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Radiation transport in laser-matter interactions Silva, Luiz Da 1988

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RADIATION TRANSPORT IN LASER-MATTER INTERACTIONS By Luiz Da Silva B. A. Sc. University of British Columbia M. A. Sc. University of British Columbia A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA November 1988 © Luiz Da Silva, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of P H M C l t S  The University of British Columbia Vancouver, Canada Date [Sec- 1*5. tf#& DE-6 (2/88) Abstract The details of x-ray emission and transport in laser-matter interactions are currently of great interest. We have irradiated aluminum targets with 0.53 /xm laser light in a ~ 2 ns ( F W H M ) pulse at an absorbed irradiance of $L = 2.3 x 10 1 3 W / c m 2 . The temporal evolution of the ii"—shell photoabsorption edge of shock compressed aluminum was mea-sured with an x-ray streak camera. The experimental results were found to be in reason-able agreement with the predictions of our one-dimensional hydrocode ( H Y R A D ) which includes detailed calculation of radiation emission and transport. The effects of a finite focal spot on the A'-edge profile were also assessed using a simplified two-dimensional hydrocode. The results suggest that the measured edge broadening is a consequence of non-uniform target conditions. Further numerical simulations using H Y R A D verified the importance of Ka emission spectrum in studying the effects of radiation transport. How-ever, these predictions were found to be in significant disagreement with our experimental measurements of the time resolved Ka emission spectrum. One possible explanation for this discrepancy is an underestimate of the Ka fluorescent yield for certain ion species of aluminum. The reason for this is the generation of excited states which can not decay through the Auger process. Nevertheless, the experimental results still show character-istics of the radiatively heated zone predicted by the hydrocode. n Table of Contents Abstract ii List of Tables vi List of Figures vii Acknowledgement xii 1 Introduction 1 1.1 Laser-matter interactions 1 1.2 Present Work 5 1.3 Thesis outline 8 2 Radiation emission and transport in Laser-irradiated targets 9 2.1 Laser-target interactions 9 2.2 Radiation Calculation 14 2.2.1 Atomic physics 14 2.2.2 Level population 17 2.2.3 Radiation emission and absorption 22 2.2.4 Inner-shell photoionization 28 2.2.5 Radiation transport 34 2.3 Summary . . . 40 3 Numerical Simulations 41 iii 3.1 Calculation of state population 41 3.1.1 Rate coefficients 44 3.2 CRE results for aluminum plasma 48 3.3 Photo-excitation and line transport 51 3.4 Continuum transport 57 3.5 Hydrodynamic code including radiation transport 57 3.5.1 Standard Calculation 63 3.5.2 Testing of HYRAD results 65 3.6 Two-dimensional hydrocode 75 4 Experimental Facility 78 4.1 Laser facility 78 4.2 Irradiation conditions 80 4.3 X-ray Streak Camera 83 4.3.1 General Design 83 4.3.2 Photocathodes 89 4.3.3 Streak camera test 90 4.3.4 Sweep speed measurement 90 5 K-edge shift in shock compressed aluminum 96 5.1 Shock trajectory measurements 96 5.2 Measurement of /f-shell photoabsorption edge for aluminum 104 5.2.1 X-ray fiducial 107 5.2.2 Aluminum K-edge measurement 110 5.3 Comparison of experimental results with 1-D simulations 112 5.4 Comparison of experimental results with 2-D simulations 120 5.5 Summary 122 iv 6 Ka Emission from laser irradiated aluminum 124 6.1 Measurement of the Ka emission spectrum 124 6.2 Simulation of the Ka measurements 126 6.2.1 Ka emission model 128 6.2.2 Ka simulation results 129 6.2.3 Comparison of simulated and measured Ka emission 133 7 Summary and Conclusions 137 7.1 Summary 137 7.2 Significant Contributions 138 7.3 Future Work 139 Bibliography 141 v List of Tables 3.1 Emitted line radiation (W/cm 3 ) 54 3.2 Absorption edge energy for various ionization stages of aluminum. . . . 65 4.3 Streak Camera Operating voltages (in kilovolts) 95 5.4 Measured and Calculated K-edge shift 122 6.5 Transition energies and fluorescence yields for the various ionization stages of aluminum included in the Ka emission model. Ionization stage refers to the ion before photoionization, that is, A l + 0 . Ka is produced when an inner electron of an A l + 0 atom is removed 129 v i List of Figures 1.1 A schematic diagram of a solid target irradiated by a laser at high inten-sities 2 2.2 A schematic diagram of a solid target irradiated by a high intensity laser. p is the target density, Tx and Te are the ion and electron temperatures respectively 10 2.3 a) Shock front propagation, b) Schematic diagram of Hugoniot (solid), isentrope (dashed), and zero degree isotherm (dotted) 13 2.4 Criteria for applicability of the various ionization models. Shaded region is parameter space for laser-produced plasmas 23 2.5 if-shell photoabsorption edge for aluminum as a function of volume ratio for a plasma temperature of 1 eV; a) A l + 3 , b) A l + 4 35 2.6 Zf-shell photoabsorption edge for A l + 3 as a function of volume ratio for different isotherms: solid line 0.1 eV, chain-dashed 2.0 eV, and dashed 4.0 eV 36 2.7 Radiation transport in planar geometry 37 3.8 Simple three-level diagram depicting the atomic processes included in our CRE model. 5 is ionization, a is radiative and three-body recombination, az>i is dielectronic recombination, X and X~l are collisional excitation and deexcitation respectively, and A is spontaneous decay. . . . . . . . . 43 vn 3.9 Average ionization as a function of a) temperature (ion density=102 0 c m - 3 ) and b) density (temperature=100 eV). Solid line our model , dashed line S E S A M E , dot-dashed line Lee, dotted line Salzmann 49 3.10 Radiation emission as a function of temperature (ion density=102 0 c m - 3 ) a) line b) bound-free c) total. Solid line our model , dashed line Salzmann, dotted Lee 50 3.11 Excited and ground state densities of a) Helium-like aluminum ( A l + n ) b) Hydrogen-like aluminum ( A l + 1 2 ) ; solid line optically thin calculation, dashed line optically thick calculation. 55 3.12 Lagrangean mesh as it evolves during the course of a calculation. Features of the laser-target interaction are labelled 59 3.13 Mesh zoning scheme used in H Y R A D calculations. The laser is incident from the right 64 3.14 Inner-shell photoionization cross section for aluminum at normal density and temperature 66 3.15 Numerical solution of a Riemann shock tube problem. Solid curves are exact solutions, symbols are H Y R A D calculations 67 3.16 Calculated profiles for 25 //m aluminum foil irradiated at = 2.3 x 10 1 3 W / c m 2 with a 0.53 fim and 2.3 ns ( F W H M ) gaussian laser pulse. a) without radiation b) with radiation 69 3.17 Temporal evolution of (a) temperature, and (b) density throughout the target. H Y R A D simulation including radiation transport. Laser is inci-dent on boundary at position 0 72 3.18 Thermodynamic profiles for 19 /im A l on 13 ^m Au multi-layered target. a) Before shock reaches interface, b) After shock reaches interface. . . . 73 3.19 Two-dimensional shock propagation simulation for 25 aluminum foil. 76 viii 4.20 Schematic of the laser used in the experiments, and a typical laser pulse. 79 4.21 Time integrated intensity distribution 81 4.22 Time integrated intensity profile averaged over the central 10 fj,m in the a) X-direction and b) Y-direction (Refer to previous figure) 82 4.23 Cross-sectional view of x-ray streak camera 84 4.24 Schematic of deflection plate driver and deflection pulse predicted by SPICE for final version of ramps 87 4.25 Cross-sectional view of streak camera assembly. (A) Film camera, (B) Intensifier, (C) Intensifier mount, (D) Taper mount, (E) Fibre optic taper, (F) Mounting flange, (G) Streak camera housing, (H) Streak Camera, and (I) Mounting collar 88 4.26 Static mode image of the entrance slit 91 4.27 Setup used to measure sweep speed. Streak record shows the two nitrogen laser pulses with 0.5ns separation 92 4.28 Sweep linearity plot, crosses are measured points, solid line is best fit second order polynomial 93 5.29 Experimental setup for shock trajectory measurement 97 5.30 Streak records of shock breakout emission and fiducial in (a) 38.4 /xm and (b) 53 aluminum target 99 5.31 a) Luminous intensity of the target rear surface as a function of time for a 38.4 fim thick aluminum foil irradiated at $£, = 2.3 x 10 1 3 W / c m 2 b) Fiducial pulse, the shock transit time for this shot is TS = 1.1 ns 100 ix 5.32 Shock trajectory in aluminum at a laser irradiance of *3>£, .= 2.3 x l O 1 3 W / c m 2 ; open circles are the experimental results; an d solid line represents simula-tion results from H Y R A D . Time zero corresponds to the peak of the laser pulse 102 5.33 Simulated trajectories: solid line *3>£ = 10 1 2 W / c m 2 , dashed line $L — 10 1 3 W / c m 2 , chain-dashed line $ L = 10 1 4 W / c m 2 . 103 5.34 The calculated ablation pressure pulse from H Y R A D (solid) and gaussian laser pulse (dash) 105 5.35 a) Average ionization vs temperature, b) pressure vs internal energy; solid line ion density 1 0 1 8 c m - 3 , dashed line ion density 1 0 2 0 c m - 3 , dotted line ion density 1 0 2 2 c m - 3 106 5.36 a) Experimental setup used to calibrate x-ray fiducial, b) streak records for 16 /zm aluminum foil 108 5.37 Onset time of transmitted x-ray emission at 1598 eV 109 5.38 Experimental setup used to measure the temporal evolution of the K-edge of aluminum I l l 5.39 Streak records for a) 9 fim aluminum target, b) 25 /zm aluminum target. 113 5.40 Principal Hugoniot of aluminum (dashed curve) and the corresponding shifts in the /('-shell photoabsorption edge derived from solid state model (solid line), plasma model (triangles) 115 5.41 Measured and simulated time history of K-edge position, solid line mea-sured, dashed simulated 90% level, dash-dot simulated 10% level ; a) 9 /zm aluminum target, b) 25 /zm aluminum target 116 5.42 Simulated density profiles in 25 /zm aluminum target, a) t = —1.5 ns, b) t = 0.0 ns, c) t = 1.5 ns 118 x 5.43 Trajectory of the region being probed, (dashed line) for a 50 i z m aluminum target , and the shock trajectory (solid line) for comparison 119 5.44 Temporal evolution in the aluminum K-edge energy. Results of measure-ments (solid curve) and 2-dimensional calculations (dashed curve - 90% in-tensity level; dot-dashed curve - 50% intensity level; dot-dot-dashed curve - 10% intensity level) 121 6.45 Experimental setup for Ka spectrum measurement 125 6.46 Ka spectrum for 25 pm aluminum target irradiated at = 2.3 x 10 1 3 W / c m 2 ; a) using P E T crystal, b) using R A P crystal 127 6.47 Forward Ka emission as a function of time for a 25 /xm aluminum target irradiated at = 2.3 X 10 1 3 W / c m 2 : a) including radiation transport, b) no radiation transport 131 6.48 Backward Ka emission as a function of time for a 25 /im aluminum target irradiated at = 2.3 x 10 1 3 W / c m 2 : a) including radiation transport, b) no radiation transport 132 6.49 Temporal history of rear side Ka emission for different laser intensities : a ) $ L = 10 1 3 b ) $ L = 10 1 4 134 xi Acknowledgement I would like to thank my supervisor Dr. Andrew Ng for his support and his neverending campaign to obtain the resources required for our experiments. As well, I thank my supervisory committee, Dr. B. Ahlborn, Dr. A.J . Barnard, Dr. R. Barrie, Dr. B. Bergersen, and Dr. F. Curzon for their time and effort. I am indebted to Gordon Chiu for his help in performing the experiments. I deeply value the time and effort spent by Dr. B. Godwal on the solid state model calculations and his many suggestions regarding this work. Jim Waterman's editorial comments and interesting ideas were greatly appreciated. The many discussions with Peter Celliers, Dean Parfeniuk and Dr. F. Cottet before they left our group were of great help. I would also like to thank Alan Cheuck for his technical support and great efforts to obtain all the necessary supplies. My stay at the University of Rochester would not have been possible without the efforts of Dr. M . Richardson and Dr. P. Jaanimagi. I will always be indepted to Paul for teaching me some of the many secrets of x-ray streak cameras. Finally, I would like to thank my wife Angela whose encouragement, and help, throughout the course of this work made it all possible. xn Chapter 1 Introduction 1.1 Laser-matter interactions Motivated largely by the prospect of using lasers as drivers for inertial-confinement fusion, research on laser-matter interactions has been extremely active over the past decade'l]-[3]_ These investigations have led to the study of many fundamental physics problems relevant to understanding phenomena in plasma physics, geophysics, astronomy, and condensed matter physics. The multi-disciplinary nature of laser-matter interactions stems from the wide range of plasma conditions occuring in the laser-driven ablation process. The laser-driven ablation process is illustrated schematically in figure 1.1. When laser light strikes the surface of a solid target it is initially absorbed within a skin depth of the surface. As the intensity increases, material is evaporated from the solid surface and at intensities greater than 1 0 9 W/cm 2 the vapour is ionized. The laser light is then absorbed by free electrons which transfer part of their energy to the ions through collisions. This rapidly forms a low density, high temperature plasma called the corona which expands into the surrounding vacuum. Laser absorption occurs in this plasma up to the critical density layer where it is reflected (the critical density is the electron density at which the plasma frequency is equal to the laser frequency). Energy transport by thermal conduction from the coronal plasma to the cold target material drives the ablation process. The ablated material flows outward becoming part of the coronal plasma. The pressure generated by the heating process and the momentum of this plasma 1 Chapter 1. Introduction 2 Figure 1 . 1 : A schematic diagram of a solid target irradiated by a laser at high intensities. Chapter 1. Introduction 3 flow generates a large ablation pressure which drives a shock wave into the target. This wave produces a region of high density, low temperature, shock compressed material. Most of the previous experiments on laser-matter interactions have concentrated on the study of one of these regions. For instance, in the coronal plasma processes such as laser-induced parametric instabilities'^ -'^, electron thermal conduction'^ -'^, and x-ray emission'^""have been extensively investigated. Similarly, the use of laser-driven shocks to study materials at high density and pressure has been widely reporte d t l 3 ] - [17] . More recently, however, significant interest has arisen in studying the coupling of x-ray radiation generated in the coronal plasma to the compressed target. There are two primary reasons for this interest. The first stems from the need in laser-fusion studies to accurately assess the effects of x-ray radiation transport on the overall target hydrodynamics. This knowledge is essential for optimizing the laser-driven implosion of spherical shell targets. A more fundamental reason for investigating this coupling is to study the possibility of using the x-ray emission in the coronal plasma to probe the state of high-density matter in the adjacent shock compressed material. The first theoretical study of the effects of radiation transport on target hydrody-namics was performed by Duston et a l . ' ^ ' In a detailed computational study of the atomic physics involved in laser heating of thin foils they found that an ionization wave can propagate through the plasma, causing shifts in inner-shell photon absorption edges and increasing radiation transport through the target. Subsequent numerical investiga-tions by Schmalz et al. and Salzmann et al. showed that radiation transport can give rise to an important region which they refer to as the radiatively heated zone. In their approximate treatment they find that the radiatively heated zone, located between the ablation front and the shock compressed material, can extend over a significant region of the target and be relatively uniform in temperature and density. Recently, however, Chapter 1. Introduction 4 some discrepancies in theoretical predictions have arisen. Marchand et al. ' in calcu-lating the change in the ablation mass due to radiation, have suggested that Salzmann et al. overestimate the extent of the radiatively heated zone due to their neglect of x-ray absorption in the corona. These initial calculations stimulated numerous experimental s tud ies^ l - ^] on the transport of x-ray radiation through high-density matter. A strong motivating factor for studying radiation transport is to assess its effect on laser-driven shock waves. For instance, under what conditions does x-ray preheat (heating of the target material ahead of the shock front) perturb the temperature and density of the material significantly? This is an important issue in equation of state measurements where it is often assumed that the material ahead of the shock front is at normal density and temperature. If the degree of preheat can be accurately calculated its effects may be mitigated by using an appropriate experimental technique. For instance, one approach for effectively eliminating x-ray preheat in high-27 targets is to use multi-layered targets in which a layer of low-27 material (e.g. CH) is placed over the high-27 layer. Ablation occurs in the low-27 material thus reducing the x-ray emission from the coronal plasma. Moreover, if the dynamic impedance (product of density and shock velocity) of the high-27 layer is higher than that of the low-27 ablator, then to attain a given shock pressure in this layer, the necessary laser intensity can be significantly reduced thus further lowering the x-ray emission. The reason for this is that shock reflection at the interface of the two layers enhances the pressure applied to the high-27 layer. This impedance-mismatch technique'28]-[31] has been recently investigated both numerically'"^' and experimentally'"^'-'"^'. On the other hand, x-ray preheat, if well characterized, could be used to heat a layer prior to the arrival of the shock wave. This would allow the possibility of studying shock compression of matter under different initial conditions. It is also crucial in laser-driven shock wave experiments to determine Chapter 1. Introduction 5 the effect of x-ray absorption behind the shock front. This is important because it can significantly modify the temperature and density conditions (spatial profiles) in the shock compressed material. Aside from assessing the effects of radiation on shock hydrodynamics, an accurate understanding of radiation emission and transport is important for diagnostic purposes. For instance, the spectrum of the plasma emission can be properly evaluated by taking into account the plasma opacity. This can provide the basis for plasma diagnostics using spectroscopic measurements. These diagnostics are crucial for current investigations of x-ray lasing s c h e m e s ' ^ - O f more importance to our study, the x-ray emission from the coronal plasma can be used as an effective probe of the high density shock-compressed material in the target. Several interesting experiments have been reported recently which demonstrate this possibility. Bradley et a l . ' ^ used the x-ray emission from a Bi layer to probe the /f-shell photoabsorption edge of a radiatively heated and shock-compressed chlorine plasma. Hall.et a l . ' ^ used x-ray emission from a uranium backlighting source to record the extended x-ray absorption fine-structure (EXAFS) spectrum of the aluminum A'-shell photoabsorption edge, which yielded the first experimental observations of ion correlation effects in a dense plasma. 