength and irradiation conditions, this term is not important. The ponderomotive pressure is only important at very long wavelengths or very high irradiances. Energy loss due to hot-electron generation has also been neglected. This is reasonable since the energy that is carried away by hot electrons in our experiments has been measured to be less than 0.1% of the incident laser energy43. Energy loss due to stimulated scattering processes has also been neglected. Stimulated Brillouin 3 ' 4 scattering is the only significant reflection mechanism in our experiment50. This was accounted for by measuring the absorption of the laser light 5 0 (using a Ulbricht integrating sphere) and using the absorbed laser intensity in the simulation. This is required since the simulations predict that essentially 100% of the incident laser light is absorbed. The measured absorption was always greater than 85%. A detailed x-ray emission and x-ray transport calculation is not included in the simulation. X-ray transport into the dense target material is particularly important for high Z targets due to the high emission level of x-rays for these targets 1 7 - 2 0 . The spectrally integrated x-ray conversion was measured to be of the order of ~ 10% for copper in our experiment51. The measurements presented in this thesis indicate that energy transport from the coronal plasma is the dominant heating mechanism of the rear surfaces of thick (~ 20fim) copper and molybdenum targets. A target is considered to be thick if the transit time of a sound wave through the target is of the order of or longer than the laser pulse length. To model our measurements on high Z targets, a simple transport model was developed and incorporated into the hydrodynamic code. This will be described later. CHAPTER 2: THEORY 15 Figure 2-2 shows profiles of the hydrodynamic variables Ins after the time of peak laser intensity as predicted from the simulation for an aluminum target using a laser pulse which is Gaussian in shape with a pulse length of 2ns (FWHM). The peak laser intensity was 3 x 1013W/cm2 and the wavelength of the incident laser was 0.53/im. The equation of state and the thermal conductivity were obtained from the SESAME data library. Radiation transport was not included in this calculation. Initially the target front surface was located at x = 0. The coronal plasma, ablation zone and the shock front are indicated in the figure. Note that the ion and electron temperatures are equal behind the shock wave. This occurs since the relaxation time in the shock compressed material is of the order of O.lps, as estimated from the electrical conductivity in the SESAME library. Most of the measurements described in this work are measurements at the target rear surface as a shock wave emerges from this surface. To obtain high spatial and temporal resolution of the calculated plasma profiles at the target rear surface for comparison to these measurements, we only simulated the hydrodynamics of the shock unloading material. In these calculations, the profiles were calculated from the instant of shock wave arrival to later times. For these simulations a single temperature code PEC (Plasma Expansion Code), which incorporated a numerical scheme with flux corrected transport3 9'5 2, was used. The details of these calculations are discussed later. Some results from the hydrodynamic code L A S N E X 5 3 are also given in the thesis. This code was developed by American scientists to simulate inertial confine-ment fusion processes. 2.4 The SESAME equation of state for aluminum The SESAME data library provides equation of state and other material properties in tabular form on a temperature and density grid. Computer programs are also available to retrieve the data and interpolate between grid points. SESAME 91 AU03HI midVHJ CHAPTER 2: THEORY 17 data and computer programs are available on request from Los Alamos National Laboratory2 9. Seven different theoretical models were used to obtain an equation of state for aluminum over a large range in density and temperature54. Figure 2-3 gives the density-temperature regimes of each model as well as the areas where these models were replaced with an interpolation method to ensure a smooth global equation of state (shaded regions). Region 1, the ionization equilibrium regime, was treated with Roger's quantum-statistical-mechanical theory55 (ACTEX). This theory is valid in the presence of moderate to strong plasma coupling. (A plasma is strongly coupled if the potential energy of the plasma is much greater than its kinetic energy.) It is based on a many-body perturbation expansion. For temperatures below 0.8eF and densities below liquid density, the equation of state was calculated by Young using his soft sphere model for metals56. This is identified by region 2 in Figure 2-3. This is a semi-empirical model in that certain input parameters were adjusted to ensure the reproduction of experimental isobaric data. Region 3, between the soft sphere and A C T E X equations of state, is a narrow band of the ionization equilibrium equation of state. This was filled with calculations based on a Saha method57. The Saha region of the equation of state is based on the Planck-Larkin partition function57 for a single excited electron. This was modified with Debye-Htickel corrections57 to the ionization potentials. The region below temperatures of leV and densities between liquid density and a compression of 2 (region 4 of Figure 2-3) was treated with a semi-empirical model developed by Grover, Royce and Young called G R A Y 5 8 . The melting tran-sition is in this regime as well most of the experimental data. The GRAY model treats the solid equation of state with the Dugdale-McDonald form of the Gruneisen-Debye theory58. For the liquid, the solid Gruneisen equation of state is corrected with terms depending on the ratio of the temperature to the melting temperature CHAPTER 2: THEORY 18 10 -3 10"' 10' DENSITY (g/cm3) 10' Figure 2 - 3 The density-temperature regimes of each equation of state model used for the global equation of state for aluminum are given (From K.S. Trainor, H-Division Quarterly Report, Lawrence Livermore National Laboratory, UCID-18574-82-2 (1982), pages 20-23.) The shaded regions indicate the areas where a numerical interpolation scheme was required to join the equation of state models smoothly. CHAPTER 2: THEORY 19 at that density. The melt transition is calculated according to a Lindemann law 5 8. The experimental data used by GRAY are the normal-state data (solid density, sound velocity, Gruneisen gamma, and the electronic specific heat coefficient), the melting temperature at 1 atmosphere and the cohesive energy. Also used are ana-lytic fits for the Gruneisen gamma and the shock velocity as a function of density along the Hugoniot curve. The zero degree contribution to the pressure and energy for high compres-sion (p > 5g/crn3) was calculated by McMahan with band theory based on the self-consistent, augmented plane wave (APW) method59. The cold curve is repre-sented by region 5. For nonzero temperatures and densities greater than 5g/cm s the equation of state was calculated with the TFNUC code6 0-6 1. TFNUC generates a thermal electronic equation of state using the Thomas-Fermi-Kirzhnits model and a thermal nuclear equation of state with a Gruneisen-like theory at low temperatures and one-component plasma theory at high temperatures. This regime is region 6 in Figure 2-3. For the remaining region (region 7), Ross calculated the equation of state for the dense, partially-ionized liquid with the variational formulation of liquid metal perturbation theory62 which was generalized to account for the electronic excitation of conduction and core electrons. The equation of state in the SESAME data library was obtained by smoothly joining these separate equation of state theories. The shaded regions in Figure 2-3 indicate the regions where an interpolation routine was required to join these theories. At high temperatures, the models merge relatively smoothly whereas at low temperatures the physics is more complicated and there were some substantial mismatches between adjacent theories. In these cases, portions of each equation of state subsurface on either side of the boundary were replaced with a numerical interpolation routine to ensure a smooth and continuous final equation of state. For the joining procedure the authors of the equation of state demanded that: (i) all CHAPTER 2: THEORY 20 equation of state functions be continuous (except at a phase transition), (ii) the isothermal bulk modulus be positive and (iii) the heat capacity be positive. Figure 2-4 gives the final equation of state isotherms for both the pressure and internal energy. The solid-vapor and liquid-vapor transitions are evident in the figure. Melting is obscured by the relatively weak temperature dependence of pressure and internal energy near the melting transition. However, it is visible when one examines the thermal part of the equation of state. The shaded portions of the figure indicate, approximately, the regions relevant to the measurements presented in this thesis. 2.5 T h e S E S A M E e l e c t r i c a l c o n d u c t i v i t y m o d e l This model depends entirely on a particular application of the "average-atom" approximation30 that the material is a homogeneous medium in which there is distributed a collection of spherical scattering centres. The homogeneous medium is a theoretical model which represents a region of constant potential between scat-tering centres, through which the electrons may propagate freely as plane waves. The location of any two scattering centres is related by the two-body correlation function. Higher order correlations are not considered for simplicity. It is assumed that there is a certain density of free electrons which are free to propagate in the interstitial region. Only single-site scattering is considered for simplicity. This is essentially the mean free path approximation. The mean free path depends on the length scale set by the ionic separation distance. In Ziman theory30, this scale length arises from the two-body correlation function g{r), which is the Fourier transform of the ion-ion structure factor S (q) where q is a momentum. The ion-ion structure factor enters the theory explicitly. The electron-ion collision cross-section ot(q) and the density of the free electrons Zi/Uo are computed from some suitable prescription appropriate for the material of interest where, e is the kinetic energy of the incident electron, Z{ is the average number of free electrons CHAPTER 2: THEORY 21 10"3 1CT2 10"1 10° 101 10 2 103 VOLUME (cm 3/g) Figure 2-4 E q u a t i o n of s tate i s o t h e r m s for a l u m i n u m f r o m the S E S A M E l i b r a r y : a) pressure i s o t h e r m s ; b) energy i s o t h e r m s . T h e s h a d e d region indicates the region r e l e v a n t t o o u r m e a s u r e m e n t s . CHAPTER 2: THEORY per ion and Ho is an average atomic volume. In the SESAME library, these are estimated using a detailed calculation of the electronic structure of the material under consideration. Using the above physical assumptions, Evans et a l . 6 3 obtained what is known as the t-matrix formulation of the Ziman theory where n is the electrical resistivity, q is the momentum transfer in a collision, p is the original momentum of the electron and a ~ 1/137. //^(e) is the Fermi-Dirac distribution function fPll{e) = (exp[/3(e - //)] + l ) \ [2 - 12] where f3 is the inverse temperature and /J, is the chemical potential. The momentum, p, is related to the energy, e, through the usual relativistic dispersion relation p2 = (2mc2+e)e [2-13] for the free electrons where m is the rest mass of an electron. For strongly coupled degenerate systems one must question the validity of the single-site scattering approximation64. One possible improvement is to attempt to generalize the Ziman theory to include multiple scattering. However, this has not been done successfully. Another approach is to renormalize the electron mass to approximate the effects of higher order scattering65. However, the SESAME data were derived with the electron mass set equal to the electron rest mass and this question was left open. It is interesting to note that the measurements discussed in this thesis probed this regime. To evaluate equation 2-11 for a given temperature and density we need: (i) the free electron density Zt-/fin (ii) the chemical potential /i, (iii) the electron-ion CHAPTER 2: THEORY 23 scattering cross-section cr€(t7) and (iv) an appropriate ion-ion structure factor S(q). There is no formal reason that these cannot be chosen independently of each other, since there is no rigorous requirement in the theory that they be internally consis-tent. In the SESAME library, these quantities were not calculated self-consistently. The free electron density, the chemical potential and the electron-ion scatter-ing cross-section were calculated from the self consistent field for a single scattering centre. For the isolated atom, the most successful theory at present is the self consistent Hartree-Fock-Slater mean field approximation (MFA) 6 6 . The ionic po-tentials used for the calculations in the SESAME library were required to approach these potentials in the low density, high temperature limit. For the high density and high temperature limit, the potentials were required to approach the Thomas-Fermi-Dirac potentials (TFD) 2 5 . For the intermediate regions of density and temperature, neither of these approximations is appropriate. A more sophisticated model has been developed by Liberman6 7 in an attempt to incorporate the desirable features of both. This was essentially a TFD calculation with an improved (but still local) exchange approximation, with shell corrections obtained by exact solutions of the Dirac equation for the important electronic states. On the other hand, this model could be called a MFA calculation corrected for finite-volume boundary conditions. This model effectively provides a physical interpolation between the MFA and TFD limits and has been called AIJ (atom in jellium). The models given above are used to determine the electronic density of states dNe/de and the cross-section o€(q) as a function of energy. The cross-section was determined using a partial wave analysis68. From the density of states, the chemical potential was determined be requiring charge neutrallity in a ion sphere. [2-14] CHAPTER 2: THEORY where Z is the atomic number of the material. The number of free electrons per atom was then computed from /•OO JO where the free electron density of states is — 1 = _|p{mc2 + e) 2 - 16 de TIZ The atomic volume was calculated using n » = j ^ I 2 " 1 7 ! where A is the atomic mass, No is Avogadro's number and p is the material density. The principal virtue of the approach used in the SESAME library is that it leads to a internally consistent calculation of the chemical potential, free electron density and the electron-ion scattering cross-section. The ion-ion structure factors used for the calculations in the SESAME library do not have an explicit connection to the electron-ion potentials. At the material melting point, the Percus-Yevick structure factor69 was used. This structure fac-tor can be accurately fit to neutron diffraction data. For high temperatures, this structure factor is inappropriate. The Debye-Huckel structure factor was used in this regime where 2 ( 3n„ ) 2 / 3 CHAPTER 2: THEORY is the Debye radius, with the ion-ion coupling constant T, given by r = (3H 0 ) 2 / 3 [2 - 20] A third choice for the structure factor was also used. This was based on the two-body correlation function given by 3 0 where rc is the core exclusion radius, taken as a adjustable parameter, and B is the normalization constant required to conserve probability. The structure factor, S(q) is obtained from the Fourier transform of g(r). This third choice of the structure factor has the virtue that it approaches the Debye-Huckel limit as r c —> 0 and the hard sphere limit as —> oo. The third structure factor is also close to the Percus-Yevick solution near the melting point of the material. Finally, some of the calculations were done using one component plasma struc-ture factors 7 0 , 7 1. With this information, equation 2-11 was integrated for the electrical resis-tivity. The electrical conductivity from the SESAME library is given in Figure 2-5. Also indicated in the figure is the approximate region of parameter space relevant to the measurements presented in this thesis. In this region the system is strongly coupled and degenerate, thus, the single-site scattering approximation used in this conductivity model breaks down. However, the authors of the SESAME library feel this should lead to an error of at most a factor of two 3 0. It is interesting to note s ( r ) = o, [2-21] CHAPTER 2: THEORY Figure 2-5 E l e c t r i c a l c o n d u c t i v i t y f r o m the S E S A M E l i b r a r y . T h e s h a d e d region i n d i c a t e s the a p p r o x i m a t e r e g i o n relevant to the m e a s u r e m e n t s presented in th is thesis . CHAPTER 2: THEORY 27 that our measurements indicate that the conductivity in the SESAME library is of the order of a factor of two too high. 2.6 L e e a n d M o r e ' s e l e c t r i c a l c o n d u c t i v i t y m o d e l This conductivity model is relatively simple, in fact, it is almost analytic28. The transport coefficients are obtained from moments of the electron distribution function. The electron distribution function was obtained from solutions of Boltz-mann's equation in the relaxation time approximation assuming that the distribu-tion function differs little from an equilibrium distribution function. This theory is valid when electrons scatter off individual scattering centres (single-site scattering approximation) as in the model in the SESAME library. However, when strong ion-ion correlations exist, (solid and liquid phases) the theory breaks down. This simple theory can be used to give reasonable transport coefficients by using analytic expressions for the electron mean free path which contain some empirical param-eters. Our measurements probed the material accurately enough that suggestions can be made about the choice of these parameters. In this conductivity model, the transport coefficients were obtained from solu-tions of the Boltzmann equation in the relaxation time approximation using Fermi-Dirac statistics for the equilibrium electron distribution function28. The electrical conductivity can be written as where fp^ is the Fermi-Dirac distribution function as defined in equation 2-12 and e is the kinetic energy of an electron of velocity v, [2 - 22 1 2 [2 - 23 t = -2 mv CHAPTER 2: THEORY e is the electron charge, TC is the relaxation time, p is the electron momentum and h is Planck's constant. The relaxation time is calculated with contributions from both electron-ion and electron-neutral scattering according to Matthiessen's72 rule. This gives 1 - 1 + 1 where rei = — — [2 - 24] and n0voen are the electron-ion and electron-neutral collision times. Here, n, is the ion density, no is the neutral density, oei is the electron-ion momentum transfer cross-section and oen is the electron-neutral momentum transfer cross-section. In order to calculate the transport coefficients, the chemical potential and average ionization state as a function of temperature and density are required. The average ionization was obtained from the screened hydrogenic model7 3 and the chemical potential was obtained from the Thomas-Fermi model2 5. The relaxation time is calculated by using measured values for the electron-neutral momentum transfer cross-section, oen, and the Coulomb cross-section for the electron-ion momentum transfer cross-section, oe{. The electron-neutral cross-section is a function of the size of the atom. The Coulomb cross-section is given by aet- = 4 7 r ( Z , - ) V ! ^ [2-25] CHAPTER 2: THEORY 29 where Z{ is the ionization state and ln A is the Coulomb logarithm. To obtain coefficients valid over a large parameter space, several prescriptions for the Coulomb logarithm were used. In the regime where the material is a plasma the Coulomb logarithm is ap-proximated by lnA = i l n ( l + &Lx/*LJ (2-26] where bmax and bmin are the upper and lower cutoffs on the impact parameter for Coulomb scattering. These cutoff parameters were chosen to account for the physical effects of electron degeneracy, Debye-Huckel screening and strong coupling. In the plasma regime, the maximum impact parameter is determined by the Debye-Huckel screening length. The screening length is 7 4 1 = 4nnee2 47rnj(eZ,-)2 _ bmax + r | ) V » + kT 1 J where Tp is the Fermi temperature. The minimum impact parameter was set by the classical distance of closest approach. This is Z{€ r -j Omtnl = o 2 ~ 2 8 However, at high energy, bmin is set by the uncertainty principle. This is bmim = A/2 = [2 - 29] 2mv where A is the de-Broglie wavelength. To simplify the calculation, 6 m i n was eval-uated at the thermal velocity, v = (3kT/'m)1/2, or in the degenerate limit at the Fermi velocity, v = (2Ep/ra)1/2. The minimum impact parameter is taken as bmin — max\bmini ,bmini] [2 - 30] CHAPTER 2: THEORY 30 These impact parameters were used in the temperature and density regime marked (l) in Figure 2-628 for the transport calculation. This is the high temperature, low density regime. In strongly coupled plasmas, the Debye-Huckel treatment of the screening breaks down due to ion-ion correlation effects. The screening length calculated using Debye-Huckel theory becomes less than the interatomic distance, 2ro- However, to be consistent with results from simulations of a one component plasma75, Lee and More 2 8 set the screening length to the interatomic distance. Region 2 in Figure 2-6 gives the temperature density regime where these conditions were used. In this model the Coulomb logarithm was required to have a value greater than or equal to 2.0. This minimum value of the Coulomb logarithm agrees with the numerical simulations of Green and Lee 7 6. Region 3 in Figure 2-6 gives the temperature density regime where this condition applies. The Coulomb logarithm in equation 2-27 is applicable only in plasmas. For solids and liquids, strong ion-ion correlations can have a dramatic effect on the electron relaxation time. Above the melting temperature there exists a region in which the calculated mean free path becomes less than the interatomic distance. Without a detailed analysis of the collision mechanism appropriate to this region, a minimum conductivity was adopted following the arguments developed for an amorphous semiconductor by Mott 7 7 . The electron transport was calculated with the relaxation time, r c = ro/v. We have used two versions of this transport model in our analysis: one with the relaxation time set equal to rc — TQ/V and one with the relaxation time set equal to r c = 2ro/v where 2ro is the interatomic spacing and v is the average electron velocity. Region 4 in Figure 2-6 gives the temperature-density regime where this condition applies. Note that the region shown in Figure 2-6 was calculated using the first version of the model, the size of region 4 increases when the second prescription is used for the relaxation time. CHAPTER 2: THEORY 31 DENSITY (g/crr.3) F i g u r e 2-6 The density-temperature regimes of each mean free path model used in the calculation of the electrical conductivity in Lee and More's conductivity model. (From Y.T. Lee and R.M. More, Phys. Fluids 27, 1273(1984). CHAPTER 2: THEORY For the liquid and solid, Ziman 7 8 developed a theory for electron transport which takes into account ion-ion correlations using the ion-ion structure factor. Given an ion-ion structure factor, the integral for the electron mean free path is evaluated numerically. To simplify the computation, Ziman 7 8 calculates the ion-ion structure factor in the long wavelength limit. Using this approximation together with Lindemann's melting law, he analytically evaluated the integral for the electron mean free path. The result for a metal below its melting temperature Tm is A = 5 0 r 0 ^ T<Tm [2-31] This equation is also known as the Bloch-Gruneisen formula for metallic electrical conductivity. Above the melting temperature, the mean free path was taken to be A = 5 0 r 0 ^ - T>Tm [2-32] where 7 is chosen (empirically) for each metal correctly to give the increase in resistivity at melting. In calculating the electron mean free path, the following formula was adopted for the melting temperature which was derived from Thomas-Fermi theory79. Tm = 0..32[e/(l + 0}4t2b-2/3(eV) [2 - 33] where b — 0.6.Z1/9, £ — 9.0Z03p/A. Z is the nuclear charge, p is the mass density and A is the atomic weight. The formula is based, in part, on the Lindemann melting criterion. This formula gives melting temperatures which agree reasonably well with experimental data for many normal metals. In general, it is found that the electrical conductivity calculated using equations 2-31 and 2-32 for the electron mean free path agrees adequately well with experimental data for many monovalent CHAPTER 2: THEORY 33 metals. Region 5 in Figure 2-6 gives the temperature density regime where these conditions apply. The information given above is sufficient to obtain the electrical conductivity over a wide parameter range in density and temperature by numerically integrating equation 2-22 to yield the electrical conductivity. This was done by Lee and More 2 8. Two versions of the transport model were obtained: one was with the relaxation time set to r c = TQ/V in region 4 of Figure 2-6 and the other with the relaxation time set to TC = 2ro/v. Both were used in the simulations of our measurements (described in Chapter 5) which are sensitive to this parameter. In fact, it appears that simulations based on the second version of the conductivity model are in better agreement with the measurements. Figure 2-7 shows the electrical conductivity as a function of density and tem-perature for aluminum based on r c = TQ/V in region 4 of Figure 2-6. Also shown in Figure 2-7 is the approximate region relevant to the measurements presented in this thesis. There are two noteworthy differences between the model in the SESAME library and the calculations of Lee and More. In the SESAME library, the electron-ion collision cross-section was calculated using the detailed electronic structure in a partial wave analysis whereas in Lee and More's model the Coulomb cross-section was used. Lee and More justified their assumptions by comparing their calculations to their own version of a partial wave analysis. They concluded that their model is accurate to within a factor of two of the true conductivity as do the authors of the SESAME data library in the regime where the electron mean free path approaches the interatomic spacing. The conductivity values from these models were found to be similar, within a factor of 2 to 4 in this regime. It can be seen from Figures 2-6 and 2-7 that the electrical conductivity predicted by the two models are decades apart in the low-temperature, low-density limit. This results since electron-neutral collisions become important. In Lee and More's model, the electron-neutral collision CHAPTER 2: THEORY -Figure 2-7 EJe^ t • j r e gion indicate tu ° ° n d u c t ) v ' t y from Lee and M ~ • « « a p p r o x i m a t e region r e | e v a ^ ca,c„ l a t i o n s. T h e ° o u r measurements. CHAPTER 2: THEORY 35 cross-section was included in the calculation of the relaxation time, whereas, in the SESAME library these collisions are neglected. This process is only important for Zi -C 1. In the plasma regime relevant to our measurements Z t ~ 1. 