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X-ray sources and shock compression schemes for photoabsorption edge spectroscopy Dyke, Steven R. 1995

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X - R A Y S O U R C E S A N D S H O C K C O M P R E S S I O N S C H E M E S F O R P H O T O A B S O R P T I O N E D G E S P E C T R O S C O P Y . B y S T E V E N R. D Y K E B.Sc.(Hon.) , Physics, University of Vic to r i a , 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR THE DEGREE OF M A S T E R O F S C I E N C E in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA September 1995 © S T E V E N R . D Y K E , 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t i sh 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 writ ten permission. D E P A R T M E N T O F P H Y S I C S The University of Br i t i sh Columbia 2075 Wesbrook Place Vancouver, Canada V 6 T 1W5 Date: Abstract A scheme to probe the temporal evolution of the K-shel l photoabsorption edge in shock compressed aluminum is presented in two components. In the first component, a one dimensional hydrodynamic code coupled to a non-local thermodynamic equilibrium cal-culation is developed and used to model shock propagation and target preheat in laser irradiated C H - A l - S i multilayer targets. Single and multiple shocks are used to create con-ditions in the aluminum by which density, temperature and ionization effects on K-shell photoabsorption can be isolated. The final result consists of the optimal laser parameters and target designs for creating well characterized plasma states, necessary for accurate atomic modeling of the K-edge. The second component describes the results of x-ray spectra observed from 12 elements across the periodic table, in search of an emission source to backlight the K-edge measurement. Spectral flatness and high emission levels in the region of 1520-1600 eV around the K-edge of aluminum are the criteria for a suitable backlighter. Lead, gold and y t t r ium seem promising as backlighter sources, and the emission lines observed in silver provide a convenient spectral calibration source. i i Table of Contents Abstract ii Table of Contents iii List of Tables vii List of Figures viii Acknowledgement xv 1 Introduction 1 1.1 Laser-Matter Interactions 1 1.2 Photoabsorption Edge Research 2 1.3 Present Work 6 1.3.1 Strategy to Unfold K-Edge Dependencies . . 7 1.3.2 Backlighter Sources : 7 1.4 Thesis Organization 9 2 Physical Processes in Laser-Matter Interactions 10 2.1 Laser Driven Shock Waves 10 2.1.1 Shock Wave Formation 10 2.1.2 Shock Compression of Solids and the Rankine-Hugoniot Relations 13 2.1.3 Impedance-Mismatch Technique 18 2.2 Atomic Physics . . . : . 21 iii 2.2.1 Introduction 21 2.2.2 Atomic Processes 22 2.2.3 Non-Local Thermodynamic Equi l ib r ium Mode l 25 2.3 Radiat ion Emission, Absorption and Transport . 28 2 :3.1 Bremsstrahlung Emission 28 2.3.2 Recombination Emission '. . 29 2.3.3 Line Emission . . 30 2.4 Radiat ion Transport 31 3 Numerical Simulations 35 3.1 Introduction . . 35 .3.2 Laser-Target Code ( L T C ) 36 3.2.1 Physical Content in L T C 36 3.2.2 Numerical Methods in L T C 41 3.2.3 Comment on L T C 42 3.3 Physics Content in the N L T E Model 42 3.3.1 Calculat ion of State Population • • 42 3.3.2 Rate Coefficients . . 43 3.3.3 Photoexcitation 47 3.3.4 Line Transport 48 3.3.5 Continuum Transport . 50 3.4 Modifications for Present Work 50 3.5 Comparison of N L T E Calculat ion wi th R A T I O N Code 53 3.5.1 Energy Levels and Level Coupl ing 53 3.5.2 Rate Coefficients 54 3.5.3 Results of Comparison . 55 iv 3.6 Survey of Simulation Parameters . 61 4 Results of Radiation-Hydrodynamic Simulations 64 4.1 General Results . .'. . ' 64 4.1.1 Hydrodynamics . 64 4.1.2 Radiation Transport 64 4.1.3 Laser Absorption 67 4.1.4 Flux Limiter : . 69 4.1.5 Final Choice of Absorption Routine and Flux Limiter . . . . . . . 71 4.2 Optimized Shock Compression Schemes 72 4.2.1 Laser Pulse and Target Design .72 4.2.2 Density and Temperature Conditions 74 4.2.3 Discussion of Individual Compression Schemes 74 5 Experimental Facility 95 5.1 Laser Facility 95 5.2 Focal Conditions . . . . . . 97 5.3 X-Ray Spectrometers 99 5.4 Targets . 102 5.5 Processing of Data 102 6 Results and Discussion of X - R a y Spectra 110 6.1 X-Ray Spectra 110 6.1.1 Bismuth, Lead and Gold 114 6.1.2 Tungsten 118 6.1.3 Tantalum . . . 118 6.1.4 Samarium 122 v 6.1.5 T i n 122 6.1.6 Silver 122 6.1.7 Niobium 125 6.1.8 Y t t r i u m 127 6.1.9 Germanium 127 6.1.10 Magnesium 131 6.2 Discussion , 131 7 Summary and Conclusions 136 7.1 Summary of Present Work 136 7.2 Conclusions 137 7.3 Future Work 137 Bibliography 139 List of Tables 3.1 Photoabsorption edge energies for the different ionization stages of carbon: 51 ' 3.2 Ionization potentials for the ground and excited states included in the N L T E model. . 52 3.3 Transitions and oscillator strengths used in the N L T E calculation. . . . . 53 4.1 Pressures and speeds characteristic of single shock waves in the C H layer. 66 4.2 Pressures and speeds characteristic of single shock waves in the A l layer. 66 4.3 Pressures in the C H and aluminum layers when different laser absorption routines are used 68 4.4 Dependence of peak pressure wi th in and shock arrival t ime at the alu-minum layer on the value of the flux l imiter used. Simulation parameters are described in the text. . 71 4.5 Compression and temperature conditions obtained in the shock compres-sion schemes, along wi th final target and laser pulse parameters. 76 5.1 Summary of sample elements used in this work in order of decreasing atomic number. . . . 105 5.2 Aluminum wavelengths used in deriving the spectral dispersion relation. . 108 6.1 Experimental and theoretical energies of lines of an aluminum spectrum when the spectral calibration equation is applied 115 v i i List of Figures 1.1 Qualitative picture of a laser-target interaction. 3 1.2 , A general compression (density )-temperature space used to illustrate the method of decoupling density, temperature, and ionization effects on K -edge position. 8 2.1 Schematic of the hydrodynamics in a laser irradiated target, p is the target density, and T e and Tj are the electron and ion temperatures. The critical density layer is denoted by ncr . . . 11 2.2 Illustration of the shock build up process in a laser driven shock wave. . . 14 2.3 Schematic of the shock discontinuity. P is pressure, p is density, E is total internal energy, D is shock speed and u is particle speed. Variables subscripted with a 0 represent the undisturbed solid. Particle speed ahead of the shock front is zero. . . . 17 • 2.4 Hugoniot (solid), isotherm (dot-dash), and isentrope (dash) 19 2.5 Schematic of the impedance mismatch technique, pi, Pi, Ei,Usi, and UPi are the density, pressure, total internal energy, shock speed, and particle speed in the two regions. At the interface, P0 4- P x — P 2 and Upi = UP2- • 20 2.6 Shaded region represents the parameter space accessible in laser-produced plasma experiments. Applicability of certain models is shown. For exam-ple, the line labeled L T E Z=5 corresponds to the range of validity of the local thermodynamic equilibrium model for Boron 26 2.7 Radiation transport in planar geometry. 32 vii i 3.1 Level coupling scheme used in this work. S is ionization, a is radiative and three-body recombination, aoi is dielectronic recombination, X and X~l are collisional excitation and de-excitation, respectively, and A is the spontaneous decay 44 3.2 Ionizations produced by the present N L T E calculation and R A T I O N for a carbon plasma wi th an electron density of 10 1 9 c m - 3 . The dashed lines wi th circles correspond to R A T I O N calculations, whereas the solid lines wi th squares correspond to N L T E calculations 56 3.3 Ionizations produced by the present N L T E calculation and R A T I O N for a carbon plasma with an electron density of 10 2 0 c m - 3 . The dashed lines wi th circles correspond to R A T I O N calculations, whereas the solid lines wi th squares correspond to N L T E calculations. 57 3.4 Ionizations produced by the present N L T E calculation and R A T I O N for . a carbon plasma wi th an electron density of 10 2 1 c m - 3 . The. dashed lines with circles correspond to R A T I O N calculations, whereas the solid lines wi th squares correspond to N L T E calculations 58 3.5 Ionizations produced by the present N L T E calculation and R A T I O N for a carbon plasma wi th an electron density of 10 2 2 c m ~ 3 . The dashed lines wi th circles correspond to R A T I O N calculations, whereas the solid lines with squares correspond to N L T E calculations 59 3.6 Ionizations produced by the present N L T E calculation and R A T I O N for a carbon plasma with an electron density of 10 2 3 c m - 3 . The dashed lines with circles correspond to R A T I O N calculations, whereas the solid lines wi th squares correspond to N L T E calculations. 60 3.7 Five-zone scheme used in target design. . . 