1.2 Present Work The main objective of this work was to study x-ray radiation transport in laser-matter interactions, both numerically and experimentally, focussing on the use of x-ray spec-troscopy to probe the dense plasma produced by laser-driven shock compression. The first phase of this study was to develop an appropriate atomic physics model to be incorporated into an existing one-dimensional hydrodynamic code. The numerical modelling of the radiation emission and transport is similar to that proposed by Duston et Chapter 1. Introduction 6 alJ 1 8 - 4 21 in their pioneering work. Briefly, a collisional-radiative equilibrium model with detailed configuration accounting was coupled with our one-dimensional hydrocode to calculate the ion-level populations and radiation emission. The transport of line emission was implemented by a probabilistic scheme, whereas continuum radiation was transported by a multi-group approach. Although we were unable to match the accuracy in the atomic rate coefficients used in the work of Duston et al., we improved significantly on their treatment of the hydrodynamics. Firstly, we used the piecewise parabolic method of Colellaf44' 4 5 ' to solve the fluid equations rather than the flux-corrected-transport technique used by Duston. This enabled us to more accurately model the spatial structure of the shock wave inside the target as well as that of the rarefaction wave which propagates from the free (back) surface after the emergence of the shock wave. Secondly, we used a well established equation of state obtained from the SESAME data library'4*^ instead of assuming an ideal gas equation of state. This was crucial for accurate calculations of temperatures and densities in the compressed solid. Using this coupled radiation-hydrodynamic code (henceforth called HYRAD) we first investigated the effects of x-ray radiation transport on the temperature and density pro-file in the target. The results indicated that at laser intensities of ~ 101 3 W/cm 2 x-ray preheat can be neglected in an aluminum target at depths greater than 10 pm . The radiatively heated zone predicted in earlier works^' 4"^ was also observed in our simula-tions. More importantly, HYRAD was used in conjunction with a new solid state model, wliich calculates the inner-shell photo-absorption edge of a plasma at different tempera-tures and densities, to predict the change in the aluminum /C-shell photoabsorption edge as the target is compressed by the laser-driven shock wave. The additional modelling of Ka emission in HYRAD suggested that the Ka spectrum may be a useful diagnostic for determining the dominant ionization states in the radiatively heated zone. Chapter 1. Introduction 7 In order to measure the if-edge energy and Ka emission experimentally it was first necessary to build an appropriate x-ray streak camera. Commercial cameras available at that time were not suitable as they lacked several features we considered vital. A l l of these units had a short photocathode (< 25 mm) which restricts the x-ray spectrum which can be recorded. Furthermore, the width of the photocathode was limited to 100 pm . This significantly reduces the x-ray collection aperture of the camera. These concerns led us to build a x-ray streak camera which was at the time being designed by Paul Jaanimagi at the University of Rochester. The unit has a 40 mm by 1 mm photocathode which made it ideal for our experiments. In the experiments, this x-ray streak camera was combined with various diffraction crystals to spectrally and temporally resolve the x-ray spectrum transmitted through the target. In the first experiment, the temporal evolution of the if-shell photo-absorption edge in aluminum was measured as the target underwent shock-compression and subse-quent rarefaction. The results indicated a red shift of the absorption edge as the density of the compressed target increased. This was in good qualitative agreement with H Y R A D calculations. The large change in opacity across the photo-absorption edge makes this measurement extremely sensitive to a narrow region of the target with the largest red shift. This suggests that the if-edge energy may provide an unique way of tracking the temporal development of laser-driven shocks in opaque targets. The effects of the finite focal spot size were also assessed by ray tracing the x-ray emission in simplified two dimensional simulations. The results suggest that two-dimensional effects can lead to an observed edge profile significantly broader than the actual edge profile. Unfortunately, this made any detailed measurement of edge broadening at these extreme conditions impossible. Motivated by H Y R A D simulations which indicated that Ka emission could be used to determine the dominant ionization states in the radiatively heated zone, the Ka spectrum Chapter 1. Introduction 8 in aluminum was measured in a second series of experiments. The results indicate an anomalous abundance of A l + 7 ions which could not be accounted for in the one-dimensional simulations. Several possible reasons for this discrepancy are considered. 1.3 Thesis outline In chapter 2, we first present a detailed description of the laser-driven ablation process. Following this we introduce the various processes which must be considered in the mod-elling of radiation transport in laser matter interactions. This includes a description of the solid state model we used to calculate the if-shell photo-absorption edge of alu-minum. In chapter 3 the collisional-radiative equilibrium model we used for calculating the level populations and how it is coupled to our one-dimensional hydrocode is described in detail. The effects of radiation transport on the temperature and density of the target conditions are also summarized. A description of the experimental facility and the x-ray streak camera built in the course of this work is given in chapter 4. The time-resolved measurement and interpretation of the aluminum /('-shell photo-absorption edge is given in chapter 5. Chapter 6 presents the measurement of the Ka emission from laser irra-diated aluminum along with results of simulations. A summary of the main results and conclusions is given in chapter 7 as well as suggestions for further investigations. Chapter 2 Radiation emission and transport in Laser-irradiated targets In this chapter, we first give a brief qualitative description of the important processes in laser-target interactions. This is followed by a review of the atomic physics models used to determine level populations and subsequently radiation emission. The mechanisms affecting radiation absorption and transport are also discussed. 2.1 Laser-target interactions The important processes governing laser-target interactions differ significantly depending on irradiation conditions. For our discussion we consider a solid target irradiated by a high intensity ( $L > 1 0 1 1 W/cm 2 ), short wavelength (\L < 1 fim ) and long pulse (TL > 2 . 0 n s FWHM) laser. The long laser pulse restriction allows us to consider the ion state population to depend only on the instantaneous plasma conditions. This is important in the atomic physics models discussed later. Figure 2 .2 shows a schematic diagram of the different regions in the target at a time when the ablation process is well established. The focussed laser light propagates and is absorbed in the coronal plasma (Region 1 ) . For our laser conditions laser absorption is dominated by inverse bremsstrahlung a b s o r p t i o n I n this process, the electrons, oscillating in the electric field of the laser, collide with the ions and transfer part of their energy. Since the electron-ion collision frequency increases with electron density, laser absorption is more efficient at higher densities. The laser light penetrates up to the critical density layer where it is reflected. The critical density, ncr oc \/\ 2L (A^ is the laser 9 Chapter 2. Radiation emission and transport in Laser-irradiated targets 10 Figure 2.2: A schematic diagram of a solid target irradiated by a high intensity laser, p is the target density, Tt and Te are the ion and electron temperatures respectively. Chapter 2. Radiation emission and transport in Laser-irradiated targets 11 wavelength), is the electron density at which the plasma and laser frequency are equal. Although resonance absorption'4'''! of the laser light can also occur near the critical layer, its contribution to the total absorption is negligible at short laser wavelengths and low intensities (< 1014 W/cm 2 ). The absorbed laser energy produces a region of high temperature (~ 1 keV), low density plasma near the target surface. As a consequence, a very large temperature gradient is produced leading to thermal transport towards the dense target material. This heat flux continuously vaporizes and heats the target material as it expands into the vacuum. The region where the large temperature gradient exists is called the ablation zone (Region 2), and the point where it joins onto the cold solid is called the ablation front The momentum of the high speed plasma flow into the vacuum generates large pressures at the surface of the target. This drives a strong shock into the target (Region 3) which propagates ahead of the ablation front. The importance of laser driven shocks makes a brief review of shock wave physics appropriate. Compression shock waves are generated when the local sound speed, Cs, increases with material density. This is generally the case and for an ideal gas, Cs ~ p1^2, where p is the density. Hence a compression disturbance or wave will propagate faster in regions of higher densities than in those of low density, thereby causing the density perturbation to steepen into a sharp wave front or shock wave which propagates faster than the speed of sound in the unperturbed material. The locus of thermodynamic states obtained from shock compression of a material is uniquely determined by its equation of state. This locus, usually called the shock adiabat, the Rankine-Hugoniot curve or simply the Hugoniot, can be calculated in a completely general manner by considering the conservation of mass, momentum, and energy across the shock front. Specifically, PoUs = Pl(Us - Up) (2.1) Chapter 2. Radiation emission and transport in Laser-irradiated targets 12 Pi-Po = PoUsUp (2.2) Ej - E0 = l / 2(Px + P 0)(l/po - 1/Pi) (2-3) where p, P, and £ are the density, pressure and internal energy respectively. The sub-scripts 0 and 1 denote the unshocked and shocked regions, while Us and Up are the shock wave velocity and the particle velocity in the compressed region (refer to figure 2.3). This yields three equations, with five parameters associated with the shocked state: Us, Up, Pi, pi and E\. If the equation of state, P = P(p,E), is known for the material, then it may be added as a fourth equation leading to a system of equations which can be reduced to a single equation with one free "parameter. This equation gives the locus of points (i.e. Hugoniot) reached by a single shock for specific initial conditions (p0, P 0 , E0). In figure 2.3 the Hugoniot curve, isotherm and isentrope passing through the initial state are shown for comparison. Alternatively, if the equation of state (EOS) of the material is not known it can be measured by shock wave experiments. From the three conservation equations and simultaneous measurement of two shock parameters (e.g. Us and Up) the EOS parameters Pi,pi and Ei (along the Hugoniot) can be inferred!50'. The coronal plasma is also an intense source of radiation. In particular, for high Z targets experimental results by Mochizuki et a l . ' ^ indicate that as much as 40% of the incident laser energy escapes the plasma as x-ray radiation. This radiation represents a significant energy transport mechanism and can lead to several effects. First of all, x-rays can be deposited near the ablation front to enhance the ablation process. More important, however, are the effects of x-ray preheat on the target material ahead of the shock front. If a significant amount of x-ray preheating exists, equation of state measurements'^ can be adversely affected. In the following section we shall review some of the theory necessary for investigating the effects of radiation more carefully. Chapter 2. Radiation emission and transport in Laser-irradiated targets 13 Figure 2.3: a) Shock front propagation, b) Schematic diagram of Hugoniot (solid), isentrope (dashed), and zero degree isotherm (dotted). Chapter 2. Radiation emission and transport in Laser-irradiated targets 14 2.2 Radiation Calculation 2.2.1 Atomic physics The first phase in calculating the radiation emission is the development of an accurate model to calculate the various level populations. This in turn requires a knowledge of all the atomic processes and associated rate coefficients important in the regime of interest. In the following, we consider the physical processes necessary in our study. Collisional ionization The process of collisional ionization Nz(j) + e=> Nz+1(k) + e + e (2.4) consists of an ion Nz(j) of charge Z in a state j losing an electron through a collision with an electron. Although ion-ion collisions can also lead to ionization, they are expected to only be important at proton energies'5- '^ of ~ 20 keV. Several general theoretical formulae (scalable to different species) are available: (i) the semi-empricial Lotzt 5 2 ' formula com-monly used in plasma simulations, (ii) the distorted-wave exchange approximation'5"^', (iii) the scaled Coulomb Born approximation'54' and (iv) the exchange classical impact parameter (ECIP) method!55'. The cross-sections calculated with these theories are gen-erally within a factor of two of experimental results. Three-body recombination Three-body recombination Nz(j) + e-re=> Nz-\k) + e (2.5) is the inverse process of electron ionization (the non-recombining electron is necessary for conservation of energy and momentum). As a consequence, the rates for this process are Chapter 2. Radiation emission and transport in Laser-irradiated targets 15 calculated by using the principle of detailed balance. The resulting rate coefficients are proportional to the square of the electron density (oc n 2), and as a result this process is only important at high densities. In fact for plasma densities below 1016 c m - 3 three-body recombination can generally be neglected'5^. Radiative recombination Radiative recombination NZ(j)-re => N^W + hv (2.6) consists of an ion capturing a free electron and a photon being emitted. This process represents the major source of x-ray radiation over a large range of plasma conditions. For complex ions, radiative recombination cross-sections are generally obtained through a detailed balance with the inverse process'5'''] of photo-ionization. Dielectronic recombination Dielectronic recombination is a three step process. In the first phase an incident electron is resonantly captured by the recombining ion in a level with large principal quantum number, the execess energy being spent in the excitation of a bound electron Nzti) + e^Nz-\j'ik')- (2-7) If the electron energy is equal to the excitation energy of state j' minus the binding energy of state k', the capture process takes place without radiation. The two-electron excited state, however, is unstable since it is above the ionization limit of the recombined ion. It can either undergo the inverse process, auto-ionization, or the excited electrons can radiatively decay to lower levels, that is, Nz~\j\k') Nz-\k) + hv. (2.8) Chapter 2. Radiation emission and transport in Laser-irradiated targets 16 Generally, however, the stabilizing radiative process is the decay of level j' into an excited state below the ionization limit. The presence of the outer electron results in the radiated photon wavelength being longer than that for the unperturbed resonance transition. This accounts for the satellite lines commonly observed. Collisional excitation and de-excitation Collisional excitation, and its reverse process de-excitation, Nz(j) + e & Nz(k) + e, (2.9) occurs when collisions (pre-dominantly electron-ion), induce a transition in the state of the bound electrons. Contrary to ionization processes, approximate formulae of general validity for excitation are unavailable. As a consequence, one has to rely on individual calculations. An excellent review of the current theory is given by Henry'5 8]. Radiative transitions Radiative transitions Nz(j) Nz(k) + hv, (2.10) are the source of line emission. As is the case for collisional de-excitation, theories applicable to a wide range of transitions are largely inaccurate. Wiese'5^ has reviewed both experimental and theoretical evaluations of transition probabilities, also discussing the systematic trends of oscillator strengths along an iso-electronic sequence. Photo-excitation and photo-ionization When the plasma density is high enough so that re-absorption of radiation is important, one must consider the effects of photo-excitation and photo-ionization Nz{j) + hv Nz(k), (2.11) Chapter 2. Radiation emission and transport in Laser-irradiated targets 17 Nz(j) + hv =j> Nz+1(k)-re. (2.12) These processes which are the inverse of radiative transition and radiative recombination affect both the population distribution and the emitted spectrum. In many models the plasma is assumed to be optically thin to its own radiation and these effects can be neglected. 2.2.2 Level population Detailed calculation of the radiation emission by plasmas requires knowledge of the pop-ulation of all the atomic levels for each ion. In principle, this requires the solution of a complex system of rate equations describing the population and de-population of levels by the various processes described in the previous section. If the plasma conditions are non-steady state one has to solve a time-dependent system. It is, however, often suffi-cient to consider the steady state problem. In this section we shall discuss three models commonly used to determine the level populations. Coronal model In low density plasmas collisional ionization and radiative recombination dominate over the other atomic processes. The coronal model, therefore, assumes that the ionization balance is determined by a balance between collisional ionization of an ion from the ground state and radiative recombination to that level. The population distribution is then given by Nz(g)neSz(g,Te) = Nz+lneaz+1(g,Te) (2.13) or Nz(g) Sz(g,Te) *z+1(9,Te) (2.14) Chapter 2. Radiation emission and transport in Laser-irradiated targets 18 where n e and Te are the electron density and temperature, Nz(g) and Nz+1(g) are the population densities of ions with charge Z, Z +1 in their ground levels, and Sz(g, Te) and afi+1(Te) are the rate coefficients for coUisional ionization and radiative recombination respectively. It has been pointed out'^ 0 , ^ \ however, that this model should be modified to include dielectronic recombination. This process can be included in equation 2.14 by replacing the term a | + 1 ( 7 ; ) by a | + 1 ( 5 , Te) + afjl^tot, T e ) , where azDY(tot, Te) is the dielectronic rate coefficient summed over all relevant levels. K e y ^ ^ suggest that the effects of dielectronic recombination dominate at low nuclear charge Z^ whereas radiative recombination begins to dominate for Zfj > 15. In the coronal model the population densities of excited levels are determined by a balance between the rate of coUisional excitation from the ground level and the rate of spontaneous radiative decay. Thus, ,t7,., X(g,j)neNz(g) . Nz(j) = e .\y} (2.15) where X(g,j) is the coUisional excitation rate from the ground state g, to the excited state j, and A(j, k) is the radiative decay rate. This model is applicable where the radiative decay rate of a state dominates over coUisional decay, that is, ^A(j,k)»neY:XU,k) (2.16) For the coronal model to be valid, this criterion should be satisified for all excited states j. Unfortunately, however, there is always a value j above which it cannot be satisfied. This is the case because with increasing quantum number the probability of spontaneous decay decreases whereas that of coUisional de-excitation increases. McWhirtert 6 4 ! gives a criterion for validity of equation 2.14 for levels with principal quantum number n < 6, Chapter 2. Radiation emission and transport in Laser-irradiated targets 19 namely n e < 5.9 x 10 l o(Z + l) 6r e 1 / 2exp ^ [cm-3;eV] (2.17) Laser produced plasmas are generally too dense for equation 2.17 to be valid, nevertheless, a coronal description may be appropriate for lower values of n (subject to equation 2.16). The local thermodynamic equilibrium (LTE) model At sufficiently high densities collisional processes dominate. As a consequence, each process is accompanied by its inverse, and these pairs of processes occur at equal rates by the principle of detailed balance. The population distribution of the states is, therefore, the same as it would be in a system in complete thermodynamic equlibrium. For a Maxwellian distribution of free electrons, the charge state is determined by the Saha-Boltzmann equation^5' " ' j ^ ) 1 2 ^ ) ^ ] e x p l H T j ( 2 ' 1 8 ) where uiz(j), u>z+1(g) are statistical weights and xZ(J) *s ^ he ionization energy. This is equivalent to setting the collisional ionization rate equal to the three-body recombination rate. From this relation it is clear that the level populations of an ion are then determined by the Boltzmann equation NfU) = '^U) (xz(k)-xzU) Nz(k) u>z(k)CXP 1, kTe The requirement that collisional processes dominate imposes a lower bound on the electron density. McWhirtert 6 4' gives a necessary, but not sufficient, condition for the LTE model to apply, namely, that the electron density should satisfy ne > 1.8 x 1014 Ty\(j, kf [cm-3;eV], (2.20) where x(j, k) is the largest energy level difference in the level scheme of the ion considered. At lower densities the Saha-Boltzmann equation can still be used in a restricted sense. As (2.19) Chapter 2. Radiation emission and transport in Laser-irradiated targets 20 was mentioned in the previous section, there is always a principal quantum number above which coUisional transitions dominate and hence equation 2.18 applies. This principal quantum number, commonly referred to as the collision limit n c , can be estimated from the hydrogenic relation where x ' s the hydrogen ionization potential. The small amount of atomic physics data necessary for this model makes it an at-tractive alternative. It is therefore frequently used in order to provide an estimate of the charge-state distribution in laser-produced plasmas. Nevertheless, care must be taken to verify that criterion 2.20 is satisfied. CoUisional radiative equilibrium (CR or CRE) model For laser-produced plasmas where electron density and temperature vary by many orders of magnitude, neither of the two previous approximations can be safely used. As a consequence, laser-produced plasmas require the use of a full collisional-radiative model where all the coUisional and radiative processes previously discussed are included. This model reduces to a coronal model at low densities and an LTE model at very high densities. In principle a full CRE model includes a large number of rate equations since for each ion all excited states must be considered. Fortunately, since the collision limit in laser-produced plasmas is low, the number of excited levels which have to be considered is significantly reduced. Often the objectives of specific CRE calculations help to reduce the number of states. For instance, if one is interested in line emission within a particular ion species it is often possible to reduce the number of states considered in other ions without any significant effects. Alternatively, if the primary goal is in modelling the plasma hydrodynamics, radiation energy considerations may determine which states are (2.21) Chapter 2. Radiation emission and transport in Laser-irradiated targets 21 included. For example, Duston et al.