2.7 H u g o n i o t r e l a t i o n s For aluminum, we have measured several shock parameters: the shock tem-perature, shock velocity and the shock pressure. A plot of one shock parameter as a function of an other is called an Hugoniot curve. These curves can also be calculated from an equation of state as follows. Consider the case of a steady shock wave where the shock front connects an undisturbed state with a uniform shocked state. It can be shown using the conservation of mass, momentum and energy that across a shock front the following relations are obeyed PQUs = Pi{Us - Up) P\-PQ = PoUsUp [2 - 34] E l - E0 = \{Pi + P0)(±-- ±) . 2 Po p\ Here Us is the shock velocity and Up is the particle velocity in the compressed region. The frame of reference is such that the particle velocity is zero in front of the shock wave. The suffixes 1 and 0 represent the quantities in the shocked and unshocked regions. Given an equation of state, P = P(p,T) and E = E(p,T), these equations can be solved. Note, these relations are valid at the shock front even for an unsteady shock and are independent of the curvature of the shock front. These relations are known as the Rankine-Hugoniot equations. Given the initial conditions and the value of one of the shock parameters, all the other shock parameters can be uniquely determined. The collection of all such points is known as the Rankine-Hugoniot curve or the shock adiabat or simply the Hugoniot. The chapter 2: theory principal Hugoniot curves are the Hugoniot curves with the initial state of the material at standard conditions (room temperature and density). Figure 2-8 shows the shock pressure versus shock velocity principal Hugoniot curve for aluminum as well as the shock temperature on the Hugoniot curve as a function of shock velocity. Hugoniot curves based on three different equation of state models are given: the SESAME equation of state, a Thomas-Fermi equation of state25 and a equation of state based on the Saha model 5 7. These curves will be compared to measurements in chapter 4. 2.8 The role of two dimensional effects The shock wave measurements presented in this thesis were interpreted using one dimensional models. The validity of this analysis is discussed in this section. Due to the finite size of the laser focal spot, the size of the planar part of the shock wave is also finite. For uniform laser irradiation over a disc at the target plane a simple criterion has been derived79 to describe the size of the planar region of the shock wave. As the shock wave propagates through the target, the edge rarefaction wave propagates towards the centre of the shock compressed region reducing the size of the planar region of the shock front. Since the shock speed and the speed of sound in the shock compressed region are comparable in the laboratory frame of reference, one would expect that the shock wave will retain its planarity for a target thickness of half the spot diameter. However, since the target material is compressed significantly by the shock wave, and the ablation process maintains a constant pressure at the ablation surface, a more appropriate criterion is that the spot diameter should be larger than twice the thickness of the compressed target. The maximum compression achieved in our experiment was ~ 3 for the maximum shock speed of ~ 2.5 x 106cm/s in aluminum. Thus, for a 40>m diameter spot size with uniform laser irradiation and an initial foil thickness of 50>m, the size of the planar region at shock breakout would be expected to be approximately 15/xm in CHAPTER 2: THEORY 1 10 SHOCK PRESSURE (Mbar) Figure 2-8 H u g o n i o t curves for a l u m i n u m : a) shock pressure versus shock v e l o c i t y : a n d . b) shock t e m p e r a t u r e versus shock ve loc i ty . S o l i d curves are based on the S E S A M E e q u a t i o n of s tate , the d a s h e d curves are based on the T h o m a s - F e r m i m o d e l a n d the d a s h - d o t curves are based on a S a h a e q u a t i o n of s tate . CHAPTER 2: THEORY diameter. However, the measured planar region of the shock wave was found to be of the order of 40fim in diameter. This has been attributed to the Gaussian laser spot distribution used in the measurements. In our experiment, the spatial intensity distribution of the laser is approxi-mately Gaussian with 60% of the laser energy within a 40/xm disc and 90% of the energy within a 80//m disc (discussed in chapter 3). This distribution is expected to launch a shock wave which has a diameter much greater than the 40/zm central region; however, it is expected to be nonplanar outside the central region. The weaker part of the shock wave at the edge may maintain the planarity of the shock wave in the central region by reducing the radial pressure gradients. This allows thicker targets to be used in the experiments. To verify this argument it would be necessary to simulate the measurements using a two dimensional hydrodynamic code; however, this is not available. In addition to the finite spot size, there are also small fluctuations in intensity about the smooth Gaussian profile (to be discussed in chapter 3). These spatial intensity modulations can also reduce the planarity of the shock front. However, the spatial scale of these modulations is of the order of < 4/xm, thus, for thick foils (~ 50/xm) the nonuniformities in the shock front due to these modulations are expected to dissipate due to two dimensional effects before shock breakout. CHAPTER S: EXPERIMENTAL FACILITY 39 C H A P T E R 3 E X P E R I M E N T A L F A C I L I T Y 3.1 In t roduc t ion In this chapter, the laser facility, beam diagnostics and laser irradiation con-ditions at the target plane are given. The laser facility and the beam diagnostics are described in section 3.2. In section 3.3, the characterization of the laser beam intensity distribution at the target plane is presented. 3.2 Neodymium-g lass laser system and beam diagnostics The experimental facility is based on a Quantel neodymium-glass laser sys-tem, which includes a Nd-YAG (neodymium yttrium aluminum garnet) oscillator, a Nd-YAG preamplifier and two Nd-glass amplifiers. The aperture of the final am-plifier rod is 25mm in diameter. The laser oscillator is passively Q-switched with a dye-cell and provides a single laser pulse at 1.062/xm in the TEMQQ mode. The temporal pulse shape is approximately Gaussian with a full width half maximum (FWHM) of 2ns. For the series of experiments described in this thesis, the laser output was frequency doubled to give 0.532/im laser light. This was accomplished with a KD*P (deuterated potassium dihydrogen phosphate) crystal. The system is capable of delivering 12 joules at 1.064/tm and 7 joules at 0.532//m. A schematic diagram of the beam diagnostics and the experimental set up used to measure the focal spot distribution is given in Figure 3-la. The collimated laser CHAPTER S: EXPERIMENTAL FACILITY Figure 3-1 a) Schematic diagram of the experimental setup showing the beam diagnostics and the setup used to measure the focal spot distribution at the target plane, b) Temporal pulse shape of the incident laser beam. CHAPTER S: EXPERIMENTAL FACILITY 41 beam (25 mm diameter) is brought to the target using a series of dichroic mirrors. The use of dichroic mirrors, together with the chromatic aberration of the focussing lens (//10 optics), reduces the intensity of infrared laser radiation on target to a negligible level (less than one part in 105). As shown in Figure 3-la, a beam splitter was used to reflect part of the incident laser light towards a Gentec laser energy meter (model ED-200) which measures the incident laser energy, and part to a fast photodiode (Hamamatsu model R1193U, 350 ps rise time) which monitors the pulse shape. Shown in Figure 3-lb is a typical pulse shape measured with the photodiode. The same beam splitter was used to collect part of the backscattered light, which was monitored with similar energy and power detectors. The best focus was obtained by maximizing the backscattered laser light as a function of focussing lens position5 0. 3.3 Irradiation conditions The laser beam was focussed onto the target plane with //10 optics. Charac-terization of a laser-plasma interaction requires the measurement of the spatial and temporal distribution of the laser energy at the target plane. This was accomplished by imaging the laser focal spot onto the slit of a streak camera (Hamamatsu Tem-poral Disperser System, model C1370-01) which yielded either a time resolved, one dimensional image or a time integrated, two dimensional spatial distribution of the laser intensity at focus. The spatial resolution of the measurement was better than 4/xm. Figure 3-2 shows a time integrated, two dimensional image of the focal spot distribution at the best focus. Detailed analysis of this and similar measurements showed that 90% of the laser energy was contained in a spot of 80/zm diameter and 60% of the energy was within a spot of 40//m diameter. Time resolved mea-surements (30ps resolution) showed spatial intensity modulations of less than 30%, and time integrated measurements showed spatial intensity modulations of less than 10%. Spatial intensity modulations are defined as the ratio of the amplitude of the CHAPTER S: EXPERIMENTAL FACILITY F i g u r e 3-2 Time integrated focal spot distribution at the best focus. CHAPTER S: EXPERIMENTAL FACILITY 43 small scale intensity fluctuations (small scale with respect to the laser spot size) to the overall Gaussian intensity profile. For the shock wave experiments, it was found that the size of the shock break-out region at the target rear surface was consistent with the focal spot size contain-ing 60% of the incident laser energy. These results are given in chapter 4. Thus, this spot size was used to characterize the laser intensity using ^ 0 = 4 — - ^ [3-1] where Eaf,s is the absorbed laser energy, is the laser pulse length and DQO is the focal spot size containing 60% of the laser energy. The absorption as a function of incident laser intensity was determined using Ulbricht integrating sphere50. The energy of the laser beam was changed by changing the pumping voltage of the flash lamps in the last two laser amplifiers. This resulted in slight changes in the focal spot size and pulse length. This was accounted for in the analysis of the data. The laser parameters at best focus are given in Table I. Table I. Laser parameters D 6 0 £ 9 0 (f)Qo(max) E^max) 0.53/xra 2ns 40/xm 80jj,m 2 x 10uW/cm2 7J CHAPTER 4: LUMINESCENCE MEASUREMENTS 44 C H A P T E R 4 L U M I N E S C E N C E M E A S U R E M E N T S 4.1 Experimental setup The speed and temperature of shock or heat waves in solids can be estimated from measurements of the luminous radiation emitted from the rear surface of the solids as the shock or heat wave emerges at this surface. Opaque targets were used in these experiments, thus, the emission starts at the time of the wave breakout and continues as the plasma expands into the vacuum. The experimental setup is given in Figure 4-1. The target front surface was irradiated by the focussed laser beam at 10° off target normal. This tilt should not change the uniformity of the irradiation since the focal depth of the laser at the target plane is ~ 700/im; thus, a planar shock wave may be generated. The target rear surface was imaged onto the slit of a high speed streak camera with f/4 achromatic optics at an observation angle of 12° relative to target normal. Two modes of operation were used with this experimental setup: the target rear surface could be imaged directly onto the slit of the streak camera to measure the spatial distribution of the luminous radiation; alternatively, a 60° quartz prism could be inserted in the optical system to obtain time resolved spectra of the rear surface emission. An aperture was used to limit the image size at the streak camera slit. To determine the relative timing of the shock wave arrival at the target rear surface compared to the incident laser pulse, a fiducial (reference beam) was recorded on the streak camera as shown in Figure 4-1. L A S E R Figure 4-1 Experimental setup for the luminescence measurements. CHAPTER 4: LUMINESCENCE MEASUREMENTS 4.2 M e a s u r e m e n t o f s h o c k i n d u c e d l u m i n o u s e m i s s i o n For this measurement the target rear surface was imaged directly onto the slit of the streak camera. The optical system was focussed by imaging a fine wire mesh placed at the object plane onto the streak camera. The mesh spacing (measured using an optical microscope) was used to determine the magnification of the system. The timing of the laser fiducial was calibrated by measuring the relative timing between the fiducial pulse and the incident laser pulse through the object plane. In this experiment two identical 100 A bandpass interference filters with centre wavelengths of 5700 A were placed at the entrance of the streak camera. This eliminated any stray 0.53/xm laser light (rejection ratio of approximately 108 at 5300 A) from entering the streak camera. The spatial resolution of the optical system was approximately 3 pm as determined from the diffraction limit of f/4 optics at 5700 A. The dynamic range of the camera was measured to be approximately 100. The streak time and the streak uniformity was checked by passing a short pulse dye laser (~ 60ps) through an etalon to give a train of evenly spaced pulses. The streak speed was found to be uniform to the resolution of the measurement (~ 60ps). The test results from the manufacturer indicate that the streak time per channel was uniform to within 1.3%80. The temporal resolution of the streak camera was not measured, however, the test results from the manufacturer indicate that the ultimate camera resolution was measured to be 1.6ps. Figures 4-2a and 4-2b show the streak records for 30pm and 50pm thick alu-minum foils respectively both irradiated at (J>QQ ~ 1014W/cm2. The corresponding spatial profiles of the luminous emission from the target rear surface are given in Figures 4-3a and 4-3b. These profiles were obtained by integrating the raw data from the time of shock breakout to 100 ps after breakout. This integration time gave a reasonable signal to noise ratio. The camera slit width used for these mea-surements was 150 pm which yields a spatial resolution of approximately 11 pm. Convoluting this resolution with the resolution of the optical system gives an overall CHAPTER 4: LUMINESCENCE MEASUREMENTS 0 1 H in c LU 34 100pm i i 0 l/> c UJ M a) f SHOCK 1 FIDUCIAL 100pm i i • > » L . + • • ~ r- m f m m * • b • t SHOCK t FIDUCIAL b) Figure 4-2 Spatially resolved measurements of the luminous emission at 5700.4 due to shock wave breakout for 30 /jm and 50 /xm thick aluminum foils irradiated at 6GO ~ 10}4W/cm2: a) 30 nm: and, b) 50 (tm. CHAPTER 4: LUMINESCENCE MEASUREMENTS Figure 4-3 Spatial profiles of the luminous emission from the target rear surface for the 30 pm and 50 pm foils, integrated from the time of shock breakout to \00ps after shock breakout: a) 30 pm; and, b) 50 p.m. CHAPTER 4: LUMINESCENCE MEASUREMENTS spatial resolution of the order of 11 p.m. The apparent structures seen on the profiles have been attributed to fluctuations due to the low signal level. The diameter of the emission region was 48 fim (FWHM) for the 30 pm thick foil and 42 nm (FWHM) for the 50 nm foil at the time of shock breakout. The size of the luminous region was typically between 40 and 50 pm (FWHM) in all experiments at the time of shock breakout and usually remained confined to 80//m at late times. The shock breakout region is approximately equal to the focal spot size containing 60% of the laser energy; subsequently, <J>QQ was used to characterize the laser intensity. 4.2a Measurement of shock speed in aluminum The shock transit time is defined as the time difference between the time of shock breakout and the time of the peak incident laser intensity. Figure 4-4a shows the luminous intensity, spatially integrated over the central 35 pm of the breakout region, as a function of time for the 50 nm thick foil. The rise time of the luminous intensity for this shot was approximately 90 ps. This rise time is defined as the time required for the signal to increase by a factor of ten. The temporal resolution of the diagnostic was determined from the streak camera resolution (~ 35ps corresponding to a slit width of 150pm), surface roughness of the target (~ 50ps for < 1p.m. roughness and a shock speed of the order ~ 2 x 10Qcm/s) and a finite rise time of the luminous emission due to the optical depth of the unperturbed foil (~ lOps resulting from a 2000 A optical depth81 and a shock speed of ~ 2 x 106cm/s). This gives an overall temporal resolution of the diagnostic of approximately 60 ps. This estimate does not include the effect of non-planarity of the shock front which further reduces the spatially averaged rise time. Thus it appears that the measured rise time of the luminous emission is instrument limited. Typically the one decade rise time was of the order of 100 ps for all measurements. The fiducial pulse shape is shown in Figure 4-4b. The relative delay between the fiducial and luminous emission CHAPTER 4: LUMINESCENCE MEASUREMENTS 50 Figure 4-4 a) Luminous intensity of the target rear surface as a function of time for the 50 /im thick aluminum foil irradiated at 4>QQ ~ 1014W/cm2. b) Fiducial record T$ is the shock transit time through the foil. CHAPTER 4: LUMINESCENCE MEASUREMENTS 51 was -40 ps± 100 ps. Evidently, the shock transit time for this shot was 1.3 ± 0.1 ns. The foil roughness was estimated from the optical quality of the foil surface. The material density quoted by the manufacturer82 was the standard density for aluminum 2.7g/cm3. The purity of the foils was always better than 99.9%82. Foil thicknesses were calculated from the measured weight and the areal dimensions of the foil using the quoted foil densities. The estimated foil thickness has an uncertainty of approximately 2%. The shock trajectory was determined by repeating the transit time measure-ment as a function of foil thickness at constant peak laser irradiance. Figure 4-5 shows the measured shock trajectory in aluminum for an absorbed irradiance of $60 ~ 10 1 4 W/cm 2 . Each measured point is an average of at least 5 measurements with the error bars defined as the standard deviation of a sample. Also given in the figure is the shock trajectory from LTC simulations. For the code calculation a Gaussian laser pulse of 2ns (FWHM) and maximum intensity of 10i4W/cm2 were used. For all the LTC simulations presented in section 4.2, an analytic approxima-tion to the T F K model for the electron pressure and an ideal gas law for the ion pressure50 were used. Evidently, the measured points lie in a reasonably straight line which indicates that the shock wave has reached a quasi-steady state after prop-agating 19/xm into the target and remains steady and one dimensional to at least 50 iu,m into the target. This is further verified by the excellent agreement between the experiment and the simulation. The measured shock speed was obtained from the slope of the straigth line which was a least squares fit to the data. The accuracy of the measurement was estimated from the uncertainty in the fit. For this trajectory the measured shock speed was 2.2 ± 0.3 x 106cm/s. The uncertainty in the fiducial timing does not affect the uncertainty in the velocity measurement since the slope of the shock trajectory gives the velocity. Furthermore, the uncertainty in the foil I ' / * / CM 0 10 20 30 40 50 TARGET THICKNESS (jjm) Figure 4-5 Shock transit time as a function of foil thickness at. a laser irradiance of ~ 10 1 4W/cra 2 for aluminum open circles are the experimental results; and the dashed curve are simulation results from L T C . The transii time is defined as the time between shock breakout and peak laser intensity. CHAPTER 4: LUMINESCENCE MEASUREMENTS 53 thickness (~ 2%) does not increase the uncertainty in the velocity measurement significantly. This technique was used to obtain shock velocity as a function of incident laser intensity. The absorption of the laser light as a function of peak incident laser intensity has been characterized in an earlier work5 0 using an Ulbricht integrating sphere. The results are presented in Figure 4-6. The shock speed in aluminum as a function of absorbed laser intensity is presented in Figure 4-7. Also shown are results from LTC simulations, which are in good agreement with the data. However, a much more useful comparison of theoretical equations of state to experimental results is on a Hugoniot curve, since Hugoniot curves only depend on the equation of state of the material and not on the details of laser-target interactions. 