62 ix 4.1 Snapshot of hydrodynamic profiles wi th in the target at 200 picoseconds of simulation time. The shock wave is shown propagating through the C H layer, imminent upon the CH-A1 boundary. This particular target consists of 27 fim C H - 3 pm A l - 6 //m S i . The laser irradiance is 3.75 x 10 1 4 W / c m 2 . 65 4.2 Plot of reflectivity as a function of time for both the electromagnetic wave solver and inverse bremsstrahlung absorption routines. . 70 4.3 Visua l aid to the important parameters in laser pulse design. The rise and fall times between levels is fixed at 100 ps .. 73 4.4 Illustration of the densities and temperatures achieved in the shock com-pression schemes . 75 4.5 Compression (i.e. density) as a function of time wi th in the aluminum layer for the reference state. The decompression prior to shock arrival is due to radiative preheat. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. . 78 4.6 Temperature as a function of time wi th in the aluminum layer for the ref-erence state. Note the temperature increase prior to shock arrival due to radiation preheat. The time interval over which quasi-steady state condi-tions are achieved is indicated by the horizontal line. . . . . . . . . . . . 79 4.7 Pressure as a function of time within the aluminum layer for the refer-ence state. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. .' ; 80 x 4.8 Compression (i.e. density) as a function of time within the aluminum layer for the high average ionization state. Note the much higher level of decompression caused by radiative preheat in this case as opposed to other simulation results. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line 83 4.9 Temperature as a function of time within the aluminum layer for the high average ionization state. Radiat ion preheat is clearly an important effect for this laser irradiance. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line 84 4.10 Pressure as a function of time within the aluminum layer for the high average ionization state. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line 85 4.11 Average ionization as a function of shock pressure for aluminum along the single shock Hugoniot. These values are obtained from the Q E O S model. 86 4.12 Compression (i.e. density) as a function of t ime wi th in the aluminum layer for the isochoric state: The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line 88 4.13 Temperature as a function of time within the aluminum layer for the iso-choric state. The low laser irradiance used for the first pulse leads to a situation where radiation preheat is not a significant factor. The time in-terval over which quasi-steady state conditions are achieved is indicated by the horizontal line 89 4.14 Pressure as a function of time within the aluminum layer for the iso-choric state. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line 90 x i 4.15 Compression (i.e. density) as a function of time within the aluminum layer for the isothermal state. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line 4.16 Temperature as a function of time within the aluminum layer for the isothermal state. The low laser irradiance used for the first pulse leads to a situation where radiation preheat is not a significant factor. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line 4.17 Pressure as a function of t ime within the aluminum layer for the isother-mal state. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line 5.1 Schematic of the laser facility. O S C is the oscillator, P A is the preamplifier and A l - 4 are the amplifiers. I R M is an infrared mirror and SF1-6 are the vacuum spatial filters. A P is the hard aperture and S A is the saturable absorber. P l - 8 are polarizers, B S R is a beam splitter-reflector, N D is a neutral density optical absorber, IF is an interference filter, and G E N T E K is the piezoelectric beam energy monitor. S H G is the second harmonic generation crystal 5.2 Sample laser pulse. The horizontal scale is 1 nanosecond per division. . . 5.3 Experimental arrangement for scanning the focal distribution. N D is a neutral density optical absorber, and IF is the interference filter described in the text. x i i 5.4 Images of the laser spot at best focus. Attenuation of the top picture is . 100 times greater than the attenuation of the bottom picture, and 10 times greater than the attenuation of the center picture. The magnification of these images is 165 times . 101 5.5 Target chamber and P E T crystal spectrometer. The diameter of the target chamber is approximately 90 cm. , 103 5.6 F i l m cassette. The film plane is indicated by the dotted line 104 5.7 Arrangement of the x-ray spectrometer. The path length R = a + b of the longest wavelength reference line is shown. 107 6.1 A print of a sample aluminum and tantalum exposure. Some of the known lines are labeled (refer to Figures 6.2, 6.3, and 6.8) I l l 6.2 Sample aluminum reference spectrum. Some common lines are labeled. . 112 6.3 Enlarged aluminum reference spectrum showing the clearly resolved satel-lite lines qr, a-d, and klj of A l X I 113 6.4 Sample bismuth spectrum. The position of the aluminum K-edge is shown by the dotted line 116 6.5 Sample lead spectrum. The position of the aluminum K-edge is shown by the dotted line 117 6.6 Sample gold spectrum. The position of the aluminum K-edge is shown by the dotted line ' 119 6.7 Sample tungsten spectrum. The position of the aluminum K-edge is shown by the dotted line 120 6.8 Sample tantalum spectrum. The position of the aluminum K-edge is shown by the dotted line 121 x i i i 6.9 Sample samarium spectrum. The position of the aluminum K-edge is shown by the dotted line 123 6.10 Sample t in spectrum. The position of the aluminum K-edge is shown by the dotted line 124 6.11 Sample silver spectrum. The position of the aluminum K-edge is shown by the dotted line. . 126 6.12 Sample niobium spectrum. The position of the aluminum K-edge is shown by the dotted line. . . . " 128 6.13 Sample y t t r ium spectrum. The position of the aluminum K-edge is shown by the dotted line 129 6.14 Sample germanium spectrum. The position of the aluminum K-edge is shown by the dotted line 130 6.15 Sample magnesium spectrum. The position of the aluminum K-edge is shown by the dotted line 132 6.16 Moseley plot used to aid in line classification 133 xiv Acknowledgement I would like to thank Dr. Andrew N g for his unending enthusiasm and support for this project. His mentorship has taught me many valuable lessons about how science should be done. I would also like to thank Andrew Forsman, Kristin Smith, and George Pinh'o for all of their expert assistance in performing experiments and debugging computer code. The unbounded love and support of my family and friends has carried me through many rough spots over the last two years.- For this I am truly grateful and appreciative. xv Chapter 1 Introduction 1.1 Laser-Matter Interactions The study of laser-produced plasmas and laser-matter interactions is closely connected to the development of laser technology [1]. It was the advent of the Q-switching tech-nique [2] that set the stage for creating conditions previously inaccessible within matter. Pressures of many tens of megabars, accompanied by temperatures of millions of degrees and compressions of several times solid density, are achievable when a high-power laser beam is focussed onto a target. Traditionally, only much lower pressure regimes had been accessible, and these were obtained through the use of high velocity impacts or explo-sive detonations. Underground nuclear explosions [3] have created pressures greater than those obtained with any other means, but experiments of this type are hardly practical. Now that extreme pressures can be routinely generated in the laboratory wi th commer-cially available laser sources, the opportunity exists to map out the behaviour of matter to conditions approaching those inside stellar objects. In addition to providing an understanding of fundamental plasma physics, research involving laser-produced plasmas has found application in many other fields, including atomic physics, astrophysics, inertial confinement fusion, x-ray lasers and x-ray lithogra-phy. A detailed account of the use of laser-produced plasmas can be found in the work of Hora [4]. A brief overview of the processes involved in laser-target interactions at this point is 1 Chapter 1. Introduction 2 instructive. Figure 1.1 shows a schematic of the laser-target interaction. Light incident from a high-power laser is focussed onto a solid target, and ini t ia l ly absorbed within the skin depth of the front surface. As the intensity of the laser radiation increases, material from the target surface evaporates and expands outward (ablation), reaching speeds in excess of 10 7 cm/s . A t laser irradiances greater than 10 9 W / c m 2 , the ablated material is ionized [5], and forms a low density, high temperature plasma called the corona. The laser light penetrates the coronal plasma only up to the critical density. A t this point, the electron density yields a plasma frequency equal to the laser frequency and the incident laser light is reflected. Electron thermal conduction, x-ray radiation, and suprathermal electrons are possible mechanisms for carrying absorbed laser energy from the critical density layer to the ablation front, enhancing the ablation process. The momentum of the expanding plasma is balanced by the formation of a shock wave which propagates into the target and produces a dense, strongly coupled plasma. These plasmas typically have electron densities of ~ 10 2 3 c m - 3 , temperatures of 1-40 eV, and ion-ion coupling constants much larger than one [6]. The range of plasma densities and temperatures attained in the different regions of a laser-heated solid can easily span several orders of magnitude. 