^^l used experimental line emission spectrum to determine the states that are energetically important. In the collisional-radiative model an atomic rate equation of the form ^ = ~EWfkN(j) (2-22) k k is written for each ground and excited state included in the model, where Wkj is the rate for populating state j , and is the rate for depopulating state j. If the plasma density and temperature change on a time scale slower than that of the important atomic processes, equation 2.22 reduces to the steady state form Y,WfkN(3)=j:Wk,N(k). (2.23) fc k McWhirter et a l . ' ^ suggest that a good estimate of the time needed to reach steady state is given by 10 1 2 <„ = — [s;cm-3]. (2.24) n e In the case of laser produced plasmas most of the emission originates near the critical density layer, hence, ne ~ ncr(<x A£ 2) and tst = 8.98 x 10- 1 0 X2L. [s; /xm ] (2.25) For a laser wavelength of — 0.53 /xm this condition implies that the thermodynamic time scales must be longer than tst ~ 200 ps in order for the steady state C R E model to be valid. , In principle, C R E models can be used for any plasma conditions if accurate atomic rates for all the processes are known. Unfortunately, cases arises when accurate atomic rates are not available. This is particularly the case at low temperatures and high den-sities where the assumptions of binary collisions and electron non-degeneracy cannot be applied. Burgess '^ points out that an obvious breakdown of the binary assumption Chapter 2. Radiation emission and transport in Laser-irradiated targets 22 occurs if particle separations are comparable with orbital dimensions of bound electrons. For hydrogenic ions and principal quantum number n this implies where ao is the Bohr radius. Electron degeneracy is important when the Fermi energy ep is comparable to the thermal energy, that is, In figure 2.4 we illustrate the limits of validity of the coronal, LTE, and CRE models for an aluminum plasma. Also shown is the range of plasma parameters which may be created in laser-produced plasmas. As expected, the CRE model is the most widely applicable. 2.2.3 Radiation emission and absorption In this section, we shall discuss the spectral emission of plasmas with emphasis on x-ray emission. This includes continuum emission, both bremsstrahlung (free-free) and recombination (free-bound), and line (bound-bound) emission. The inverse absorption processes are also presented. Bremsstrahlung emission Bremsstrahlung emission is a result of acceleration of a charge in the Coulomb field of another. In most cases, however, this interaction is between electrons and ions with electrons being the primary emitter. The power radiated per unit volume and frequency interval as a consequence of the this interaction is given by Ro — (2.26) (2.27) (2.28) Chapter 2. Radiation emission and transport in Laser-irradiated targets 23 TEMPERATURE leVl Figure 2.4: Criteria for applicability of the various ionization models. Shaded region is parameter space for laser-produced plasmas. Chapter 2. Radiation emission and transport in Laser-irradiated targets 24 where dajf(E, v) is the differential cross-section for emission of bremsstrahlung radiation, ve is the electron velocity, and f(E) is the electron distribution function (Maxwellian in most cases). An accurate evaluation of dojf requires a quantum mechanical treatment, since pho-tons of energies comparable to that of the emitting particle can be produced. However, a classical treatment of the cross-section gives the correct functional dependence for most of the physical parameters. For this reason it is common to express dajf as the product of the classical cross-section and of a correction factor gjj(v, Z, 2£), called the Gaunt factor. A complete derivation of the dajj can be found in Rybicki et a l . ' 6 8 ' . Integration of equation 2.28 over a Maxwellian distribution of temperature Te gives 6 8 x 1 0 - 4 7 hA») = ' m l / 2 ne^Z2Nzgff(v,Z,Te)exp(-hv/Te) (2.29) Te Z [W c m - 3 Hz - 1 ; eV,cm~ 3 ] . The weighted average Gaunt factor g^j has been calculated for hydrogenic ions by Karzas and Latter! 6 9!. The total power per unit volume emitted by thermal bremsstrahlung can be easily calculated by integrating over frequency. The result is Ij} = 1.53 x 10 - 3 2 Tl'2n.Y.Z2NzgTJS{Z,Tt) (2.30) z [W c m - 3 ; e V , c m - 3 ] . The frequency averaged Gaunt factor J^y is in the range of 1.1 to 1.5. Choosing a value of 1.2 will give an accuracy to within about 20%. The inverse process of bremsstrahlung is collisional absorption Or inverse bremsstrahlung absorption whose importance in laser absorption was previously discussed. In laser pro-duced plasmas this processes is important for low photon energy and high density where it can cause saturation of emission at the blackbody limit. The absorption coefficient K / / ( ^ ) can be evaluated by applying Kirchhoff's law! 6 8] *//(") = «Jf(v)B(v) (2-31) Chapter 2. Radiation emission and transport in Laser-irradiated targets 25 where ejj is the emission per unit solid angle and B{v) is the Planck blackbody function *(,) = — • (2-32) With tjj calculated from equation 2.29 the bremsstrahlung absorption coefficient is KfJ(v) = 3.4 x 106 Te-l/2neu-3(l - exp(-hp/Te))J2gj}Z2Nz [cm"1]. (2.33) z Recombination continuum For moderately high-Z elements with ionic ionization potentials \Z > kTe, the dominant source of continuum at photon energies hp > xZ lB radiative recombination (refer to equation 2.6). In this process a free electron of energy E is captured in a bound level of ionization energy xf _1> a n ^ a photon of energy hv = (E + xf-1) i s emitted. The photon energy is, therefore, a function of the electron energy, and only photons with energy greater than xz~l are generated. This leads to the recombination edges observed in free-bound continuum spectrum. If oT2_x n(E) is the radiative recombination cross section into the bound level n of the recombined ion, then the total radiated power PJh is -pn neNz aTzT_ln{E)vMf{E)dE (2.34) Jo or alternatively, P^ = h^neNza^in(E) (2.35) where ctrzT_1 n is the radiative recombination rate. The radiated power per unit frequency interval can then be obtained by equating expression 2.34 to / » = / , _ , ( 2 - 3 6 ) Chapter 2. Radiation emission and transport in Laser-irradiated targets 26 and summing over the different shells for which hu > Xn 1 (that is n > nmin). A convenient expression is obtained if we use the hydrogenic ion cross- sectionf63' 6.4 x I P ' 4 7 ^ATz72 (~hv\ _ Uxl'1- ( „ x fXz-i\ IMU) = =175 n e ^ J V Z exp — ^ — — - gfb(v, Z,n)exp I — - J (2.37) [W c m - 3 Hz _ 1 ;eV,cm - 3 ] , where <7yb is the average free-bound Gaunt factor'69], and £ n is the number of available states for recombination into shell n. Aside from the edge discontinuities and small variations in the Gaunt factors, the spectral variation exp( — hv/kTe) is the same for both bremsstrahlung and recombination continuum. This result is often used to get an estimate of the electron temperature in a plasma. The inverse process of recombination radiation is photoionization absorption. There-fore, applying the principle of detailed balance in conjunction with the Kirchhoff-Planck relation, the absorption coefficient for ground-state ions of charge Z — 1 is given K,M = 1.3 X 10" N'-*^-£2-(f->V-> [ l - exp ( i £ [cm - 1], where Uz is the partition function. (2.38) Line emission Line radiation is the result of a radiative transition from an upper level u to a lower level /-. The radiated power per unit volume and frequency interval is simply given by Ibb(u) = NuAulhvQ<P^) (2.39) where Au\ is the transition rate (Einstein A-coefficient) and <f>e(v) is the normalized emission line profile function. Although the absolute emissivity of a spectral line can be used to determine Nu, and indirectly, electron temperature or density, it is much more Chapter 2. Radiation emission and transport in Laser-irradiated targets 27 common to use line ratios as a diagnostic of Te and ne. This has motivated extensive analysis^3 , 5 ^ of the intensities of //-like and He-like ion resonance lines, their fine structure, and forbidden or dielectronic satellites. The absorption coefficient can be easily expressed in terms of the two Einstein B-coefficients which account for both absorption (Biu) and stimulated emission (negative absorption,Bui). Specifically, Khh{v) = ^ (NlBluMv) ~ NuBul<f>e(v)) (2-40) where <f>a is the normalized absorption line profile function. Employing the relationship between the Einstein coefficients g,Blu = guBul (2.41) and Aui = ^ r B u l 2.42 c1 c2 , Qu Nu we can rewrite equation 2.40 as where we have also assumed (j)a = 4>e = (j). A detailed derivation of the absorption coefficient and Einstein coefficients can be found in Rybicki and Lightman'^'. One source of line radiation which is an important diagnostic of the cold target is Ka emission. Spectral lines identified as Ka emission are often seen in experimental spectra when high energy photons or particles are incident on cold target material. For short wavelength lasers, Ka radiation is predominantly the result of x-ray photoioniza-tion of an inner-shell electron. In this process a primary photon having an energy greater than the binding energy removes a /f-shell electron leaving the atom in an excited state. Chapter 2. Radiation emission and transport in Laser-irradiated targets 28 The atom can return to the unexcited state by various processes of which two are dom-inant. The first of these is where an electron from one of the upper levels (i.e. a higher principal quantum number n than that of the ionized level) falls to the excited level and is accompanied by the emission of a photon. This leads to Ka emission when the upper level is in the L-shell. The second process is similar in its initial stage to the first, but in this case, the radiation produced following the transference of an electron from an upper level is not emitted, but goes into ionization of an electron from a lower energy shell. This process is known as the Auger process'^ and can lead to the production of two (or more) vacancies in the upper levels. This double vacancy production is an important factor in the formation of satellite lines. The quantity of radiation from a specific level will be dependent upon the relative efficiency of the two opposing de-excitation processes involved. This relative efficiency is usually expressed in terms of the fluorescent yield. The fluorescent yield ui is defined as the number of x-ray photons emitted divided by the total number of vacancies formed in the associated level. A general trend is that the fluorescent yield increases with increas-ing atomic number. In the case of aluminum the if-shell fluorescent yield is less than 10%. As a consequence Ka emission can usually be neglected as an energy transport mechanism. However, since it is the only emission which can escape the cold target mate-rial it is an extremely important diagnostic. In particular, one approach to investigating the effects of radiation transport which we will discuss in chapter 6 is to measure the Ka emission spectrum. 2.2.4 Inner-shell photoionization The absorption mechanisms discussed in the previous sections deal with photon absorp-tion by valence electrons through photoexcitation or photoionization. However, in the cold, dense plasma, inner-shell photon absorption is the dominant absorptive process. Chapter 2. Radiation emission and transport im Laser-irradiated targets 29 Neglect of this process can lead to a gross underestimate of the x-ray deposition and, as a consequence, energy transport. Moreover, it accounts for the inner-shell vacancies which ultimately lead to Ka emission as described in the previous section. At below solid densities the inner-shell photoionization depends only on the electronic configuration of the ion, namely, on the number of bound electrons. The temperature and density dependence is reflected only through the population of the various ionic species. Therefore, the total absorption coefficient is given by «f»(") = E E J V J < n K i ) , ( 2- 4 4) j n where «" n is the absorption coefficient for sub-shell n for ionic species j. Numerous papers have been published on the calculation of K™n for various atoms^ 2 , The main result is that ionization of outer shell electrons affects the photoionization cross-section of inner electrons only in causing a shift of the absorption edge. Moreover, when the ionization occurs from a partially populated shell, the cross-section is reduced rel-ative to the full-shell case by a factor proportional to the number of electrons in the shell. From these results one would expect that given the neutral atom value rz™n(v, 0), the absorption coefficient of an electron from shell n of state j can be obtained from 0 hv < En(j) { «?nl>,0fe hv>En(j) (2.45) h,, ~> P. (n\ were £ n j and £ n 0 are the number of bound electrons in shell n of the respective level, and En(j) is the corresponding binding energy. The primary factor affecting the value of En(j) is the the ionization stage. As the ion-ization stage increases, so does the binding energy. For instance the if-shell photoabsorption edge energy of A l + 3 is 1588 eV whereas for A l + 4 it has a value of 1634 eV. To illustrate the importance of this edge shift in accurately calculating energy transport consider the transport of x-rays in the range 1588 eV-1634 eV through an aluminum plasma. When Chapter 2. Radiation emission and transport in Laser-irradiated targets 30 the plasma is predominantly A l + 3 these x-rays are above the photoabsorption edge and significant absorption occurs. As the material heats up and is ionized to a state of A l + 4 it becomes optically thin to this radiation. This process leads to an ionization burn wave first described by Duston et a l . ' ^ . This will be discussed in detail in the next chapter. Calculating En(j) as the binding (ionization) energy of an isolated atom is generally adequate at high temperture and low density where interactions with free electrons and neighbouring ions are a small perturbation. It is also a valid approximation when the objective is to model energy transport. The reason for this is that the edge shift due to these secondary effects is comparatively small. Nevertheless, an accurate calculation of the photoabsorption edge position and profile is important because it leads to a crucial diagnostic for probing the electronic state of cold dense materials. Conversely, experi-mental measurements of the absorption edge in plasmas of known conditions allows us to assess the validity of various theoretical models. In the following section we discuss the important mechanisms which govern the photoabsorption edge. Photoabsorption edge energy The photoabsorption edge energy En(p, T) can be viewed as the sum of three terms: En(p, T) = E{ON{P, T) + AECL(P, T) + AEDEG(p, T). (2.46) The first term , E^ON, represents the n-shell ionization energy of a free ion whose ion-ization state is determined by the plasma density and temperature. The effects of free electrons and neighbouring ions on the bound states are included in the continuum low-ering term AECL. The effects of these interactions on the free electrons are included in the degeneracy term, AEDEG which can be be expressed in three parts AEDEG(P, T) = AEee + AEei + AEde. (2.47) Chapter 2. Radiation emission and transport in Laser-irradiated targets 31 The first term represents the electrostatic interaction of the electrons with each other leading to a positive potential energy AEee. The free electrons also interact with the ions in the plasma, leading to a negative potential energy AEei. Since the plasma as a whole is quasi-neutral an electron will see a net positive charge in its vicinity, and the overall potential energy will be negative. The final energy term AEde accounts for the Fermi nature of the electrons. As a result, the lower-energy continuum states will be occupied by free electrons, up to approximately the Fermi energy. In literature AECL, and AEDEG are often combined and referred to as continuum lowering. We make the distinction simply because different models can be used for calculating each term. The calculation of EION, AECL, and AEDEG, can be performed by various tech-niques. For instance, Bradley et a l . ' ^ used a temperature dependent Thomas-Fermi theory and a modified ion-sphere model of the plasma for their calculation of the pho-toabsorption edge in chlorine. More recently, Godwal' 7^ used a solid state approach to calculate the different energy terms. A description of the techniques used in this solid state model are given in the following section. One problem with this phenomenological model of the photoabsorption edge energy is that it does not calculate the electronic structure of the bound and free electrons in a fully self-consistent manner. Clearly a better approach would be to calculate the electron binding energy EB of an atom in a self-consistent field. The edge energy would then simply be En(p,T) = EB(p,T) + p (2.48) where p is the chemical potential and accounts for the Fermi nature of the electrons. Some preliminary results of two such models (HOPE! 7 5 ! and INFERNO' 7 6 !) will be discussed in chapter 5. Chapter 2. Radiation emission and transport in Laser-irradiated targets 32 Solid state model calculation The first step in the calculation of the photoabsorption edge is to determine the ioniza-tion state for the given plasma temperature and density. For this we used the CSCP-I E E O S ' 7 7 ' 7 8 ' (complete screened coulomb potential ionization equilibrium equation of state) form of the modified Saha ionization theory. This theory, which includes high density effects such as continuum lowering, partition function cutoff, and pressure ion-ization, is justified by its previous success in predicting normal and high pressure material properties' 7 9 '. The ionization energy term E^ON is then obtained for the dominant ions from detailed Hartree-Fock' 8 2 ' calculations (including relativistic corrections). Alterna-tively, these values can be obtained from tabulated results. For example, Clementi and and Roetti ' 8^' have published E^N for a variety of elements. The interaction terms AEee, AEei are estimated indirectly using second order pseudo-potential theory. Specifically, AEDEG(p, T) = AEde -Ut-U2 (2.49) where U\ and U2 are the first and second order correction terms. The first order correction term U\ is determined by the zero pressure condition (energy minimization) at normal density and temperature. For U2 (band structure energy) we use the expression' 8 3 ' U2 = J2\S(g)fv(9fx(9H9) (2-50) 9 where 5(g) is the structure factor at the reciprocal lattice vector g, v(g) is the form factor, x(sO i s the Lindhard dielectric function, and e(g) is the screening factor. The form factor is evaluated by introducing an appropriate bare ion pseudopotential. For our study of aluminum we used Ashcroft's ' 8 4 ' empty core pseudopotential with core radius rc = 1.12 a.u. and face-centered-cubic structure. In calculating AEde we recall that for Chapter 2. Radiation emission and transport in Laser-irradiated targets 33 a free electron the probability, p, that a state of energy e is filled is given by 1 P = (2.51) 1 + exp((e - n)/kT) where p. is the chemical potential. For degenerate electrons p > 0 and we have that p < 1/2 for e > p. Therefore, a good estimate of AEde is simply A Ed, = < p for/! > 0 0 p < 0 (2.52) For the chemical potential of a free electron gas we used the approximate expression [85] P = «F 7T 2 fkT\ ( for kt < eF) (2.53) where tp is the Fermi energy (~ 11.7 eV for aluminum). It is important to note that for non-zero temperatures there are electron vacancies with energies less than the chemical potential. This is one of the mechanisms which must be considered in determining the edge profile. Finally, to calculate AECL the energy of the conduction band electrons in the lattice, EB, was computed with the linearized-muffin-tin orbital (LMTO) routines of Skriver^ 6 ' . These routines evaluate the electron eigenvalues of an ordered solid within the local density framework of Kohn- Sahm! 8 7 l . For our study of aluminum Barth-Heidin exchange correlation was used and the calculations were carried out for the 3s3p configuration in the face-centered-cubic structure by including the angular momentum states up to / = 3. AECL was then determined from the expression AE CL EB — EA. (2.54) where EA is the binding energy of the same electron in the isolated atom. In figure 2.5 the if-shell photoabsorption edge for aluminum is plotted as a function of specific volume Chapter 2. Radiation emission and transport in Laser-irradiated targets 34 ratio for A l + 3 and A l + 4 . Clearly, the most important mechanisms for modifying the edge position is the ionization state. On the other hand, at temperatures below the Fermi temperature the photoabsorption edge is mainly a function of density. This is evident in figure 2.6 which shows the edge position for three different isotherms. Comparison of these predictions with experimental measurements are made in chapter 5 . 2.2.5 Radiation transport If we neglect the reabsorption of radiation then the measured spectrum is simply gov-erned by the line of sight integration. However, as we have alluded to in our discussion of absorption coefficients, this approximation is often invalid. This is particularly the case in high denisty laser-produced plasmas where opacity effects have two important conse-quences. Firstly, the relation between the measured intensity and the emission coefficient is no longer simple. For instance line intensities and line shapes are significantly modi-fied by absorption in the plasma. Secondly, the atomic level populations are modified by photo-excitation and ionization. This in turn alters the spectral emission and introduces a coupling between the radiation field and atomic level populations. If we assume that the radiation field adjusts instantly to any change in the tempera-ture and density then the equation of radiative transfer can be written' 8 8 ' which simply states that the intensity increment dlv(s) over the ray path ds is due to the increase e„(s)ds due to emission and to the decrease til/(s)I1/(s)ds due to absorption, and KU are the emission coefficient and absorption coefficient respectively. In planar geometry (figure 2.7) the transfer equation takes a particularly simple form if, instead of s, we introduce the optical depth T„ defined by drv — K,l/(s)pds = K„(x)dx Chapter 2. Radiation emission and transport in Laser-irradiated targets 35 1590 > ~ 1580H CD CC UJ g 1570 -I UJ o w 1560H 1550-1 gJON • (a) 0.5 0.6 0.7 0.8 V / V . 