4.2b Measurement of shock pressure in a l u m i n u m The shock pressure was estimated from ion expansion measurements. These are temporally and spatially integrated measurements, which are discussed in detail in earlier works 1 2 ' 5 0 ' 8 3 . A brief description of the measurements will be given below. The blow off plasma is assumed to be cylindrically symmetric. An array of differen-tial ion-calorimeters (energy meters) and Faraday cups (ion current detectors) are used to measure the energy density and velocity of the blow off plasma as a function of angle from target normal. The momentum of the blow off plasma perpendicular to target normal is estimated using where E{6) and V{6) are respectively the measured energy (per steradian) and velocity distributions in the blow off plasma. The angle 0 is measured from target normal. The ablation pressure is estimated by assuming the interaction time is of the order of the length of the laser pulse (FWHM) and the blow off plasma [4-1] QUA P TEH 4: L UMINES GEN CE ME A S UREMENTS 54 T - T T T 1 1 . — r — r — i 1 8 10 1 J 2 4 6 8 1 0 U 2 INCIDENT LASER INTENSITY (W/cm2) Figure 4-6 Laser absorption as a function of incident laser intensity for aluminum (from D . Pasini , P h . D . thesis, University of Br i t i sh Columbia , 1984). ABSORBED LASER INTENSITY (10 1 3 W/cm 2 ) F i g u r e 4-7 Shock speed in aluminum as a function of absorbed laser intensity: open circles are the experimental results; and, the dashed line are results from LTC simulations. =0 > ri > CHAPTER 4: L UMINESCENCE ME AS UREMENTS 5 6 originates from an area of the order of the focal spot size. D90 is used since the ablation measurements are spatially integrated. The ablation pressure is given by PM ~ ^ |4 - 2a] where 77 is the interaction time and Aj is interaction area. The ablation pressure as a function of absorbed laser energy is given in Figure 4-812. Also shown in the figure are the results from the LTC simulations which are in reasonable agreement with the data. However, the good agreement may be fortuitous since the measured value represents the pressure averaged over the laser focal spot whereas the simulation is one-dimensional. The experimental results can be approximated by , i < - 2 » ) This was obtained by fitting a power law to the data. <f>Qo rather than 0go was used in this expression for convenience, since </>Q0 is used throughout this thesis. These results are compared to the results of other laboratories in references 12 and 50. Our scaling factor was found to agree well with all other results; however, the numerical factor varied by approximately a factor of 2. This has been attributed to the measurement technique as well as the details of the irradiation conditions between the different measurements12'50. 4.2c Shock speed and pressure Hugoniot curve for aluminum The principal Hugoniot curve of shock velocity as a function of shock pressure can be constructed from the data in Figures 4-7 and 4-8 assuming that the abla-tion pressure and shock pressure are equal. This was verified using hydrodynamic simulations. The measured Hugoniot can then be compared to the predictions of theoretical equations of state. This is particularly convenient since the calculation of principal Hugoniot curves does not require a hydrodynamic code. Figure 4-9 shows our experimental results as well as results from the experi-ments of Mitchell and Nellis 3 6. Principal Hugoniot curves derived from an ideal gas equation of state (Saha), Thomas-Fermi equation of state and from the equation of state in the SESAME data library are also given in the figure together with results CHAPTER 4: LUMINESCENCE MEASUREMENTS 1 1 l l l l l l 1 0 U A B S O R B E D L A S E R INTENSITY (W/cm ; Figure 4-8 A b l a t i o n pressure as a f u n c t i o n of a b s o r b e d laser i n t e n s i t y for a l u -m i n u m : o p e n circles are the e x p e r i m e n t a l results; a n d . the d a s h e d l ine are s i m u l a -t i o n resu l ts f r o m L T C . £ 1 O CD 1 10 | SHOCK PRESSURE (Mbar) | > j Figure 4-9 Shock speed a n d pressure H u g o n i o t curves for a l u m i n u m : sol id circles are the measured d a t a ; s o l i d >j l ine are the m e a s u r e m e n t s of M i t c h e l l a n d N e l l i s ; dashed line is the S E S A M E H u g o n i o t ; dash-dot l ine is the 'r> T h o m a s - F e r m i H u g o n i o t ; dashed-dot-dot is the ideal gas H u g o n i o t ; stars are results f r o m L A S N E X s i m u l a t i o n s ; g. a n d , the d a s h - d o t - d o t - d o t curve are results f rom L T C s i m u l a t i o n s . CHAPTER 4: LUMINESCENCE MEASUREMENTS 5 9 from the hydrodynamic codes L A S N E X 5 3 and L T C . Our experimental results are in good agreement with the data of Mitchell and Nellis and all the other equations of state except for the ideal gas law which is known to be inadequate for our exper-imental conditions. However, the data agrees well with the Thomas-Fermi model which is also known to be inadequate in this regime. Although this measurement was the first reported Hugoniot measurement for aluminum above 1.5 Mbar16, the measurement accuracy is insufficient for use-ful equation of state studies. The accuracy required for equation of state studies is of the order of 1% on a shock pressure versus shock velocity Hugoniot curve whereas the accuracy of the measurements reported here are approximately 20%. The experimental uncertainty can be attributed to shot-to-shot fluctuations in the temporal and spatial characteristics of the laser beam. Furthermore, the pressure measurement is a space and time integrated measurement where as the shock ve-locity measurement is spatially and temporally resolved. Thus the good agreement of the experimental results with the theoretical calculations may be fortuitous. In order to study the equation of state, a Hugoniot curve which is much more sensitive to the equations of state should be used for laser generated shock wave experiments. 4.3 Measurement of shock temperature in aluminum By coupling a prism dispersive element to a high speed streak camera, we have obtained the first time resolved measurement of the complete visible spec-trum (4200^ — 6900A) of the shock induced luminescence from the rear surface of a laser irradiated planar target 3 8. This provided simultaneously both temporal and detailed spectral information on the luminescent spectrum. Assuming the emis-sivity of a greybody, it further yielded time resolved measurements of the spectral temperature of the target rear surface. As compared with brightness temperature measurements, this eliminated the need for an absolute intensity calibration of the detection system as well as an accurate measurement of the area of the emitting CHAPTER 4: LUMINESCENCE MEASUREMENTS surface at all times. The latter consideration is of particular importance in experi-ments where the shock heated area does not remain constant in time. Measurements of the entire visible spectrum would also improve the accuracy in determining the spectral temperature since any significant deviations from the greybody assumption would be evident even without an absolute calibration. Details of the technique will be discussed first. 4.3a Spectral temperature measurement The prism was operated at minimum deviation at 5550A, the band centre. The spectral dispersion of the optical system, including the imaging optics and the streak camera, was calibrated by imaging a region of 150/xm in diameter (limited by the aperture shown in Figure 4-1) and illuminating the image plane with a tung-sten filament. The intensity profiles transmitted through calibrated 100A bandpass interference filters of centre wavelengths of 4300A, 4700A, 5300A, 5700A, 6300A and 6900A were recorded by the streak camera operated in the focus (or static) mode. The measured dispersion is shown in Figure 4-10 as well as a third order polynomial fit to the data. The dispersion was also calculated, for a prism to cam-era distance of 63cm, using an empirical refractive index85 (for fused silica). (This curve is shown in Figure 4-10). The calculation was matched to the data at 5550.4 (centre of the spectrum between 4200A - 6900A). The useful wavelength range in the measurement was 4200A to 6900A. For wavelengths less then 4200A the dispersion is very strong resulting in a low signal to noise ratio and for wavelengths longer then 6900A the response of the streak camera drops off rapidly. Furthermore the optica] quality of the achromats deteriorates at short wavelengths. The minor discrepencies between the measured and the calculated dispersion can be attributed to the accuracy in determining the prism orientation, that is, the minimum devia-tion at 5550A. The fit to the measured dispersion yields the camera pixel number as a function of wavelength, n(X). This is used in the data analysis. CHAPTER 4: LUMINESCENCE MEASUREMENTS 61 CO o E o x »— o z u LLl o > < o w so" \ V V O MEASUREMENT - - FIT — CALCULATED 120 160 PIXEL NUMBER 200 Figure 4-10 Dispersion of the optical system used for the spectral measurement: open circles are the measured data; dashed curve is a third order polynomial fit to the data; and, the dash-dot curve is a calculated dispersion (matched at 5550J1) . CHAPTER 4: LUMINESCENCE MEASUREMENTS The spectral intensity I[X,t) was calculated using /(A,o = / 'KO dn dX 5(A) 4-3] where n(X) is the pixel number, I'(n,t) is the intensity recorded by the streak camera and S(X) is the combined spectral response of the imaging system 8 5 , 8 6 and the streak camera80. However, to obtain a reasonable signal to noise ratio, the signal must be time integrated over some period At. This time scale has a significant effect on the interpretation of the measurement since the plasma expands to spatial scale lengths larger then the optical depth of the plasma in a very short time 7 8. For our experiment a 200ps integration time gave a reasonable signal to noise ratio. The measured spectral intensity becomes where At = t<i — t\ and t\ is the time of shock breakout. The spectral temperature, Ts{At), is obtained from a greybody fit (a greybody fit is required since the optical system was not calibrated for absolute response) to the spectrum lE{X,At). The form of the curve fit to the data is where e{At) is the relative emissivity of the plasma, h is Planck's constant, c is the speed of light and k is the Boltzmann constant. In the analysis described above, it has been assumed that the plasma emission spectrum is Planckian in form. Planckian emission has been previously observed from similar plasmas87. Furthermore, the spectra presented in this work (sections 4.3b and 4.4a) are consistent with the greybody form. 4 - 4] IF{X,At) = e(At) 2hc2 1 [4-5] A5 exp\hc/XkTs{At)} - 1 CHAPTER 4: LUMINESCENCE MEASUREMENTS A streak record of a time resolved spectrum is presented in Figure 4-11 for a 38 fim thick aluminum foil irradiated at <J>QO ~ 5 X 10^3W / cm2. Note that the spectrum shifts to longer wavelengths after shock breakout. This means that the detector observes a cooler plasma as time increases. Figure 4-12 shows the measured spectrum as well as the greybody fit to this spectrum. To obtain this result the data shown in Figure 4-11 was integrated from the time of shock breakout to 200 ps after shock breakout. The temporal resolution of the optical system was similar to the resolution in the shock speed measurement ~ 60ps as determined from the camera slit width, foil roughness and the optical depth in the unperturbed foil. The greybody fit to the data yields a temperature of 1.2eV. The observed modulations were random fluctuations due to the low level of luminous intensity recorded and were not correlated with spectral lines. Such fluctuations can be reduced at the expense of temporal resolution. The measured spectrum indeed shows a Planck form. To determine the temperature of the unloading plasma as a function of time, the spectral intensity IE was calculated in contiguous time intervals of 200 ps and greybody curves were fit to these spectra. The temperature as a function of time for this shot is given in Figure 4-13. To obtain an estimate of the uncertainty of the measurement, Figure 4-14 gives an average temperature as a function of time, ob-tained by averaging 5 shots on 38 nm thick foils irradiated at 060 ~ 5 x 10 1 3W/cm 2. The shock breakout time was used as a reference to overlay the shots. The error bars are defined as the standard deviation of a sample. The shock temperature is taken to be the peak measured temperature. For this laser intensity the "shock temperature" was 1.1 ± 0.2eV. (Shock temperature is in quotations since the mea-sured temperature is not the true shock temperature as explained below.) The shock speed was measured in exactly the same manner as described in section 4.2a. The shock speed for this measurement was 1.8 ± 0.2 x 106cm/s. This method was used to obtain the "shock temperature" as a function of shock velocity. CHAPTER 4: LUMINESCENCE MEASUREMENTS WAVELENGTH (jjm) 0.7 OA Figure 4-11 A streak record of a time resolved spectrum of the rear surface lumi-nous intensity of a 38 /im thick aluminum foil irradiated at 4>QQ ~ 5 X 1013W/cm2. • J i l l OA 0.5 0.6 0.7 WAVELENGTH ( um) F i g u r e 4-12 Rear surface luminescent spectrum integrated from the time of shock breakout to 200ps after shock breakout: the open triangles are the experimental data; and, the dashed curve is a greybody fit to the data. The greybody fit gave a temperature of 1.2eV. 2 1 0 0 1 2 TIME (ns) Figure 4-13 Rear surface temperature as a function of time for a 38 pm thick aluminum foil irradiated at 0 6 O5 x 10 1 3 W/cm 2 . Figure 4-14 Average rear surface temperature as a function of time for 38 [im thick foils irradiated at 5 x 1013W/cm2 (5 shots averaged). CHAPTER 4: LUMINESCENCE MEASUREMENTS The systematic uncertainty in the temperature measurement was not deter-mined. An estimate of this uncertainty requires knowledge of the uncertainty of the cathode calibration curve (cathode efficiency as a function of wavelength) pro-vided by the manufacturer as well as dispersion of the other optical components in the system. This information was not available. For a definitive equation of state measurement this problem should be addressed. As seen in Figures 4-11, 4-13 and 4-14, the rear surface temperature decreases rapidly as a function of time. It has been pointed out that measurements of shock temperatures in shock heated solids can be severely affected by the material that is released by the rarefaction wave that develops when the shock wave emerges from the solid81. The unloading process produces a plasma which expands and cools. In a short time (~ 2ps) the gradient scale length of the unloading plasma becomes larger then its optical depth81. Thus, the source of the observed emission originates from a progressively cooler plasma as time increases. Any experimental measurement requiring observation times longer than this time scale (~ 2ps) will underestimate the true shock temperature. Hence, the measured spectral temperature depends on the integration time used in the analysis. Consequently, it is important to include the integration time in the simulation of the results. 4.3b Reliability of the spectral temperature measurement The reliability of the measurement was checked in several ways. The spectral resolution of the system was calculated as well as measured. The spectral resolution depends on the image size at the camera slit as well as the prism dispersion. Thus, the spectral resolution depends on the size of the shock breakout region. Previous spatially resolved measurements have shown that the luminescent region on the tar-get rear surface was approximately 40/xm in diameter at the time of shock breakout and remained confined to approximately 80/xm at later times. These are consistent CHAPTER 4: LUMINESCENCE MEASUREMENTS with the laser focal spots containing 60% and 90% energies respectively. To min-imize the effect of image size on the spectral resolution, the overall magnification used in this experiment was limited to approximately three so that the image size corresponding to the shock heated region of 40 — 80//m in diameter was confined to only a few pixels in the camera readout (full screen is 256 pixels). Using an aperture (Figure 4-1), the maximum area of the target rear surface observed was limited to 150//m in diameter. The aperture reduced the noise level due to stray laser light in the measurements. Furthermore, it was used to measure the spectral resolution of the diagnostic as described below. The resolution was calculated for image sizes of 40/xm, 80/im and 150/im in diameter, using an empirical index of refraction for the prism 8 5. The results are shown in Figure 4-15. The size of the shock breakout region was measured to be between 40/xm and 50/zm in diameter (section 4.2), hence the curve for the 40/xm breakout size represents the resolution of the measurement. As the image size was limited to 150/xm, the curve for this size represents the worst possible resolution. These calculations were verified for the 150/iim image size by deconvoluting a tung-sten emission spectrum transmitted through the calibrated interference filters88. These results are in good agreement with the calculation. Since the entire visible spectrum was used to calculate the spectral temperature, the wavelength resolution achieved is more than adequate even for the lowest resolution indicated in Figure 4-15. Furthermore, the tungsten light source was used to verify, in-situ, the overall performance of the optical system. The temperature of the tungsten filament was measured using an optical pyrometer and its emission spectrum was recorded by the streak camera in focus (static) mode. As an example, Figure 4-16 shows the measured spectrum for a tungsten temperature of 2000 ± 200ii . A greybody curve (equation 4-5) fit to the spectrum yielded a spectral temperature of 1900 ± 200K, CHAPTER 4: LUMINESCENCE MEASUREMENTS 70 Figure 4-15 Spectral resolution of the optical system: measured resolution for 150/xm image size are the open circles; calculated resolution for the 150 p.m image size is the dashed line; calculated resolution for the 80 pm image size is the dash-dot line; and, the calculated resolution for the 40 pm image size is the dash-dot-dot line. Figure 4-16 The measured tungsten spectrum for a temperature of 2000/f: experimental data are the open triangles; and the dashed line is a greybody fit to the data. This fit gave a temperature 1900K. CHAPTER 4: LUMINESCENCE MEASUREMENTS 72 in excellent agreement with the pyrometer measurement. In the curve fitting proce-dure the relative emissivity is assumed to be independent of wavelength. This is a reasonable assumption since the relative spectral emissivity of a tungsten filament at 2000X differs by less than 5% in the spectral range of interest89. 4.3c Brightness temperature measurement The spectral temperature measurement was further compared with an inde-pendent brightness temperature measurement. The imaging optics and the streak camera were calibrated in-situ and dynamically (in streak mode) for absolute re-sponse through two identical 100A bandpass interference filters with centre wave-lengths of 5700A. A tungsten filament of 250>m diameter was placed at the target plane. A disk of 150//m in diameter on the surface of the filament, was imaged onto the slit of the streak camera. The luminosity of the tungsten filament at temper-atures between 1500 — 3400.K' were recorded by the streak camera operated in the streak mode. (The curvature of the emitting surface was accounted for in the anal-ysis.) The temperature of the filament was measured using an optical pyrometer. Using puplished data for the emissivity of tungsten89 an absolute calibration of the entire optical system was obtained. This calibration yields a calibration constant, A. The temperature could then be expressed in terms of the measured luminous intensity and the streak camera slit width, using equation 4-5 to obtain, where Tg is the brightness temperature at wavelength A (A = 5700A in this measure-ment), A is the calibration constant, D is the slit width used for the brightness tem-perature measurement, Dcai is the slit width used during calibration and Ig is the measured luminous intensity. Equation 4-6 was derived assuming A A/A ~ 0.02 <C 1 CHAPTER 4: LUMINESCENCE MEASUREMENTS where AA is the bandwidth of the interference filter and A is the centre wavelength of the interference filter used at the streak camera input. The emissivity, e, was set equal to 1 when calculating the brightness temperature of aluminum (blackbody assumption). Using the same laser irradiance as the measurement presented in Figure 4-14 (<^ 60 ~ 5xl013W/cm2) the shock induced luminescence was measured. The intensity was averaged over a 200 ps period to obtain a reasonable signal to noise ratio as in the spectral temperature measurement. This yielded a brightness temperature of 1.3 ± 0.7eV, in good agreement with the spectral temperature. The experimental uncertainty was estimated from the uncertainty in the calibration constant, A. 4.3d Simulation of the temperature measurements To assess the effect of shock unloading on the temperature measurements from surface luminescence, a one dimensional hydrodynamic code, PEC (Plasma Expansion Code), was used to determine the time evolution of the rear surface rarefaction wave subsequent to the shock breakout. The equation of state used in the simulations was obtained from the SESAME data library. This equation of state has been described in chapter 2. The spectral emission as seen by a detector viewing directly the released material was then calculated. From this, both the spectral and brightness temperatures were estimated as a function of time. Furthermore, the time integrated spectral temperature was computed, corresponding to that observed in the experiment. The one dimensional hydrodynamic code PEC treats only the hydrodynamics of the compressed aluminum rarefying into the vacuum. The process of laser matter interactions is not included although the shock wave could be taken as generated by laser-driven ablation. The experiment is modelled from the instant of shock breakout to later times. At t = 0, the shocked material is considered to be effectively semi-infinite with a step discontinuity at the plasma-vacuum interface. Thus, the CHAPTER 4: LUMINESCENCE MEASUREMENTS shock front was considered as planar with zero thickness. The initial state of the aluminum (shock compressed) was determined from the principal Hugoniot derived from a SESAME equation of state. The calculations had a spatial resolution of at least 0.02/ira. In the simulations, the hydrodynamic equations for mass, momentum, and total energy where solved using a numerical scheme with flux corrected transport for a one temperature plasma 3 9 ' 5 2. For the high density and low temperature plasma considered here, a single temperature model is appropriate since the relaxation time of these plasmas is of the order of O.lps, as estimated from the SESAME data library, which is much smaller than the time scale of interest, ~ 200ps. The electron thermal conductivity and the equation of state were also obtained from the SESAME data library. For the following discussion, we consider a case with initial conditions of p = 6.9g/cmz and T = 4.3eV which corresponds to a shock pressure of 8.7Mbar and a shock speed of 2.3 x 106cm/s in aluminum. Spatial profiles from the simulation of density and temperature are given at different times in Figure 4-17. Time zero represents the time that the shock wave arrives at the back of the foil (solid-vacuum interface). To calculate the spectral emission reaching a detector, the unloading ma-terial was assumed to radiate locally as a blackbody. The intensity of the emission at wavelength A emitted at time t by the plasma at position x with temperature T(x, t) is given by Ix{x,t) 2hc2 1 4-7] A 5 exp\hc/XkT{x,t)} - 1 Accordingly, the total luminosity at A "observed" by a remote detector is [4-8] CHAPTER 4: LUMINESCENCE MEASUREMENTS 75 POSITION (fim) POSITION (ym) I.IOOps c) 6-E W •N. S i ->-in \ \ \ \ \ 1 : M 2 3 < z U l Q b l X u i 4 5 6 POSITION lf>m) u E < 2 a u i a . u i t t>200ps ii 6 E >-\ \ i i 1"! 