1.2 Photoabsorption Edge Research The shock waves generated by laser-matter interactions provide a test-bed for high pres-sure equation of state research. In the past, however, this research has focussed on shock speed [7, 8, 9, 10] and temperature [10] measurements, which yield l i t t le information on the electronic structure, ionization state, ionization potential, ion-ion, or ion-electron correlation strength [11, 12] of dense plasmas. These properties are mutually dependent, and are important for opacity and equation of state calculations. Chapter 1. Introduction Figure 1.1: Qualitative picture of a laser-target interaction. Chapter 1. Introduction 4 One of the first indications that K-shel l photoabsorption edge effects could play a role in x-ray transmission was reported in the experiment of N g et al. [13]. In this work, a significant increase in x-ray transmission through shock compressed aluminum targets (as compared to cold aluminum targets) was attributed to a blue shift of the K-shel l photoabsorption edge of the aluminum. Such a shift would allow a larger set of photon energies to pass through the target. However, the x-ray transmission measurement was crit ically dependent upon knowledge of the shock breakout time, and it was eventually realized that the shock breakout time had not been correctly determined. Thus, the increased x-ray transmission was not due to a blue shift of the K-shel l photoabsorption edge as had been thought. The first experiment to probe the atomic physics of a dense plasma was that reported by Bradley et al. [14]. In this pioneering work, the structure and position of the K -shell photoabsorption edge of chlorine was measured as multilayer B i - K C l targets were radiatively heated and subsequently shock compressed by laser ablation. The observed red shift of 8 ± 4 eV during the compression phase suggested pressure ionization of the 3p level, also evidenced by the appearance of the ls-3p absorption line. This experiment was also the first demonstration of the depression of ionization potential [14] as the plasma density increased past solid density. The edge shift was modeled as the competition between an energy increase in the edge from ionization effects and an energy decrease due to continuum lowering. The edge broadening, however, was attributed to Stark broadening and gradients in temperature and density wi thin the shock compressed region. Whi le this experiment represents the first successful effort in unraveling the atomic physics of strongly coupled plasmas, it is not without problems. The use of emission from the front side plasma for the absorption measurement precludes a determination of the true position of the unshifted photoabsorption edge. In the experiment, the best Chapter 1. Introduction 5 measurement of the unshifted edge position was st i l l red-shifted by 3 eV from high res-olution measurements on cold material [15], possibly a result of higher levels of x-ray heating than had been anticipated. More importantly, however, is the appearance of the ls-3p absorption line on top of the K-edge. As this line appears, it becomes difficult to determine whether the absorption is due to the K-shel l or due to the Sp vacancy, thus adding another level of uncertainty to the measurement. Final ly , in an effort to characterize the shock, a one dimensional hydrocode which did not include any type of radiation transport physics was used. Moreover, questions regarding the accuracy of the high pressure equation of state of chlorine could expose problems wi th interpreting the edge shift as due to x-ray or shock heating processes. Shock induced red shifts of 7 ± 2 eV in the K-shell photoabsorption edge of aluminum were later reported by DaSi lva et al. [16] and interpreted with the solid state model of Godwal et al. [17] and the density functional theory approach of Perrot [18]. The use of a relatively thick, uniform target in the experiment by DaSi lva et al. [16] served two purposes. Target uniformity allowed plasma states to be characterized by shock speed measurements, while target thickness was chosen to reduce the level of radiative preheat. Emission from the front side plasma passing through the target provided the absorption measurement. Unfortunately, the l imited amount of emission at the start of the laser pulse left the ini t ia l unshifted edge position unobserved. Moreover, the l imited level of transmitted x-rays forced the data to be recorded only on Polaroid fi lm, leading to only a semi-quantitative assessment of intensity levels and edge position. More definitive measurements of the K-shel l photoabsorption edge profile are needed. A l u m i n u m represents the ideal material for these experiments, since its high pressure equation of state is well established [19] and thus the shock compressed system can be accurately modeled. Wel l characterized plasma states are necessary to test atomic physics models. Chapter 1. Introduction 6 1.3 Present Work The structure and position of \he K-edge in aluminum (1560 eV in undisturbed material [20]) depends upon density, temperature, and ionization state [21], as well as effects such as continuum lowering [22] and electron degeneracy. The dense plasma behind the shock front is not hot enough to emit an appreciable flux of x-rays, and hence emission spectroscopy is not a viable technique for probing this region. A promising alternative is to use absorption spectroscopy and study the temporal evolution of the spectral profile of an inner shell photoabsorption edge. Ideally, the x-rays for the absorption measurement would be provided by a backlighter, a laser-produced plasma situated behind the sample which is to be probed. Wel l characterized plasma states, necessary for accurate atomic modeling, are produced by coupling tailored laser pulses to specific targets. The aim of this thesis research is to provide the necessary background to carry out such an experiment. Target and laser pulse design is done with a one-dimensional radiation-hydrodynamics computer code. A non-local thermodynamic equilibrium ( N L T E ) model is employed to calculate ion level populations and emitted radiation. Continuum radiation is transported through the target using multigroup diffusion [23], whereas line radiation is transported using the concept of photon escape probabilities [23, 24]. The piecewise parabolic method [25, 26] is used in conjunction wi th equations representing the conservation of mass, momentum, and energy, to model shock propagation and hydrodynamics within the target. The targets consist of C H - A l - S i sandwiches, where the C H and Si layer thicknesses are varied but the A l thickness is fixed at 3 microns. The C H layer on the laser deposition side provides pressure enhancement of the shock wave as it passes from the C H to the aluminum layer, as well as preventing the aluminum from releasing out the front side Chapter 1. Introduction 7 prematurely. The silicon layer allows quasi-steady state conditions to be achieved in the aluminum before the shock emerges from the rear surface. This quasi-steady state is required for ~100 picoseconds, which, given the temporal resolution of modern streak cameras, should provide sufficient time for measurements. 1.3.1 Strategy to Unfold K - E d g e Dependencies The routes to decoupling the effects of density, temperature and ionization state on K -edge energy are represented as three branches originating from a reference state in a density-temperature space (refer to Figure 1.2). The term 'reference state' is merely a label for the state wi th which conditions on the three branches are compared in order to deduce the effects of different hydrodynamic conditions on the K-edge position. The branch for studying density effects includes the reference state plus one point at higher density (but the same temperature), created by launching multiple shocks into the target. Likewise, multiple shock compression is used to create a point of the same density as the reference point, but lower temperature. A greater average ionization resultant.from the higher temperatures created by large amplitude single shock waves allows the study of ionization state effects on K-shel l photoabsorption. Hence, the simulations have been undertaken to provide the.optimal pulse and target characteristics to achieve the above four states necessary for unfolding the individual dependencies. 