0.9 > >-CD CC UJ z UJ a a UJ 1635 1625 1615 H 1595 1585-JplON E K \ (b) 0.5 0.6 0.7 0.8 V / V . 0.9 Figure 2.5: /f-shell photoabsorption edge for aluminum as a function of volume ratio for a plasma temperature of 1 eV; a) A l + 3 , b) A l + 4 . Chapter 2. Radiation emission and transport in Laser-irradiated targets 36 1558 v/v0 Figure 2.6: -shell photoabsorption edge for A l + 3 as a function of volume ratio for different isotherms: solid line 0.1 eV, chain-dashed 2.0 eV, and dashed 4.0 eV. Chapter 2. Radiation emission and transport in Laser-irradiated targets 37 Figure 2.7: Radiation transport in planar geometry. Chapter 2. Radiation emission and transport in Laser-irradiated targets 38 or and the source function Specifically, ^(s) = p / Kl/(s')ds', (2.56) Jo S„(s) = ^f\. (2.57) dIV{Tv,p) . P - = S„ - (2.58) where p = cos 8. Integration of this equation leads to the formal solution of the transfer equation = /,(0,/x)exp (^) + Texp (iL^lA S ^ ^ d r ' v . (2.59) \ H ) Jo \ ft J n The first term is simply the initial intensity incident on the plasma boundary diminished by absorption. The second term is the integrated source again diminished by absorption. In the special case of a constant source function S„, equation 2.59 gives the solution L(TU,P) = S. + exp (^^j (7,(0,/x)- S„). (2.60) The two limiting cases of this approximation are the optically thick (T„ >> 1) 1v{TV,H) = SV (2.61) and the optically thin (T„ = nvps << 1) IV(TV,P.) = (\-TV)IV{Q,H) + IVS (2.62) approximations. This simple example illustrates that only in homogeneous and optically thin plasmas can you determine the emission coefficient from the emitted intensity. In calculating the energy transfer in planar geometry it is convenient to consider that IU consists of two components (the forward component) and I~ (the backward component) with the boundary conditions 7+(0,/z) = 0 and / ; ( r j ,xx) = 0 (2.63) Chapter 2. Radiation emission and transport in Laser-irradiated targets 39 (assuming no external radiation). The radiative flux (the energy flow per unit time per unit area perpendicular to the T axis) is then given by F„(T„) = Ft ~ F~ (2.64) where F+(T„) = 2TT / ' 1+p.dp. and F~(TV) = 2TT I~p.dfi. (2.65) Jo Jo For the special case of an isotropic source function (S^ independent of p) and no external radiation we have that F^r,) = 2TT T SV{TI)E2{Tv - r'v)dr'v - f" SV(TI)E2(T'v - rv)dr'v Jo JT„ (2.66) where E2 is the second order exponential integral. The absorbed power per unit volume is then given by Eabs(x) = Ip-. (2.67) dx For the purposes of numerical calculations it is more appropriate to calculate the energy absorbed in a cell n (refer to figure 2.7) using the expression. E?3 = F„(T?) - F^T?-1). (2-68) Hence, the energy absorbed in cell n due to emission from cell 0 (and including absorption in cell 0) is approximately E? = | A X 0 [E2(T:- r°/2) - £ 2 « + 1 - r°/2)] (2.69) where IQ is the radiated power per unit volume which can include bremsstrahlung , recombination, and line radiation and the factor of 2 arises from our assumption of isotropic (equal forward and backward) emission. This expression is used extensively in the numerical calculations discussed in the following chapter. Chapter 2. Radiation emission and transport in Laser-irradiated targets 40 2.3 Summary It is clear from this discussion that a detailed analysis of radiation transport involves the solution of the radiative transfer equation coupled with a collisional-radiative model of the plasma. In cases where detailed spectroscopic information is not necessary it may be adequate to neglect the effects of photo-excitation and photo-ionization on the level popu-lation. In fact this approximation has been made in several recent publications^' 4 3 ' ^ where modelling of target hydrodynamics was the primary consideration. The validity of this approximation in the study of laser target interactions is further discussed in the following chapter. Chapter 3 Numerical Simulations In this chapter, a description of the hydrodynamic computer codes used to simulate the experiments is given. We first discuss our methods of calculating the ion state population and radiation transport. How these routines were incorporated into our one-dimensional hydrocode is then described. Finally, the two-dimensional computer code, which considers only hydrodynamic effects, is briefly discussed. 3.1 Calculation of state population The model we used for calculating the ion state populations is based on the collisional-radiative equilibrium model introduced in chapter 2. Recall that the state population of a given ion state Nz(j) is governed by a rate equation of the form dNz\ dt ( ; ) neNz-\j")S Zz'L^ + neNz+\]')a Zz'lu, + £ Nz(i)Al3 + ne £ Nz(i)Xj + ne £ Nz(i)X, (3.70) where SZ,i1 3„ is the collisional ionization rate from Nz to Nz(j), a is the sum of the radiative , dielectronic and three-body recombination rate, that is , a = an + OLOI -f nea3B, (3.71) 41 Chapter 3. Numerical Simulations 42 Aij is the spontaneous decay rate from levels i to j, X(i,j) is the collisional excitation rate, and X_1(i,j) is the collisional deexcitation rate. Although the model allows cou-pling between all states, we consider only the coupling between each excited level and the ground state of the next ionization stage (refer to figure 3.8) . This is a good approx-imation for electron densities less than 1022 c m - 3 where coupling to the excited states of the next ionization stage is negligible'89'. Moreover, when the characteristic time scales for changes in the plasma temperature and density are long compared to the equilibrium time scale we can simplify equation 3.70 by setting the time derivative to zero. This is the case for the ~ 2 ns (FHWM) laser pulse considered in this study. This version of the CRE model neglects photoionization and photoexcitation, limiting its validity to the analysis of optically thin plasmas. Nevertheless, it was used extensively in hydrodynamic simulations because of its reduced computation time. The modifications made to this model to include opacity effects will be detailed in section 3.3. The accuracy of any CRE model in predicting the actual plasma state depends pri-marily on how well the rate coefficients for the various processes reflect the true reaction cross sections. A variety of rate coefficients were used to assess the effects on the state population and radiated power. In most cases, the average ionization computed var-ied by less than 10% for the various coefficients. On the other hand, the relative state populations within a specific ion stage could change significantly. This is particularly the case for the less abundant ion species where order of magnitude differences in level population were predicted. Similar results were observed by Salzmann'91-*' who concluded that parameters sensitive to the average ionization, such as internal energy, free-free ra-diation and to a lesser extent bound-free radiation are insensitive to the rate coefficients. As expected, however, line emission is greatly influenced by the accuracy of the rate coefficients. The rate coefficients we used in our calculations are a compromise between accuracy in the emission spectrum and computational speed. Chapter 3. Numerical Simulations 43 N Z + 1 U ) X N z+i A'" 1 ) A X NZ(J) X - 1 ) A Figure 3.8: Simple three-level diagram depicting the atomic processes included in our CRE model. S is ionization, a is radiative and three-body recombination, api is dielec-tronic recombination, X and X'1 are coUisional excitation and deexcitation respectively, and A is spontaneous decay. Chapter 3. Numerical Simulations 44 3.1.1 Rate coefficients Ionization-rate coefficient The ionization-rate coefficient was obtained from Landshoff and Perez 's '^ results, Sif = 6 . 7 x l O - » f ( ^ 2 e x p ^ ( W a ( 0.915 0.42 \ 3 i ( i + o.o64Jbr./x/)2 + ( i + o.bkTjxff ) [ c m / s e c J ' ( j where ( is the number of outer shell electrons. Three-body recombination rate coefficient. The three-body recombination rate is related to the coUisional ionization rate by detailed balance, therefore, , \z-i,3 ( 27r f r 2 \3/2g(Z-l,j) ( x f ' 1 ] ^ r 6 / i , , 7 ^ ( a 3 B ) ^ ° = 25 (z,o) e x p l ^ J 5*- , J' [ c m / s e c ] ' ( } where g(Z — is the multiplicity of the state formed by recombination and g{Z, 0) is the multiplicity of the ground state of the core ion. Radiative recombination rate coefficient The radiative recombination rate is given by S e a t o n ^ ( a * ) i , o 1 J : = 5.2 x 10- 1 4 Z<£ 1 / 2(0.43 + 0.5ln</> + 0.47(£- 1 / 3 ) , [cm3/sec] , (3.74) where cf> = xf /\kTe). Chapter 3. Numerical Simulations 45 Dielectronic recombination rate coefficient The dielectronic recombination rate coefficient is calculated using the Burgess and Mer ts ' 9 3 , 9 4 ' formula 9 4 x I D - 9 / F\ ( « ™ ) i . o 1 , 0 = r 3 / 2 B(Z)j:hA(Z,j)exP [ — ) [cm3/sec;eV] , (3.75) where Z1,2(Z 4- l W 2 B(Z) = -—V ' , Z< 20 (3.76) K ' ( Z 2 + 13.4)1/2 - V ' E = (3.77) 1 + 0 . 0 1 5 Z 3 / ( 2 T + l ) 2 v ; and xxl2l(l + 0.105x + 0.015x2) A n = 0 A(Z,j) = { (3.78) 0.5x 1 / 2 / ( l+0.21x +0.30a;2) A n / 0 x = ^ ( 1 + Z). (3.79) Xtf Here Xij a n d / U a r e the excitation energy and the oscillator strength respectively of the i —> j transition of the recombinig ion of charge Z. Although the sum is over all the excited states j it is usually sufficient to only consider a few states' 9 5 ' for which the principal quantum number n is below the collision limit n c . Radiative and Collisional transition rate coefficients The coefficient for spontaneous emission is just the Einstein coefficient and the values were obtained from a variety of sources'96'-'I*-""*'. The rate coefficients for collisional excitation were calculated using the semi-classical method of impact parameters'*^' and are related to the oscillator strength by the relation X(i,j)= 1.578 x 10- 5 / t J < y exp ( ^ & ) [cm3/sec;eV] (3.80) XijT' \ le J Chapter 3. Numerical Simulations 46 where < gtJ > is the thermally averaged Gaunt factor (i.e. averaged over a Maxwellian distribution) and the oscillator strength / t J assumes the / value of the allowed transition to the level with the same principal quantum number. The value of the Gaunt factor was taken from the work of Mewe' 1 0 2 ] in which he proposed a four parameter interpolation formula < gl3 >=A-r(B-rC(j>-D<j>2)e*E^) + C<j> (3.81) where <)> = Xij/{kTe) , E^ is the exponential integral and the parameter values for the various transitions are: allowed transitions A n = 0 "" A = 0.60, B = C = 0, D = 0.28 allowed transitions A n ^ 0 A = 0.15, B = C = 0, D = 0.28 (3.82) forbidden monopole or quadrupole A = .15, B — C — D = 0 spin flip transitions A = 0, B = 0, C = 0.1, D = 0 For H — ,He — ,Li— and Ne—like isoelectronic sequences, these parameters give rates within a factor of two of the available theoretical and experimental data. For other isoelectronic sequences the accuracy is reduced to within a factor of three. Although this is acceptable for total radiation transport calculations it is only a rough approximation for line emission calculations. The deexcitation rate coefficient is obtained from the detailed balance relationship: X-\],i) = (3.83) Continuum lowering At high ion and electron densities local electrostatic fields develop in the plasma which modify the various processes. The main effect, which is continuum lowering, is accounted for by simply reducing the ionization potential of each state xf by a n amount Axf as Chapter 3. Numerical Simulations 47 calculated by Stewart and Pyatt'-"-04' to be = [3(Z> + l)K + ir»-l X> 2{Z' + 1) K ) Z' = (3.85) < Z > v ; K = 1.937 x 10- 1 0 (Z + 1 ) ^ Z ' * (3.86) where A\f and Te are in eV and the ion density Ni is in c m - 3 . Once the rate coefficients had been selected the solution of the steady state CRE mode] equation (neNz-\3")SZzU,,, + neNz+1(j')aZz-ilt, + £ N^An + ne Nz(i)X^ + ne £ i V ^ z ) ^ ) i>J i>j s<J / -Nz(j) (n e 5f J 1 J" + n e a | - 1 J " + £ ^ + n e £ X ' 1 + n e £ J ^ ) = 0 (3.87) \ t<J i<J i>3 ' simply involved matrix inversion. However, the electron density dependence of the equa-tion makes it necessary to iterate until the solution converges. The number of iterations were usually less than 10 when an initial estimate of the average ionization as calculated by a coronal equilibrium model was used. It should be noted that the rate coefficients as described are independent of the ion species. This dependence is introduced through the input data which specifies the ionization energy, multiplicity, and principal quantum number for all the states to be included. Moreover, the input data must include oscillator strengths (or Einstein A-Coefficient) and type of transition for all the transitions to be considered. In the following section we present results obtained using this model for an aluminum plasma. Chapter 3. Numerical Simulations 48 3.2 C R E results for aluminum plasma As discussed in section 2.2.2 the number of states included in the CRE model depends primarily on the objectives of the simulation. For our purposes of accurately modeling radiation transport in laser irradiated aluminum we choose an atomic level structure similar to that proposed by Duston and Davis'4 2!. It includes 104 states, with a majority in the higher ionization stages, and 128 transitions. A simpler version which considered only excited states in He—like and H—like aluminum was also used. It was found to accurately predict the emission spectrum above 1.5 keV making it an adequate model for investigating the effects of radiation preheat. In order to test the validity of our CRE model and the computer program which implements it we performed a variety of calculations. In figure 3.9 we compare the average ionization computed by this model to the values obtained from S E S A M E ' 1 0 6 ! and those by Salzmann'105] and Lee' 8 9! using different CRE models. Although the actual model used in the SESAME routine is not known, it is a version of the LTE model discussed in chapter 2. This accounts for the higher average ionization predicted since the model underestimates (possibly neglects) radiative and dielectronic recombination. On the other hand, the agreement with the other CRE models is very good. Furthermore, as the density decreases we observe that the model approaches a coronal equilibrium and that the average ionization is weakly dependent on density. These results suggest that our model is accurately calculating the average ionization. The next step is to study the radiation emission. In figure 3.10 we compare the calculated radiation emission of the three CRE models. The agreement between our model and that of Lee is very good. On the other hand, Salzmann's neglect of line emission from A l + 3 - A l + 9 accounts for his underestimation of the radiated power at lower temperatures. The two peaks in the radiation emission are Chapter 3. Numerical Simulations 49 Figure 3.9: Average ionization as a function of a) temperature (ion density=1020 cm~3) and b) density (temperature=100 eV). Solid line our model , dashed line SESAME, dot-dashed line Lee, dotted line Salzmann. Chapter 3. Numerical Simulations 50 Figure 3.10: Radiation emission as a function of temperature (ion density=1020 cm 3) a) line b) bound-free c) total. Solid line our model , dashed line Salzmann, dotted Lee. Chapter 3. Numerical Simulations 51 manifestations of the L—shell and if-shell of aluminum. The increase in ionization energy of an electron from a closed shell leads to significant population of excited states in the less ionized ions and consequently higher line emission. In neglecting photo-excitation and ionization this model is only valid when the plasma can be treated as optically thin. Nevertheless, it has several obvious advantages. First of all, it is computationally faster. Secondly, it is local; that is the level populations only depend on the local density and temperature. The benefit of this is that the required level populations can be calculated for a range of plasma densities and temperatures and subsequently used as tabular data in hydrodynamic simulations. As we will see in the following section this is impossible once we introduce photo-excitation. 3.3 Photo-excitation and line transport To incorporate photo-excitation into our model, we used the concept of photon escape-probability. This approach has been used extensively in recent years'18' 1 0 7 J . Consider a finite region divided into a number n of smaller regions. Line photons emitted in each region will have a finite probabilty of being absorbed in any other region, as well as being reabsorbed in the local region or escaping from the plasma. Let Cdl be the probability that a line photon emitted in region j is absorbed in region i. Furthermore, let N? be the total upper level population of region i and Aui be the spontaneous transition probability for the line of interest. The rate equation governing the upper level population in region i is then dNu n -IT = J2(N?R^-N?W*) + Y,N?AuiC,>- (3.88) The first two terms account for all processes populating and depopulating the upper level. This includes all the coUisional and radiative processes discussed in the previous section. The final term represents the photo-excitation of the level by photons emitted Chapter 3. Numerical Simulations 52 from all the cells j into cell i. As before, we will confine our study to the steady state case where dNf/dt = 0. It is evident from equation 3.88 that the added complexity arises in the calculation of C J t . For a photon emitted at a point in cell j, towards cell i , the probability that it is absorbed in cell i is simply given by P3l = Pe(r:i)-Pe(TJ^Arl). (3.89) P e ( r ) is the angle-averaged probability that a photon traverses an optical depth r without being absorbed or scattered and is defined b y ' 7 ^ Pe(r) = d v Jld^^{^J^) (3-90) 1 r°° = - / du^E2(r<f>u) (3.91) 2 Jo where z/v and <\>v are the normalized emission and absorption line profiles. In equation 3.89, A T ; is the line center optical depth and is related to the line center absorption coefficient K0 by the expression AT; = KoAa:,-, (3.92) where A x , is the width of region i. For the case of a Doppler line profile we have . ''To)'0"'0 W - N") l ^ v - - 3 l - < 3 ' 9 3> Furthermore, is the line-center optical depth from cell j to the boundary of cell i. The applicability of this method is dependent upon having efficient techniques for evaluating Pe. For this reason we adopted the following approximate expressions Pe(T0) (1 + 1.861607T0 + 0.817393TO2)-X T 0 < 3 v ~ (3.94) 0.286(ToV/ln(1.95To))- 1 T 0 > 3 Chapter 3. Numerical Simulations 53 which were calculated assuming doppler profiles and are accurate'10'''! to better than 7% (at high densities line broadening due to lifetime of the upper state may make a Voigt profile calculation necessary). In calculating C:i for cells of finite width, P e must be averaged over the cell emitting the photons. This is especially important when the emitting and absorbing cells are adjacent since r can vary significantly from the front to the back of the emitting cell. This is easily accomplished by analytically integrating the expressions for Pe across the emitting cell j so that 1 / + _ y-T + T , + A T _ \ C * = 9 A ( / P ^ D T - / P ^ D T • ( 3 - 9 5 ) 2 Tj \JT}, JT^+T, J The factor of 1/2 in the above expression accounts for the assumed isotropic photon emission. As detailed above this transport model does not calculate the actual opacity-broadened line shape. If the actual line shape is required one has to post-process the results using a more sophisticated multi-frequencey ray trace mode l ' 1 0 8 ' . Nevertheless, its simplicity and economy make it suitable for the use in coupled radiation-hydrodynamic problems where accurate energy transport is necessary. To illustrate the effects of photo-excitation on level population and line emission we consider a uniform 1.5 mm thick plasma slab at a temperature of 600 eV and a total density of 10 1 9 ions per cm 3 . For this problem we only consider excited states of A l + n and A l + 1 2 which are dominant for these plasma conditions. In figure 3.11 we present the level densities as a function of position for helium-like and hydrogen-like aluminum. Solutions for both the optically thin and optically thick calculations are shown. The most obvious feature is that the excited state densities are raised well above their optically thin values by photon pumping. Also evident are the spatial gradients in state population despite a uniform plasma. The gradients are more significant in the excited state which Chapter 3. Numerical Simulations 54 A l + n (//e-like) Line Optically Thin Optically Thick l 5 2 - ls2p 3P 3.08 x 10 1 0 3.25 x 10 1 0 ls2p IP 8.90 x 10 1 1 5.50 x 1 0 n l s 2 - ls3p IP 1.29 x 10 1 1 9.07 x 10 1 0 l s 2 - IsAp lP 4.40 x 10 1 0 2.64 x 10 1 0 A1+1 2 ( / / - l ike) Is -2p 2.28 x 1 0 n 2.43 x 10 1 1 Is -3p 2.86 x 10 1 0 3.14 x 10 1 0 Is — Ap 9.57 x 109 9.72 x 109 Is — bp 4.00 x 109 3.87 x 109 Table 3.1: Emitted line radiation (W/cm 3 ) . are controlled by radiative pumping and are a result of reduced pumping radiation near the boundary. It is interesting to note, however, that the line emission from the plasma can be insensitive to this pumping (Table 3.1). For instance, the line center opacity of the A l + 1 1 l 5 2 — 1S2JD \P line is ~ 50 yet it is effectively thin since the emitted power is only attenuated by 30% from its optically thin value. The reason for this of course is that line emission can propagate through the plasma through a series of absorptions and subequent re-emission. This mechanism allows photo-excitation to be neglected without adversely affecting the energy transport calculation. It is important to note, however, that the line shape can be significantly different from that of the equivalent optically thin line. " Line photons are also absorbed by non-resonant processes. These mechanisms, which include inverse-bremsstrahlung and inner shell absorption, were discussed in chapter 2 and must be included in the transport calculation. In fact these absorption mechanisms dominate in the cold compressed target. The method used to include this in our radiation transport is now outlined. The optical depth in each cell due to non-resonant absorption Chapter 3. Numerical Simulations 55 i o 1 S J 8 >_ 10 4 | 1016 UJ UJ 10 03 10 ,13 io ,w-» (a) 1 c 9 r 1 P -750 -500 -250 0 250 POSITION [microns] 500 750 10 .19 ? 