5 2-a 3 4 5 6 POSITION {ym) 2 >-10 F i g u r e 4-17 Spatial profiles of density p (solid curve) and temperature T (dashed curve) as well as the "observed" luminous density (integrand of equation 4-8) at A = 5700.4° (dash-dot curve is for the bremsstrahlung opacity, dash-dot-dot curve is for the SESAME opacity) at different times after shock wave arrival at the rear surface: a) t = 0; b) t = 25ps; c) t = lOOps; and, d) t = 200ps. Time zero corresponds to the time of shock breakout. CHAPTER 4: LUM1NESGENCE MEASUREMENTS 76 where a is the plasma opacity. For a degenerate plasma near solid density and low temperature (~ leV), the only available opacity data is the frequency averaged Rosseland opacity from the SESAME data library. This opacity model is valid for an optically thick plasma in which the local radiation field is only a small perturbation away from that expected in thermodynamic equilibrium at the local temperature. This is equivalent to the diffusion approximation for radiation transport90. The Rosseland mean opacity is obtained by averaging the photon mean free path with respect to frequency employing a weighting function90. The frequency dependent mean free path is weighted to yield the correct integrated energy flux in the radiation field due to temperature gradients in the plasma90. The frequency dependence of the opacity can only be assessed using the brems-strahlung absorption coefficient which is valid for an Maxwellian plasma90 where u is the frequency, m is the electron mass, Z{ is the ionization, and ne are the ion and electron densities. This model is also valid in the diffusion approxima-tion. The bremsstrahlung absortion coefficient was obtained using the bremsstrah-lung emission coefficient (derived for a Maxwellian plasma90) along with Kirchhoff's law 9 0. The ratio of the absorption coefficient to the emission coefficient is a unique function of the plasma temperature and the photon frequency according to Kirch-hoff 's law. Both of these opacities were used in the simulations while noting their limitations. The integrand in equation 4-8 represents the "observed luminosity" at A reach-ing a remote detector due to the plasma at position x. To identify the "observed" emitting region, we have also plotted this luminous density for A = 5700A in Figure 4-17. Evidently, as the surface unloads, the cooler material rapidly shields the hot plasma behind it. The surface luminosity also decreases rapidly. Compared with [4-9] CHAPTER 4: LUMINESCENCE MEASUREMENTS the Rosseland mean opacity, the bremsstrahlung opacity yielded an emitting layer slightly deeper inside the unloading plasma because of the lower absorption at this wavelength. For the emission at any wavelength, A, the "observed" brightness or absolute temperature T\(t) can be determined from its luminous intensity I'^{t) using equa-tion 4-7. The "observed" brightness temperature as a function of time at A = 5700A is given in Figure 4-18a. Calculations based on the frequency dependent opacity gave a slightly higher brightness temperature since a higher temperature emitting layer was "observed". The spectral temperature Ts(t) can be obtained by comparing the relative spectral intensity I'x(t) with the Planck form, h(t) = e(t) for 4200.4 < A < 6900^ 1 as in the experiment. e(t) is the "effective" emissivity adjusted to obtain a fit of the Planck form Ix to the calculated luminous intensity 7 .^ For a blackbody e = 1. This constant is necessary to simulate the experimental determination of Ts from the measured spectrum where the absolute intensity of the emission was not known. The calculated Ts(t) is given in Figure 4-18b where the "effective emissivity" varied from 1 to 0.9 for the Rosseland opacity case and from 1 to 0.6 for the bremsstrahlung opacity case for t = 0 to t — 200 ps. The decrease in t(t) is due to the inhomogeneity of the plasma; the density gradient scale length increases as a function of time. Thus, the plasma radiates approximately as a greybody. (This will be elaborated further in the discussion on the time integrated spectrum.) It should also be noted that the small "bump" at / ~ 20ps in the Ts{t) curve for the frequency dependent opacity case resulted from the inability of the numerical code to provide a perfect transition from the initially steep gradients to 2hc2 A 5 exp[hc/\kTs(t)} - 1 4 - 101 - I — l l l l l l I 1 I — I — I I I I I 10 100 TIME (ps) > uj 3' OC Z> < cc 2 uJ a. UJ b) \ \ V \ • i i 1 1 1 1 • i i i 1 10 1 T T T l 11 100 TIME (ps) Figure 4-18 a) Calculated brightness temperature at A = 5 7 0 0 A (solid curve for the bremsstrahlung opacity, dashed curve for the SESAME opacity), b) Calculated spectral temperature Ts{t) (solid curve for the bremsstrahlung opacity, dashed for the SESAME opacity). CHAPTER 4: LUMINESCENCE MEASUREMENTS smooth profiles. However, this transient effect causes significant distortion of the results only at very early times. For comparison with the experimental measurements of the luminous spectra where a measurement time At is required, the time integrated spectrum is calculated An "observed" time integrated spectral temperature Tg(At) can then be obtained by fitting this relative spectral intensity to the Planck form given in equation 4-10. Figure 4-19a shows the dependence of Tg(At) on At. In these calculations, the "effective emissivity" e varied from 1 to 0.7 for the Rosseland opacity case and from 1 to 0.6 for the bremsstrahlung opacity case for At = 0 to At — 200ps. The results presented in Figures 4-18a, 4-18b and 4-19a clearly show that the measurement of the true temperature of the shock heated solids would require a temporal resolution better than a few picoseconds. Figure 4-19b shows the calculated time integrated spectra for At = 200ps. In-terestingly, both spectra showed excellent agreement with the Planck form in which an "effective emissivity" was used. This behavior was observed in the experiment. Small deviations can be seen when a perfect blackbody spectrum (e = 1) is fit to the calculated spectrum (Figure 4-19b). This yielded 13 — 25% lower spectral temperatures. Such comparisons can be made for measurements which include an absolute intensity calibration. However, the lack of an absolute intensity calibration presents no ambiguity in the interpretation of the experimental data since numer-ical simulations are required to account for the effect of the unloading plasma and the analysis of the data can be treated accordingly. Also evident from Figure 4-19a is the effect of the frequency dependent bremsstrahlung opacity. The reduced ab-sorption at shorter wavelengths (equation 4-9) allows such radiation to be observed from a plasma region of higher density and temperature. This leads to a skewing of F i g u r e 4-19 a) Calculated time integrated spectral temperature Tg(At) as a function of the integration time At (solid curve is for the bremsstrahlung opacity, dashed curve is for the SESAME opacity), b) Calculated time integrated spectrum for At — 200ps. Open circles are for the bremsstrahlung opacity; dash-dot-dot curve is the fit to the Planck form; dash-dot-dot-dot curve is the fit to the blackbody form. Solid circles for SESAME opacity; solid curve is the fit to the Planck form; dash-dot curve is the fit to the blackbody form. CHAPTER 4: LUMINESCENCE MEASUREMENTS 81 the spectrum towards shorter wavelengths, yielding a higher spectral temperature than predicted by the simulations based on the Rosseland mean opacity. 4.3e Shock temperature a n d shock speed H u g o n i o t curve for a l u m i n u m Figure 4-20 shows the measured temperatures as a function of shock speed. Each datum point for the brightness temperature corresponds to a single mea-surement whereas that for the spectral temperature represents the average of five or more measurements. The brightness temperature shows reasonable agreement with the spectral temperature. Also shown in Figure 4-20 are principal Hugoniot curves from L A S N E X 8 4 simulations and from shock calculations using the ideal-gas5 7, Thomas-Fermi24, and SESAME equations of state. The SESAME Hugo-niot curves, modified for the 200ps integration time of the measurement for both the SESAME and bremsstrahlung opacities, are also shown in the figure (spectral temperature). For temperatures < leV, the modified Hugoniot curves are almost identical to the uncorrected curve. For higher temperatures, the simulations using the bremsstrahlung absorption coefficient yielded higher spectral temperatures than those using the frequency average SESAME opacity. This is because of the lower opacity at all wavelengths and the increased absorption with wavelength according to the bremsstrahlung absorption calculation. In the regime of interest here, how-ever, the differences in the calculated temperatures are relatively small and both of the corrected SESAME Hugoniot curves are in reasonable agreement with the data. The Hugoniot curve from LASNEX simulations, being almost identical to the uncorrected SESAME Hugoniot curve, is therefore in excellent agreement with the measurement. The data clearly rejects the Hugoniot curves based on either the ideal gas or Thomas-Fermi equations of state. 4.3f D i s c u s s i o n of the H u g o n i o t measurements for a l u m i n u m The limited availability of theoretical Hugoniot data in the literature has restricted the comparison of our results to ideal gas and Thomas-Fermi equation of CHAPTER 4: LUMINESCENCE MEASUREMENTS 82 1 2 SHOCK S P E E D ( 1 0 6 c m / s ) Figure 4-20 Open circles, spectral temperature measurement; triangles, brightness temperature measurement. Shock temperature and shock speed principal Hugoniot curves for aluminum: solid circles, LASNEX simulations; dashed line, SESAME; dash-dot line, Thomas-Fermi; and, dash-dot-dot line, ideal gas. SESAME Hugoniot curve corrected for the integration time of the measurement: SESAME opacity, dash-dot-dot-dot; and, bremsstrahlung opacity, dash-dot-dot-dot-dot. CHAPTER 4: LUMINESCENCE MEASUREMENTS 83 state models, as well as SESAME data and LASNEX simulations. The advantage of using the shock speed versus shock temperature Hugoniot curve for equation of state studies is evident. Although the experimental accuracy is not sufficient to generate benchmark equation of state data, these measurements have demonstrated the feasibility of obtaining useful Hugoniot curves in a conventional laboratory ex-periment. The significant point to note is that, excluding nuclear explosions, the pressure attainable using laser-driven ablation is much higher than that attain-able using standard techniques. Furthermore, the shock temperature versus shock velocity measurement probes the thermal part of the equation of state, E(p,T), reasonable directly whereas the standard experiments probe the pressure part of the equation of state, P(p, T), since only mechanical variables are measured such as shock pressure and shock velocity. It may be possible to determine important ther-modynamic properties such as the melting temperature on the principle Hugoniot curve using this diagnostic21. In the lower pressure regime it should be possible to apply the shock temperature, shock velocity measurement technique to shock waves generated using a two-stage light gas gun. This would result in a very accu-rate measurement of the temperature since the area of the emitting surface would be ~ 1cm 2 for the gas gun measurement compared to an area of ~ 1 x 10~ 4 cm 2 in laser generated shock wave experiments. This gives a signal level which is ~ 10 4 times higher allowing a much shorter integration time and better signal to noise ratio in the spectral data. To further improve the interpretation of these measurements a better opacity model is required. On the other hand, the accuracy of the measurement could be improved by using a larger laser focal spot size since this would give a better signal to noise ratio for a given integration time. Moreover, it may be useful to calibrate the optical system absolutely to measure the brightness temperature as a function of wavelength. This would allow the measurement of the plasma emissivity as a function of wavelength. CHAPTER 4: LUMINESCENCE MEASUREMENTS The ultimate utility of the shock speed and shock temperature Hugoniot curve will depend on the ability to improve the accuracy of the temperature measurements as well as the opacity data. 4.4 Luminescence measurements in M g , A l , C u , and M o In this section, we will discuss rear surface luminescence measurements as a function of target Z in laser generated shock wave experiments. For the high Z targets, copper (Z=29) and molybdenum (Z=42), it was found that the rise time of the luminous emission was long compared to the temporal resolution of the measurement. This indicated that the heating of the target rear surface was not due to shock heating. The source of this effect was attributed to x-ray transport through the target. This effect was present for all target thicknesses and irradiation conditions in the high Z targets. 4.4a Th ick foils at a moderate laser i rradiance In this subsection, the rear surface luminous intensity for targets of different Z was measured under the same laser conditions. The thicknesses of the targets were chosen such that the heating of the rear surfaces occurred at approximately the same time relative to the incident laser pulse. This insured that the hydro-dynamics was similar in each case. Figure 4-21 shows streak records of the target rear surface luminous intensity for a 50>m thick magnesium target, a 42//m thick aluminum target, a 20/zm thick copper target, and a 15 fim thick molybdenum target, all irradiated at <J>QQ ~ 1013W/cm2. Note, that the magnesium and the alu-minum targets exhibit a rapid cooling after the initial heating of their rear surfaces. This is indicated by the shift of the spectra to longer wavelengths at later times. This observation is consistent with plasma expansion at the target rear surfaces subsequent to shock breakout as described in section 4.3. However, the copper and molybdenum measurements indicate that the expansion at the target rear surface CHAPTER 4: LUMINESCENCE MEASUREMENTS WAVELENGTH ( j jm) 0.7 OA a) WAVELENGTH (um) 0.7 OA b) Figure 4-21 Time resolved spectra of the rear surface luminous emission for various target materials irradiated at <f>60 ~ 10lzW/cm2: a) 50>m magnesium target; and. b) 42pm aluminum target (continued on the following page). Figure 4-21 Time resolved spectra of the rear surface luminous emission for various target materials irradiated at 4>G0 ~ 10uW/cm2: c) 20>m copper target; and, d) 15/xm molybdenum target. CHAPTER 4: LUMINESCENCE MEASUREMENTS 87 is almost isothermal. This is inconsistent with simple shock wave unloading. The luminous intensities as a function of time, obtained by integrating the raw data from A ~ 4700,4 to 6300,4, are shown in Figure 4-22. The time of shock breakout is clearly observed in the magnesium and aluminum data; however, this is not the case for the copper and molybdenum measurements. Note also that the magnesium and aluminum data exhibit a fast rise time in the luminous intensity whereas the copper and molybdenum results show a much slower rise time. The rise time is estimated by plotting the luminous intensity on a logarithmic scale as shown in Figure 4-23. For convenience, we have defined the rise time of the signal to be equal to the time required for a factor of ten increase in the signal level. For the measurements: T\Q ~ llOps for magnesium, rio ~ 120ps for aluminum, T\Q ~ 800ps for copper and T\o ~ lOOOps for molybdenum. The target rear surface heating time (due to a shock wave or other energy transport mechanisms) is defined as the difference between the time of peak laser intensity and the time at which the luminous emission rises to half the peak luminous intensity. The observed rear surface heating times are: ~ 1.5ns for magnesium, ~ 1.7ns for aluminum, ~ 1.1ns for copper and ~ 1.0ns for molybdenum. The difference between these times for the different targets is small compared to the laser pulse length. Thus, all the target rear surfaces expand into the vacuum at approximately the same time relative to the incident laser pulse. This ensures that the energy transport processes (such as shock waves or heat waves) are at the same stage of development at the time of wave breakout; this leads to a meaningful comparison of the data. The rise time of the luminous intensity due to the wave breakout was measured to be of the order of lOOps for both the magnesium and the aluminum. This is approximately equal to the temporal resolution of the optical system (best expected resolution ~ 60ps for a decade rise in the signal level). Hence, it is reasonable to suggest that the rise time of the luminous emission for the magnesium and aluminum CHAPTER 4: LUMINESCENCE MEASUREMENTS 88 600 T . 0 1 2 3 A 5 TIME (ns) Figure 4-22 The luminous intensity as a function of time (integrated from A = 4700A to 6 3 0 0 A ) for the targets irradiated at </>6o ~ 1013W/cm2: a) 50>m magne-sium target; and, b) 42/zm aluminum target (continued on the following page). CHAPTER 4: LUMINESCENCE MEASUREMENTS 89 TIME (ns) Figure 4-22 The luminous intensity as a function of time (integrated from A = 4 700A to 6 3 0 0 A ) for the targets irradiated at 0 6o ~ l013W/cm2: c) 20>m copper target; and, d) 15pm molybdenum target. CHAPTER 4: LUMINESCENCE MEASUREMENTS 1000 >-y— (J) z LU 100: TIME (ns) TIME (ns) F i g u r e 4-24 The luminous intensity as a function of time (integrated from A = 4 700A to 6300A) for the targets irradiated at </>60 ~ 1013W/cm2 (logarithmic scale) a) 50/xm magnesium target; and, b) 42/xm aluminum target (continued on th< following page). CHAPTER 4: LUMINESCENCE MEASUREMENTS Figure 4-24 The luminous intensity as a function of time (integrated from A = 4 700A to 6300A) for the targets irradiated at 0 6o ~ 1013W/cm2 (logarithmic scale): c) 20/ira copper target; and, d) 15/im molybdenum target. CHAPTER 4: LUMINESCENCE MEASUREMENTS targets is T\Q < lOOps. This is characteristic of a planar shock front emerging from the rear surfaces of these targets. Although, the measured rise time for this data is not small compared to the 200 ps integration time used in the analysis, the measured shock temperature is not very sensitive to the exact placement of the time window used in the analysis. Hence, this does not pose a serious problem. For the copper and molybdenum targets the rise time of the luminous intensity is always long compared with the temporal resolution of the measurement. The low count level preceding the signal due to the on-set of the luminous emission (Figure 4-23) in these targets is likely noise. For the magnesium and aluminum targets there is certainly no measureable signal preceding the shock breakout. This indicates that the preheat is less than 0.2eV; the limit of detection. The measured rear surface temperatures as a function of time are given in Figure 4-24. The temperatures were calculated using 200 ps integration windows. The rise time in the temperatures, for the copper and molybdenum targets, could not be accurately determined from the measurements due to the low signal levels at the lower temperatures. The spectra for these shots at the times indicated in Figure 4-24 are given in Figure 4-25 (200 ps integration time) along with greybody fits to the data. Significant deviations of these spectra from a Planck form are not observed. However, there are large fluctuations in the data due to the low signal level. The measurements can be compared with predictions from hydrodynamic simulations. The hydrodynamic code (LTC) was run for aluminum, copper and molybdenum targets. An equation of state was not available for magnesium. The equations of state for copper and molybdenum were obtained from the SESAME data library. These equations of state incorporate the same models as the aluminum equation of state described in chapter 2. As pointed out earlier, in the aluminum targets the material through which the shock propagates is not heated before the shock arrival. Thus, the aluminum results can be modelled using a hydrodynamic CHAPTER 4: LUMINESCENCE MEASUREMENTS > LU cr < cr LU o_ LU 1 2 3 TIME (ns) 1 2 TIME (ns) Figure 4-24 Temperature as a function of time for the previous shots calculated using a 200p.s integration time: a)50//m magnesium target; and, b) 42/zra aluminum target (continued on the following page). The spectra at the points marked 'S' are given in Figure 4-25. CHAPTER 4: LUMINESCENCE MEASUREMENTS TIME (ns) TIME (ns) Figure 4-24 Temperature as a function of time for the previous shots calculated using a 200ps integration time: c) 20pm copper target; and, d) 15pm molybdenum target. The spectra at the points marked 'S' are given in Figure 4-25. CHAPTER 4: LUMINESCENCE MEASUREMENTS 95 1.5-< 1.0H >-to 0.5H 2 LU »— Z T = 0.87eV t = 1.7 ns 0.0-4000 5000 6000 WAVELENGTH (A) a) 7000 1.0-1 • ZD O J B -< 0.6->-h - 0.4-LO 7 LU 0.2-I 0.0-4000 T = 0.71 eV t = 1.9 ns 4 » 4 # * * b) 5000 6000 WAVELENGTH (A ) 7000 Figure 4-25 Spectra for the points marked 'S' in Figure 4-24: a) 50>m magnesium target; and, b) 42/zm aluminum target (continued on the following page). The points are the experimental data and the dashed line is a greybody fit to the data. The temperature obtained from the fit as well as the time of the spectrum is given in the upper right hand corner. CHAPTER J,: LUMINESCENCE MEASUREMENTS 96 1.5-< 1.0-£ 0.5H LU T = 0.63 eV t = 1.4 ns c ) 0.0-4000 5000 6000 WAVELENGTH (A) 7000 0.6-0.5-0.4-0.3-0.2-0.1 4000 T = 0.44 eV t =1.5 ns d) 4 ' 4 5000 6000 WAVELENGTH (A) 7000 Figure 4-25 Spectra for the points marked 'S' in Figure 4-24: c) 20pm copper target; and, d) 15pm molybdenum target. The points are the experimental data and the dashed line is a greybody fit to the data. The temperature obtained from the fit as well as the time of the spectrum is given in the upper right hand corner. CHAPTER 4- LUMINESCENCE MEASUREMENTS code without energy transport from the coronal plasma to the dense target material. The aluminum measurements then provided a standard to obtain an accurate esti-mate of the experimental laser intensity. This reduces the uncertainty in estimating the laser intensity since the laser focal spot size does not need to be known. This was done by varing the intensity used in the simulation until the results matched the aluminum data. The laser intensity which yielded agreement with the aluminum results was 1.25 x \0l3W/cm2. The laser pulse length used in the simulation was 2ns {FWHM). The code was then run for other materials. This is reasonable since the exper-iments on the other materials were done under identical conditions as the aluminum measurements. Furthermore, the absorption at this intensity is essentially 100% for all target materials. The aluminum absorption was measured to be ~ 97% at this laser intensity50 as shown in Figure 4-6. For the higher Z targets the absorption is expected to be higher10. Thus, a correction is not needed for the absorption. The profiles of the hydrodynamic variables derived from the simulations are shown in Figures 4-26, 4-27 and 4-28, plotted in Lagrangian coordinates. The shock trajecto-ries are given in Figures 4-29, 4-30 and 4-31. The formation of the shock wave can be seen when the density profile sharpens up into a discontinuity. As can be seen in Figure 4-29, this occurs at approximately 20/um into the target for the aluminum. As noted above, the calculated shock transit time in the aluminum target is set to match the measured transit time (Figure 4-29). A shock wave was also observed to form in the copper target, as indicated in Figures 4-27 and 4-30. However, it forms late in the pulse and near the target rear surface. For the molybdenum target (Figures 4-28 and 4-31) the simulation indicates that a strong shock wave is never formed during the laser pulse. The temporal profiles of the rear surface luminous intensity and