1.3.2 Backlighter Sources The ideal backlighter would be spectrally flat (i.e. no spectral lines or emission edges), and as bright as possible in the region surrounding the K-edge, 1520-1600 eV. W i t h these characteristics in mind, a search for a suitable backlighter has been carried out. Spectral emission from 12 elements (Mg, Ge, Y , Nb, A g , Sn, Sm, Ta, W , A u , Pb , and B i ) has been recorded with a flat pentaerythritol ( P E T ) crystal spectrometer. A n aluminum Chapter 1. Introduction 8 1 1 1 1 I 1 1 1 1 | 1 1 — i i — | — i 1 — i — i — | — i — i — i — i — | 1 — i 1 1 — | 1 — i r 0.0 1 • i i I i i i i I i i i i ' I i i i i I . . . -. I 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Compression Figure 1.2: A general compression(density)-temperature space used to illustrate the method of decoupling density, temperature, and ionization effects on K-edge position. Chapter 1. Introduction 9 spectrum superimposed upon the sample spectrum was used for energy calibration with an accuracy of ± 1 eV in the region around 1560 eV. Atomic number differences between the samples was less than ten, providing adequate coverage in the region of the periodic table studied. 1.4 Thesis Organization Chapter 2 reviews the important processes in laser-matter interactions, notably shock propagation and the emission, absorption and transport of radiation. A description of the non-local thermodynamic equilibrium model is also given. Chapter 3 describes the computer codes used to model the ablation process and optimize the laser pulse and target designs. Chapter 4 summarizes the results of the numerical simulations. Details of the experiment are given in Chapter 5, and the x-ray spectra are discussed in Chapter 6. Concluding remarks are made in Chapter 7. Chapter 2 Physical Processes in Laser-Matter Interactions This chapter presents a brief review of the physics involved in the formation of shock waves when laser light is focussed onto a solid target. Shock propagation in multilayered targets is discussed in terms of the Rankine-Hugoniot equations and the impedance-mismatch technique. As well, emphasis is placed on describing the physics involved in the emission, absorption and transport of radiation through the plasma. 2.1 Laser Driven Shock Waves 2.1.1 Shock Wave Formation Features of the ablation process along wi th associated density and temperature profiles appear in Figure 2.1. For short wavelength (A < 1/um), medium intensity (I ~ few x 10 1 5 W / c m 2 ) laser radiation , the dominant absorption mechanism is inverse bremsstrahlung, or free-free absorption [28]. In this process, electrons oscillating in the electric field of the incident light wave collide wi th ions, and thus convert their directed energy of motion into random thermal energy. The atoms of the target material are quickly heated and ionized, forming a dense plasma on the target surface. Dur ing this ini t ia l plasma formation, there exists a dense plasma bounded by solid target on one side and vacuum on the other. This step profile in density quickly decays into a rarefaction wave, and ions are accelerated into the vacuum at the ion acoustic speed [29]. The ejected material forms a low density 10 Chapter 2. Physical Processes in Laser-Matter Interactions 11 POSITION Figure 2.1: Schematic of the hydrodynamics in a laser irradiated target, p is the target density, and T e and are the electron and ion temperatures. The critical density layer is denoted by ncr. Chapter 2. Physical Processes in Laser-Matter Interactions 12 (ne < 10 2 3 c m - 3 ) , high temperature ( T e > 100 eV) region called the coronal plasma (in analogy with the coronal layer of stellar objects), labeled as region 1 of Figure 2.1. The laser light penetrates the coronal plasma only to the critical density layer , the point at which the electron density yields a plasma frequency resonant with the laser frequency and the incident light is reflected. Analyt ical ly, the critical density is given by where eo is the permit t ivi ty of free space, m e and e are the electron mass and charge, respectively, and u t is the laser frequency. The reflection process can take place over spatial scales as short as a few laser wavelengths [30]. Note that the crit ical density is proportional to the inverse square of the laser wavelength, and hence shorter wavelength radiation propagates further into the target before being reflected. Energy transport beyond the crit ical density layer is accomplished by electron thermal conduction, x-ray radiation emitted by the hot corona, and suprathermal electrons[27] generated at the cri t ical density layer. The steep temperature gradient which exists between the cold solid and the hot coronal plasma gives rise to a large thermal flux towards the interior of the target. This occurs in region 2 of Figure 2.1, which is known as the conduction or ablation zone. The ablation zone joins the cold solid at the ablation front. Energy carried to the ablation front is balanced by a return flux of accelerated and heated target materials into the corona. The ablation front moves through the target at a constant rate provided that the mass ablation rate remains constant. Once the energy flux between the crit ical density layer and the solid target is established, shock formation arises as a consequence of conservation of mass, momentum and energy. The momentum of the ablated material must be balanced, and in this way a shock wave is produced (region 3 of Figure 2.1). The amplitude of the shock wave wi l l depend Chapter 2. Physical Processes in Laser-Matter Interactions 13 upon the laser irradiance, and as the irradiance increases (e.g. the rising edge of a laser pulse) a series of shock waves of increasing amplitude are launched into the target. The shock wave is a compressional disturbance, and the first of this series wi l l travel at slightly over the local sound speed, compressing the target. The next shock wave in the series wi l l travel at the sound speed of the compressed material, and hence propagates faster than the previous. (Recall that the sound speed is given by cs = )• This effect leads to a pile up zone, where multiple shocks of increasing amplitudes and velocities are propagating into the target. The latter waves wi l l catch up with the earlier ones, leading to a steepening of the shock profile, as illustrated in Figure 2.2. The many small shock waves have thus coalesced into as ingle well-defined shock front, characterized as a propagating discontinuity in density, temperature and pressure. Constant ablation pressure sustains the shock, so that further acceleration of the shock front requires an increase in the ablation pressure. 2.1.2 Shock Compression of Solids and the Rankine-Hugoniot Relations Shock compression of solids is sufficiently different from the case of liquids and gases to warrant a brief review. A lucid account may be found in Zel'dovich and Raizer [31]. The essential difference between a solid and a l iquid (or gas) is the strong interparticle interaction present in a solid. Atoms in condensed matter are bound together by an elastic potential which resists expansion and contraction, while at the same time minimizing the potential energy of the system. This interparticle. force is effective over distance scales of only Angstroms, and yet accounts for the abili ty of a solid to support shear stresses, whereas a l iquid or gas cannot. The interparticle forces present in a solid are usually accounted for by the inclusion of pressure and energy terms arising solely from this elastic force. In this way, the .total Chapter 2. Physical Processes in Laser-Matter Interactions Figure-2:2: Illustration of the shock build up process in a laser driven shock Chapter 2. Physical Processes in Laser-Matter Interactions 15 pressure and internal energy is given by P = Pc + Pt (2.2) E = EC + Et (2.3) where Pc and Ec are the elastic force terms and Pt and Et are the thermal terms. Recall that pressure arises from the transfer of momentum of particles participating in thermal motion. The non-thermal pressure and energy terms arise from the interparticle binding forces alone, and remain finite even as the temperature approaches absolute zero. An. impor tan t aspect in the shock compression of solids is the distinction between strong and weak shock waves. In the weak shock l imi t , the principal contribution to the internal energy is Ec, and the shock speed is related to the compressibility of the material at standard conditions. This approaches the l imi t of an acoustic wave, and the compression of the material is of the order of a few percent. The pressure range for which shocks in solids are weak is up to ~0.1 M B a r . In the strong shock l imit , however, the energy imparted to the material by the shock wave far exceeds the material's elastic potential. The resulting shock heated material behaves as an ideal gas, and the ideal gas equation of state may be applied. In this case, the thermal term Et dominates. Pressures of the order of ~100 M B a r are needed to generate strong shock waves in solids. For pressures intermediate between these l imits , the thermal and elastic terms contribute approximately equally. The