10 E s_ 10 g 1016 UJ Q is UJ 1016 CO 10 14 10 .13 (b) -750 • -500 -250 0 250 POSITION [microns] 500 750 Figure 3.11: Excited and ground state densities of a) Helium-like aluminum ( A l + n ) b) Hydrogen-like aluminum (Al + 1 2 ) ; solid line optically thin calculation, dashed line optically thick calculation . Chapter 3. Numerical Simulations 56 at a photon energy hv0 is given by . A T - = (Kff(v0) + K f > 0 ) ) A x t (3.96) where expressions for KJJ and Kin are given in section 2.2.3. In this equation we consider the absorption coefficients to be constant over the line profile. This approximation is justified considering that the absorption coefficients are at most proportional to (hv)~3. If we neglect photo-excitation then the number of photons absorbed in cell j due to line emission from cell i can be evaluated using equation 2.69. The resulting expession is i / A r c n A r c n \ it* = + -j-) - E ^ + ~ir+Arr) (3-97) where the factor of 1 /2 accounts for the assumed equal probability of photon emission in either direction from cell i. If we define the continuum escape probability as A r c n P - ( r ) = E2(T* + -f- + r) (3.98) then we have that N? = \N?Aul (P,?(0)P^(0) - P - ( A r r ) P ^ ( A r O ) (3.99) when all absorption processes are included. The coupling coefficient Cji, which only considers resonant absorptions, is then approximated by the expression A PPE C = N? — (3.100) 3 APPe + A P c n where ApPe = p j 5 ( 0 ) - P / / (Ar , ) (3.101) A P c n = P^(0) - P ^ ( A r / n ) . (3.102) Chapter 3. Numerical Simulations 57 Consequently, the number of photons absorbed due to all other processes is A P C N APPe + APm J V f = M — - 7 — . (3.103) In our transport calculation this simply represents an absorbed energy of El£ = hvoNg'. (3.104) This assumption is justified since most inner-shell vacancies are filled by Auger transi-tions. The excess energy carried off by the ionized electron in this process is quickly thermalized through collisions. 3.4 Continuum transport Bremsstrahlung and recombination radiation are transported using a multi-group ap-proach. The photon energy range of interest is divided into a number of bins. The bin boundaries are selected such that the opacity and emitted spectrum within a bin are approximately constant. This allows us to transport the energy within each bin in a way analogous to that used for line transport. Specifically, for bin k extending from to ^T(fc) = \UQ**i [E2(T^ + ^ f.) - E2(T™ + + AT-) (3.105) where /,(£) is given by li(k) = P + 1 dv (I„(v) + J/6c») (3.106) (refer to equations 2.29 and 2.37). 3.5 Hydrodynamic code including radiation transport The hydrodynamic code which was modified to incorporate our CRE model and radiation transport has been described extensively in previous work'^ 9 ' . Therefore, we will only Chapter 3. Numerical Simulations 58 present a brief description emphasizing our changes. The one-dimensional hydrocode HYRAD solves in a Lagrangean formalism the dif-ferential equations describing a single fluid, two temperature plasma. In this model, electrons and ions are assumed to have the same fluid velocity, implying no charge sepa-ration. 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. In the Lagrangean formalism the calculation follows the time evolution of individual fluid elements. Moreover, the computational mesh evolves during the course of the cal-culation and extends between the target boundaries, i?t(t) and R0(t) (laser illuminates the boundary R0 > RA. A plot of the Lagrangean mesh as it evolves during the course of a typical calculation is shown in figure 3.12. Each line represents a fluid element (mesh-point) and is fixed to the local fluid reference frame. The important regions characteristic of laser plasma interactions are labelled in this figure and will be refered to in subsequent discussions. The time, t, and Lagrangean coordinate, m, are regarded as the independent vari-ables rather than t and space coordinate (R) as used in an Eulerian description. The Lagrangean coordinate m is defined in terms of the density profile in laboratory coordi-nates, r and t, by, m(r,t) = [ p(r',t)dr' [4 - 1] where i?t(2) is the position of the left hand (inner) free surface of the target and p(r,t) is the density profile. In terms of t and m the differential equations which represent the conservation of mass, momentum and energy are: dV du « = 0 ( 3 - 1 0 7 ) dt dm v ' OI>*pter3 JU, 59 Chapter 3. Numerical Simulations 60 (3.111) The principal dependent variables that are solved for are specific volume, V — 1/p, fluid velocity, u, and the material internal energies, Ee and Ei, for the electrons and ions respectively. A number of secondary quantities are required to complete the description, and these can be obtained once the principal dependent variables are defined. Most important is the equation of state, which is expressed in the form, Te = Te(V,Ee), Pe = Pe(V,Ee) (3.112) Ti = Ti(V,Ei), Pi = Pi(V,Ei) (3.113) where jTe, 2, and Pe, Pi are the electron and ion temperatures and pressure components respectively. Other important quantities include the ion density, . ni = p/Amp, (3.114) where A is the average atomic mass number of the target material, and mp is the proton mass; the electron density and ionization state, ne =< Z > rii (3.115) and, the Coulomb logarithm'*'''2' In A = 16.34 - k(r e 3 / 2n7 1 / 2£ _ 1)- (3-116) Although the average ionization < Z > can be obtained from the appropriate S E S A M E data table, for our calculations it was determined by the C R E model previously described. Chapter 3. Numerical Simulations 61 The energy source terms which enters into the conservation of energy equations are given by Qi = H,-KEI (3.117) QE = HE + KEI + Alas + XRAD (3.118) where H represents the heat flow due to thermal conduction, KE{ is the rate of energy transfer between the ions and electrons, Aias is the rate of absorption of laser light and Xrad accounts for radiation transport. The heat source term is given by H=-V-KVT (3.119) P where K is the thermal conductivity. The electron thermal conductivity is obtained from the S E S A M E data library whereas for the ion thermal conductivity, the Spitzer ' 1 1 ^ expression was used. The electron-ion energy exchange term is given by[U2] KEI = 0.59 x 10- 8 n e(r t - TE)T3L2M~L < Z2 > In A (3.120) where M is the atomic mass number. Absorption of the laser energy is assumed to occur via inverse bremsstrahlung up to the critical density. The absorbed power is given by XaUr,t) = ~?ir- (3-121) p or where the local laser intensity is related to the laser intensity at the plasma boundary, r = R0, by *L(r,<)= $L(R0,t)exp(-a(R0-r)). (3.122) The absorption coefficient, a, is that derived by J.R. Stallcop and K . W . Bil lmant 1 1 1 ! . Anomalous absorption is modelled by depositing a legislated fraction of the laser energy Chapter 3. Numerical Simulations 62 reaching the critical density surface into the first overdense zone. This rather crude approach is adequate in the regime of laser intensites of interest here since anomalous absorption accounts for less than one percent^"7' of the total absorption. The radiation source term XTai for each cell is calculated in the radiation transport calculation. Specifically, XUt) = -T,{E^ + E^) (3.123) r l where El£ and E™ account for line and continuum x-ray depostion and are evaluated using equations 3.104 and 3.105 respectively. Solution of the equations is accomplished in two phases for each timestep. In the first phase the hydrodynamic motion of the mesh is calculated by advancing the coordinates, r, and velocities, u, using the piecewise parabolic method developed by Colella and Woodward' 4 4 ' 4 5 ' . This defines the variables r (hence V and p), and u at the advanced time level. In the second phase the energy equation is solved implicitly for Ee, Et, and iterated to convergence. Iteration is necessary to account for the nonlinear dependences of the source terms and equation of state on the thermodynamic variables. Usually 1 to 3 iterations are sufficient for convergence to accuracies of the order of 1%. In developing this model several mechanisms were ignored. Hot-electron generation and transport, which for our laser conditions have been measured'^ 4 ' to account for less than 1% of the incident laser energy, were neglected. The numerous parametric processes which can develop in the corona are only indirectly considered. Specifically, the experimentally measured absorbed laser intensity (rather than the incident laser intensity) is used in the simulations. In this way the energy which is lost to these other processes is accounted for. Recent improvements to electron thermal transport algorithms'6', which accurately account for electron thermal flux inhibition have not been incorporated. Fortunately, Chapter 3. Numerical Simulations 63 however, for our experimental conditions the laser intensity and the plasma temperature gradient scale lengths are well within the classical heat transport regime. 3.5.1 Standard Calculation A l l the simulations discussed in this thesis share a number of common features. Specif-ically, for all the calculations a mesh of 100-150 cells was used. The mesh was finely zoned near the surface where the laser light is absorbed, and more coarsely zoned deeper in the target where the compression takes place. This was done by dividing the mesh into two regions as shown in figure 3.13. Region 1 at the target surface was zoned into 20-40 cells of uniform mass which matched the first cell of region 2. In region 2 the ratio of adjacent cell masses was set to a value greater than 0.85 (depending on target thickness) to provide a smooth increase in cell mass. The equation of state and electron thermal conductivity were obtained from the S E S A M E data library! 4 6]. In many of the runs the one-temperature version of the model was used because of its better stability. In the case of aluminum this was justified based on a comparison of results obtained using the one-temperature and two-temperature ver-sions of the model. This is expected since the electron and ion temperatures only differ significantly in the low density corona, where for high Z targets, the electrons account for most of the thermal energy. As a consequence, the error in transport coefficients resulting from the use of the plasma temperature rather than the electron temperature is negligible. The average ionization and radiation emission for aluminum were calculated using the C R E model previously described. Furthermore, in the standard calculation photo-excitation and photo-ionization were neglected. Although this reduced the accuracy of the emission spectrum, the effect on energy transport and the temperature and density of the compressed target was negligible. For the purpose of radiation transport the continuum spectrum was divided Chapte r 3. Numerical Simulations 64 R E G I O N 2 R E G I O N 1 50-150 C E L L S 20-50 C E L L S LASER 0 Figure 3.13: Mesh zoning scheme used in HYRAD calculations. The laser is incident from the right. Chapter 3. Numerical Simulations 65 Ionization K-sheU Z-shell M-shell Stage energy (eV) energy (eV) energy (eV) A l + 0 1549.94 80.709 4.874 A I + 1 1559.50 89.808 17.987 A1+2 1572.32 102.92 28.623 A1+3 1588.42 117.60 A1+4 1634.22 190.29 A1+5 1684.60 191.53 A1+6 1739.45 266.17 A1+7 1798.56 276.63 A1+8 1861.94 322.33 A1+9 1929.05 394.91 A1+10 1991.66 443.80 A l + n 2073.50 A 1 + 1 2 2304.58 Table 3.2: Absorption edge energy for various ionization stages of aluminum. into 30 energy bins extending from 100 eV to 10000 eV. The inner-shell photoionization cross section for neutral aluminum is taken from the results of Biggs and Lighthill '^ 5 ' and is shown in figure 3.14. The positions of the various absorption edges were obtained from the tabulated results of Henke et a l J ^ 6 ' and are summarized in table 3.2. 3.5.2 Testing of HYRAD results The accuracy of HYRAD without radiation transport has been checked extensively in previous work'^^. For example, in figure 3.15 we present the HYRAD solution of the classical Riemann shock tube problem. In this test a sharp discontinuity separates two regions of material at different initial pressures. At t=0 the material is allowed to move thus simulating the rupture of the diaphram in shock tube problems. It is evident from the results that the details of the rarefaction wave and shock front are accurately modelled by HYRAD. The code predictions of various scaling laws in the laser-driven Chapter 3. Numerical Simulations 66 10 -f i i i i i i i 11 i i i i i i i 11 i i i i i i i i T 10 100 1000 10000 ENERGY [eV] Figure 3.14: Inner-shell photoionization cross section for aluminum at normal density and temperature. Chapter 3. Numerical Simulations 67 RAREFACTION CONTACT SHOCK FAN SURFACE I x (jjm) Figure 3.15: Numerical solution of a Riemann shock tube problem. Solid curves are exact solutions, symbols are HYRAD calculations. Chapter 3. Numerical Simulations 68 ablation process were also found to be in good agreement with the results of analytical models ' 4 8 , ^ ' 7 ' and other laser-target interaction co des ' 1 1 8 ' . What now remains to be investigated is the validity of H Y R A D when radiation transport is considered. Single layer targets The effects of radiation transport are illustrated in figure 3.16 which shows a snapshot of the profiles of various hydrodynamic and thermodynamic variables calculated with and without radiation transport. The simulation corresponded to a 25 pm aluminum foil irradiated at a peak laser intensity of $L = 2.3 x 10 1 3 W / c m 2 with a 0.53 pm laser. The laser pulse was taken to be gaussian with a duration of 2.3 ns (FWHM ) . These irradiation conditions (which we will refer to as our standard irradiation) were chosen because they correspond to the experimental parameters discussed in the following chapter. Radiation transport has several distinct effects on the target. First of all, hard x-rays (energy > 1 keV) propagate deep into the target and preheat the material ahead of the shock wave. At these modest laser intensitites this effect is negligible beyond ~ 10 pm in aluminum. Meanwhile, soft (Sub-keV) x-rays are transported into the ablation zone and help in the ablation process. However, electron thermal conduction is still the dominant energy transport from the coronal plasma into the ablation zone. As a result, radiation emission is predominantly an energy loss mechanism which leads to a slight reduction in the ablation pressure. At higher intensities, however, when electron thermal transport is inhibited, soft x-ray transport may be more efficient in driving the ablation process. A more significant effect of radiation transport, which is evident in figure 3.16, is the heating of the shock compressed material. In fact the temperature profile behind the shock front is more uniform when radiation transport is included. To understand this we must recall that as the ablation pressure increases a compression wave of increasing amplitude is driven into the target. The wavefront steepens into a single shock as it Chapter 3. Numerical Simulations 69 LASER 20 15 10 5 0 LAGRANGE AN COORD, (jum) Figure 3.16: Calculated profiles for 25 pm aluminum foil irradiated at $ L = 2.3 x 10 1 3 W/cm 2 with a 0.53 pm and 2.3 ns (FWHM) gaussian laser pulse, a) without radiation b) with radiation. Chapter 3. Numerical Simulations 70 propagates, which continues to strengthen as the laser intensity increases. As a conse-quence, material deep inside the target is compressed primarily by a single strong shock. This is important because the degree of shock heating increases with shock strength. On the other hand, material near the ablation front is compressed almost isentropically by a sequence of weak shock waves. At similar material pressures this results in higher densities and lower temperature. This accounts for the temperature gradient in the com-pressed material when radiation transport is neglected in the calculation. However, when radiation transport is included the temperature gradient is significantly reduced. This is a result of decreasing x-ray energy deposition at increasing depth in the target giving rise to greater heating near the target front side. A unique feature which is characteristic of radiation transport is the generation of a relatively uniform temperature region ahead of the ablation front. There are two basic mechanisms which account for this structure. First it is the result of a nonlinear radiation heat wave driven by the diffusion of radiation into the cold material. Simplified descriptions in the form of self-similar solutions have been given by M a r s h a l and, more recently, by Pakula and Sigel^ 2 ^' 121] This wave structure is similar to that observed for the ablation front. Another mechanism which helps in the generation of the radiatively heated zone (RHZ) is the shift of the K-sheW absorption edge as the material is ionized. Specifically, as material at the leading edge of the wave is heated and ionized the absorption edge shifts to higher energy hence reducing the opacity to x-rays below the edge (refer to section 2.2.4). This leads to an ionization burn wave which propagates through the target. This mechanism can be particularly important in the case of aluminum where the A l + n resonance line (1598 eV), which can account for as much as 10% of the radiation, lies above the 1560 eV edge of aluminum at normal density but below the edge of A l + 4 (1634 eV). This line, therefore, is strongly attenuated until the aluminum plasma is ionized to at least A l + 4 . An interesting feature of this Chapter 3. Numerical Simulations 71 mechanism is that it is self-regulating. That is, if the shift in edge position significantly reduces the energy deposition then the material temperature may decrease. This in turn lowers the ionization thus reducing the shift in the edge position. This mechanism combined with soft x-ray diffusion accounts for the nearly uniform temperature in the radiatively heated zone. Since the hydrodynamics tend to maintain a relatively uniform pressure, the constant temperature in the RHZ region implies a similar constant density profile. In figure 3.17 the temporal evolution of this RHZ region is clearly evident in the temperature profile. Multi-layered targets In recent years the use of multi-layered targets in laser-driven shock wave experiments has increased significantly. There are two primary reasons for this increased popularity. Firstly, low Z materials can be used as the ablator to reduce the x-ray production and resulting preheat. Secondly, with any given laser pulse the induced shock pressure can be greatly increased by stacking low and high dynamic impedance materials (impedance-mismatched tagets). Such pressure enhancement was first demonstrated by Vesser et a l . [ 1 3 ] for 1.06 pm laser radiation and aluminum-gold targets, and Cottet et a l . ' 3 4 ' for 0.26 ixm radiation and aluminum-gold targets. Motivated by these experiments as well as a series of experiments recently performed by our group on similar targets we decided to investigate the effects of radiation transport on these targets. In figure 3.18 we present the thermodynamic conditions at two times during the sim-ulation of a multi-layered target, consisting of a 19 /zm aluminum on. 13 pm gold, irradi-ated under standard conditions ( $ L = 2.3 x 10 1 3 W / c m 2 , \L = 0.53pm, TL = 2.3 ns). A n interesting feature is evident at the A l - A u interface prior to shock arrival. Although the 19 pm layer of A l significantly reduces the x-ray transmission, the increased opacity of the Au leads to the remaining x-rays being absorbed in a very narrow region. This in Chapter 3. Numerical Simulations 72 Figure 3.17: Temporal evolution of (a) temperature, and (b) density throughout the target. HYRAD simulation including radiation transport. Laser is incident on boundary at position 0. Chapter 3. Numerical Simulations 73 Figure 3.18: Thermodynamic profiles for 19 pm Al on 13 pm Au multi-layered target, a) Before shock reaches interface, b) After shock reaches interface. Chapter 3. Numerical Simulations 74 turn results in a significant, increase in temperature and pressure of this thin gold layer. Consequently, a weak compression wave propagates away from the interface. Using an in-terferometer in conjunction with a streak camera it may be possible to see evidence of this wave by looking for a displacement of the free gold surface. In order to do this, however, the thickness of the gold layer would have to be reduced to eliminate the possibility of this compression wave being overtaken by the shock wave. For instance, in figure 3.18(b) this overtake has already occurred. Also evident in this figure is the enhancement of the shock pressure at the interface. When the initial shock reaches the interface a shock wave is transmitted into the Au layer and a second shock wave is reflected back into the A l . This backreflected wave propagates into material which has already been shock com-pressed, thus further increasing its pressure. This accounts for the pressure enhancement at the interface which then drives a high pressure shock into the Au layer. For these target and laser conditions the pressure upon reflection increases from ~ 3.5 Mbar to ~ 6.8 Mbar. In order to achieve comparable pressures in a single-layer gold target the laser intensity would have to be approximately twice as high. This would result in higher x-ray radiation and greater preheat. In principle, one could further reduce the x-ray effects by increasing the thickness of the A l layer. Unfortunately when the aluminum layer becomes too thick, shock decay due to decreasing laser intensity reduces the pressure of the shock before it reaches the interface. Ultimately, in multi-layered target design it is the competing objectives of reduced x-ray preheat and maximum shock pressure which determine the thickness and composition of the various layers. Chapter 3. Numerical Simulations 75 3.6 Two-dimensional hydrocode In the detailed analysis of our experimental results it was sometimes necessary to assess the contributions of two dimensional effects. To do so we used a modified version of a two-dimensional hydrocode (SHYLAC2) developed by Marty and Cottet l l 2 2 ] . Briefly, SHYLAC2 is a two dimensional Lagrangean code which solves the fluid equations but neglects any thermal transport or laser absorption. Our modifications consisted simply of incorporating the SESAME EOS into the code. The original version of SHYLAC2 used a Mie Gruneisen equation of state which is ac-curate over a limited parameter space. Moreover, extracting material temperatures from this EOS was difficult. Instead of modelling the laser ablation process, SHYLAC2 assumes that a pressure pulse is applied at the target boundary. The validity of such an approximation in predict-ing the compressed target conditions in one dimensional simulations has been previously demonstrated'36^. For our purposes the temporal evolution of the assumed pressure pulse can be determined from experimental measurements of the ablation pressure scaling with laser intensity. Furthermore, the spatial profile can be approximated by using the inten-sity distribution of the laser focal spot and the pressure-intensity scaling relationship. An example of the results obtained for a gaussian (temporally 2.3 ns FWHM and spatially 80 pm FWHM) presure pulse is shown in figure 3.19. The grid represents the compu-tational grid with each cell representing a fluid element. The non-planar shock front is clearly visible. The inability of this code to incorporate the laser driven ablation process has other obvious problems. First of all, no account is made of the evolution of the coronal plasma and ablation front. This can modify the effective irradiation area and in turn make the Chapter 3. Numerical Simulations Figure 3.19: Two-dimensional shock propagation simulation for 25 pm aluminum foil. Chapter 3. Numerical Simulations 77 simple intensity scaling of the ablation pressure inaccurate. Furthermore, in neglect-ing thermal transport, the effects of thermal smoothing on the pressure profile are not modelled. Chapter 4 Experimental Facility A description of the laser facility and the x-ray streak camera used in the experiments is given in this chapter. The target irradiation conditions are also discussed. 