temperature from the simulations and from the measurements for the copper and molybdenum targets are shown in Figures 4-32 and 4-33. The calculated profiles were determined j- CHA P TER 4 •  L UMINESCENCE ME A S UREMEN Tfi 98 O . F i g u r e 4-26 Density and temperature profiles from the simulation for aluminum irradiated at 1.25 x 10nW/cm2 with a gaussian 2ns FWHM laser pulse. The equation of state and thermal conductivity were obtained from the SESAME data library. The laser is incident on the plasma at x = 0. CHAPTER 4: LUMINESCENCE MEASUREMENTS Figure 4-27 Density and temperature profiles from the simulation for copper irra-diated at 1.25 x 10 1 3W/cm 2 with a gaussian 2ns F W H M laser pulse. The equation of state and the thermal conductivity were obtained from the S E S A M E data library. The laser is incident on the plasma at x — 0. CHAPTER 4: LUMINESCENCE MEASUREMENTS 100 equation of state and the thermal conductivity were obtained from the SESAME data library. The laser is incident on the plasma at x = 0. CHAPTER 4: LUMINESCENCE MEASUREMENTS 50 TIME (ns) F i g u r e 4-29 Trajectories of density layers from the simulation for the aluminum. The open circle is the measured transit time for the aluminum target. CHAPTER 4: LUMINESCENCE MEASUREMENTS 102 25 CH 1 1 1 1 1 - 3 - 2 - 1 0 1 2 TIME (ns) Figure 4-30 Trajectories of density layers from the simulation for the copper. CHAPTER 4: LUMINESCENCE MEASUREMENTS 1 0 3 25 'Z ' ""' 1 T 1 - 3 - 2 - 1 0 1 2 TIME (ns ) Figure 4-31 Trajectories of density layers from the simulation for the molybde-num. CHAPTER 4: LUMINESCENCE MEASUREMENTS Figure 4-32 Luminous intensity and temperature profiles from the measurement and simulation for the copper: a) Luminous intensity profiles for the copper where the solid line is the measured profile and the dash-dot line is from the simulation; b) temperature profiles where the circles are the measured temperature and the solid line is the simulated temperature profile. CHA P TEH 4: LI-MINES CENCE ME A SI'HEMEN TS 105 > Q) LU or cr LU CL L U O.H TIME (ns) F i g u r e 4-33 Temperature profiles for the molybdenum where the circles are the measured temperature and the solid line is the simulated temperature. CHAPTER 4: LUMINESCENCE MEASUREMENTS from the temperature in the last few cells in the unloading wave in the simulations. The temperature was converted into a luminous intensity by assuming the rear sur-face radiated as a blackbody. A detailed opacity calculation was not performed. This should not affect the estimation of the rise time of the luminous intensity. The results of the simulations for the copper and molybdenum targets are in disagree-ment with the measurements. This discrepancy can be attributed to x-ray transport from the coronal plasma to the dense target material. This will be discussed later. The shock temperature obtained from the simulation for the aluminum tar-get was ~ l.OeV. The measured temperature, 0.7 - 0.8eV, is consistent with the simulation if the unloading of the plasma is taken into account as seen in Figure 4-20. However, the measured copper temperature is 0.2 eV above that predicted by the simulation. Furthermore, the calculated rise time of the luminous inten-sity is much faster than that observed in the experiment. In the simulation, the profiles of the hydrodynamic variables were recorded in lOOps intervals. This lim-ited the calculated rise times of the luminous intensity and temperature profiles to ~ lOOps. The intensities at the peak emission level were arbitrarily matched since in the experiment an absolute calibration was not obtained. For the molybdenum target, the simulation indicates that only a very weak shock wave (shock temper-ature of ~ 500K) propagates through the target whereas in the measurement the temperatures reaches values of ~ 0.5eV. The luminous intensity derived from the simulation for the molybdenum target was not plotted since the luminous intensity from a 500K blackbody is approximately 17 decades below the detector sensitivity in this measurement. Thus, as well as having a slow rise time in the luminous emis-sion, the measured temperature of the target rear surface is higher than expected for a shock wave breakout. Suppose there exists strong energy transport from the shock heated material to the material in front of the shock wave which causes a shock wave with a finite thickness. (However, there is no energy transport from any other source.) A shock CHAPTER 4: LUMINESCENCE MEASUREMENTS 107 wave with finite thickness will cause a slow rise time in the luminous intensity. Since energy is only transported from the shock heated material, and the shock front has a finite scale length, there exists a time independent initial and final state in front of and behind the shock wave. Thus, the shock wave satisfies the Rankine-Hugoniot relations (equations 2-35) and the state of the material behind the shock wave is on the principal Hugoniot. The shock speed as well as the shock temperature will not be affected by this process. In fact, the measured temperature would be expected to be lower than that predicted from the principal Hugoniot due to the opacity of the unloading plasma as the shock wave emerges. However, if the source of preheat is from the coronal plasma, then additional energy is transported to the shock front, thus, the state of the shocked target is no longer on the principal Hugoniot. One would expect a higher temperature for the shocked target since energy is being supplied to the material. On the other hand, heating by direct x-ray shine-through may be the dominate heating mechanism rather than shock heating. In the measurement we have observed temperatures higher than that expected for shock heating in the copper and molybdenum targets. Such processes must be modelled using a hydrodynamic code which includes energy transport from the coronal plasma. 4.4b Mechanisms for energy transport to the target rear surface Possible mechanisms of energy transport to the dense target material are: fast electrons, thermal conduction, photon transport from directly behind the shock front, and x-ray transport from the corona plasma. In an extensive study of laser plasma interactions at 0.53pm, Mead et a l . 9 1 , 9 2 have shown that less than ~ 1% of the incident laser light was converted to hot elec-trons even at a target irradiance of ~ 2 x 10 1 5 lV/cm 2 . The level of hot electron pop-ulation as well as hot electron temperature decreases rapidly at lower irradiances. To assess the level of hot electron production in this experiment, we have measured CHAPTER 4: LUMINESCENCE MEASUREMENTS 1 0 8 the x-ray continuum emission using a set of four filtered PIN diodes placed at the target front side at angles of ±5° and ±30° off target normal. These results were presented in an earlier work50. The relative sensitivity of the diodes was calibrated using the laser produced plasma as a source. (The spectral response of the diodes93 were also taken into account in the data analysis.) Aluminum foils were used as x-ray absorption filters. The measured x-ray intensity as a function of the filter cutoff energy (l/e transmission) for an absorbed irradiance of 4>QQ ~ 8 X 1013W/cm2 is presented in Figure 4-34. An electron sweeping magnet with field strength of ~ 0.1T was placed in front of the PIN diode at —5° off target normal to discrimi-nate against signals caused by fast electrons reaching the detector. (A lower signal was observed compared to the PIN diode without a magnet at +5°.) The mea-sured x-ray continuum showed that less than 0.1% of the absorbed laser energy was deposited in a 1.5 — 2keV resonantly heated94, suprathermal electron distribution, consistent with the result of Mead et al. For the higher Z targets, the ionization state in the corona is higher than that for the aluminum targets. This means that inverse bremsstrahlung absorption is stronger; thus, less laser light is absorbed resonantly. Consequently, the hot electron population is smaller in these targets. Evidently, such a small hot electron component with a limited range95 (~ 0.1/^ra) cannot account for the observed heating. Heating of the target rear side by thermal conduction has been shown to be important only for very late times96 (after the termination of the laser pulse) even for thin targets. The thermal heat front lags behind the shock front as evident from the hydrodynamic simulations. The emission from the shock heated material directly behind the shock front is of relatively low frequency since the temperature behind the shock wave is of the order of leV. In fact the emission is predominately in the ultra-violet spectral range. The photons emitted from these cold plasmas have a very short range in solids81, < 0.2/xm. Since the shock speed in copper and molybdenum targets is of the order CHAPTER 4: LUMINESCENCE MEASUREMENTS 109 CUTOFF ENERGY (KeV) Figure 4-34 The x-ray spectrum from aluminum targets at an absorbed irradiance of 060 ~ 8 x 1013W/cm2. The filtered PIN diodes were placed at ±30° and ±5° off target normal and an electron sweeping magnet was used to eliminate signals caused by fast electrons (from D. Pasini, Ph.D. thesis, University of British Columbia, 1984). CHAPTER 4: LUMINESCENCE MEASUREMENTS 110 of ~ 106cm/s this energy transport mechanism predicts a rise time in the luminous emission of the order of 20 ps. This is almost two orders of magnitude faster than the measured rise time for the copper and molybdenum targets. Furthermore, this effect should be strongest in aluminum targets since the shock temperature is the highest for a similar laser irradiance. Thus, this mechanism cannot explain the slow rise time of the rear surface luminous emission in the high Z targets. Energy transport by x-rays is particularly important for high Z targets irradi-ated with short wavelength lasers due to very high x-ray conversion efficiencies1 7 - 2 0. These x-rays may provide a source of preheat. Most of the experiments that have been reported 8 7 , 9 7 ' 9 8 are concerned with radiative preheat in thin foils. For these targets, prompt x-ray shine through into the cold solid dominates. For thick targets, the process of radiation transport becomes much more complex. X-ray deposition and re-emission in the dense target may lead to ionization burn-through99. In the presence of a strong shock wave, as in our experiment, the photoabsorption cross-section of the dense target can also be greatly modified as a result of compression and ionization by the shock wave 1 0 0. X-ray transport from the hot corona into the dense target represents the most plausible preheating mechanism. Figure 4-35 shows the x-ray emission from aluminum, copper and molybdenum as a function of absorbed laser irradiance. The measurements were made using differential calorimeters with x-ray filters (2000A aluminum on 2 nm polycarbonate51). The l/e cutoff energy of these filters was ~ 0.8keV. As shown in Figure 4-35, x-ray emission was highest in the copper targets and lowest in the aluminum targets. This is in qualitative agreement with the observation of preheat fronts in the copper and molybdenum targets. Only the hard x-rays (> 5keV) are expected to heat the rear surface of the copper and molybdenum targets, however, since this measurement is spectrally integrated (discussed in section 4.4c) it cannot be used to assess the fraction of laser energy-converted to these x-rays. On the other hand, assuming isotropic x-ray emission, l?1 Figure 4-35 X-ray energy flux from aluminum, copper and molybdenum as a function of absorbed laser £ irradiance. The circles are the experimental data and the solid lines are fits to the data. CHAPTER 4: LUMINESCENCE MEASUREMENTS 112 this measurement indicates that the conversion efficiency of the incident laser light into the x-ray spectrum above 0.8keV is up to 20%. These results support the idea that x-ray transport to the target rear surface is the mechanism that causes the slow rise time of the luminous intensity and the anomalously high rear surface temperatures for the copper and molybdenum targets. 4.4c M o d e l f o r r a d i a t i o n t r a n s p o r t A simple x-ray transport model is described in this section. The target material is assumed to be directly heated by x-ray shine-through from the coronal plasma. The x-ray opacity is assumed to be independent of the state of the material. The comparison of the model predictions to the experimental results is strictly qualitative since the parameters required for the model have not been measured. The purpose of this study is to see if this model can reproduce the experimental observations with reasonable assumed parameters. In particular, we estimate the x-ray energy required to penetrate the target to produce the observed heating and the required conversion efficiency of the incident laser light into these x-rays. The x-ray flux is assumed to be some fraction, fx, of the instantaneous laser flux. This assumption is weak. However, in order to get a better estimate of the x-ray intensity, one would have to use a hydrodynamic code which included a detailed radiation dynamics calculation". This is not available yet. A single effective opacity for the x-ray radiation was used in the simulation for simplicity. The results of simple calculations used to estimate the required x-ray con-version efficiencies as a function of x-ray opacity for the copper and molybdenum targets are given below. The density of the foils were assumed to be the density under standard conditions for these calculations. The energy required to heat the target rear surface to the temperature measured in the experiment is calculated as a function of foil opacity. For the copper target the rear surface temperature was taken to be 0.65ey and for the molybdenum target 0.5eV as determined from CHAPTER 4: L UM1NESCENCE ME AS UREMENTS 113 Figures 4-24c and 4-24d. This x-ray energy was then compared to the total energy available in the laser pulse. The required x-ray conversion efficiency (conversion efficiency, fx, is the ratio of the energy required to heat the target to the energy available in the laser pulse) as a function of x-ray opacity is shown in Figures 4-36 for both the copper and molybdenum targets. Note that the efficiency curves have a fairly sharp cutoff towards the high opacity side of the curves. Furthermore, there is a minimum in the conversion efficiency as evident in Figures 4-36. The opacity corresponding to this minimum conversion efficiency will be denoted by op. This most efficient opacity for target rear surface heating corresponds to oD=^r • [4 - 12] T P This condition means that the photon mean free path is equal to the foil thickness. For the copper target, the opacity which gave the minimum required x-ray conver-sion efficiency was 56cm2 jg where the foil thickness was 20pm and the foil density was taken to be 8.93^ /cms (standard conditions). For the molybdenum target, the opacity which gave the minimum required x-ray conversion efficiency was 65cm2/<7 where the foil thickness was lbfim and the foil density was taken to be 10.2g/cm3 (standard conditions). These values for the opacity were used in the simulations. The simulation was run for the copper and molybdenum targets with the x-ray transport model included self-consistently in the code. The conversion efficiency, fx, was adjusted until the late time temperature of the target rear surface predicted in the simulation agreed with that observed in the experiment. In the simulations, the rear surface temperature was given by the temperature in the last few cells. The material opacity was not folded into the "observed" temperature estimation. However, the opacity effects should be relatively small since the inclusion of radia-tion transport results in an expansion which is almost isothermal. The profiles for the luminous intensity from the code and the experiment could then be compared. CHAPTER 4: LUMINESCENCE MEASUREMENTS 114 0.1 1 10 100 1000 ABSORPTION COEFFICIENT (cm^/g) 0.1 1 10 100 1000 ABSORPTION COEFF IC IENT (cm^/g) Figure 4-36 The required x-ray conversion efficiency to heat the target rear surface of the foil to the measured temperature as a function of x-ray opacity: a) copper; and, b) molybdenum. CHAPTER 4: LUMINESCENCE MEASUREMENTS 115 The temperature profile was converted to a luminous intensity profile (integrated from 4700.4 to 6300A) by assuming blackbody emission. The results from the simulations for the copper targets irradiated at 1.25 x 1013W/cm2 are given in Figure 4-37. The maximum intensity of each curve was arbitrarily matched, since an absolute calibration was not obtained in the experi-ment. The x-ray conversion, fx, was 1.2% and the x-ray energy was 8keV. This x-ray energy corresponded to the opacity op resulting in the most efficient heating of the target rear surface. This value for the opacity was used in the simulation since a better estimate of the appropriate opacity would require a calculation based on a detailed radiation dynamics code. This is not available. The x-ray opacity as a function of x-ray energy (standard conditions)101 is shown in Figure 4-38. Also indicated in the figure is the x-ray opacity used in the simulations, op = 56cm2 jg. This value for the opacity intersects the curve at ~ 8keV (near the K-edge) and at ~ llkeV. For our coronal plasma conditions (Te ~ 500eV)50 much stronger x-ray emission is expected near ~ 8keV rather than ~ 17keV. However, the x-ray energy does not enter the simulation; only the x-ray opacity is important. The simula-tions which included x-ray transport appear to be in better agreement with the measurement than the simulation without transport. However, as seen in Figure 4-37b, a shock wave propagates through the foil and reaches the back of the foil after the initial heating. The appearance of the shock wave is delayed because the foil expands before the shock reaches the back of the foil. Thus, the shock wave travels through an effectively thicker foil of reduced density. This was not observed in the experiment. Furthermore, the heating of the target rear surface was found to occur later than predicted by the simulation. The discrepancy between the results from the x-ray heating model and the measurements are significant. For this direct x-ray heating model, a change in the opacity used in the simulation has little effect CHAPTER 4: LUMINESCENCE MEASUREMENTS 116 TIME (ns) Figure 4-37 Luminous emission and temperature profiles for copper irradiated at 1.25 x \013W/cm2: a) luminous intensity profiles; and, b) temperature profiles. The measured luminous intensity profile is the solid line and the measured temperature profile are the circles. The code results with no x-ray transport are represented by the dash-dot-dot curves and the results with x-ray transport included in the calculation are represented by the dash-dot curves. CHAPTER 4: LUMINESCENCE MEASUREMENTS 117 ENERGY (keV) F i g u r e 4-38 X-ray opacity as a function of photon energy for copper. The horizon-tal line is the value of the opacity which gives the most efficient heating of the target rear surface for a 20>m thick foil at a density of 8.93^/cm3 (standard conditions). CHAPTER 4: LUMINESCENCE MEASUREMENTS 118 on the calculated temporal characteristics of the luminous intensity since the x-ray emission has been assumed to be proportional to the laser intensity. Perhaps a more sophisticated x-ray transport model is required for modelling the copper measurements. As pointed out by Duston et al. 9 9 , radiation transport via the process of absorption and re-emission and ionization burn-through can cause heating of the target rear surface. In thick targets, one would expect such radiation heating to lag behind the shock front99. This transport mechanism, however, can be greatly modified if the dense target becomes sufficiently ionized as a result of heating and pressure ionization due to the shock wave. The photoabsorption cross-section, including the absorption edges, of the shocked material will be modified according to its degree of ionization100. This may enhance radiation transport through the target. To improve the interpretation of the data, we require time resolved spectra of the x-ray emission. These measurements used in conjunction with the lumines-cence measurements would present a powerful tool for the study of x-ray transport through compressed solids since the luminescence measurements give a fairly direct measurement of the energy transported to the target rear surface whereas the spec-tral measurements can be used to assess the frequency of the x-rays which heat the target rear surface. The results from the simulation for the molybdenum irradiated at 1.25 x 1013W/cm2 are shown in Figure 4-39. The x-ray conversion was 0.25% and the energy of the x-rays was llkeV. This x-ray energy corresponded to the x-ray opacity giving the most efficient heating of the target rear surface. The x-ray opacity as a function of x-ray energy (standard conditions)101 is given in Figure 4-40. Also indicated in the figure is the x-ray opacity used in the simulation, op — 65cm2fg. This value of the opacity intersects the curve near ~ llkeV and near ~ 22keV (near the K-edge). As stated earlier, stronger x-ray emission is expected at the lower x-ray energies. As seen in Figure 4-39, the results are in good agreement with the CHAPTER 4: LUMINESCENCE MEASUREMENTS 119 1000q TIME (ns) F i g u r e 4-39 Luminous emission and temperature profiles for molybdenum irra-diated at 1.25 x 10 1 3 W/cm 2 : a) luminous intensity profiles; and, b) temperature profiles. The measured luminous intensity profile is the solid line and the measured temperature profile are the circles. The code results with no x-ray transport are rep-resented by the dash-dot-dot curves and the results with x-ray transport included in the calculation are represented by the dash-dot curves. OH A P TER 4: LI 'MINES CENCE ME A SI 'REM EN TS 120 ENERGY (keV) Figure 4-40 X-ray opacity as a function of photon energy for molybdenum. The horizontal line is the value of the opacity which gives the most efficient heating of the target rear surface for a 15/im thick foil at a density of 10.2g/cm3 (standard conditions). CHAPTER 4: LUMINESCENCE MEASUREMENTS 121 measurement. In this case, there is very little shock heating (small bump in Figure 4-39) since shock heating is insignificant even in the case of no x-ray transport. 