locus of thermodynamic states which can be reached in the shock compression process is determined by the equation of state (EOS) of the material. The E O S is gener-ally represented as a two-dimensional slice of the three-dimensional surface of pressure, density and temperature. Reversible thermodynamic processes are represented as contin-uous paths between ini t ia l arid final states on the E O S surface. The irreversible process Chapter 2. Physical Processes in Laser-Matter Interactions 16 of shock compression is represented as a jump discontinuity from ini t ia l to final states on the E O S surface. For a given ini t ia l state, the properties of the final state are constrained by the conservation of mass, momentum and energy across the shock front (refer to Figure 2.3). Mass and momentum conservation are expressed as V D-u V0 D (2.4) ^ ^ Du , „ s P - P 0 = — (2.5) where D is the shock speed and u is the particle speed behind the shock front. The equation for the conservation of energy across the shock front relates the internal ener-gies of the ini t ial and final states Eo, E , to the pressure and volume jumps across the discontinuity E - E 0 = ±x(P-P0)(V0-V). (2.6) Equations 2.4- 2.6 form the Rankine-Hugoniot relations, and can be derived from com-pletely general arguments. They are independent of the aggregate state of the material through which the shock propagates, and are valid for both spherical and planar shock waves [32]. N o assumptions of continuity of flow variables is made in their derivation. These relations can be obtained in a mathematically more formal manner by considering the shock front discontinuity as the l imi t ing case of very large but finite gradients. One then integrates the hydrodynamic equations of motion across this layer as the thickness tends to zero. The Rankine-Hugoniot relations constitute a system of three equations wi th five un-knowns. If the equation of state, P = P(p, E) is known for the material, it may be added as a fourth equation. This allows the system to be reduced to one equation with one free parameter, which yields the locus of thermodynamic states (i.e. the Hugoniot Chapter 2. Physical Processes in Laser-Matter Interactions 17 A P i P 1 E 1 - D Po P o E 0 Figure 2.3: Schematic of the shock discontinuity. P is pressure, p is density, E is total internal energy, D is shock speed and u is particle speed. Variables subscripted with a 0 represent the undisturbed solid. Particle speed ahead of the shock front is zero. Chapter 2. Physical Processes in Laser-Matter Interactions 18 curve) reached by a single shock for specific ini t ia l conditions. In Figure 2.4, a qualitative example of the Hugoniot, isotherm and isentrope are shown for comparison. 2.1.3 Impedance-Mismatch Technique The impedance-mismatch technique was first demonstrated for laser-target interactions by Vesser et al. [33]. Subsequent studies by Holmes et al. [34] and Cottet et al. [35] have verified the usefulness of this technique for pressure enhancement of shock waves at the interface of dissimilar materials. The important parameter characterizing the process is the material-dependent shock impedance, pUs. When the shock wave reaches the interface of the two media, reflected and transmitted waves are generated. Whether these waves are reflections or rarefactions depends upon the relative shock impedances of the two materials. Specifically, i f P 0 is the amplitude of the incident pressure pulse, and Pi, P 2 those of the reflected and transmitted pulses (refer to Figure 2.5) then [36] p _ {pUsh ~ (pUs)l p , x • 1 ~ 7~TT\ 1 I TT\  X r ° \ Z - ' ) {pUs)2 + {pUs)i 2{pUs)2 {pUs)2 + [pUs)i where the subscripts 1 and 2 denote the regions on either side of the interface. Con-servation of momentum and energy require that the pressure and particle velocity be continuous across the interface. Pressure enhancement can be achieved by propagating a shock into a material which has a higher shock impedance than the material from which the shock originated. Con-versely, impedance mismatch explains the low pressure release wave associated wi th shock breakout. Whi l e technically the impedance mismatch technique refers to shock propagation Chapter 2. Physical Processes in Laser-Matter Interactions A Figure 2.4: Hugoniot (solid), isotherm (dot-dash), and isentrope (dash). Chapter 2. Physical Processes in Laser-Matter Interactions 20 INTERFACE Pi P 1 E 1 P2 P 2 E 2 Figure 2.5: Schematic of the impedance mismatch technique. ph Pi, Eh USi, and UPi are the density, pressure, total internal energy, shock speed, and particle speed in the two regions. At the interface, PQ + Px = P 2 and Upl = Up2. Chapter 2. Physical Processes in Laser-Matter Interactions 21 through dissimilar materials, the idea can also be applied to uniform targets. For ex-ample, consider the case of two shock waves propagating through a uniform target. The second shock travels in the previously compressed and heated material, and is just about to overtake the first. Just after coalescence, a shock wave of a new pressure and speed is formed, which then traverses the undisturbed material. A t this point a counter-propagating rarefaction is generated. The rarefaction is due simply to the fact that there now exists a shock wave of a new pressure, and hence speed, traveling in a mate-rial wi th different density. Arguments using this idea of shock impedance can be used to explain the ' r inging' phenomena associated wi th shock compression of multi-material targets, and wi l l be evident when inspecting the hydrodynamic profiles inside targets presented in Chapter 4. 2.2 Atomic Physics 2.2.1 Introduction Radiat ion from laser produced plasmas can be identified as originating from essentially three different regions within the target [37]. Emission from the high temperature, low density coronal plasma consists mainly of faint continuum emission. This is due primarily to the fact that the most abundant ionic species are either fully ionized or are ground state H-like or He-like ions. The greatest part of the radiation originates from the region between the cri t ical density layer and the ablation front. Here, the plasma density and temperature are high enough to cause intense x-ray emission. The remaining target material, including the plasma state behind the shock front, is not hot enough to emit appreciable amounts of x-ray radiation, but rather, emits as a blackbody source [37]. For medium to high atomic number elements, the emission is dominated by bound-bound transitions [38, 39], whereas for low atomic numbers, the emission tends to be Chapter 2. Physical Processes in Laser-Matter Interactions 22 composed primari ly of recombination continuum [40]. The relatively low temperatures needed to fully ionize low atomic number materials leads to fewer bound electrons from which line emission can originate. As well, in low atomic number plasmas, bound-free and free-free emission dominate over bound-bound line radiation due to electron collisional quenching of the line radiation. The population density contributing to continuum radi-ation lies below.the collisional quenching l imi t [40], and therefore continuum radiation is unaffected. Furthermore, in high atomic number elements as much as 50% of the total incident laser radiation can be converted to x-rays [42, 72], making the plasma an intense source of radiation. This radiation represents a significant energy transport mechanism and can lead to several effects. X-rays can be deposited near the ablation front, enhancing the ablation process. X-ray radiation also represents an energy loss mechanism, which can lead to a reduction in ablation pressure and hence shock speed. More importantly, however, x-fays propagating into the target can preheat the material before the shock arrives, complicating the characterization of the shocked state. The first step to quantitatively describe the radiation emission is the development of a model to calculate the level populations of the ions in the plasma. This requires knowledge of al l the atomic processes and associated rate coefficients important in the regime of interest. The processes considered important in this work are described below, followed by a discussion of the non-local thermodynamic equilibrium model used. 2.2.2 Atomic Processes Collisional Ionization Coll is ional ionization Nz(j) + e=* Nz+1(k) + e + e (2.9) Chapter 2. Physical Processes in Laser-Matter Interactions 23 occurs when an ion of charge Z, state j, becomes ionized through a collision with a free electron. State k is some excited state or ground state different from state j. Three-Body Recombination As the inverse process of collisional ionization, three-body recombination consists of an electron being captured by an ion. Nz(j) + e + e=> Nz-\k) + e. (2.10) The non-recombining electron satisfies momentum conservation. Three-body recombina-tion rates are calculated using the method of detailed balance with collisional ionization rates and the resulting rate coefficients are proportional to the square of the electron density. This process can generally be