4.1 Laser facility A schematic of the laser facility is shown in figure 4.20. It consists of a Quantel solid-state laser chain, second harmonic conversion crystal (potassium dihydrogen phosphate, K D P ) , beam steering optics, target chamber and associated beam monitoring diagnostics to record the incident pulse shape and energy. The laser chain is made up of a Nd-YAG (neodymium-yttrium aluminum garnet) oscillator and preamplifier followed by four Nd-glass amplifiers. Three vacuum spatial filters are also present in the chain to improve beam quality. The oscillator is passively Q-switched with a dye cell and produces a gaussian pulse of 2.5 ns F W H M (nominal) at a wavelength of 1.064 /zm in a TEMoo spatial mode. The output beam energy could be varied up to 15 J (by varying the amplifier pumping). The 1.064 /xm fundamental wavelength was converted to 0.532 /zm with a K D P type II conversion crystal. Output energy at the 0.532 /zm wavelength reached 7.0 J . The laser beam was steered into the target chamber by a series of 3 dichroic mirrors, and was focussed onto target with a 300 mm (nominal focal length), f/6 lens. 78' Chapter 4. Experimental Facility 79 SHG = SECOND HARMONIC GENERATOR Figure 4.20: Schematic of the laser used in the experiments, and a typical laser pulse. Chapter 4. Experimental Facility 80 4.2 Irradiation conditions The laser focal spot at the target plane was measured for a range of lens positions in order to determine the position for optimum irradiation uniformity. This was done by imaging the laser spot with an f/4 achromat onto an optical streak camera''''2 3'. The streak camera was operated in focus (non-streak) mode, and hence was functioning essentially as a video camera to record the intensity distribution. The laser was fired at full energy and attenuated by using a dichroic mirror (reflecting ~ 99%) and neutral density (ND) filters placed in the beam path ahead of the main focussing lens. The overall magnification of the imaging system was found by placing a grid of known spacings in the target plane and comparing it to the streak camera image. Figure 4.21 shows the time integrated intensity distribution of the laser focal spot. The spatial resolution in this measurement is ~ 3 pm . It shows the laser spot to be confined in a region of approximately 90 pm diameter, with intensity modulations less than 30%. In figure 4.22 the intensity distribution averaged over a 10 /zm wide region across a diameter of the laser spot in the X and Y direction is presented. In simulating the experiments with a one-dimensional hydrocode it is necessary to determine the effective laser intensity. This is done by calculating the average intensity within a region characteristic of the shock breakout region. The shock breakout region, which is defined as the area of the target rear surface over which the shock breaks out within ~ 200 ps, was measured to be ~ 80 /zm in diameter. Numerical integration of the focal spot distribution indicates that ~ 80% of the laser energy is contained within the shock breakout area. Therefore the effective intensity $L corresponding to the area of shock breakout is given by, i . - . < - , Elaser $L = 0.8r)abs—— Chapter 4. Experimental Facility 81 Figure 4.21: Time integrated intensity distribution. Chapter 4. Experimental Facility 82 Figure 4.22: Time integrated intensity profile averaged over the central 10 pm in the a) X-direction and b) Y-direction (Refer to previous figure). Chapter 4. Experimental Facility 83 Here TL is the full width at half maximum of the laser pulse shape, A80 is the shock break-out area which contains 8 0 % of the total laser energy, E i a s e r , and rjabs is the absorption fraction. This empirical formula for $L has been successfully used to specify the effec-tive intensity in one-dimensional hydrocode simulations of laser-generated shock waves in aluminum targets' 1 7 , 1 2 4 ] r p j ^ p r e { j j c ^ e ( j ablation pressures and shock trajectories were in good agreement with those measured thus providing justification for the use of this formula. For our experiments E l a s e r = 4 . 0 ± 0 . 2 J , rL - 2 . 3 ± 0 . 1 ns , nabs = 9 0 % ' 1 1 4 , 1 2 5 ] , the average absorbed irradiance is therefore 2.3 ± 0.2 x 1 0 1 3 W / c m 2 . 4.3 X-ray Streak Camera A streak camera is a device which converts the time history of a light (or x-ray) signal to a variation in intensity with position in a streak image. The signal to be measured strikes an entrance slit which forms a photocathode. The incident photons are thus converted to electrons which are then accelerated towards a phospor. Deflection plates in front of the phosphor sweep the electron image of the slit onto the phosphor. This gives an image which can be recorded on film with one cartesian axis representing time and the other cartesian axis space. 4.3.1 General Design The primary diagnostic device in these experiments was an x-ray streak camera. The unit we built is based on a design developed by Paul Jaanimagi of the University of Rochester' 1 2 6^ in which the front end electron optics of an R C A 73437 image tube' ' were replaced with a new design. This camera offered several advantages over what was commercially available at the time, specifically, ~ 20 picosecond resolution and a 40 mm long photocathode. Chapter 4. Experimental Facility 84 PHOSPHOR PLATE FOCUSSING ELECTRODE FOCUS/SLOT MACOR SPACER SLOT EXTENSION CURVED SLOT SLDT/CATHODE MACOR SPACER CURVED CATHODE ORIGINAL RCA BACK END MODIFIED FRONT END Figure 4.23: Cross-sectional view of x-ray streak camera. Chapter 4. Experimental Facility 85 Figure 4.23 shows a cross-sectional view of the complete streak camera assembly. The cathode and accelerator electrodes form a spherical diode of radii 10.1 cm and 10.0 cm respectively . The entrance slit, 1 mm wide by 40 mm long is cut directly into the removable cathode plate and positioned on the tube axis directly opposite the 2 mm x 64 mm slot in the accelerator plate. This front end electrode configuration has many desirable features. The curvature of the photocathode is a major factor in correcting the iso-chromism problem of large aperture systems. Iso-chromism is defined as the difference in transit time from cathode to phosphor between on-axis and off-axis trajectories. The more conventional approach to reducing iso-chromism is to limit the usable length of entrance slits to a maximum of 20 mm. Unfortunately, for x-ray streak cameras where the input signal cannot be demagnified to fit the photocathode dimensions in a simple manner this can be a significant limitation. Another important advantage of a curved cathode is that the imaging quality along the slit is improved due to the reduction in spherical aberrations. The slot aperture rather than a fine mesh is used in the accelerator for a number of reasons. First of all, the slot aperture is mechanicaly more rigid. Moreover, photoelectron transmission is 100% compared with 40% - 70% for meshes. The noise ghost image caused by secondary electron emission generated by x-rays and photoelectrons impinging on the accelerating electrode surfaces is also eliminated. A further benefit of the slot aperture is the fact that the magnifications parallel and perpendicular to the photocathode slit are different. Perpendicular to the slit the image is demagnified by a factor of four to eight (depending on operating voltages), thereby allowing the use of very wide (1 mm) input slits without comprimising the time resolution. For our purposes, this was extremely important since it yielded an order of magnitude increase in signal compared to the more common 100 pm slit widths. The rear section of the streak tube, comprising of the deflection plates and phosphor, is Chapter 4. Experimental Facility 86 that of a R C A 73435 image tube. The fact that the phosphor plate has a fibre optic output allowed the use of a fibre optic taper (50 mm diameter tapering to 40 mm diameter) to couple the phosphor to the image intensifier. Fibre optic rather than lens coupling was chosen in order to maximize light collection. The image intensifier is an ITT F4113 (40 mm diameter) which offers fibre optic input and output coupling. Moreover, it has a maximum luminous gain of 4000 with 9% uniformity and spatial resolution of 25 line pairs per mm. Measurements of the spatial resolution for the complete unit indicate that the spatial resolution is in fact limited by the intensifier. Polaroid as well as Tri-X film were put in contact with the intensifier to record the signal. The deflection plates used in sweeping the electron beam across the phosphor are driven by a double stack of 2N5551 avalanche transistors. The schematic of the ramp design is given in figure 4.24 along with the calculated deflection pulse. The deflection pulse for different component values was calculated by the computer program S P I C E [ 1 2 9 ] . The method used was suggested by Thomas et a l . ^ 2 7 ' . The streak camera unit is mounted in a stainless steel housing which attaches to one of the target chamber ports. In figure 4.25 we show a cross-sectional view of the complete assembly. A primary objective in the design was to have the photocathode as close to the target as possible. Although the photocathode slit is positioned on a horizontal plane at the laser beam height, the streak camera axis is oriented at 5 degrees to the horizontal. The reason for this is to prevent any particles which pass through the photocathode slit from striking the phosphor. The anode aperture, positioned 12.5 cm from the photo-cathode and 5 mm in diameter, determines the minimum angle. Unfortunately, this does not eliminate the problem of secondary electrons being generated by particles hitting the inside walls of the streak camera. There was evidence that some of the background noise observed was caused by this. In its present housing, the streak camera is open to the target chamber, therefore, it Figure 4.24: Schematic of deflection plate driver and deflection pulse predicted by SPICE for final version of ramps. Figure 4.25: Cross-sectional view of streak camera assembly. (A) Film camera, (B) Intensifier, (C) Intensifier mount, (D) Taper mount, (E) Fibre optic taper, (F) Mounting flange, (G) Streak camera housing, (H) Streak Camera, and (I) Mounting collar . Chapter 4. Experimental Facility 89 was necessary to pump the chamber down to an operating pressure of less than 8 x 10 - 6 Torr before using the streak camera. Typically, the streak camera was operated at a pressure of 4 x 1(T6 Torr. These low pressures are necessary to eliminate arcing and reduce electron-ion collisions within the streak camera which can lead to pinching of the electron beam. 4.3.2 Photocathodes As mentioned in the previous section the cathode plate is easily removed from the streak camera. This allows quick exchange of the photocathode since the photocathode sub-strate mounts directly on the concave surface of the cathode plate. The photocathodes used in this work were made by evaporating thin layers of photocathode material on a substrate which was then mounted on the cathode plate with a conductive p a i n t ^ 3 t \ In the preliminary test of the streak camera an ultra-violet (UV) sensitive photocathode was used. This consisted of a 30 //m mica substrate (~ 30% transmission at 0.26 pm ) coated with 200A of aluminum. Although gold photocathodes have been used extensively in x-ray measurements, we chose to use cesium iodide (Csl) because of its greater sensitivity. In fact at a photon energy of 1500 eV the electron yield per photon is 7 for 3000A of Csl compared to 2 for 300A of gold ' 1 3 1 ] . The drawback of using Csl is that it deteriorates faster, especially if it is exposed to moisture. Two substrates were tested for use with Csl . The first was a polycarbonate f o i l ^ 3 2 ! (B-10) coated on both sides with a 2000A layer of aluminum. The advantage of this substrate is that it has excellent transmission (~ 95%) in the 1540 eV to 1560 eV photon energy range. Unfortunately, it is extremely delicate, making it difficult to mount smoothly on the cathode plate. The second substrate used was 12.5 pm berrylium; it has a lower transmission 60%) but it is considerably easier to mount. Chapter 4. Experimental Facility 90 4.3.3 Streak camera test In this section we discuss two tests performed to prepare the streak camera for operation. The first was made with the streak camera in static mode. In this mode the deflection plates are both grounded and the intensifier gated for several milliseconds at a rate of 50 Hz. We then performed a dynamic test of the complete system to measure the true sweep speed. Imaging test The first test of the streak camera was to evaluate its imaging characteristics in static mode. To do this an ultraviolet mercury l a m p ' 1 3 3 ' was placed in front of the 1 mm by 40 mm entrance slit of the camera. The operating voltages were then adjusted until an optimum image could be seen on the phosphor. The results presented in figure 4.26 show a clear image of the slit with the best imaging quality at the center. The position of the best image can be easily moved to the extremes by simply adjusting the operating voltages. Furthermore, the imaging quality can be improved in the future by replacing the flat phosphor with a curved phosphor which matches the focussing characteristics of the streak camera. Also evident in the image is the demagnification of the slit width by approximately a factor of four as previously described. 4.3.4 Sweep speed measurement The final step in making the streak camera operational was to measure the sweep lin-earity, specifically, to determine the relation between time and position on the streak record. Ideally the sweep speed should be linear and dt/dx — constant. However, for the comparatively slow sweep speeds required this was difficult to achieve. A nitrogen laser was used to generate a 700 ps (FWHM) pulse at a wavelength of Chapter 4. Experimental Facility 91 Figure 4.26: Static mode image of the entrance slit. Figure 4.27: Setup used to measure sweep speed. Streak record shows the two nitrogen laser pulse with 0.5ns separation. Chapter 4. Experimental Facility 93 Figure 4.28: Sweep linearity plot, crosses are measured points, solid line is best fit second order polynomial. Chapter 4. Experimental Facility 94 0.337 pm . The beam was then bisected and a fixed time delay inserted. A small fraction of the main pulse was also used to trigger the streak camera voltage ramps. The complete setup is presented in figure 4.27 along with a streak record showing the two pulses. By changing the delay between the streak camera trigger and laser pulse, the position of the two pulses on the streak record could be changed. Figure 4.28 shows a plot of the measured dt/dx as a function of the distance from the top of the intensifier. A second order polynomial fit to the data is also presented. The bias and ramp voltages applied to the ramps were adjusted so that the sweep was most linear near the center. Integration of the polynomial leads to the relation t = 0.24x + 0.0052a;2 + 1.0 x 10~V where x is the position measured from the top of the intensifier and t is the elapsed time. In quantitative anlysis of streak records it is important to correct for the effect of sweep non-linearity in the signal intensity, 1 signal, that is, dx In Table 4.3 we summarize the operating voltages determined to be optimum for our requirements. Clearly, the bias and ramp voltages will have to be adjusted if a different sweep speed is required. The maximum sweep speed achievable was found to be approximately 0.25 ns per centimeter (on phosphor). We operated at approximately 1.2 ns per centimeter. If slower sweep speeds are desired it may be necessary to modify the ramp circuit. Chapter 4. Experimental Facility 95 Operating voltages Cathode Slot Focus Bias Ramps -19.88 -17.95 -17.15 ± 0.80 ± 2.10 Table 4.3: Streak Camera Operating voltages (in kilovolts). Chapter 5 K-edge shift in shock compressed aluminum In this chapter we describe the measurement of the shift in the K-shell photoabsorption edge of shock compressed aluminum. First, we will discuss the measurement of the shock trajectory. This was necessary in order to obtain an accurate estimate of the target conditions. The K-edge measurement is then described in detail. A comparison of the experimental and predicted results is then presented. 5.1 Shock trajectory measurements Shock trajectory measurements offer a unique approach for determining the temperature and density of compressed solids for which accurate equations of state are available. For this reason they enable us to assess the validity of the target conditions predicted by H Y R A D by comparing the experimental and calculated shock trajectories. For opaque targets, the arrival of the shock wave at the rear surface is signalled by a sudden onset of luminous emission. The shock trajectory can, therefore, be obtained by measuring the onset time of the emission for targets of various thicknesses. The experimental setup used in this measurement is given in figure 5.29. The target front surface was irradiated by the focussed laser beam at target normal. The rear surface of the target was imaged onto the entrance slit of a Hamamatsu C979 streak camera with f/1.4 optics placed at target normal. The optical system was focussed by imaging a fine wire mesh placed at the object plane corresponding to the target rear surface onto the streak camera. The luminous emission from the target rear surface due 96 Chapter 5. K-edge shift in shock compressed aluminum 97 Figure 5.29: Experimental setup for shock trajectory measurement. Chapter 5. K-edge shift in shock compressed aluminum 98 to shock heating was observed through an 100A bandpass interference filter centered at 4300A. This eliminated any stray laser light from being collected. The shock transit time through the target was determined by comparing the time of shock breakout to the laser fiducial pulse which was simultaneously recorded by the streak camera. 