4.4d Copper and molybdenum at lower intensities In the simple x-ray transport model, the radiative heating of the target rear surface is coupled to the laser pulse since the x-ray intensity has been assumed to be proportional to the laser intensity. Thus, the temporal profile of the heating depends on the temporal profile of the laser pulse; but is weakly dependent on the peak laser intensity. The measured luminous intensity and temperature as a function of time are given in Figure 4-41 and Figure 4-42 for copper and molybdenum irradiated at 7.5 x 10l2W / cm2. The results from the simulation are also shown in these figures. For the copper simulations, the x-ray conversion used was fx = 0.98%. This x-ray conversion factor was also chosen so that the temperature profile agreed at late times. The heating of the target rear surface was found to occur later than predicted by the simulation incorporating the x-ray transport model. Furthermore, in the simulation, a shock wave was found to propagate through the target whereas this was not observed in the measurement. It is unlikely that the experiment can be modelled accurately using such a simple transport model. As stated earlier, the purpose of this study is to determine if heating of the target rear surface by x-rays is a reasonable assumption. The x-ray conversion requirements, ~ 1%, may not be unreasonable. For the molybdenum, the x-ray conversion used in the x-ray heating model was fx = 0.42%. The results are similar to those for the higher intensity. The transport model reproduces the measurement reasonably well. 4.4e Discussion of the copper and molybdenum luminescence measurements The slow rise time in the luminous emission of the rear surfaces of laser irradiated copper and molybdenum foils has been attributed to heating of the rear CHAPTER 4: LUMINESCENCE MEASUREMENTS 1 2 2 1000; => 100: < z UJ 10= SHOCK ARRIVAL —-I a) CODE WITH X-RAYS / / CODE WITHOUT X-RAYS 1 / L SHOCK ARRIVAL 1 TIME (ns) 0.6 0.5' J 0.4 H LU * 03 < a 0.2H LU Q. UJ 0.1" 0.0 b) of. CODE WITH X-RAYS 4\ EXP. X o I / "° o / | CODE WITHOUT J L X-RAYS - 1 0 1 2 3 A 5 TIME (ns) Figure 4-41 Luminous emission and temperature profiles for copper irradiated at 7.5 x 10]3W/cm2: a) luminous intensity profiles; and, b) temperature profiles. The measured luminous intensity profile is the solid line and the measured temperature profile are the circles. The code results with no x-ray transport are represented by the dash-dot-dot curves and the results with x-ray transport included in the calculation are represented by the dash-dot curves. CHAPTER 4: LUMINESCENCE MEASUREMENTS 123 F i g u r e 4-42 Luminous emission and temperature profiles for molybdenum irra-diated at 7.5 x 1 0 1 3H 7 / c m 2 : a) luminous intensity profiles; and, b) temperature profiles. The measured luminous intensity profile is the solid line and the measured temperature profile are the circles. The code results with no x-ray transport are rep-resented by the dash-dot-dot curves and the results with x-ray transport included in the calculation are represented by the dash-dot curves. CHAPTER 4: LUMINESCENCE MEASUREMENTS 124 surfaces by x-ray transport rather than shock waves 4 3' 4 4. A simple x-ray transport model was incorporated into the hydrodynamic code LTC. Some of the qualitative features of the copper measurements could not be reproduced using the x-ray heating model. The slow rise time of the luminous in-tensity was observed in the simulations. However, the heating occured much earlier than that observed in the experiments. Furthermore, the simulation predicted that a shock wave propagated through the target; this was not observed in the measure-ment. Such disagreement with experimental results may be caused by the neglect in the simple transport model of important effects such as the change in the x-ray opacity of the target material due to target compression and the dependence of the foil opacity on x-ray energy. For the molybdenum target, the x-ray heating model could reproduce all the features of the measurement. The required x-ray energy was ~ llkeV and the conversion of the incident laser energy into x-rays at this energy was 0.2 — 0.5%. The purpose of the simple model was to determine if heating of the target rear surfaces of copper and molybdenum targets could be due to x-ray transport from the corona. The results of the simulations appear to support this idea. How-ever, for a quantitative measurement of x-ray transport through compressed solids, time resolved spectral measurements of the x-ray emission is required. These mea-surements, with the luminescence measurements at the target rear surface, form a powerful combination for the study of radiation transport since the luminescence measurements yield a fairly direct measure of the energy transported to the target rear surface and the x-ray spectral measurements can be used to assess the frequency of the x-rays which penetrate to the rear surface. Furthermore, the interpretation of the data would be improved using simulations including a self-consistent radiation dynamics calculation. CHAPTER 5: REFLECTIVITY MEASUREMENTS 125 C H A P T E R 5 R E F L E C T I V I T Y M E A S U R E M E N T S 5.1 E x p e r i m e n t a l s e t u p The validity of theoretical electron transport models for dense plasmas can be tested using the measured reflectivity of laser light from the expanding free surface of a shock compressed solid. Figure 5-1 shows the schematic of the experimental setup used for the measurement. The rear surface of the target foils were carefully polished to near optical quality with a residual roughness of < 0.2/zm. The rear surface of the target was imaged onto the entrance slit of the streak camera at normal incidence through f/2 optics. Normal incidence was used so that the polarization of the probe laser was irrelevant. This simplified the measurement and the simulation of the measurement. High magnification (xl30) was used in order to obtain the highest possible spatial resolution (~ 3/xm). The probe laser was a Rhodamine 6G dye laser which was developed in-house. This was optically pumped using leakage light (A = 5300A) from one of the turning mirrors in the main laser beam. The wavelength of the probe beam was 5700A. The probe beam was injected into the imaging system using a beam splitter. Two identical lOOA bandpass interference filters of centre wavelength 5700A were placed at the entrance of the streak camera to eliminate stray light from entering the streak camera. A fiducial (reference beam) was recorded on the streak camera as a timing reference. Figure 5-1 E x p e r i m e n t a l se tup for the rear surface ref lect ivity measurements . CHAPTER 5: REFLECTIVITY MEASUREMENTS 127 5.2 M e a s u r e m e n t of the reflectivity of shocked a l u m i n u m Figure 5-2 shows a streak record of the probe laser light reflecting off the rear surface of a 50/xm thick aluminum target irradiated at (foci ~ 7 x 1013W/cm2. The spatial resolution in these measurements was approximately 3/zm. The shock arrival time is the time at which the target rear surface becomes absorbing as indicated in the figure. (The rear surface is absorbing in the white region of the figure and reflecting in the black region.) It should be noted that the grey scale for the video display shown in the figure has only seven levels whereas the actual data is stored with eight-bit resolution. The dynamic range of the camera was of the order of 100. The shock is observed to emerge in a region of approximately 40/xm in diameter with < 2% variation in the shock speed across this breakout region as estimated from the shock transit time. This is consistent with the earlier measurements. This indicates that the shock wave is reasonable planar. To obtain the spatial and temporal dependence of the rear surface reflectiv-ity of the target, a vertical normalization window and a horizontal normalization window are used. The horizontal window is chosen to span a time interval where the rear surface of the target is undisturbed (before the shock or heat wave arrival time). The intensity profile in this window ('A' in Figure 5-2) is used to eliminate the spatial modulations of the probe laser from the streak record. The vertical window is chosen in a region where the signal is not affected by the shock wave arrival at the target rear surface. The intensity profile in this window ('B' in Figure 5-2) is then used to eliminate the temporal modulations of the probe laser from the streak record. This then yields the relative reflectivity as a function of time in one spatial dimension as shown in Figure 5-3. The absolute reflectivity is esti-mated by comparing the reflectivity of the unperturbed target to the reflectivity of a first surface mirror placed at the target plane. The reflectivity as a function of time is obtained by averaging the central 10/xm of the shock breakout region for similar shots. The signal is averaged over a 10/xm region to obtain a good signal to CHA P TER 5: REFLEC T1VITY ME A S UREMEN TS 128 Figure 5-2 A streak record of the reflectivity of the rear surface of a 50//m thick aluminum target for an absorbed irradiance of 0(lO ~ 7 x 1 0 1 3 H 7 / c m 2 . ' A ' indicates the horizontal normalization window and ' B ' indicates the vertical normalizing win-dow. Tg is the time of shock arrival. The time scale is relative in this figure. The fiducial is delayed by 1.8ns. 40 POSITION (jjm) 2 so IS I Figure 5-3 The same data as in Figure 5-2 after normalization with respect to windows 'A* and ' B ' . The t i scale is absolute in this figure. T ime zero corresponds to the peak laser intensity. CHAPTER 5: REFLECTIVITY MEASUREMENTS 130 noise ratio. As the shock breakout is planar in this region, the experiment can be modelled using one dimensional calculations. For the quantitative results presented below a faster streak speed was used (1.25ns streak time) in the measurements, which gave a temporal resolution of ~ lOps. The results of several rear surface reflectivity measurements are shown in Fig-ures 5-4, 5-5 and 5-6. The shock speeds for the data presented in these figures were 1.7±0.2 x 106cm/s, 2.2±0.2 x 106cm/s and 2.5±0.3 x 106cm/s respectively as de-termined from the measured shock trajectories (such as Figure 4-5). The measured reflectivities were overlayed by matching the shock breakout times. (The onset time of the decrease in the reflectivity.) Although the temporal resolution of the measurement is of the order of lOps, the shock breakout time (or equivalently, the position of time zero on the time scale) has an uncertainty of ~ lOOps. This results in an uncertainty of the shock speed of the order of 10% which is consistent with the earlier measurements (section 4.2a). When the shock wave emerged, the rear surface reflectivity was observed to decrease almost exponentially in time. As can be seen from the figures, the measurements are reasonably reproducible. The shot to shot fluctuations give an indication of the accuracy of the measurement. Surface roughness of the target rear side (typically < 0.2/xm) would lead to a variation in the shock breakout time of < lOps. This was much shorter than the observed time constants (of the order of 100 — 200ps) for the decrease of the reflectivity. It should also be noted that similar results are obtained for the reflectivity averaged over the central 3/xm region at the expense of the signal to noise ratio. This again demonstrated the planarity of the shock wave observed. 5.3 S i m u l a t i o n of the reflectivity measurements To understand the experimental results, one needs to examine the process of reflection of electromagnetic radiation by an inhomogeneous plasma. This problem has been treated in detail by Ginzburg 1 0 2. The measurement was simulated from 1 >-> LU -J Li_ LU cr 0.01 -0.1 T 0 — r 0.1 0.2 —r~ 0.3 0.5 TIME (ns) F i g u r e 5-4 Temporal evolution of the reflectivity of the rear surface of a 25pm thick aluminum target for a shock speed of 1.7 > 10 ( 'cm/«. Time zero corresponds to the peak laser intensity. 3 5 ! S: io ta ta > 1.0 TIME (ns) F i g u r e 5-5 Temporal evolution of the reflectivity of the rear surface of a 50//m thick aluminum target for a shock speed of 2.2 x .1Vf'cm/'s. Time zero corresponds to the peak laser intensity. 0.01 0.3 0.4 0.5 TIME (ns) 0.6 0.7 0.8 Figure 5-6 Temporal evolution of the reflectivity of the rear surface of a 50jum thick aluminum target for a shock speed of 2.5 x \0ucrn/s. Time zero corresponds to the peak laser intensity. CHAPTER 5: REFLECTIVITY MEASUREMENTS 134 the instant of shock arrival at the target rear surface to later times. First, consider an ideal situation of a planar shock wave of zero thickness propagating toward the rear surface of the target. Before the shock wave arrives, the reflectivity of the target rear surface is given by the Fresnel formula1 0 2 Rs(») = 4na/" + 1 - 2 ^ ^ [5-1] 4no/u + 1 + 2v/27r<7/a; where u is the frequency of the probe laser and a is the electrical conductivity of the unperturbed target. At the instant of shock arrival the temperature and density of the target rear surface change to values determined by the principal Hugoniot for aluminum. This results in a different value for the electrical conduc-tivity and accordingly a different reflectivity as determined by equation 5-1. This formula is valid when the gradient scale length of the index of refraction is such that [(l/n)dn/dx] _ 1 <C \/2ir, where n is the refractive index of the plasma and A is the wavelength the probe laser. Figure 5-7 gives the SESAME data for p and T as a function of shock velocity for aluminum. Also plotted in the figure is the corresponding d.c. electrical conductivity, er, from the SESAME data library. The reflectivity of the shocked homogeneous aluminum target, Rs, is given in Figure 5-8. As is evident in the figure, Rg varies weakly with shock velocity. In fact, for shock speeds less than 3 x 106cm/s or correspondingly, for shock pressures less than 20Mbar, the reflectivity of the shocked aluminum surface is a relatively insensitive parameter for inferring the electrical conductivity of the plasma. The reflectivity of aluminum (with a perfect surface) at standard conditions is Ro ~ 0.98 6. The reflectivity of the target rear surface would be expected to drop from RQ to Rs at the instant of shock arrival. Subsequent to shock breakout, the plasma starts to expand into the vacuum. Thus the incident probe radiation propagates up a density gradient. The hydrodynamics of the unloading target were treated using the one-dimensional hydrodynamic code PEC incorporating flux corrected TEMPERATURE (eV) i n o o 7^  LO TJ m m o cn o O o J I l i l t J I I I ' ' ' ' DENSITY ( g / c m 3 ) i 1 co L_ —r~ C O — T -ELECTRICAL CONDUCTIVITY (10 1 6 s" 1 ) SI M3PV3 HIS V3IM AIIAII031J3H :S HSIJVHL) CHAPTER 5: REFLECTIVITY MEASUREMENTS 136 Figure 5-8 Reflectivity calculated for a shocked aluminum surface as a function of the shock speed. CHAPTER 5: REFLECTIVITY MEASUREMENTS 137 transport and SESAME equation of state data 4 1 , 5 2 . The density and temperature profiles were computed as a function of time. A mesh size of < 0.02/xm was used in the calculations. Since the characteristic unloading speed (isentropic sound speed) is much less than the speed of light, the reflectivity of the plasma at any instant can be determined by solving the Schrodinger equation at a fixed time 1 0 2, t, d2E u2 + -Te{u,x,t)E = 0 dx2 c 2 The dielectric function e is approximated by the collisional, cold plasma dispersion relation ul(xA) r 1 - i e{u,,x,t) = 1 - P V ' — r — [5 - 3] V ' 1 u2 LI + ii/ei(x,t)/u>\ 1 J where u>p is the electron plasma frequency and t/et is the electron-ion collision fre-quency which is derived from the SESAME data library 2 9 (or from Lee and More's conductivity model28) using the electrical conductivity o and the average ionization 102 utl(x,t) = [5-4] The plasma frequency, up, is given / 4 7 r n e e 2 \ 1 / 2 up = e— . 5 - 5 \ m / An exact solution of equation 5-2 is not possible. However, when the density gradient scale length of the unloading plasma becomes larger than the wavelength of the incident light, propagation of the incident radiation in the plasma can be treated using the approximation of geometric optics (WKB) and the reflected ra-diation taken as a perturbation of this solution 1 0 2. This condition is given by CHAPTER 5: REFLECTIVITY MEASUREMENTS 138 \{\/n)dn/dx]~l » X/2n. The reflectivity is then, R = tanh | (~~c~/ niy)dy)dx\ [5-6] where the refractive index n = -^ /e. This reflectivity calculation is coupled to the hydrodynamic simulation. Figure 5-9 shows some profiles of density and temperature at various times for the case corresponding to shocked aluminum where the shock speed was 2.2 x Evidently, reflection occurs near the critical density layer for the probe laser. In the region of lower density, the probe radiation is strongly absorbed by collisional (inverse bremsstrahlung) absorption. In this regime, where the temperature is of the order of leV and the density is of the order of lg/em 3 , the electron mean free path approaches the interatomic distance. 5 . 3 a Sensitivity of the reflectivity diagnostic To assess the sensitivity of the reflectivity diagnostic we have calculated the reflectivity using the simulations described above for different probe laser wave-lengths and electrical conductivities. Figure 5-10 shows the reflectivity as a func-tion of time for probe laser wavelengths ranging from 0.25/j.m < X < 0.78/xm for a shock wave with a speed of 2.5 x 106cm/s, using data from the SESAME library. These wavelengths are representative of probe light from the ultraviolet to the near infrared. Time zero represents the time of shock breakout. The gap between the shocked reflectivity R$ at t = 0 and the reflectivity determined using equation 5-5 is due to the limited regions of validity of the calculations ( W K B approximation). Compared with the reflectivity of aluminum under normal conditions (RQ ~ 0.9), 10Gcm/s. To identify the plasma regions where significant reflection of the probe radiation (A = 5700A) occurs, we have also plotted the spatial profiles of dR/dx. CHAPTER 5: REFLECTIVITY MEASUREMENTS 139 Figure 5-9 Spatial profiles of density (solid line), temperature (dashed line) and dRjdx (dashed-dot line) for the case of a shock speed of 2.2 x 10 6cm/s: a) t = 0; b) t •= 50ps; c) t = lOOps; and, d) t - 200ps. The location of the critical density layer (ncr) for the probe laser is indicated in the figures. CHAPTER 5: REFLECTIVITY MEASUREMENTS 140 TIME (ps) Figure 5-10 Reflectivity curves calculated for probe radiation of different wave-lengths for a shock strength of 2.5 x 10 6cm/s using data from the S E S A M E data library. Time zero corresponds to the time of shock breakout. CHAPTER 5: REFLECTIVITY MEASUREMENTS 141 determination of the shocked aluminum reflectivity, (Rs), would require measure-ment accuracy of the order of ±10%. On the other hand, the reflectivity of the unloading plasma appears to decrease approximately exponentially with time. The calculated time constants can be easily verified experimentally using a streak camera with temporal resolution of 2 — lOps. To test the sensitivity of the calculated reflectivity to the electrical conduc-tivity, the simulations were repeated using electron conductivity values which were arbitrarily reduced to 1/2, 1/4 and 1/8 of the SESAME values. The results of the simulations are shown in Figure 5-11 for a shock wave with a speed of 2.5 x 106cm/s and probe laser wavelength of 5700A. In addition to the reduction in the reflec-tivity, Rg, a greater change can be seen in the time constant for the decrease in the reflectivity. This suggests that the reflectivity measurements provide a sensitive test for electrical conductivity models. 5.3b Limitations on the detection of Rs The reflectivity of the shocked aluminum, Rs, was not observed in the ex-periment as can be seen in Figures 5-4, 5-5 and 5-6. The results from Figure 5-8 show that for shock speeds up to 3 x 106cra/s, the reflectivity is only reduced by < 20% (calculated using data from the SESAME library) compared with the initial reflectivity of the foil, ~ 90%, under normal conditions. This may be within the range of the accuracy of the measurement. However, the possibility of measuring Rs was further impeded by the finite temporal resolution of the measurement due to camera resolution (~ lOps), residual foil roughness (~ lOps) as well finite shock rise time. Furthermore, the initial foil reflectivity was less than that expected for an ideal aluminum surface due to the residual foil roughness. To see if these effects are sufficient to wash out the measurement of the reflec-tivity Rs of the shocked aluminum, a simple calculation is presented. An estimated theoretical curve for reflectivity as a function of time was constructed as follows. CHAPTER 5: REFLECTIVITY MEASUREMENTS 142 Figure 5-11 Dependence of calculated reflectivity curves on the electrical conduc-tivity for a shock strength of 2.5 x lO^'cm/s and a probe laser wavelength of 5700A using data from the S E S A M E library. Time zero corresponds to the time of shock breakout. CHAPTER 5: REFLECTIVITY MEASUREMENTS 143 The reflectivity is assumed to be constant at RQ before the shock breakout. At the time of shock breakout the reflectivity is assumed to drop instantaneously to R s and remain at this value until the density gradient scale length of the plasma is equal to X/2n. This time, ts, is determined from unloading isentropes calculated from the SESAME data. For times greater than ts the reflectivity is assumed to decay exponentially with time constant TS as determined from the measurement. The reflectivity as a function of time is then R{t) = Ro t<0 R{t) = R s 0 < t < ts [5-7] R{t) = R s expf-* ~ t s ) t s < t \ TS > For the calculation the parameters used where Ro — 0.9, R s — 0.7, 15 = 5ps and TS — 65ps where ts was determined from SESAME isentropes for a shock speed of 2.2 x 106cm/s and Ts was determined from the data presented in Figure 5-5, the measured decay constant for this shock speed. Since the foil is rough, the shock wave will emerge from various local regions of the target rear surface over a time interval IR (the shock wave emerges at the target rear surface at different times for different positions). The roughness of the foil is of the order of 0.2/xra which is much smaller than the spatial resolution of the diagnostic (~ 3//m), thus the effect of the foil roughness is to smear out the measured reflectivity profile. It will be assumed that as the shock wave breaks out of the target rear surface the total area of the shocked region increases linearly with time between —IR/2 to tp/2. At t = — tfi/2 the shock wave starts to emerge at the target rear surface and at t = tf{/2 the breakout is complete. The actual spatial and temporal distribution of the shock wave breakout is unknown, however, we are not interested in the fine details of the breakout, but rather the effect of the width CHAPTER 5: REFLECTIVITY MEASUREMENTS of the distribution, tp. This effect is similar for other distributions. The change in the reflectivity signal due to the foil roughness is then, 144 R'(t) where tp has been estimated to be lOps in our measurement (as estimated from 0.2/im roughness and a shock speed of 2 x 106cm/s). The finite temporal resolution of the measurement is mainly due to the streak camera slit width used in the measurement. This causes the signal at any instant in time to be spread over several pixels in the camera readout. This distribution will be assumed to be Gaussian with a width of Tp. Thus, the observed reflectivity signal can be written J-oo VnTF K TF ' where Tp has been estimated to be lOps in our experiment. The results of the calculation are shown Figure 5-12. It is evident that even without the observed fluctuations in the reflectivity included in the calculation, the measurement of R$ is impossible. However, the exponential decay at later times is essentially unaffected, thus, the measured decay of the reflectivity should provide useful information for comparison to theoretical electron transport models. 5.3c C o m p a r i s o n o f t h e s i m u l a t i o n s w i t h e x p e r i m e n t Two conductivity models were used in the simulations. They were the data from the SESAME library and two versions of Lee and More's conductivity model. The experimental results and the predictions of the simulations are given in Figures 5-13, 5-14 and 5-15. The predictions based on the SESAME conductivity data appear to be too high whereas that using Lee and More's conductivity model with CHAPTER 5: REFLECTIVITY MEASUREMENTS 145 Figure 5 - 1 2 Estimated reflectivity as a function of time: a) ideal curve (solid line): b) curve the effect of foil roughness (dashed line) and c) curve including both the effect of foil roughness and temporal resolution (dashed-dot line). TIME (ns) Figure 5-13 Measured reflectivity of the rear surface of a 25/um aluminum target for a shock speed of 1.7 x l O ' c m / s (solid curves). Time zero corresponds to the peak of the incident laser pulse. Calculated reflectivity R$ of the shocked free surface using S E S A M E conductivity data (open circle). Calculated reflectivity of the unloading plasma using S E S A M E conductivity data, (dashed line), Lee and More's model with A m m = ro (dash-dot line) and Lee and More's model with A „ , M - 2r n (dash-dot-dof line). 