neglected for plasma densities below 1016 cm - 3 [43]. Radiative Recombination Radiative recombination is the emission of a photon upon the capture of a free electron by an ion. Nz\j) +.e => Nz~\k) + hu (2lll) This process represents the major source of x-ray radiation emitted over a large range of plasma conditions. Radiative recombination rates are generally obtained through detailed balance with photo-ionization rates. i Chapter 2. Physical Processes in Laser-Matter Interactions 24 Photoexcitation and Photoionization As the plasma density increases to the point where reabsorption of radiation becomes important, the plasma is known as optically thick. In this scenario, population distribu-tions in the various excited levels of the different ionization stages are altered by pho-toexcitation and photoionization, which in turn affects the spectral emission [96]. This introduces a coupling between the radiation field and the population distribution which must be considered to obtain accurate spectral information. Physically, the processes of photoexcitation and photoionization are represented as Nz(j) + hu =» Nz(k) (2.12) and Nz(J) + hi/=> Nz+l{k) + e , (2.13) respectively. A n y model which ignores these effects is only treating the optically thin case, and may not provide an accurate calculation of radiation emission or transport levels. Collisional Excitation and De-excitation Collisional excitation and its inverse process of de-excitation, Nz(j) + e & Nz\k) + e (2.14) occur when electron-ion collisions induce a transition in the state of the bound electrons. Detailed balance may be used to calculate the de-excitation rate when the excitation rate is known. Chapter 2. Physical Processes in Laser-Matter Interactions 25 Radiative Transistions Also termed bound-bound transitions, radiative transitions Nz(j)=>Nz{k) + hv (2.15) are responsible for the line emission of atoms. Bound electrons make transitions to lower energy levels, releasing the excess energy in the form of radiation. Dielectronic Recombination The three-step process of dielectronic recombination starts wi th the capture of an electron by an ion into a level of large principal quantum number. The excess energy excites a bound electron Nz(j)+e^Nz-l(f,k') . (2.16) The state consisting of the two excited electrons is unstable, and can undergo radiative decay Nz~1(f, k') Nz~1(k) + hu . (2.17) Generally, the stabilizing radiative process is the decay of level j' into an excited state which lies below the ionization l imi t . The outer electron causes the radiated photon wavelength to be longer than that of the unperturbed resonance, giving rise to satellite lines in atomic spectra. 2.2.3 Non-Local Thermodynamic Equilibrium Model The shaded region of Figure 2.6 shows the extent of parameter space accessible in laser-produced plasma experiments. It is immediately apparent that only a very general model may be used to adequately describe the atomic processes present over the entire range Chapter 2. Physical Processes in Laser-Matter Interactions 26 TEMPERATURE [«V] Figure 2.6: Shaded region represents the parameter space accessible in laser-produced plasma experiments. Appl icabi l i ty of certain models is shown. For example, the line labeled L T E Z=5 corresponds to the range of validity of the local thermodynamic equi-l ibr ium model for Boron. Chapter 2. Physical Processes in Laser-Matter Interactions 2 7 of density and temperature. T he coronal model, which assumes that ionization is deter-mined by a balance between collisional ionization and radiative recombination, is only applicable where the radiative decay rate is much greater than the collisional decay rate 52Ajk».ne'EfXjk.. ( 2 . 1 8 ) k<j :k<j Many atomic processes are neglected with this model, making it a crude approximation except for very low density plasmas. The local thermodynamic equilibrium ( L T E ) model, on the other hand, is valid only at sufficiently high densities [44]. In this model, the charge state is determined from the Saha-Boltzmann equation [45] from which the ion level populations are calculated using the Boltzmann relation Here, wf and xf a r e the statistical weight and ionization energy, respectively, of level i. The high electron densities needed to make the assumptions of the L T E model valid preclude its use for modeling the physics of most plasmas. . Instead, a collisional-radiative equilibrium ( C R E ) or non-local.thermodynamic equi-librium ( N L T E ) model must be employed. This model is applicable over all density and temperature regimes, and gives equivalent results to the coronal and local thermodynamic equilibrium models in the limit of low and high electron densities, respectively. In this 'non-local thermodynamic equilibrium model, an atomic rate equation of the form ^L = Y,WkjNk-Y:W.l}Nj . ( 2 . 2 0 ) U l k k is written for each ground and excited state included in the model. Here, Wkj and 'W~k are the rates for populating and depopulating, respectively, the level Nj. These coefficients include all of the physical processes described in the previous section. If the Chapter 2. Physical Processes in Laser-Matter Interactions 28 hydrodynamic characteristics of the plasma change on a time scale greater than that of the important atomic processes, Equation 2.20 reduces to the steady state form o = E ^ * - E w 7 * 1 ^ • • (2-21) k k A n estimate of the time required to reach steady state has been suggested [46], tss = 8.98 x 1 0 - 1 0 A £ , (2.22) where the laser wavelength XL is in microns. For 0.53 pm light, tss ~ 200 picoseconds. This estimate for tss allows the rate equation 2.20 to be used in for few-nanosecond time scale simulations, but could pose problems for the shorter time scale simulations (~ 1 nanosecond) presented in Chapter 4. In principle, the non-local thermodynamic equilibrium model can be used for any plasma conditions provided accurate atomic rates for all processes are known. There are cases, however, where accurate atomic rates are unavailable. In this model, the processes of collisional excitation and collisional ionization are included despite possibly large uncertainties in their atomic rates. Fortunately, Salzmann [47] has concluded that parameters sensitive to the average ionization , such as internal energy, free-free and free-bound radiation, are generally insensitive to the rate coefficients. 2.3 Radiation Emission, Absorption and Transport 2.3.1 Bremsstrahlung Emission Thermal bremsstrahlung emission arises from the acceleration of an electron in the elec-trostatic field of an ion, with the electron as the primary emitter. The power radiated per unit volume per unit frequency interval is given by /•oo IfJ{y) = hvY,neNz I vef(E)doffdE (2.23) ry J 0 Chapter 2. Physical Processes in Laser-Matter Interactions 29 where ve and f(E) are the electron velocity and distribution functions, respectively. The differential cross-section for bremsstrahlung emission, doff, is usually expressed as the classical cross-section times a correction factor, gff, called the Gaunt factor. Integration of Equation 2.23 over a Maxwell ian electron distribution of temperature T e yields 6 8 x 10~ 4 7 7//W = — ^ 1 / 2 ne £ z2N%f e*p(-hv/Te) (2.24) le Z where / / / has the unit of [W cm"" 3 H z - 1 ] . The inverse bremsstrahlung absorption coefficient, Kff(u), can be evaluated by ap-plying KirchhofT's law [48] Zff(v)=*ff(y)B(v) (2.25) where the bremsstrahlung emission per unit solid angle, Cff(v), is calculated from Equa-tion 2.23 and B(u) is the Planck blackbody function „ , N 2 / u A 1 = — eMhv/kT) - 1 ' ( 2 ' 2 6 ) 2.3.2 Recombination Emission In the process of recombination, a free electron of energy E is captured by an ion into a bound level of ionization energy x f - 1 , and a photon of energy hv = (E + Xn~l) *s emitted. The energy of the emitted photon is thus a function of the electron energy E, and only photons with energy E > Xn~l a r e generated. This is the source of the recombination edges observed in free-bound continuum spectra [49, 50]. Using the hydrogenic ion cross-section [51], the power radiated per unit volume per unit frequency interval by free-bound radiation is 6 8 x 1 0 - 4 7 f -v-Z-i W = neJ2Z>NzeM~hv/Te) £ k^-g^Mx^/Te) (2.27) le Z n > n m i n neIe Chapter 2. Physical Processes in Laser-Matter Interactions 30 in units of [W c m - 3 H z - 1 ] . In Equation 2.27, gfb is the average free-bound Gaunt factor [52] and £ n is the number of available states for recombination. The sum n > nmin is taken over the different shells for which hv > Xn'1 • The photoionization absorption coefficient Hfb(u) can be calculated using detailed balance in conjunction with Kirchhoff's law and the Planck function. This leads to TJZ f v Z - l \ 2 -,40 TKTZ- 1 u \X ) C Z - 1 , ,-3. = 1.3 x i o « ^ - > ^ ^ ? f - v where Uz is the parti t ion function and /?