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. The accuracy in fiducial timing is better than 100 ps. Figure 5.30 shows spatially resolved streak records of the emission from the shock breakout region in 38.4 /zm and 53 /zm aluminum targets. Time zero corresponds to the peak of the laser pulse. The shock is observed to emerge at the free surface in a region of ~ 80±10 /zm diameter. Moreover, there was no evidence of any spatial modulation in the breakout region which would indicate the presence of hot spots (localized regions of very high intensities). Figure 5.31 shows the luminous intensity as a function of time, spatially integrated over the central 50 /zm of the breakout region, for a 38.4 /zm aluminum foil. The rise time of the luminous intensity for this shot was approximately 100 ps. The temporal resolution of this diagnostic was limited by the streak camera resolution (~ 40 ps corresponding to a slit width of 200 /zm ), surface roughness of the target (~ 50 ps for < 0.5 /zm roughness and a shock speed of the ~ 1 x 106 cm/s) and the finite rise time of the luminous emission due to the optical depth of the unperturbed foil (~ 20 ps for an optical depth of 2000A). This gives an overall temporal resolution of approximately 70 ps. This would suggest that the observed rise time of the luminous emission is mainly due to the measurement technique. The shock trajectory measured with this technique is shown in figure 5.32. Also shown are the shock trajectory predicted by the H Y R A D simulations for identical irra-diation conditions, namely gaussian laser pulse of 2.3 ns (FWHM) and intensity = 2.3 x 10 1 3 W / c m 2 . Each experimental data point is an average of at least 5 shots with Figure 5.30: Streak records of shock breakout emission and fiducial in (a) 38.4 fim and (b) 53 ixm aluminum target. Chapter 5. K-edge shift in shock compressed aluminum 100 woo •£ 600 500 *5> c £ 400 J= 300 200 TOO 0 900 <& 500 1 400 £ 500 200 100 0 (a) 1 1 \ -2 -1 0 , 1 T I M E ( n s e c s ) 1 • 2 3 •fr . • » "1A M -1 0 1 T I M E ( n s e c s ) Figure 5.31: a) Luminous intensity of the target, rear surface as a function of time for a 38.4 M m thick aluminum foil irradiated at $ L = 2.3 x 1013 W/cm 2 b) Fiducial pulse, the shock transit time for this shot is TS = 1.1ns. Chapter 5. K-edge shift in shock compressed aluminum 101 the error bars representing one standard deviation. It is clear from the results that the shock wave has reached a quasi-steady state after propagating 22 //m into the target and remains steady and one dimensional to a depth of at least 53 pm . Moreover, the excel-lent agreement between the measured and calculated trajectories suggests that the shock is well modelled by H Y R A D . The importance of this is that it implies that the target conditions calculated by H Y R A D are a good estimate of the actual conditions. This is only justified because the S E S A M E equation of state for aluminum is known to be very accurate up to pressures of 4.25 M b a r ' 1 3 4 ' . This is not the case for other materials (e.g gold). It is interesting to note that the simulated trajectory shows evidence of two distinct waves whose velocities are 0.6 x 106 cm/s and 1.5 x 106 cm/s. The first wave is attributed to an elastic wave which is launched at low ablation pressures. When the elastic limit is exceeded (~ 100 kbar) a slower plastic wave is generated which follows the initial disturbance. As the ablation pressure increases, however, the plastic wave strengthens and begins to catch up with the elastic wave. The time of coalescence of the two waves depends on the peak laser intensity and the rise time of the laser pulse. For a peak laser intensity less than ~ 10 1 2 W / c m 2 the second wave is never generated, while for laser intensities exceeding ~ 10 1 4 W / c m 2 the coalescence occurs at very early times. Following the coalescence a single strong shock propagates, accelerating as the driving pressure increases. These various phases are evident in figure 5.33 where the shock trajectories for different laser irradiances are shown. Another factor which affects the coalescence time and, more importantly, the target conditions at low intensities, is the time history of the ablation pressure. In figure 5.34 we plot the laser intensity and ablation pressure as a function of time for the standard irradiation conditions. It is evident from the results that the ablation pressure does not have a simple scaling with laser intensity. This is particularly true at early times in the Chapter 5. K-edge shift in shock compressed aluminum 102 0 10 20 30 AO 50 POSITION (Aim) Figure 5.32: Shock trajectory in aluminum at a laser irradiance of = 2.3 x l O 1 3 W / c m 2 ; open circles are the experimental results; and solid line is simulation results from H Y R A D . Time zero corresponds to peak of the laser pulse. Chapter 5. K-edge shift in shock compressed aluminum 103 Figure 5.33: Simulated trajectories: solid line = 10 1 2 W / c m 2 , dashed line $ L = 10 1 3 W / c m 2 , chain-dashed line $ L = 101 4 W / c m 2 . Chapter 5. K-edge shift in shock compressed aluminum 104 ablation process when the plasma temperature is less than 100 eV. In the coronal plasma the density and temperature are such that an ideal gas equation of state is a good approximation. Hence, the ablation pressure is determined by the temperature and average ionization of the plasma. However, at low tempertures the internal energy of the material does not have a simple dependence on temperature. The reason for this is that the atomic shell structure of the material effects the partitioning of energy. When the material is fully ionized, the laser energy goes into thermal energy of the plasma and the pressure scales with laser intensity (Pabi oc $ £ 8 ) . On the other hand, when the material is partially ionized, a fraction of the energy goes into atomic exitation. This results in a reduced pressure scaling with laser intensity. In the coronal plasma the L-shell is effectively ionized at temperatures exceeding 100 eV. The high ionization energy for the first K-shell electron (2073 eV) leads to a large temperature range in which the average ionization is nearly constant (< Z >~ 11). This effect is evident in figure 5.35. Also shown in figure 5.35 is the plasma pressure as a function of internal energy obtained from S E S A M E ' 4 6 l The increase in excitation energy of the closed K-shell aluminum ion leads to a region of gradual pressure increase. The net effect of this temporal behaviour of the ablation pressure on the shock trajectory is to increase the coalescence time. 5.2 Measurement of K-sheU photoabsorption edge for aluminum For our experimental conditions, the temperature of the dense plasma generated by shock compression is ~ 1 eV, as a consequence, absorption spectroscopy represents the most viable technique for studying the electronic structure. In this section we describe the first time resolved measurement of the J\-shell photoabsorption edge of aluminum un-dergoing shock compression. First, however, we briefly discuss our approach to obtaining a timing fiducial. Chapter 5. K-edge shift in shock compressed aluminum 105 Figure 5.34: The calculated ablation pressure pulse from H Y R A D (solid) and gaussian laser pulse (dash). Chapter 5. K-edge shift in shock compressed aluminum 106 Figure 5.35: a) Average ionization vs temperature, b) pressure vs internal energy; solid line ion density 1 0 1 8 c m - 3 , dashed line ion density 10 2 0 c m " 3 , dotted line ion density 1 0 2 2 c m - 3 . Chapter 5. K-edge shift in shock compressed aluminum 107 5.2.1 X-ray fiducial As in the shock trajectory measurements it was necessary to simultaneously record the signal and an appropriate timing fiducial. At first we considered using a bifurcated cathodef 1 3 5! in which a section of the cathode was U V sensitive, so that we might record the frequency converted laser light (0.35 /zm ) as a time marker. The reduction in the usable cathode region along with increased complexity made this a less attractive op-tion. We chose instead to correlate the emitted (front side) and transmitted (rear side) resonance line ls2p— Is2 (1598 eV) and neighbouring satellites. Taking into account the ~ 300 ps delay between laser peak and front side peak line emission predicted by the hydrocode, the transmitted signal could then be directly related to the laser pulse. The experimental setup used for this measurement along with the streak record obtained for a 16 /zm foil is shown in figure 5.36. The emitted and transmitted x-rays were individually dispersed by two flat P E T crystals onto the streak camera cathode. The crystals were placed at identical positions relative to the streak camera to eliminate any timing error. A shield was positioned around the streak camera to prevent plasma particles from reaching the cathode. The x-ray emission passed through small openings in the shield which were covered with a layer of B-10 foil. The front, side emission was further attenuated by an additional layer of B-10 foil and a 12.5 /zm berrylium foil. This attenuates the front side emission by approximately a factor of 4 compared to the transmitted emission. The emitted and transmitted x-rays were separated with a metal plate to prevent any overlap on the cathode. At an irradiance of $ L = 2.3 x 10 1 3 W / c m 2 (standard conditions) the onset time of the transmitted x-ray emission was measured for aluminum foils of thicknesses ranging from 9 /zm to 22 /zm . The results presented in figure 5.37 indicate that the onset time is Chapter 5. K-edge shift in shock compressed aluminum 108 Figure 5.36: a) Experimental setup used to calibrate x-ray fiducial, b) streak records for 16 fim aluminum foil. Chapter 5. K-edge shift in shock compressed aluminum 109 Figure 5.37: Onset time of transmitted x-ray emission at 1598 eV. Chapter 5. K-edge shift in shock compressed aluminum 110 approximately linearly dependent on the target thickness. The shot to shot timing jitter was less than 200 ps. Hence the use of the transmitted x-ray signal as a fiducial limits our timing accuracy to ~ 200,ps. 5.2.2 Aluminum K-edge measurement The experimental setup used to measure the time-resolved absorption spectrum of the aluminum targets is shown in figure 5.38. A curved P E T crystal positioned directly behind the target collected x-rays traversing the target at 3 to 7 degrees to target normal. This corresponds to an x-ray energy range of 1535 eV to 1585 eV. The crystal formed a cylindrical section of radius 15.3 cm. Since it was impossible to operate the crystal and streak camera in a standard Von Hamos configuration^ 3^, the crystal was used in a collimating mode to reduce chromatic aberrations. Source broadening effects due to the ~ 100 fxm diameter plasma source ~ 40 cm from the slit limit the spectrometer resolution to 0.2 eV. This is significantly less than the measured resolution of ~ 1.5 eV which was inferred from the identification of two satellite lines having an energy separation of approximately 2 eV. Crystal quality turns out to be what limits the resolution. The spectral resolution measured for a flat P E T crystal was better than 1 eV. The wavelength dispersion at the streak camera was calibrated both by using the satellite line near 1575 eV and the K0 line at 1540 eV. To further confirm the normal position of the aluminum K-edge, a thin 5 pm aluminum target was irradiated with the laser and a 9 pm aluminum foil was placed in front of the x-ray streak camera. The crystal was oriented such that the normal K-edge (1560 eV) was centered on the streak camera entrance slit. The low x-ray signal made it necessary to use 20000 A S A polaroid fiW*3''') to record the streak records. In figure 5.39 we show the streak records for 9 ^m and 25 pm aluminum targets irradiated at the standard irradiance of 2.3 x 10 1 3 W / c m 2 . Note that in the case of Chapter 5. K-edge shift in shock compressed aluminum 111 LASER X-RAY STREAK C A M E R A Figure 5.38: Experimental setup used to measure the temporal evolution of the if-edge of aluminum. Chapter 5. K-edge shift in shock compressed aluminum 112 the 9 pm target, a 9 pm aluminum filter was placed in front of the streak camera to eliminate the satellite line emission. This helped to reduce the background noise level and also identify the normal edge position at 1560 eV. Two main features are evident in these streak records, particularly for the 25 pm target. First of all, a definite red shift of the absorption edge is observed before the shock reaches the back surface. In repeated measurements the maximum edge shift was consistently within the range of 6 ± 2 eV for 25 pm targets and 3 ± 2 eV for 9 pm targets. The second feature of interest is the shifting back of the edge towards its normal position as the target unloads. The reasons for this temporal behaviour are discussed in the following section. Unfortunately, a measurement of the edge at very early times was not possible because of a lack of x-ray emission from the plasma. 5.3 Comparison of experimental results with 1-D simulations In order to simulate the iv-edge measurement we needed to know the dependence of the edge position on target density and temperature for shock compressed aluminum. To do this we employed the solid state model discussed in section 2.2.4. We first calculated the average ionization using the CSCP- I E E O S ' 7 7 ' 7 8 ' (complete screened coulomb potential ionization equilibrium equation of state) form of the modified Saha ionization theory. For our experimental conditions this model gave an average ionization of 3 in the compressed material. Figure 5.40 shows the shift of the 7\-shell photoabsorption edge predicted by this solid state model as well as the plasma model described by Bradley et al. ' 4 0] as a function of compression ratio along the principal Hugoniot of aluminum is shown. For aluminum at normal density and temperature, the plasma and solid state models yielded an edge position of 1561.8 eV and 1560.8 eV respectively. This compares favourably with the accepted value' 1 1 6 ] of 1560 eV. Whether the discrepancy is an accurate indication of Chapter 5. K-edge shift in shock compressed aluminum 113 3-2 -1 -UJ 0 -t — - 1 -w c UJ 3 2 1 0 - 1 --I r -15A0 1560 M V - y > . 1540 1580 ( a ) ( b ) 1560 E(eV) Figure 5.39: Streak records for a) 9 pm aluminum target, b) 25 pm aluminum target. Chapter 5. K-edge shift in shock compressed aluminum 114 the error at higher pressure is difficult to assess. Using these results in conjunction with H Y R A D simulations, a temporal history of the TV-edge position could be calculated. Specifically, x-rays in the energy range of 1545 eV to 1565 eV were divided into 60 groups and transported through the target using the H Y R A D post-processor. The opacity in each cell was calculated as described in section 2.2.4 except that the 7\-edge position predicted with our solid state model was used. In figure 5.41 we show the time history of the measured edge position and the cal-culated 10% and 90% intensity levels for 9 pm and 25 pm aluminum targets irradiated at 2.3 x 10 1 3 W / c m 2 . The sharp edge profile calculated is a consequence of neglecting edge broadening mechanisms in the simulation. The 25 pm target shows three distinct phases which are a manifestation of the target dynamics. At early times the low pressure wave discussed in section 5.1 gives rise to a nearly constant red shift of ~ 3 eV. As the driving pressure increases and the shock wave accelerates, the edge position shifts further to the red. Once the shock wave has reached the rear surface the edge position begins shifting back towards its normal position. This is the result of target decompression caused by a rarefaction wave from the unloading free surface and a rarefaction wave from the ablation surface where the laser intensity is decreasing. This phase is also clearly ob-served experimentally. Overall there appears to be good qualitative agreement between the measured and simulated time history of iV-shell edge energy. In the 9 pm target the first shock wave reaches the free surface at relatively early times (~-2 ns). However, since the shock is comparatively weak the unloading material expands slowly. Meanwhile, the laser intensity which is still increasing continues to drive stronger shocks which recompress the target. This effectively accelerates the target as a whole and maintains an almost constant target density. Once the laser intensity begins to decrease, the target rarefies in an analogous way to the 25 pm target. Again there Chapter 5. K-edge shift in shock compressed aluminum 115 Figure 5.40: Principal Hugoniot of aluminum (dashed curve) and the corresponding shifts in the -TV-shell photoabsorption edge derived from solid state model (solid line), plasma model (triangles). Chapter 5. K-edge shift in shock compressed aluminum 116 1565 1565 Figure 5.41: Measured and simulated time history of K-edge position, solid line measured, dashed simulated 90% level, dash-dot simulated 10% level ; a) 9 pm aluminum target, b) 25 pm aluminum target. Chapter 5. K-edge shift in shock compressed aluminum 117 is good qualitative agreement between the measured and simulated time histories of the TV-shell edge energy. The earlier return of the measured edge position to its normal position is likely due to two-dimensional unloading effects which were not included in the 1-D simulations. It may be argued that the state of the target is rather inhomogenous. In fact, for a 25 pm , target the state of the compressed target varies significantly from the ablation front to the rear surface. This is illustrated in figure 5.42 where the density in the target is plotted at various times during the simulation. However, it turns out that the edge position is determined by a thin (~ 5 pm ) region behind the shock front which has a near uniform density and temperature. The reason for this is that the diagnostic is sensitive to the region which has the largest red shift. This is due to the large change in opacity (~ 400 to 4000 cm 2/g) across the absorption edge which for a 5 pm layer implies a factor of 100 difference in x-ray transmission. In figure 5.43 we show the trajectory of the center of this layer for a 50 pm aluminum target. The edge shift is weakly dependent on temperature in the compressed material. As a result, the maximum edge shift occurs in the region of maximum compression. In the long laser pulse regime considered in this work the material near the ablation front undergoes nearly isentropic compression if we neglect x-ray preheat. This would imply that the highest target density occurs very near the ablation front. However, x-ray preheat increases the temperature and consequently reduces the density, maintaining a uniform pressure profile. These competing effects lead to the region of maximum density being somewhere between the ablation front and shock front. The situation is different for nearly "square" laser pulse shapes in which the shock front becomes steady state within a few microns of the target front surface. In this case, the region of maximum target density will be at the shock front. Chapter 5. K-edge shift in shock compressed aluminum 118 LAGRANGEAN COORDINATE [micron] Figure 5.42: Simulated density profiles in 25 pm aluminum target, a) t = —1.5 ns, b) t = 0.0 ns, c) t = 1.5 ns. Chapter 5. K-edge shift in shock compressed aluminum 119 POSITION Uim) Figure 5.43: Trajectory of the region being probed, (dashed line) for a 50 pm aluminum target , and the shock trajectory (solid line) for comparison. Chapter 5. K-edge shift in shock compressed aluminum 120 5.4 Comparison of experimental results with 2-D simulations In the interpretation of the experimental K-edge shift data we also considered the two-dimensional nature of the problem' 1 3 ^. The crystal collects x-rays traversing the target from any point within the irradiation area. As a consequence the diagnostic probes both the central portion of the shock, which is approximately one^dimensional, and the outer regions where the driving pressure is lower and shock attenuation occurs. Fortunately, however, the strong intensity dependence of the x-ray emission ( 13)[139] a g c o m p a r e d to that of the ablation pressure reduces the two-dimensional effect. This is particularly the case when measuring the transmission spectrum near target normal. Nevertheless, we have performed a simple ray tracing calculation to get an estimate of the effect. The density profile was calculated by using the two-dimensional hydrocode S H Y L A C 2 described in section 3.6. A pressure pulse (2.3 ns F W H M ) with a gaussian spatial profile (80 pm F W H M ) was applied at the boundary to model the ablation pro-cess. The validity of such an approximation in predicting the target conditions have been confirmed by one dimensional simulations' 3^. The peak pressure of 3.8 Mbar was chosen to match the observed shock transit times. This was slightly higher than the 3.5 Mbar peak pressure derived from one-dimensional simulations to compensate for the two-dimensional shock attenuation. The edge position predicted through target center and viewed at target normal agrees very well with the detailed one-dimensional simulation. However, significant broadening is observed when the emission from the entire irradiation region viewed at ~5 degrees is collected (figure 5.44). Although the edge position at the 90% intensity level continues to be in good agreement with one-dimensional results, the 10% intensity level is quite different. In particular, the 10% intensity level edge position is nearly constant for a long period near the peak of the laser pulse. It is interesting to note that the broadening of the Chapter 5. K-edge shift in shock compressed aluminum 121 Figure 5.44: Temporal evolution in the aluminum K-edge energy. Results of measure-ments (solid curve) and 2-dimensional calculations (dashed curve - 90% intensity level; dot-dashed curve - 50% intensity level; dot-dot-dashed curve - 10% intensity level). Chapter 5. K-edge shift in shock compressed aluminum 122 Aluminum Calculated K-Edge Shift Thickness Density Temperature Measured calculated ( Mm ) (g/cm 3) (eV) (eV) ±2 (eV) ± 1 9 3.3 2.9 -3 -3 16 4.1 1.0 -4 -5 25 6.0 0.5 -6 -9 Table 5.4: Measured and Calculated K-edge shift. edge found in the simulation is strictly due to the lateral non-uniformity of the target, yet it would appear to account for what is experimentally observed. The experimental data would imply an upper bound on any edge broadening mechanisms of ~ 5 eV for these target conditions. In principle, by improving the crystal collection properties and streak camera sensitivity it should be possible to accurately measure this edge structure. 5 . 5 Summary In table 5.4 we summarize both the experimental and calculated K-edge shift at the peak of the laser pulse for three different aluminum targets. The temperature and density were obtained from detailed simulations for our standard irradiation conditions. The error quoted for the measurements represents the accuracy in determining the edge position from the film. The error in the calculated edge shift was simply taken as the difference between the calculated and measured edge position of uncompressed aluminum. Whether this error estimate is justified at high densities is uncertain. Although the experimental and simulation results agree within the estimated uncertainty, the model appears to consistently overestimate the red shift. This is particularly the case at higher densities. We have also obtained the shift in the K-edge position as calculated by the fully self-consistent codes H O P E ' 7 5 ! and I N F E R N O ' 7 6 ] . At two-fold shock compression, the red shift of the aluminum K-edge was calculated to be 1.7 eV by H O P E and less than 0.3 eV Chapter 5. K-edge shift in shock compressed aluminum 123 by I N F E R N O . These values are much less than what was measured experimentally, and they also disagree with the solid state model predictions. This suggests that H O P E and I N F E R N O , which are known to be accurate only at high densities (p > 2p0), predict incorrect binding energies at low densities. Chapter 6 Ka Emission from laser irradiated aluminum In this chapter we describe some preliminary measurements of the Ka emission spectrum of laser-irradiated aluminum. A comparison of the experimental results and H Y R A D simulations is then presented. We conclude by giving several possible reasons for the observed discrepancy. 