1.0 TIME (ns) Figure 5-14 Measured reflectivity of the rear surface of a 50//m aluminum target for a. shock speed of 2.2 x 10f7:m/.s (solid curves). Time zero corresponds to the peak of the incident laser pulse. Calculated reflectivity /i*,s- of the shocked free surface using S E S A M E conductivity data (open circle). Calculated reflectivity of the unloading plasma using S E S A M E conductivity data (dashed line). Lee and More's model with Xmin - r 0 (dash-dot line) and Lee and More's model with \ m r v -- 2r n (dash-dot-dot line). TIME (ns) Figure 5-15 Measured reflectivity of the rear surface of a 50pm aluminum target for a shock speed of 2.5 X 10('cm./.s (solid curves). Time zero corresponds to the peak of the incident laser pulse. Calculated reflectivity /?,<? of the shocked free surface using S E S A M E conductivity data (open circle). Calculated reflectivity of the unloading plasma, using S E S A M E conductivity data (dashed line), Lee and More's model with A M M = ro (dash-dot line) and Lee and More's model with A „ M , - 2TQ (dash-dot-dol line). CHAPTER 5: REFLECTIVITY MEASUREMENTS 149 Tc — fo/v appear to be too low. Predictions based on the version of Lee and More's conductivity model with TC = 2TQ/V agree best with the data, especially for the two weaker shock waves. It should be noted that even for late times the reflectivity of the plasma appears to remain at the level of a few percent. The cause of this is not understood and may be due to scattering processes not included in the model used in the simulations. The discrepancy between the simulations and measurements can be traced to the inability of existing transport calculations to treat the collision process when the electron mean free path, Xmfp, becomes of the order of the interatomic distance, 2rn. The densities and temperatures of the unloading plasma are given by the release isentropes as determined from the SESAME data as shown in Figure 5-16. The plasma regions probed, calculated using the integrand in equation 5-6, (for the SESAME conductivity model) are marked by the bold lines. The upper density limits correspond to the critical density for the probe radiation where as the lower density limits correspond to plasmas which give rise to a reflectivity of > 1% of the total reflectivity. Thus, for the times that the simulation is valid we are probing a dense plasma region characterized by densities of approximately lg/cm3 and temperatures of approximately leV, a regime where \mfp approaches ro. In the SESAME data, the electron transport properties were calculated in the "average atom" approximation, as in Lee and More's model, assuming two body ion-ion correlations and single-site scattering. However, in the SESAME data, detailed electronic structure was calculated in a self-consistent ionic potential. These results was used to calculate the collision cross-section, the chemical potential and the free electron density using partial wave analysis. This information was then used in the t-matrix formulation of the Ziman model6 3 along with expressions for the ionic structure factor to obtain the electrical conductivity. However, for strongly interacting systems, such as the plasma of interest here, the single-site scattering approximation breaks down. 0.01 0.1 1 10 DENSITY (g/cm3) Figure 5-16 Density temperature regimes of the plasma probed in the reflectivity measurements while equation 5-6 is valid (bold lines). The solid lines are the rcleasse isentropes from the S E S A M E data library for shock velocities of 1.7 x lO'cm/.s. 2.2 • I ()''<•?/»/.s and 2.5 .< lO'Vm/.s. The dash-dot line is the pricipal Hugoniot curve. CHAPTER 5: REFLECTIVITY MEASUREMENTS 151 In Lee and More's model, this region was modeled using a simple expression for the relaxation time, TC = A m i n / v . This minimum electron mean free path could be adjusted until agreement with measurements was obtained. We found better agreement with our measurement when this was set to Xmin = 2TQ as seen in Figures 5-13, 5-14 and 5-15. However, for the case with a shock speed of 2.5 x 106cm/s, there is still significant disagreement. The use of a cut-off for mean free path when it becomes of the order of the interatomic spacing is a crude approximation which allows the transport properties of a material to be estimated in a simple manner. A fully self-consistent transport model, which includes multiple scattering, may indeed be required to obtain agreement with the experiment at all shock strengths. On the other hand, the experimental results may be affected by radiation transport at the higher laser intensity (Figure 5-15) which may result in the disagreement between the simulations and the experiment. However, the x-ray transport will have to be studied in detail to determine its effect. Using A m t n = 2ro rather than Xmin — ro increases the conductivity by less than a factor of two in a small region of the conductivity model. Even though the change in the conductivity is small the change in the predicted reflectivity as a function of time is significant. The accuracy of the measurement could be improved in several ways. Tem-poral and spatial fluctuations of the probe laser could be reduced by using a laser with a single longitudinal mode. A larger focal spot size for the main laser beam and lower magnification of the rear surface imaging optics could also be used. This would allow a smaller streak camera slit width to be used which would improve the temporal resolution of the measurement. Note that the same effect could be achieved by increasing the probe laser intensity. However, if the probe intensity be-comes too high it will alter the plasma it is probing. Stepped targets could be used so that the reflectivity and the shock speed could be measured in one shot13. With these improvements, the reflectivity measurement could be an extremely accurate CHAPTER 5: REFLECTIVITY MEASUREMENTS 152 diagnostic which would provide data in a density and temperature regime that is extremely difficult to treat theoretically. Furthermore, the reflectivity measurement could be done for oblique incidence with a polarized probe beam. This allows the measurement of the intensity as well as the polarization of the reflected light. The index of refraction of the plasma can be determined in much more detail since the polarization of the reflected beam depends on the index of refraction103. Finally, the reflectivity measurements could be done as a function of the probe laser wavelength. These measurements could be used to study the validity of the cold plasma dispersion relation as applied to these plasmas. 5 . 4 Reflectivity of high Z targets In this section, we will discuss reflectivity measurements on high Z targets in the laser generated shock wave experiments, in particular, copper and molybdenum targets. The rear surfaces of these foils were again polished to near optical quality with a residual roughness of < 0.2//m. For the high Z targets (Z=29 for copper and Z=42 for molybdenum) it was found that the decrease of the reflectivity was slow compared to that expected for a shock wave emerging at the target rear surface. This indicated that the observed wave fronts had a finite spatial structure. The shape of the breakout region on an x-t plot was also found to be inconsistent with shock breakout. Furthermore, the measured time of onset of the decay in the reflec-tivity occurred well before the expected shock arrival time. These observations were attributed to x-ray transport through the target, as suggested in chapter 4. Such preheat effects were present for all irradiation conditions in these high Z targets. 5 . 4 a Reflectivity measurements on C u as a function of laser intensity The x-t plots of the reflectivity for 20/zm thick copper targets at laser irradi-ances of <j>&o ^ 1.4 x 10lzW/cm2, 2.5 x 1013W/cm2 and 5 x lOi3W/cm2 are presented CHAPTER 5: REFLECTIVITY MEASUREMENTS 153 in Figure 5-17. These results are different from those for aluminum in several ways. The size of the breakout region is smaller in the copper measurements than in the aluminum measurements. For the aluminum targets, the shock wave emerged es-sentially simultaneously over a disc of diameter of ~ 40//ra; the size of the spot then grew as a function of time. For the copper case, the size of the breakout region as a function of time appears to be parabolic in shape. If target hydrodynamics was the only important process, one would expect the size and shape of the breakout region for the copper targets to be similar to that for aluminum targets. The thickness of the compressed target foil as well as the arrival time of the wave with respect to the peak laser intensity are the most important parameters for determining the planarity of the wave. As the laser driven shock wave propagates into the target, an edge-rarefaction wave also propagates into the shock compressed material. As it propagates radially inward this rarefaction wave reduces the size of the planar region of the shock front. The thickness of the shock compressed target should be less than the laser spot size to ensure a region of planar shock wave breakout. Also at late times, well after the peak laser intensity, the rarefaction wave which was launched into the compressed target from the coronal plasma when the incident laser intensity began to decrease reaches the shock front. The rarefaction wave is two dimensional due to the finite size of the laser focal spot; thus, it reduces the planarity of the shock front. For the copper targets, the wave arrival times were approximately 0.6ns, —0.1ns and —0.7ns for laser irradiances of d>60 ~ 1.4 x 1013W/cm2, 2.5 x l013W/cm2 and 5 x 1013W/cm2 respectively. For the aluminum data presented in Figure 5-2 the arrival time of the shock wave was 0.6ns and the laser intensity was <foo — 7 x 1013W/cm2. The arrival times for the copper shots are in the range from times which are similar to earlier than the aluminum shot. Furthermore, the aluminum target is thicker than the copper target. Thus, a shock wave produced in copper targets should be as planar as that in the aluminum targets. However, this is not supported by the data. This CHAPTER 5: REFLECTIVITY MEASUREMENTS a) POSITION (pm) b ) 0 AO 80 120 POSITION (pm) F i g u r e 5-17 Normalized streak records for 20//m thick copper targets irradiated at: a) <j>m - 1.4 > 1 0 1 3 W / c m 2 ; and, b) qm ~ 2.5 x I 0 1 3 W / c m 2 (continued on the following page). T ime zero corresponds to the peak laser intensity. CHAPTER 5: REFLECTIVITY MEASUREMENTS 155 Figure 5-17 Normalized streak records for 20pm thick copper targets irradiated at: c) d*eo ~ 5 > 10 i l ' / c m 2 . Time zero corresponds to the peak laser intensity. CHAPTER. 5: REFLECTIVITY MEASUREMENTS effect may be due to radiation transport from the corona in the high Z targets. A simple model which describes this process in given in section 5.4b. The reflectivity as a function of time are given in Figure 5-18 for copper targets irradiated at the various intensities. This data was obtained at a faster streak speed than that presented in Figure 5-17 (the full sweep time was 1.25 ns rather than 5.64 ns). The temporal resolution of the measurement was ~ lOps. As noted earlier, the uncertainty in the time axis is approximately lOOps since the time of the peak laser intensity can only be measured to this accuracy. The error bars indicate the typical uncertainties in the measurements at different points in time. The data were obtained by averaging the reflectivity signal over the central 10p:m region of the observed wave front at the target rear surface. Over this region the experiment can likely be modelled using one dimensional models. At the highest laser intensity the reflectivity signal decreases more slowly (Figure 5-18c) than it does for the lowest laser intensity case (Figure 5-18a). This is inconsistent with a reflectivity decrease due to shock unloading. For a shock wave, the shock temperature increases with shock velocity and thus with laser in-tensity. The density gradient scale length approaches A/27T in a shorter time for a hotter plasma since the characteristic unloading speed (the isentropic sound speed) increases with temperature. Furthermore, the electrical conductivity is a weak func-tion of temperature in the density-temperature regime relevant to the measurement. Thus one would expect that the time constant for the decrease in the reflectivity to be shorter for the higher laser intensities if the change in reflectivity is due to a shock wave as in the case of the aluminum experiment. The much slower decrease in the rear surface reflectivity at the higher laser intensities would be more indica-tive of x-ray transport from the corona since one would expect the intensity of the x-ray emission to increase with laser intensity and accordingly radiation transport begins to play a more dominant role. In Figures 5-18b and 5-18c, the target rear surface appears to be melting before the peak of the laser pulse. This is likely due CHAPTER 5: REFLECTIVITY MEASUREMENTS CJ LU —J u. LU cr 0.01 r OA 0.5 0.6 0.7 T I M E (ns) T -0.2 -0.1 0 0.1 0.2 T I M E (ns) Figure 5-18 Reflectivity as a function of time for 20>m thick copper targets irra-diated at: a) 06O ~ 1.4 >•' ] 0 1 3 W/cm 2 ; and, b) <?tio ~ 2.5 x }013W/cm2 (continued on the following page). Time zero corresponds to the peak laser intensity. T 0.1 CHAPTER 5: REFLECTIVITY MEASUREMENTS 158 Figure 5-18 Reflectivity as a function of time for 20/j.m thick copper targets ir-radiated at: c) (pco — 5 X 1013W/cm2. Time zero corresponds to the peak laser intensity. CHAPTER 5: REFLECTIVITY MEASUREMENTS 159 to heating by x-rays produced in the leading edge of the laser pulse, and the higher the laser intensity the earlier the rear surface melts. In the high Z targets, the measured onset time of the decrease in the reflec-tivity also occurs at a time sooner than the predicted shock wave arrival time as determined from the hydrodynamic simulation LTC without radiation transport. The code was run for the measured laser intensities and the shock arrival times from the code were calculated using data from the SESAME library. The arrival times were 1.25ns, 0.84ns and 0.41ns for laser intensities of 1.4 x 10xzW/crn2, 2.5 x 10 1 3W/cm 2 and 5 x 1013W / cm2 respectively. These times are much later than the measured time of onset of the decrease in reflectivity: approximately 0.6ns, —0.1ns and —0.7ns respectively where time zero corresponds to the time of peak laser intensity. This would also be indicative of radiative preheat. It is interesting to compare the observation of preheat from the reflectivity measurements with that from the luminescence measurements. Unfortunately, the measurements have not been obtained for identical laser intensities. The closest comparison we can make is for </>QO ~ 1.4 x 10l3W / cm2 in the reflectivity measure-ment and 1.25 x 10l3W/cm2 in the luminescence measurements. For these inten-sities, it was shown in the chapter 4 that x-ray transport from the corona to the dense solid was required to explain the observations. The decrease in the reflectivity starts at t ~ 0.6ns (Figure 5.18a) and the increase in the luminous intensity starts somewhere in the range of 0.5 — 0.7ns (Figure 4-23c). These times are remarkably close. The luminous intensity measurement is primarily sensitive to the tempera-ture of the target rear surface. The threshold for detecting luminous emission in the measurement is in the range of 0.2 — QAeV. On the other hand, the reflectivity measurement is very sensitive to the density gradients in the plasma. The reflectiv-ity starts to decrease when the plasma expands. For copper at standard pressure the melting point is at ~ 0.09eF and the boiling point is at ~ 0.22eV. (The boiling and melting temperatures may increase under pressure.) The plasma will start to CHAPTER 5: REFLECTIVITY MEASUREMENTS expand when the temperature is of the order of or higher than the boiling point. Thus, it is not surprising that the reflectivity starts to decrease at approximately the same time the luminous intensity becomes detectable. A complete study of this data would require the measurement of time resolved spectra of the x-ray emission at both the front and back sides of the targets. This will be attempted in future work. 5.4b T w o dimensional mode l for radia t ion t ranspor t In this section, a simple two dimensional model is used to assess the effect of direct (shine-through) x-ray heating of the target rear surface. The temperature of the rear surface of the target is calculated as a function of time and radial position. The surface is assumed to become absorbing when the rear surface starts to expand into the vacuum. This is assumed to occur when the target rear surface reaches some critical temperature T*, which is unknown. However, this should occur at a temperature near the boiling point of the target material. The x-ray intensity is assumed to be proportional to the laser intensity. As noted earlier this is a weak assumption; however, a better approximation is not available. Thus, the x-ray emission is taken to be spatially and temporally Gaussian. The x-rays which heat the bulk of the target are also assumed to be monochromatic (or in a narrow band) so a single effective opacity can be used for the target. The x-ray intensity is given by Ix{r,t) ~^exp(-^ r 4log(5/2))exp(- -^4log(2)) [5-10] where IX is the x-ray power density, DQO is the diameter of the disc which contains 60% of the laser energy, A is a constant corresponding to the x-ray conversion efficiency and TL is the FWHM of the laser pulse. Target compression is neglected CHAPTER 5: REFLECTIVITY MEASUREMENTS 161 in this model. This is reasonable for low laser intensities. Furthermore, the emitting surface is assumed to be optically thin to the x-ray radiation. The target is assumed to have an effective opacity, ajr>, where ory is taken as the opacity corresponding to x-rays that will most efficiently heat the central region of the target rear surface as defined by equation 4-12. The x-ray power deposited at the target rear surface at r' and t is dERl , , f°° f2lT . . , srdrd6 ~dt~ J Jo ODlx(r,t>exp(~SoDP>~sr~ ' l 5 - 1 1 ! where 5 = (r2 + r'2 + T2-2rr'cos(f5))1/2 , [5-12] and T is the target thickness. The vectors r, r ' and S are defined in Figure 5-19. The energy per unit mass deposited at the target rear surface as a function of radial position and time is then ER(r,t) = £ ^f-dt . [13] The energy ER(r,t) is then converted to a temperature TR(r,t) using the SESAME data. The constant, A , is determined by requiring at t —> oo the final temperature at the centre of the focal spot r = 0 to be the measured tempera-ture TM as determined from the luminescence measurement. This calculation was done for copper at the laser intensity (foci ~ 1-4 x \0lzW/cm2. The result for the corresponding reflectivity measurement was presented earlier in Figure 5-17a. The temperature, TM, was taken to be 0.6beV as determined by the maximum tem-perature shown in Figure 4-24c and op was taken to be 56cm2/g as determined from Figure 4-36a. TI was taken to be 2ns, DQQ was 40/im and T was 20pm. The temperature contours as a function of time are presented in Figure 5-20. Contours CHAPTER 5: REFLECTIVITY MEASUREMENTS 162 Figure 5 - 1 9 Definition of S, r and r' for the two dimensional direct heating model. CHAPTER 5: REFLECTIVITY MEASUREMENTS 163 Figure 5-20 Radial position of temperature contours as a function of time for the two dimensional direct heating model for a 20p?n thick copper targets. TM was taken to be 0.65eV for the calculation. Also given in the figure is the experimental results, dashed line. CHAPTER 5: REFLECTIVITY MEASUREMENTS 164 are shown for T* equal to 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6eV. Also shown in the figure are experimental results taken from Figure 5-17a. The calculated contours are curved near r = 0; this feature was observed for the copper targets. Furthermore, the width of the calculated absorbing regions even Ins after the wave breakout is narrower than the laser focal spot size, DQQ, for high values of T* corresponding to T* = 0.5eV and T* = 0.6eV. A comparison of the widths of the absorbing regions Ins to 2ns after the wave arrival indicates that the contours for T* = 0.2eV and T* — 0.3eV have widths that are similar to the measured contour as shown in Figure 5-20. It is interesting to note that this temperature range is near the boiling temperature of copper under standard conditions. (The boiling temperature of the material may increase under pressure.) However, these contours occur much earlier in time than the measured contour. Perhaps this should be expected since an x-ray transport model based on x-ray shine-through could not explain the luminescence measurements on copper targets. This can be attributed to neglecting the energy dependence of the x-ray opacity and the change in the opacity due to target compression. Perhaps the x-ray heating is coupled to the compression wave. This would result in heating of the target rear surface at later times then in the direct heating case since a finite time is required for a compression wave to propagate through the target. On the other hand, this simple calculation demonstrates that the shape of the absorbing region (curved near r = 0, and the width of the absorbing region) may indeed be explained by x-ray transport from the coronal plasma. Figure 5-21 shows the borders of the absorbing region as a function of time for T* equal to 0.3eV for different opacity values. This temperature was chosen since the shape of this contour is similar to the measured contour. Note that the shape of the breakout region is very sensitive to the x-ray opacity used in the calculation. For the low values of the opacity, a < 30cm2/g, the size of the breakout region increases rapidly with time. These low values of the opacity correspond to hard x-rays (for CHAPTER 5: REFLECTIVITY MEASUREMENTS RADIAL POSITION (um) Figure 5-21 S h a p e of the opaque region as a f u n c t i o n of o p a c i t y for TM equal to 0.3eV* for the two d i m e n s i o n a l direct h e a t i n g m o d e l . CHAPTER 5: REFLECTIVITY MEASUREMENTS 166 a < 30cm2 jg the x-ray energy is > 20fceV for copper as determined from Figure 4-38). For very high laser intensities, heating of the target rear surface by direct shine-through of hard x-rays may be the dominate heating mechanism. At moderate laser intensities (such as the intensities used here) heating by x-ray shine-through may become important for thin foils. The simple model presented in this section is expected to agree with measurements on targets that are heated by direct x-ray shine-through. Thus, it may be used to determine the relative importance heating by x-ray shine-through for various laser and target conditions. Such experiments would enhance the understanding of x-ray emission and transport in laser irradiated solids. 5.4c R e f l e c t i v i t y m e a s u r e m e n t s o n M o a s a f u n c t i o n o f l a s e r i n t e n s i t y The x-t plots of the reflectivity for 25/j.m thick molybdenum targets at laser irradiances of <foo 3.4 x 10 1 3W/cm 2, 5.7 x 1013W/cm2 and 9 x 1013W/cm2 are presented in Figure 5-22. These results are similar to those for the copper targets. The breakout region is parabolic in shape as it was for the copper targets. The size of the breakout region is smaller than that for aluminum targets. This observation can also be explained qualitatively by x-ray transport from the corona similar to that described for the copper targets in section 5.4b. However, different target thicknesses and laser irradiances were used in the reflectivity measurements than those used in the luminescence measurements. Hence a quantitative comparison of the corresponding results could not be made. The reflectivity as a function of time are given in Figure 5-23 for the molyb-denum targets irradiated at the intensities given above. The streak speed used for these measurements was 1.24ns full screen. The data was obtained by averaging the reflectivity signal over the central 10^m of the breakout region. Similar to the the copper targets, the reflectivity signal shows a slow decay time in the reflectiv-ity at high laser intensities and a fast decay time at low laser intensities. This is CHAPTER 5: REFLECTIVITY MEASUREMENTS 167 a) ' i i i i i i 1 0 AO 80 POSITION (jjm) Figure 5-22 Normalized streak records for 25//m thick molybdenum targets irra-diated at: a) 4>m - 3.4 > ]0]3W/cm2; and, b) <foo ~ 5.7 x 1 0 I 3 H ' I cm2 (continued on the following page). CHAPTER 5: REFLECTIVITY MEASUREMENTS 168 CHAPTER 5: REFLECTIVITY MEASUREMENTS 169 TIME (ns) Figure 5-23 Reflectivity as a function of time for 25//m thick molybdenum targets irradiated at: a) <J>QC, — 3.4 x 10 1 3l4 7/c?n 2; and. b) <j>$o ~ 5.7 x ]0 1 3H ' ' / c m 2 (continued on the following page). CHAPTER 5: REFLECTIVITY MEASUREMENTS 170 F i g u r e 5 -23 Reflectivity as a function of time for 25p.m thick molybdenum targets irradiated at: c) ey.o ~ 9 x ] 0 K ' l V / c m 2 . CHAPTER 5: REFLECTIVITY MEASUREMENTS inconsistent with a change in the reflectivity due to shock breakout at the target rear surface, as explained earlier. Furthermore, the shock arrival times as predicted using the hydrodynamic code LTC without radiation transport are 1.4ns, 1.0ns and 0.64ns for laser intensities of 3.4 x 10l3W/cm2, 5.7 x 1013W/cm2 and 9 x 1013W/cm2 respectively whereas the measured decrease in the reflectivity started at approxi-mately 0.5ns, 0.0ns and —0.3ns respectively. The anomalous early melting of the targets may again be indicative of x-ray preheat. 