/(, is in c m - 1 . (hv 1 — exp (2.28) 2.3.3 Line Emission Line emission results from the transition of an electron between bound energy levels of the ion. Specifically, for upper level u and lower level I, the radiated power per unit volume per unit frequency interval is simply Ibbi") = NuAulhuQ(j)e(u) , (2.29) also in units of [W c m - 3 H z - 1 ] . Au\ is the transition rate (Einstein-A coefficient) between levels u and I, 4>e(v) is the normalized emission line profile function, and Nu is the population of the upper level. Using the relationships between the Einstein A and B coefficients [48], the line ab-sorption coefficient can be written •'  KM=^ A- N'(j,- !k) M'' ) (2- 30) where (j>a{v) is the normalized absorption line profile function, and gu and gi are the, statistical weights for the two levels. Chapter 2. Physical Processes in Laser-Matter Interactions 31 2.4 Radiation Transport The simplest radiation transport problem to solve is that of the optically thin plasma, where reabsorption of radiation is neglected. In high-density laser-produced plasmas, however, this approximation is often invalid. Line intensities and line shapes can be sig-nificantly modified by absorption within the plasma. More importantly, though, atomic level populations are modified by photoexcitation and photoionization, leading to a cou-pling between the radiation field and the atomic level populations. If the radiation field is assumed to adjust instantaneously to changes in the plasma temperature and density, then the radiative transfer equation [53] takes the particularly simple form where tv and KV are the total emission and absorption coefficients, and include a l l of the processes described in the preceding section. Equation 2.31 is simply the statement that the intensity increment dlv over the path ds is equal to the increase e„(s)ds due to emission minus the decrease Ku(s)Il/(s)ds due to absorption. Consider the case of radiation transport in planar geometry as depicted in Figure 2.7. Using [i = cos 6 (9 being the angle between the path of an emitted photon and the x axis) and the optical depth rv dlu{s) *«/(S) - K„(S)IU(S) (2.31) ds (2.32) along with the source function Su (2.33) allows the radiative transfer equation to be written as dIu{Tv,n) drv = SV-Il V (2.34) Chapter 2. Physical Processes in Laser-Matter Interactions 32 Figure 2.7: Radiat ion transport in planar geometry. Chapter 2. Physical Processes in Laser-Matter Interactions 33 Equation 2.34 can be integrated to obtain the formal solution of the transfer equation IU(T*, A*) = MO", p) exp + jT exp { ^ ~ ) M ^ d < . ( 2 . 3 5 ) The special case of a constant source function Sv yields the solution J „ ( T „ , p) = Sv + exp x [7,(0, p) - S„] . (2.36) Two l imi t ing cases exist for this solution. In the optically thick l imi t (T„ > > 1), the emitted intensity reduces to I„{Tv,p) = Sv , (2.37) while the optically thin l imit (T„ < < 1) yields IV(Tv,p) = (\-Tv/p)Iv{Q,p) + ivs . (2.38) In the numerical calculations presented in Chapter 4, the optically thick l imi t described above is obtained by the inclusion of photoexcitation and photoionization processes. In calculating the energy transfer in planar geometry, it is useful to consider Iv as consisting of forward (/+) and backward (/~) components, subject to the boundary conditions /+(0,AX) = 0 , 7 ; ( T J , / / ) = 0 . (2.39) The energy flow per unit time per unit area perpendicular to the r axis is then given by FV{Tv) = F+- F~ (2.40) where the radiative fluxes are defined as F* = 2TT [±l I±pdp . (2.41.) Jo The absorbed power per unit volume is then calculated with HF Eabs(x) = ^  . (2.42) Chapter 2. Physical Processes in Laser-Matter Interactions 34 For numerical calculations it is convenient to consider energy absorption as occurring in one of n finite width cells using the expression Eg" = F„{T?) - FrW1) . • (2.43) The energy absorbed in cell n due to emission from cell 0 (including absorption in cell 0) is approximated with Eanbs = TfAx0[E2(T:-T»/2)-E2(T^-T°j2)} (2.44) where I0 is the radiated power per unit volume and E2 is the second order exponential integral. The factor of 1/2 accounts for the assumed isotropic forward and backward emission. Equation 2.44 is used extensively in the numerical calculations discussed in the following chapters. Chapter 3 Numerical Simulations This chapter presents a description of the numerical codes used to simulate both the hydrodynamics of the laser ablation process and the emission, absorption and transport of radiation through the target. A survey of the relevant input parameters used for the simulations is presented at the end of the chapter. 3.1 Introduction The hydrodynamics of the laser ablation process are modeled wi th the Laser-Target Code ( L T C ) , a substantially modified version of the M E D U S A inertial confinement fusion code [54]. L T C has been described in considerable detail elsewhere [55] and therefore only a brief overview of the important physics is presented here. The emphasis in L T C is placed on the accurate modeling of the shock compression process only, and so important physical processes such as radiation emission and trans-port have been neglected. These more complex processes are treated in H Y R A D , a code which couples the accurate hydrodynamic modeling of L T C wi th a detailed calculation of the emission, absorption and transport of radiation through the target. H Y R A D uses a non-local thermodynamic equil ibrium ( N L T E ) model to determine the ion state popu-lations and emitted radiation. Line and continuum radiation can be transported as in an optically thin plasma, or can be coupled to the population distribution to produce a sim-ulation corresponding to an optically thick plasma. Detailed tests have been performed for the N L T E model [56], and thus only the important aspects of the code are reviewed 35 Chapter 3. Numerical Simulations 36 here. Computer codes are necessary in modeling laser ablation, as the many simultaneous physical processes preclude the use of simpler analytic formulae [28, 57]. Simulations, however, are only approximate, and involve assumptions and simplifications. Numerical results should not be taken as the final word, as only comparison wi th experiment can validate the level of complexity used in a simulation. 3.2 Laser-Target Code ( L T C ) 3.2.1 Physical Content in L T C The one-dimensional hydrocode L T C solves the plane-parallel fluid equations for a single-fluid, two-temperature plasma. Electrons and ions are assumed to have the same velocity, but each species maintains a characteristic temperature. The most complete description involves a radiation field at a third temperature in equilibrium wi th the electrons and ions. The fluid equations are solved in the Lagrangean formalism, where the calculation follows the time evolution of the individual fluid elements. The time (t) and Lagrangean coordinate (m) are regarded as the independent variables. Specifically, the Lagrangean coordinate m(r, t) (in k g / m 2 ) is defined in terms of the laboratory coordinates (r,t) and the density profile p(r, t) by [25] where Ri(t) is the position of the free (rear) surface and r is the position of the cell under consideration. Equation 3.1 is simply an expression for the total mass of the Lagrangean cells situated between Ri(t) and r . The physics associated with the laser ablation process and subsequent hydrodynamic (3.1) Chapter 3. Numerical Simulations 37 motion of the material is contained wi th in a set of coupled differential equations rep-resenting conservation of mass, momentum and energy. In terms of m and t , these equations are dv du _ Q dt dm (3.2) du ( d(Pe + Pi) _ Q dt dm (3-3) dt + edt (3.4) dt + 1 dt ^ (3.5) d(U/P) dV _dF + dt + J U dt ~ d m + C K p (3.6) 1 dF _ d(fU) pXF c 2 dt dm c (3.7) The principal dependent variables which are solved for are specific volume V = 1/p, fluid velocity w, material internal energies Eei Ei (electron and ion components), radiation energy density U, and radiation flux F. Equations 3.2 and 3.3 represent fluid continuity and momentum conservation, respec-tively. The momentum equation considers the total hydrodynamic pressure, as indicated by the sum of the electron and ion components. Radiat ion pressure is negligible com-pared to typical shock pressures, and is thus excluded from the momentum equation. Equations 3.4 and 3.5 express energy conservation, for electrons and ions, in terms of the first law of thermodynamics. The energy source term Q is composed of the various energy transport and deposition processes included in the model. Equations 3.6 and 3.7 describe the interaction of the radiation field with the target material. The Eddington factor / in Equation 3.6 is restricted to the range | < / < 1. The detailed derivations of Equations 3.6 and 3.7 can be found in the work of Mihalas and Mihalas [53]. The radiation energy density and flux equations are included here Chapter 3. Numerical Simulations 38 only for completeness as radiation emission and transport wi thin the target is modeled in substantially greater detail with the N L T E model described in the following section. 