6.1 Measurement of the KQ emission spectrum As discussed in chapter 3, the Ka spectrum offers a unique way to probe the ionization state of low temperature, high density plasmas. In fact, it represents the only viable tech-nique for measuring the ionization state of the material in the radiatively heated zone. The reason for this is that the normal self-emission from this region is absorbed by the sur-rounding plasma. Whereas the hot coronal plasma is optically thin to the Ka emission. Furthermore, most of the Ka emission is below the K-shell photo-absorption edge of the compressed material making it possible to record the Ka spectrum from behind the tar-get. This is advantageous in that it can improve the signal to noise ratio by eliminating the particles and x-rays generated in the coronal plasma. Two different experimental setups were used for the Ka measurement. The first was that used for the K-edge measurement and is described in section 5.2.2. The second used a cylindrically curved (5.7 cm radius of curvature) R A P crystal (2d = 26.12 A) to disperse the Ka spectrum onto the x-ray streak camera (figure 6.45). The lower dispersion achieved with this configuration allowed the spectral range extending from 124 Chapter 6. Ka Emission from laser irradiated aluminum 125 LASER Figure 6.45: Experimental setup for Ka spectrum measurement. Chapter 6. Ka Emission from laser irradiated aluminum 126 the A l + 0 Ka (1487 eV) to Ha (1728 eV) to be simultaneously recorded. Although the crystal was positioned relative to the source in the classic Von Hamos configuration'1 ^ \ it was not possible to place the streak camera slit on the axis of the cylinder as desired. Therefore, the degree of focusing at the streak camera slit varies with position. However, by optimizing the crystal orientation this effect could be made negligible over the 1 cm region of interest on the entrance slit of the camera. This is particularly the case if we recall that the entrance slit is 1 mm in width, whereas the source is ~ 100 pm . In figure 6.46 we present streak records for 25 pm aluminum targets obtained with the two different setups. The spectrum could only be recorded with 20,000 ASA Polaroid film and as a consequence accurate determination of the intensity profiles was not possible. Nevertheless, several interesting features are evident. Firstly, the intensity of the A l + 7 Ka line (1540 eV) is of comparable intensity to the unresolved A l + 0 to A l + 3 Ka lines. Secondly, there is no noticable Ka emission corresponding to A l + 4 , A l + 5 and A l + 6 ions. This implies that the intensity of their emission is at least a factor of 2-5 lower than that of the A l + 7 Ka emission if we take into account the limited dynamic range of the film and the low signal levels observed. This result suggests that a significant portion of the target is predominantly A l + 7 , which in turn implies a plasma temperature of 60 eV to 80 eV, depending on the density. In any case, this temperature is well above the ~ 1 eV shock temperature and significantly below the ~ 500 eV coronal temperature. It is, however, close to the 45 eV to 55 eV temperatures predicted for the radiatively heated zone. 6.2 Simulation of the Ka measurements In order to determine whether the experimental results are evidence of the radiatively heated zone, it was necessary to simulate the process. Since generation and transport Chapter 6. Ka Emission from laser irradiated aluminum 127 Si. • • . • 31* i . ; jf •1 -If 53 ^ 7 + 0 _ A / + 3 Figure 6.46: 25 /zm aluminum target irradiated at $i b) using R A P crystal. Ka spectrum for 2.3 x 10 1 3 W/cm 2 ; a) using P E T crystal. Chapter 6. KQ Emission from laser irradiated aluminum 128 of KA emission are negligible energy transfer mechanisms, they have little effect on the target hydrodynamics. Therefore, it was reasonable to calculate the KA emission spectrum in the post-processor phase of the simulation. 6.2.1 KA emission model Before discussing the results it is important to review the processes which can lead to KQ emission. First of all a iV-shell electron with binding energy <j>k is removed by a photon of energy hv > (fix, leaving the atom in an excited state of the next ionization stage. The atom can return to the non-excited state by various processes of which two are dominant. In the first process an electron from an upper bound level falls to the TV-shell and is accompanied by the emission of a photon. This is KQ emission if the upper level is in the L-shell. The second process is similar in its initial stage to the first, but in this case, the radiation produced following the transference of an electron from an upper level is not emitted, but goes into further ionization within the atom. This process is known as the Auger process' 7 0]. The quantity of radiation emitted will be dependent upon the relative efficiency of these two opposing de-excitation processes. This relative efficiency is usually expressed in terms of the fluorescent yield. For our calculations the iv-shell fluorescence yields, U>K, quoted by Duston ' 1 8 ' were used. These are summarized in table 6.5. The method used to calculate the KA emission is as follows. First of all, for each target cell n and photon energy (or energy bin), we compute the total opacity <7;r(n, hv) of each cell as discussed in section 2.2.4. This includes the TV-shell opacity for each ion, o~K(n, z, his). We then transport all photons with sufficient energy to produce a iV-shell vacancy. From this we can calculate the number of photons absorbed per unit time and volume in every cell, Px{n,hv). The A'-shell vacancy creation rate, VR, is Chapter 6. Ka Emission from laser irradiated aluminum 129 Ionization Transition Fluorescence Stage energies (eV) yields A l + 0 1486.99 0.0333 A l + 1 1487.52 0.0460 A1+2 1488.24 0.0450 A l + 3 1488.95 0.0434 A1+4 1499.58 0.0479 A1+5 1511.64 0.0522 A1+6 1525.40 0.0619 A1+7 1540.37 0.0700 A1+8 1557.01 0.0623 Table 6.5: Transition energies and fluorescence yields for the various ionization stages of aluminum included in the Ka emission model. Ionization stage refers to the ion before photoionization, that is, A l + 0 . Ka is produced when an inner electron of an A l + 0 atom is removed. then given by VK(n,z) = J2PT(n,hp) aK(n,z,hv) hu o-T{n,hv) This expression is valid since in the simulations the optical depth of each cell A r n = axpnAxn was significantly less than 1. The Ka emission, is then calculated from the relation IK(n) = VK(n, Z)UJK(Z)AEk(Z), where AEx(z) is the Ka transition energy for ion z. The Ka emission is subsequently transported using the radiation transport routines discussed in section 2.2.5. 6.2.2 Ka simulation results In figures 6.47 and 6.48 we present the time histories of the intensities of several Ka lines as predicted by simulations performed with and without radiation transport. The A l + 0 to A l + 3 Ka emission is treated as a single unresolved feature in the spectrum due to Chapter 6. Ka Emission from laser irradiated aluminum 130 the small energy difference (~ 2 eV) separating the four lines. The difference observed between the forward (towards laser) emission (figure 6.47) and that transmitted through the target (figure 6.48) is due to two reasons. First there is a slight difference in opacity for the various lines. More important, however, is the origin of the emission. The higher ionization stages occur near the coronal region and as a result their Ka lines undergo little attenuation in the forward direction. The case is slightly more complicated for the low ionization stages which at early times dominate most of the target but are gradually eliminated near the ablation region. The effect of radiation transport on the Ka emission is also evident in figures 6.47 and 6.48. When radiation transport is neglected, the emission is dominated by the Ka lines of low ionization stages characteristic of the cold compressed target. On the other hand, when radiation transport is included, Ka emission from A l + 4 to A l + 7 becomes significant near the peak of the laser pulse. The main source of these Ka lines is the radiatively heated zone which varies in temperature from ~ 45 eV to ~ 55 eV. Another effect of radiation transport is to modify the time of peak emission for the various Ka lines. The Ka emission is governed by both the front side x-ray emission which creates the inner-shell vacancies and the relative density of the various ion stages. This accounts for the peak of Ka emission from the radiatively heated zone occuring ~ 700 ps after the peak of the laser pulse. Another interesting feature of these results is the anomolously low A l + 6 Ka emission. In fact, from analyzing the ion population within the target, it is expected that A l + 6 and A l + 5 Ka emission should be comparable, yet they differ by almost an order of magnitude. The reason for this discrepancy in emission is the difference in A'-edge positions for these two ions. A l + 6 has a AT-edge energy of 1739 eV whereas that of A l + 5 is 1684 eV. This difference is important because the i7—like resonance transition of aluminum has a photon energy of 1728 eV. As a consequence, it can ionize inner shell electrons of A l + 5 but not Chapter 6. Ka Emission from laser irradiated aluminum 131 TIME (ns) Figure 6.47: Forward Ka emission as a function of time for a 25 pm aluminum target irradiated at $L = 2.3 x 10 1 3 W / c m 2 : a) including radiation transport, b) no radiation transport. Chapter 6. Ka Emission from laser irradiated aluminum 132 - 2 - 1 0 1 TIME (ns) Figure 6.48: Backward Ka emission as a function of time for a 25 pm aluminum target irradiated at $x, = 2.3 x 10 1 3 W / c m 2 : a) including radiation transport, b) no radiation transport. Chapter 6. Ka Emission from laser irradiated aluminum 133 A l + 6 thus accounting for the large difference in Ka emission. Clearly, in interpreting the Ka spectrum the mechanisms creating the inner-shell vacancies must be carefully considered. In the case of inner-shell photoionization the details, of the x-ray emission and absorption causing the ionization must be known. 6.2.3 Comparison of simulated and measured Ka emission The simulation results presented in the previous section show significant disagreement with the experimental results. The simulations suggest that A l + 5 Ka emission should be dominant near the peak of the laser pulse not A l + 7 Ka as observed experimentally. One possible reason for this discrepancy is beam non-uniformity which can lead to lo-calized areas of higher intensity. Although the effects of hots spots on the shock front can be minimal due to smoothing mechanisms such as thermal, radiation and hydrody-namic smoothing, their effects on the conditions of the radiatively heated zone may be more significant. In order to assess the effects of irradiation uniformity we performed the simulation at two different laser intensities. The results presented in figure 6.49 indicate that the dominant ionization stage in the radiatively heated zone does not change signif-icantly when the laser intensity is varied. In fact, for the range of intensities considered, $ L = 10 1 3 W / c m 2 to = 10 1 4 W / c m 2 , the A l + 0 to A1+3 and A1+5 Ka emission always dominate. This is to be expected from the self-regulating nature of the ionization burn wave which leads to an increase in the size of the radiatively heated zone (rather than temperature) as the laser intensity increases. Hence, it is very unlikely that hot spots in the beam can account for the observed Ka spectrum. Another possible cause for the discrepancy concerns the accuracy in the radiation transport calculation. As discussed in section 3.5.2 the conditions in the radiatively heated zone are strongly affected both by the blue shift in the iV-shell photoabsorption edge as the material is ionized and the x-ray emission from the corona. If the opacity is Chapter 6. Ka Emission from laser irradiated aluminum 134 Figure 6.49: Temporal history of rear side Ka emission for different laser intensities : a) $ L = 1 0 1 3 b ) $ L = 1 0 1 4 . Chapter 6. Ka Emission from laser irradiated aluminum 135 underestimated in the calculations the radiatively heated zone will absorb less energy and remain cooler. Unfortunately, we are unable to find any physical reason for increasing the opacity. Moreover, when we compared our Ka spectrum with predictions of Dus-ton et al.' J w e found good agreement. Recently, however, it has been suggeste d[141] that the fluorescent yields used in these calculations might significantly underestimate the Ka emission of certain ions. The reason for this is the generation of metastable states which cannot decay through the Auger mechanism. As a result, the fluorescent yield for these ions can increase from a few percent to almost 100%. If this was the case for A l + 8 ions, for instance, this could account for our experimental observations. However, Ka spectrum measured by Burnett et a l . ' 1 4 ^ for aluminum targets irradiated by 10.6 pm laser light show a comparitively smooth intensity distribution of the various Ka lines. This suggests that they do not observe any evidence of anomalous fluores-cent yields. It is important to note that in the Burnett experiment, energetic electrons rather than x-rays are the dominant energy transport mechanism into the target. As a consequence, the radiatively heated zone is insignificant. This most likely accounts for the difference between our results and those reported by Burnett. In the simulation it is also assumed that ions ionized by inner-shell photoionzation instantly recombine to be in equilibrium with the local temperature and density conditions. If this is not the case then an abundance of A l + 7 may be created through an inner-shell photoionization of A l + 6 . Unfortunately, we are unable to assess this effect since it requires solving the time dependent collisional-radiative equilibrium model. Finally, the discrepancy between the experimental results and simulations could be attributed to the generation of a low temperature plasma around the perimeter of the focal spot. For our experimental conditions, there are two processes which could generate such a region. The first involves the flow of hot, thermal plasma across the focal spot boundary. The second possible process is intense X-ray heating of the area surrounding Chapter 6. Ka Emission from laser irradiated aluminum 136 the focal spot. It seems unlikely, however, that either process would lead to a plasma predominantly A l + ? as would be necessary to account for the experimental observations. Unfortunately, further analysis requires a two-dimensional laser-matter interaction code which is not available at this time. Chapter 7 Summary and Conclusions 7.1 Summary The experiments performed in this work have illustrated the importance of radiation transport both in modifying target hydrodynamics and in probing the state of dense plasmas. In the measurement of the K-edge energy in shock compressed aluminum we ob-served a red shift in the edge position as the material density increased. The results were in good qualitative agreement with the predictions of one-dimensional hydrodynamic simulations although they suggested that the theoretical model used to calculate the K-edge energy overestimated the shift in the K-edge of the dense plasma. In contrast, the H O P E and INFERNO calculations predicted edge shifts which were significantly less than those observed experimentally. This would indicate that these codes, which are known to be accurate at high densities (p > 2po), may be inappropriate for calculating the inner-shell binding energies at low densities. Unfortunately, the limited x-ray signal prohibited an accurate measurement of the edge profile. Nevertheless, the estimated edge width of ~ 5 eV is well below that predicted by simple analytical models which neglect ion-correlation effects. It appears that the observed edge broadening can be accounted for by two-dimensional effects which lead to non-uniform target compession. In the experimental measurements, it was also found that the Ka emission spectrum 137 Chapter 7. Summary and Conclusions 138 was dominated by A l + 0 - A l + 3 and A l + 7 lines. This differed from the theoretical prediction that the dominant Ka lines should be A l + 0 - A l + 3 and A l + 5 with the A l + 5 line produced in the radiatively heated zone. There are two possible causes for this disagreement. The opacity of the aluminum plasma in the radiatively heated zone may have been underestimated in the calculations. This results in less absorption and consequently lower temperatures and ionization. Alternatively, metastable transitions in A l + 8 could increase the Ka fluorescent yields from the values assumed in our calculation. 7.2 Significant Contributions The development of a detailed coupled radiation-hydrodynamic code, H Y R A D , has given us the ability to account for the effects of radiation transport in laser-matter interactions. Although similar codes have been recently reported^ 8 ' ^ \ H Y R A D offers several unique and important features. It is the only hydrocode which uses the accurate piecewise parabolic method to solve the fluid equations. It also uses a well established equation of state obtained from the S E S A M E data library instead of assuming an ideal gas equation of state. Moreover, H Y R A D is the first code in which Ka generation and transport is treated in detail. Previous calculations by Duston et al. assumed that the plasma was optically thin to the Ka emission. It should be noted that H Y R A D has not only been used in this thesis work, but also in an original investigation of shock propagation in multi-layered (aluminum on gold) targets' 3^. In the course of this work it was also necessary to develop and construct a new x-ray streak camera originally designed at the University of Rochester. This unit offered impor-tant advantages over what was commercially available, specifically, large photocathode and ~ 20 ps temporal resolution. The large photocathode (40 mm by 1 mm) allowed us to monitor a wider spectral range with increased sensitivity. This streak camera is vital Chapter 7. Summary and Conclusions 139 to high resolution x-ray spectroscopy studies. In this study we have performed the first time-resolved TV-edge measurement in shock compressed aluminum. The use of a uniform target allowed us to accurately deter-mine the density and temperature conditions in the target by comparing the measured and calculated shock trajectories. This in turn enabled us to study the effects of com-pression on the inner-shell binding energy by correlating the temporal evolution of the TV-edge with the temperature and density predicted by H Y R A D . The strong dependence of the TV-edge energy on ionization state makes it a valuable diagnostic for determining the electronic structure of the shock compressed material. Alternatively, measurement of the TV-edge energy may provide a method of tracking the development of laser-driven shocks in opaque targets. Motivated by H Y R A D simulations, we have obtained the first measurements of TVa emission from laser-irradiated aluminum targets where x-ray radiation is an important energy transfer mechanism. The disagreement between the calculated and measured TVa emission spectrum suggests some important physical processes may have been over-looked in the numerical modelling. Nevertheless, the Ka emission spectrum represents a viable diagnostic for studying the ionization balance in the radiatively heated zone in a laser-heated solid. 7.3 Future Work The TV-edge measurement offers a unique opportunity to study the electronic structure of high density matter. As the shock pressure increases, it is expected that the solid state model developed in the course of this work will become invalid. At this point self-consistent models such as H O P E and INFERNO will be more appropriate. In order to identify the regions of validity of the various models, an accurate measurement of Chapter 7. Summary and Conclusions 140 the edge position at higher shock strengths is necessary. This will require higher laser intensity, but more importantly a few modifications to the experimental technique. Firstly, it would be advantageous to use a second laser beam to produce an indepen-dent x-ray source which can be used to backlight the target. The advantage of this is that the timing of the x-ray source could be adjusted to allow observation of the edge shifts at early times. Moreover, a material other than that being probed can be used as the x-ray source. This other material can be selected to produce uniform x-ray emission in the vicinity of the .K-edge to be studied. Secondly, in order to eliminate two-dimensional effects on the measured edge profile, spatially resolved measurements of the transmitted x-rays should be made. One way to achieve this is to simply have the target mounted on a thick substrate with a small hole (i.e. small relative to the laser focal spot). The sub-strate would then block all x-rays transmitted through the target except those near the laser axis. Finally, in order to increase the x-ray signal the curved crystal spectrometer should be operated in either a Von Hamos or conical focusing configuration. 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