5.4d Discuss ion on the reflectivity measurements on C u and M o The results of the reflectivity measurements on the copper and molybdenum targets cannot be accounted for by shock heating of the target rear surfaces. The onset of the decrease in the reflectivity was observed to occur before that expected for shock breakout. The time scale for the decrease in the reflectivity is slower for the higher laser intensities which is also inconsistent with shock wave breakout. On the other hand, these measurements would suggest preheat of the target rear surface by x-ray transport from the coronal plasma. For the copper targets, a simple two-dimensional model of x-ray preheat by direct shine-through was used to estimate the spatial structure of the heat front emerging at the target rear surface. This model could reproduce some of the quali-tative features observed in the measurements such as the shape and the width of the absorbing region on an x-t plot. However, the predicted arrival time of the heating front was much eariler than that measured indicating that the experiment cannot be modelled simply by x-ray shine-through. This conclusion is consistent with the one obtained in the luminescence measurements. On the other hand, this simple model could be used to assess if heating by x-ray shine-through is the dominant heating mechanism in other experiments. The reflectivity measurement appears to be very sensitive to target preheat. Furthermore, high spatial resolution can be achieved in this measurement. Thus, CHAPTER 5: REFLECTIVITY MEASUREMENTS 172 it may be useful for studying x-ray preheat in laser-target experiments. The high spatial resolution will also facilitate the study of lateral smoothing effects in the ablation front. Finally, the measurements of the reflectivity as a function of time could be compared to predictions based on a hydrodynamic code incorporating detailed radiation calculations to test the validity of the radiation physics. CHAPTER 6: SUMMARY AND CONCLUSIONS 173 C H A P T E R 6 SUMMARY AND CONCLUSIONS 6.1 Summary The measurements presented in this thesis have demonstrated the usefulness of laser generated shock waves for the study of the properties of compressed solids. The shock pressure versus shock velocity and the shock temperature versus shock velocity Hugoniot curves were measured for aluminum. For the shock pressure versus shock velocity curve, the accuracy of the measurements (~ 20%) is not sufficient for equation of state studies (~ 1%). However, the shock temperature versus shock velocity Hugoniot curve is much more sensitive to the equation of state. Once the finite integration time in the measurement is accounted for (using PEC to calculate the hydrodynamics), the results are in good agreement with the equation of state in the SESAME data. Accordingly, it appears that equation of state studies are possible using laser generated shock waves. On the other hand, a more appropriate opacity model would be helpful. The accuracy of the measurement could be improved by using a larger laser focal spot size. The significance of doing equation of state studies with laser generated shock waves is that the pressure that can be attained with laser-driven ablation is much higher than that using any other standard laboratory technique such as two-stage gas guns or diamond anvils. The only alternative method of attaining higher pressures then that produced in laser-target interactions is by using nuclear explosions to drive shock waves. CHAPTER 6: SUMMARY AND CONCLUSIONS 174 The reflectivity of the rear surface of a laser-irradiated aluminum foil was measured as a function of time as the shock wave emerged from of this surface. This measurement was performed for three shock strengths. These results were compared with one dimensional simulations (PEC) using the SESAME equation of state and two different electron transport models: Lee and More's model (two versions) and the model in the SESAME data library. It was found that the measurement was in best agreement with the version of Lee and More's model which used the interatomic distance as the minimum electron mean free path. The reflectivity measurement is extremely sensitive to the electrical conductivity of the plasma and the probe laser wavelength, thus, it should be a very useful diagnostic for studying electron transport in compressed solids. Both the luminescence and the reflectivity measurements on the copper and molybdenum targets appear to be affected by x-ray transport from the coronal plasma. From the luminescence measurements, the onset of the heating of the rear surfaces of these targets was found to occur earlier than that expected for shock wave arrival. Furthermore, the rise time of the luminous emission was much slower than that expected for shock breakout at the target rear surface. Finally, the measured temperature of the target rear surface was higher than the expected shock heating temperature. These observations are consistent with x-ray preheat. A simple x-ray transport model, based on heating of the target rear surface by x-ray shine-through, was incorporated self-consistently into the hydrodynamic code LTC. This simple model reproduced the molybdenum results, however, the x-ray transport in copper targets appears to be more complex. In the reflectivity measurements on the copper and molybdenum targets, the onset of the decrease in the reflectivity was observed to occur earlier than that expected for shock breakout. Furthermore, the time constant for the decrease in the reflectivity was slower for the higher shock speeds. This is also inconsistent with shock heating of the target rear surface. Finally, the shape of the absorbing region OH A P TER 6: S VMM A RY AND CONCL US IONS 175 on an x-t plot was inconsistent with the expected shape for shock breakout. The effect of x-ray shine-through on the shape of the absorbing region was assessed with a simple two-dimensional model for the copper measurement. The general shape of the absorbing region was reproduced with this model, however, the onset in the decrease of the reflectivity was found to occur much earlier than that predicted by the model. Thus, both the luminescence and the reflectivity measurements indicate that the x-ray transport process in copper may be quite complex. A complete experimental investigation of the observed preheat would require the measurement of time resolved spectra of the x-ray emission. The luminescence measurements provide a fairly direct measurement of the energy transported to the rear surface of the target. Used in conjunction with the x-ray measurements, these results would yield a powerful diagnostic for assessing the x-ray transport in compressed solids. The high spatial and temporal resolution of the reflectivity measurement could be used to study the lateral smoothing effects in the ablation front. 6.2 N e w c o n t r i b u t i o n s An important contribution of this work is the establishment of two experimen-tal techniques: the luminescence measurements38 and the reflectivity measurements40. In these investigations the luminescence measurements were used to obtain the first shock pressure versus shock velocity Hugoniot curve using laser generated shock waves and the first measurement of a shock temperature versus shock velocity Hugoniot curve 1 6 ' 3 9. This is the only known measuring technique which is primarily sensitive to the thermal part of the equation of state. For shock waves generated by two-stage gas guns, this technique may be used to obtain accurate shock tem-perature versus shock velocity Hugoniot curves. Results from these measurements would enhance the current understanding of the equation of state of dense matter. CHAPTER 6: SUMMARY AND CONCLUSIONS 176 Theoretically uncertain parameters such as the melting point2 1 and boiling point along the principal Hugoniot curve may also be measured. The reflectivity measurements on aluminum targets were the first quantitative reflectivity measurements of shock unloading material. This provided a very sen-sitive measurement of the electron transport properties of compressed solids 4 1 ' 4 2, allowing us to study electron transport in degenerate, strongly coupled plasmas. Exploitation of this measurement technique may lead to a better understanding of the limitations of the single-site scattering approximation used in the theoreti-cal calculation of the transport coefficients. A summary of the luminescence and reflectivity measurements has been published1 0 4. Anomalous heating of the rear surface of thick copper and molybdenum tar-gets was observed. This was attributed to x-ray transport through the target43. A first quantitative measurement, using optical pyrometry, of the heating of a target rear surface due to x-ray transport as a function of target Z was made44. The re-flectivity measurements were also found to be consistent with x-ray preheat of the target rear surfaces. These measurements, supplemented with time resolved mea-surements of the x-ray spectra, can be used to improve the current understanding of radiation physics in laser target interactions. 6.3 Future work The luminescence measurements will be used in future equation of state studies. The laser energy has recently been increased by the installation of two new laser amplifiers. This allows experiments using a larger focal spot size which will improve the accuracy of the measurement since a larger emitting region is observed resulting in an increased signal to noise ratio. Moreover, the brightness temperature as a function of wavelength could be measured. This would yield the plasma emissivity as a function of wavelength. For low Z targets, it appears that equation of state studies by direct ablation of the material is possible. For high Z CHAPTER 6: SUMMARY AND CONCLUSIONS 177 targets, a low Z ablator will have to be used to avoid heating of the target by the x-rays generated in the coronal plasma. The reflectivity measurements will be used to study electron transport in low and high Z targets. Again, low Z ablators will be used in the high Z targets to reduce x-ray heating of the targets. Reflectivity measurements as a function of probe laser wavelength will be used to study the optical dispersion of the dense plasma. Finally, reflectivity measurements as a function of beam polarization can be used to probe the index of refraction of the plasma. The luminescence and reflectivity measurements will be used to study the x-ray transport through low Z targets. For these targets the heating of the material is predominantly by shock compression. The target material will be in a well defined state once the shock velocity is measured since its state is on the principal Hugoniot curve. The luminescence or reflectivity measurements are used to measure the shock arrival time at the target rear surface yielding the shock velocity. An x-ray streak camera will be used to obtain a time resolved x-ray spectra of the x-rays which pass through the target. This experiment will give important information such as the the K-edge energy as a function of target compression. The luminescence measurements on high Z targets give a fairly direct mea-surement of the energy transported to the rear surface of these targets. Used in conjunction with the time resolved x-ray spectral measurements, details of the x-ray transport through these targets can be studied. The high spatial resolution of the reflectivity measurements could be used to study the lateral smoothing effects in the ablation zone. Finally, to interpret the x-ray measurements on aluminum, a detailed x-ray dynamics calculation based on the collisional-radiative equilibrium model is being developed". The model will be coupled to a one-dimensional hydrodynamic calcu-lation in a totally self-consistent manner. R E F E R E N C E S 1. J. Nuckolls, L. Wood, A. Thiessen and G. Zimmerman, Nature 239,139(1972). 2. K .A. Brueckner and S. Jorna, Rev. Mod. Phys. 4 6 , 325(1974). 3. K. Estabrook and W.L. Kruer, Phys. Fluids 2 6 , 1888(1983). 4. H.A. Baldis and C.J. Walsh, Physica Scripta. T 2 / 2 , 492(1982). 5. R.C. Malone, R.L. McCrory and R.L. Morse, Phys. Rev. Lett. 3 4 , 721(1975). 6. D.E.T.F. Ashby, Nuclear Fusion 1 6 , 623(1976). 7. W.B. Fechner and F.J. Mayer, Phys. Fluids 2 7 , 1552(1984). 8. W.C. Mead, E .M. Campbell, W.L. Kruer, R.E. Turner, C.W. Hatcher, D.S. Bailey, P.H.Y. Lee, J. Foster, K .G. Tirsell, B. Pruett, N.C. Holmes, J.T. Trainor, G.L. Stradling, B.F. Lasinski, C.E. Max and F. Ze, Phys. Fluids 2 7 , 1301(1984). 9. C.E. Max, C F . McKee and W.C. Mead, Phys. Fluids 2 3 , 1620(1980). 10. P. Mora, Phys. Fluids 2 5 , 1051(1982). 11. W.M. Manheimer, D.G. Colombant and J.H. Gardner, Phys. Fluids 2 5 , 1644(1982). 12. A. Ng, D. Pasini, P. Celliers, D. Parfeniuk, L. DaSilva and J. Kwan, Appl. Phys. Lett. 4 5 , 1046(1984). 13. L. Veeser and J. Solem, Phys. Rev. Lett. 4 0 , 1391(1978). 14. R.J. Trainor, J.W. Shaner, J .M. Auerbach and N.C. Holmes, Phys. Rev. Lett. 4 2 , 1154(1979) and R.J. Trainor, N.C. Holmes, R.A. Anderson, E.M. Campbell, W.C. Mead, R.J. Olness, R.E. Turner and F. Ze, Appl. Phys. Lett. 4 3 , 542(1983). 15. F. Cottet, J.P. Romain, R. Fabbro and B. Faral, Phys. Rev. Lett. 5 2 , 1884(1984). 16. A. Ng, D. Parfeniuk and L. DaSilva, Phys. Rev. Lett. 5 4 , 2604(1985). 17. B. Yaakobi, P. Bourke, Y . Conturie, J. Delettrez, J .M. Forsyth, R.D. Frankel, L . M . Goldman, R.L. McCrory, W. Seka and J.M. Soures, A.J . Burek and R.E. Deslattes, Opt. Commun. 38, 196(1981). 18. W.C. Mead, E.M. Campbell, K.G. Estabrook, R.E. Turner, W.L. Kruer, P.H.Y. Lee, B. Pruett, V.C. Rupert, K .G. Tirsell, G.L. Stradling, F. Ze, C E . Max, M.D. Rosen and B.F. Lasinski, Phys. Fluids 26, 2316(1983). 19. D.L. Mathews, E.M. Campbell, N .M. Ceglio, G. Hermes, R. Kauffman, L. Koppel, R. Lee, K. Manes, V. Rupert, V.W. Slivinsky, R. Turner and F. Ze, J. Appl. Phys. 54, 4260(1983). 20. H. Nishimura, F. Matsuoka, M . Yagi, K. Yamada, S. Nakai, G.H. McCall and C. Yamanaka, Phys. Fluids 26, 1688(1983). 21. B.K. Godwal, S.K. Sikka and R. Chidambaram, Phys. Reports 102, 121(1983). 22. A.K. McMahan, B.L. Hord and M . Ross, Phys. Rev. B15, 726(1977). 23. F. Perrot, Phys. Rev. B21, 3167(1981). 24. R. Latter, Phys. Rev. 99, 1854(1955) and J. Chem. Phys. 24, 280(1956). 25. R.D. Cowan and J. Ashkin, Phys. Rev. 105, 144(1957). 26. D.A. Kirzhnitz, Sov. Phys. JETP 5, 64(1957) and 8, 1081(1959). 27. R.M. More, Phys. Rev. A19, 1234(1979). 28. Y.T. Lee and R.M. More, Phys. Fluids 27, 1273(1984). 29. SESAME Library, Los Alamos National Laboratory, Los Alamos: the alu-minum equation of state was obtained from table 3712 which was last updated in November 1982; the aluminum electron transport data was obtained from table 23713 which was last updated in June 1984; the copper equation of state was obtained from table 3333 which was last updated in Febuary 1984; the copper electron transport data was obtained from table 23333 which was last updated in June 1984; the molybdenum equation of state was obtained from table 2980 which was last updated in March 1973; and, the molybdenum elec-tron transport data was obtained from table 22981 which was last updated in July 1984. 30. G.A. Rinker, Phys. Rev. B31, 4207(1985). 31. G.E. Duvall, in Physics of High Energy Density, edited by P. Caldirola (Aca-demic, New York, 1971). 32. C.E. Ragan III, M.G. Silbert and B.C. Diven, J. Appl. Phys. 48, 2860(1977). 33. C.E. Ragan III, Phys. Rev. A21, 458(1980) and 25, 3360(1982) and 29, 1391(1984). 34. Ya.B. ZePdovich and Yu.P. Raizer, Physics of Shock Waves and High Temper-ature Hydrodynamic Phenomena, (Academic, New York, 1966), pages 726-730. 35. Energy and Technology Reveiw, November 1978, Lawrence Livermore Na-tional Laboratory, ed. R.R. McGuire. 36. A.C. Mitchell and W.J. Nellis, J. Appl. Phys. 52, 3363(1981). 37. N.C. Holmes, R.J. Trainor, R.A. Anderson, L.R. Veeser and G.A. Reeves, Shock Waves in Condensed Matter, 1981, edited by W.J. Nellis, L. Seaman and R.A. Graham, (American Institute of Physics, New York, 1982) page 160. 38. A. Ng, D. Parfeniuk and L. DaSilva, Opt. Commun. 53, 389(1985). 39. L. DaSilva, A. Ng and D. Parfeniuk, J. Appl. Phys. 58, 3634(1985). 40. A. Ng, D. Parfeniuk, P. Celliers and L. DaSilva, Proc. 4th APS Topical Con-ference on Shock Waves in Condensed Matter, edited Y . M . Gupta (Plenum, N.Y., 1986) page 255. 41. A. Ng, D. Parfeniuk, P. Celliers, L. DaSilva, R.M. More and Y.T. Lee, Phys. Rev. Lett. 57, 1594(1986). 42. D. Parfeniuk, A. Ng, L. DaSilva and P. Celliers, Opt. Commun. 56,425(1986). 43. A. Ng, D. Parfeniuk, L. DaSilva and D. Pasini, Phys. Fluids 28, 2915(1985). 44. A. Ng, Y. Gazit, F.P. Adams, D. Parfeniuk, P. Celliers and L. DaSilva, Proc. 11th International Conference on Plasma Physics and Controlled Nuclear Fu-sion Research (International Atomic Energy Agency, Vienna, 1987). 45. J.P. Christiansen, D.E.T.F. Ashby and K.V. Roberts, Computer Phys. Com-mun. 7, 271(1974). 46. Rutherford Laboratory: Laser Division Annual Reports 1978, 1979, 1980. 47. S.K. Godunov, Mat. Sb. 47, 271(1959). 48. L. Spitzer and R. Harm, Phys. Rev. 89, 977(1952). 49. J.R. Stallcop and K.W. Billman, Plasma Phys. 16, 722(1973). 50. D. Pasini, Ph.D. Thesis, U . B . C , 1984 (unpublished). 51. F. Adams, M.Sc. Thesis, U . B . C , 1986 (unpublished). 52. D.L. Book, J.D. Boris and K. Hain, J. Computational Phys. 18, 248(1975). 53. LASNEX is a hydrodynamic code developed by American scientists for sim-ulating fusion processess. The details of this code have not been published. 54. K.S. Trainor, H-Division Quarterly Report, Lawrence Livermore National Laboratory, UCID-18574-82-2 (1982), pages 20-23. 55. F.J. Rogers and H.E. DeWitt, Phys. Rev. A 8 , 1061(1973); and F.J. Rogers, Phys. Rev. A10, 2441(1974). 56. D.A. Young, A Soft-Sphere Model for Liquid Metals, Lawrence Livermore Na-tional Laboratory, CA, UCRL-52352 (1977). 57. C.A. Rouse, Astophys. J. 136, 636(1962). 58. R. Grover, High Temperature Equation of State for Simple Metals, Preceedings of the Seventh Symposium on Thermodynamic Properties, ed. by A. Cezair-liyan (The American Society of Mechanical Engineers, New York, 1977), p. 67. 59. Description of the APW method is given in T.L. Loucks, Augmented Plane Wave Method (Benjamin, New York, 1967); and in L.F. Mattheiss, J.R. Wood and A.C. Switendick, in Methods in Computational Physics, edited by B. 182 Alder, S. Fernbach and M . Rotenberg (Academic, New York, 1968), Vol. 8, p. 63. The manner in which the method is made self-consistent is described by J.C. Slater and P. DeCicco, Solid State and Molecular Theory Group, MIT Quarterly Progress Report No. 50 (1963) p. 46. 60. S.L. McCarthy, The Kirzhnits Correction to the Thomas-Fermi Equation of State, Lawrence Livermore National Laboratory, Livermore, CA, UCRL-14362 (1965). 61. F. Ree, D. Daniel and G. Haggin, NUC-A Code to Calculate Ionic Contribu-tions to P and E, Lawrence Livermore National Laboratory, Livermore, CA, Internal Document HTN-291 (1978). 62. M . Ross, Phys. Rev. B21 (8), 3140(1980). 63. R. Evans, B.L. Gyorffy, N. Szabo and J .M. Ziman, in The Properties of Liquid Metals, edited by S. Takeuchi (Wiley, New York, 1973). 64. J.S. Brown, J. Phys. F l l , 2099(1981). 65. M . Itoh and M . Watabe, J. Phys. F 1 4 , L9(1984). 66. D.A. Liberman, D.T. Cromer and J.T. Waber, Comput. Phys. Commun. 2, 107(1971). 67. D.A. Liberman, Phys. Rev. B20, 4981(1979) and J. Quant. Spectrosc. Rad. Transfer 27, 335(1982). 68. G.A. Rinker, Los Alamos National Laboratory Report LA-9872-MS, January, 1984. 69. D. Henderson and E.W. Grundke, J. Chem. Phys. 63, 601(1975). 70. F.J. Rogers, D.A. Young, H.E. DeWitt and M . Ross, Phys. Rev. A28, 2990(1983). 71. Y. Rosenfeld and N.W. Ashcroft, Phys. Rev. A20, 1208(1979). 72. N.W. Ashcroft and N.D. Mermin, Solid State Physics, page 323 (Saunders College, Philadelphia, 1976). 73. W.A. Lokke and W.H. Grasberger, Lawrence Livermore National Laboratory Report UCRL-52276, Livermore, California, 1977. 74. H. Brysk, P.M. Campbell and P. Hammerling, Plasma Physics 17, 473(1975). 75. S.G. Brush, H.L. Sahlin and E. Teller, J. Chem. Phys. 45, 2102(1966) and J. Hansen, Phys. Rev. A8, 3096(1973). 76. P.H. Lee, Ph.D. Thesis, University of Pittsburgh, (1977). 77. N.F. Mott, Philos. Mag. 13, 989(1966) and N.F. Mott and A. Davis, Elec-tronic Processes in Non-Crystalline Materials, Oxford U.P., Oxford, England, 1971. 78. J .M. Ziman, Philos. Mag. 6, 1013(1961). 79. C.W. Cranfill and R.M. More, Los Alamos Laboratory Report LA-7313-MS, Los Alamos, New Mexico, 1978; and, R.M. More, Laser Interaction and Re-lated Phenomena, edited by H.J. Schwarz, H. Hora, M. Labin and B. Yaakobi (Plenum, New York, 1981), Vol. 5, pages 253-276. 80. Hamamatsu Corporation, Hamamatsu City, Japan. 81. Ya. B. Zel'dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Academic, New York, 1966), p771. 82. Goodfellow Metals Ltd., Cambridge, England. 83. P. Celliers, M.A.Sc. Thesis, U . B . C , 1983 (unpublished). 84. R.M. More, private communication. 85. D.E. Gray, American Institute of Physics Handbook, 3rd. Ed. (McGraw-Hill, New York, 1972), pp. 6-29, 6-157. 86. Laser Focus Buyers Guide, 12th. Ed., 1977. 87. E.A. McLean, S.H. Gold, J.A. Stamper, R.R. Whitlock, H.R. Griem, S.P. Obenschain, B.H. Ripin, S.E. Bodner, M.J. Herbst, S.J. Gitomer and M.K. Matzen, Phys. Rev. Lett. 45, 1246(1980). 88. Corion Corporation, Holliston, MA. 89. R.C. Weast, CRC Handbook of Chemistry and Physics, 62nd. Ed. (CRC Press, Inc., Boca Raton, Florida, 1981) page E-375. 90. Ya. B. Zel'dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Academic, New York, 1966), Vol I, Chap. 2 and Vol I, Chap. 5. 91. W.C. Mead, E .M. Campbell, K .G. Estabrook, R.E. Turner, W.L. Kruer, P.H.Y. Lee, B. Pruett, V.C. Rupert, K .G. Tirsell, G.L. Stradling, F. Ze, C.E. Max and M.D. Rosen, Phys. Rev. Lett. 47, 1289(1981). 92. W.C. Mead, E .M. Campbell, K .G. Estabrook, R.E. Turner, W.L. Kruer, P.H.Y. Lee, B. Pruett, V.C. Rupert, K .G . Tirsell, G.L. Stradling, F. Ze, C.E. Max, M.D. Rosen and B.F. Lasinski, Phys. Fluids 26, 2316(1983). 93. PIN diode 100-PIN-250N; manufactured by Quantrad Corporation, 19900 S. Normandic Avenue, Torence, California 90502. 94. K. Estabrook and K.L. Kruer, Phys. Rev. Lett. 40, 42(1978). 95. E. Segre, Nuclei and Particles (Benjamin Cummings, Reading, MA, 1977), 2nd ed. 96. J. Mizui, N. Yamaguchi, S. Takagi and K. Nishihara, Phys. Rev. Lett. 47, 1000(1981). 97. B. Yaakobi, J. Delettrez, L . M . Goldman, R.L. McCrory, W. Seka, J .M. Soures, Opt. Commun. 41, 355(1982). 98. H. Shiraga, S. Sakabe, K. Okada, T. Mochizuki and C. Yamanaka, Jpn. J. Appl. Phys. 22, L383(1983). 99. D. Dusten, R.W. Clark, J. Davis and J.P. Apruzese, Phys. Rev. A27, 1441(1983). 100. G.B. Zimmerman and R.M. More, J. Quant. Spectrosc. Radiat. Transfer 23, 517(1980). 101. Chemical Rubber Company, Handbook of Spectroscopy, CRC Press, pages 28-129 (1974). 102. V.L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas, Chap-ter IV (Pergamon Press, Oxford, 1970). 103. M . Born and E. Wolf, Principles of Optics, Chapter XIII (Pergamon Press, Oxford, 1980). 104. A. Ng, D. Parfeniuk, L. DaSilva and P. Celliers, Laser and Particle Beams 4, 555(1986). 

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