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 in the prescription is the equation of state. Generally, the E O S is expressed in the form Te = Te(V,Ee), Pe = Pe(V,Ee) (3.8) Ti = Ti(V,Ei), Pi = Pi(V,Ei) (3.9) where V is the specific volume and T e , T j and Pe,Pi are the electron and ion tempera-ture and pressure components, respectively. Other important quantities include the ion density, P Ui — AmP (3.10) where A is the average atomic mass number of the target material and mp is the proton mass. The electron density is given by ne =< Z > x m , (3-11) and the Coulomb logarithm [54] is given by I n A = 1 6 . 3 4 - l n ( T e 3 / 2 n J 1 / 2 < Z >~l) . (3.12) The average ionization < Z > can be obtained by either interpolation from an appropriate table (such as the S E S A M E [89] database) or by calculation wi th an atomic model. The ionization for certain regions wi th in the target is calculated wi th the N L T E model. Values for the remainder of the regions are obtained from a table. The laser wavelength A is needed to define the crit ical density 1.1 x 10 1 5 A 2 (3.13) Chapter 3. Numerical Simulations 39 where nCT has the unit of m - 3 and the wavelength is in meters. Other necessary data such as electrical and thermal conductivities may be obtained from tabular formats so that the hydrocode can be applied equally well to a variety of target materials. The source terms which enter the conservation of energy equations are given by Qi = Hi- Kei (3.14) Qe = HE + Kei + Alas + Xrad (3.15) where H is the heat flow due to thermal conduction, KEI is the rate of energy transfer between the ions and electrons, and Aias is the rate of absorption of laser energy. The term Xrad accounts for radiation transport. Electron and ion heat conduction, HE and H{ , are calculated with the classical heat conduction model H = -V-KVT (3.16) P where K is the appropriate thermal conductivity and / is the flux limiter. For electronic heat conduction, K can be interpolated from a table (such as S E S A M E [89]) or calculated with the Spitzer expression [60] KE = 1.995 x I O - 9 0.095 ' Z + ° - 2 4 ' (1 + 0.24Z) T 5 / 2 e Z l n A (3.17) The ion thermal conductivity is calculated wi th the Spitzer equation [60] j,5/2 m = 4.3 x 1 0 - 1 2 - f - . (3.18) Z l n A v ' Electron-ion energy exchange is governed by the term [54] KEI = 0.59 x 10~ 8 n e (Tj - TE)T^2M~VZ2 In A (3.19) where M and Z are the atomic mass number and charge state, respectively, of the ions. Chapter 3. Numerical Simulations 40 Two methods can be employed to calculate the absorption of the incident laser radia-tion. The first method is adequate for ~500 picosecond or longer time-scale simulations, whereas the second method is better suited for few-hundred picosecond or shorter time-scale calculations. Results from these two methods are discussed in greater detail in the next Chapter. In the first method, laser energy is absorbed via inverse bremsstrahlung up to the crit ical density layer. The absorbed laser power at position (r, t) is given by Alas(r,t) =--^ (3.20) where $ t ( r , t) is the laser power reaching point r at time t. Using the absorption coeffi-cient of Stallcop and B i l lman [61] 13.51 p2 Z\nA A 2 ( 1 -where /3 = ne/ncr, $ L ( r , t) is calculated wi th a = 2 (1 - /?)V2 T 3 / 2 ^ ( 3 - 2 1 ) ®L(r,t) = <l>L{R0,t) rRo 1 — exp J —a(r',t)dr' (3.22) Laser energy which is not absorbed upon reaching the crit ical density layer is reflected back out of the plasma, where it can be re-absorbed on the way out. Laser energy escaping the plasma is lost. In the second method, known as the electromagnetic wave solver, absorption of laser energy is accomplished by solving the Helmholtz equations [62] for electromagnetic waves in a plasma [63]. The ini t ial cold solid and subsequent hot plasma are characterized by a temporally and spatially dependent complex dielectric function 247T e(z, i ) = 1 + : — x . a ( c j ) (3.23) where o(u>) is the electrical conductivity. A t each time step of the calculation, the Helmholtz equations are solved numerically to satisfy boundary conditions corresponding Chapter 3. Numerical Simulations 41 to incident and reflected waves in vacuum, and an evanescent wave inside the target. The solution yields the complex electric field amplitude, from which the reflectivity and energy deposition rate < E • J > = Re(o)\E\2/2 (3.24) are calculated. The energy deposition rate is then incoporated into the hydrodynamic equations as the heat source term A[as. The radiation source term X3Tad(t) for cell j is calculated wi th the N L T E segment of H Y R A D . This is discussed in greater detail in the following sections, but briefly, The terms El£ and E™ account for line and continuum radiation deposition respectively. 3.2.2 N u m e r i c a l M e t h o d s i n L T C The zoning scheme used in L T C is similar to that of M E D U S A , consisting of a mesh of cells of mass A m . Associated with each cell is a cell-centered average value for each principal dependent variable. Solution of the fluid equations is then obtained in two phases for each time-step. In the first phase, the piecewise parabolic method [25, 26] is used to solve a Riemann shock tube problem at the interface of each cell, thus advancing the hydrodynamic motion of the mesh. This defines the quantities r and u, and hence V and p , at the advanced time level. The Crank-Nicholson [54] time-centered differencing scheme is then used to solve the energy equations impl ic i t ly for Ee and E^ Solution of these equations is done iteratively to account for the nonlinear dependencies of the source terms and equation of state on the thermodynamic variables. One to three iterations are sufficient to achieve a convergence accuracy on the order of one percent. Chapter 3. Numerical Simulations 42 3.2.3 Comment on L T C The principle emphasis of L T C is to model the basic laser ablation process. Accordingly, several mechanisms have not been treated. Hot electron generation and transport has been ignored since for irradiances < 10 1 5 W / c m 2 , this process may not account for a significant fraction of energy transport [58, 59]. The various parametric processes (e.g. stimulated Br i l lou in scattering, stimulated Raman scattering, two plasmon decay, etc.) which can develop in the coronal plasma are also not considered. Most importantly, L T C does not contain the level of sophistication necessary to model the atomic physics of the coronal plasma. For this, the N L T E package described in the previous chapter is coupled to L T C . The next section discusses the details of the N L T E model developed. 3.3 Physics Content in the N L T E M o d e l 3.3.1 Calculation of State Population The collisional-radiative equilibrium model described in the previous chapter is used to calculate the ion state populations. The population of a given ion state Nz(j) is calculated wi th a rate equation of the form = [neNz-l(j")S&d» + neNz^(J>Z4i,r + " .(3-26) £ Nz^Atj + n e ]T Nz(t)X^ + ne £ J V z ( i ) X y ] i>j i>j i<j -Nz{j)[neSzzf>J' + n e a f + £ Atj + ne £ + ne £ Xtj] i<j i<j i>j where S^_Xy, is the collisional ionization rate from Nz~x{j") to Nz(j), a is the sum of radiative, dielectronic and three-body recombination rates, a = aR + a D I + nea5B, (3.27) Chapter 3. Numerical Simulations 43 Aij is the spontaneous decay rate from level i to j, and X y - , X ^ 1 are the collisional excitation and de-excitation rates from level i to j, respectively. Coupl ing between al l states is permitted within the model, but for the sake of com-putational speed, only coupling between an excited state and the ground state of the next ionization stage is considered, as illustrated in Figure 3.1. This approximation is valid for electron densities less than 10 2 2 c m - 3 [64]. For the near solid-density plasmas encountered in the following simulations, the coupling should also include non-adjacent ionization stages. However, a coupling scheme this complete would be too computation-ally intensive to be feasible, and so the above approximation must be employed. The average ionizations predicted by this N L T E calculation and a code which couples select non-adjacent ionization stages are examined in Section 3.5. The solution of the rate equa-tions can be further simplified if the characteristic time scale for changes in the plasma temperature and density is assumed to be long compared to the time scale of the atomic processes. This allows the steady state approximation dNz(j)/dt — 0 to be used. Once the coefficients for the rate equation have been calculated, the population dis-tr ibution of the ions is determined. The electron density dependence of the equations re-quires iteration of the solution unti l convergence to within a specified amount is achieved." 3.3.2 Rate Coefficients Ionization The ionization rate coefficient is calculated from the result of Landshoff and Perez [65] (3.28) 0.915 . 0.42 where S l j 1 ' 1 L(l + 0 . 0 6 4 * T e / x f )* (1 + 0MTe/xf)2\ has units of [cm 3/s] and f is the number of outer shell electrons. Figure 3.1: Level coupling scheme used in this work. S is ionization, a is radiative and three-body recombination, a D I is dielectronic recombination, X and X~l are collisional excitation and de-excitation, respectively, and A is the spontaneous decay. Chapter 3. Numerical Simulations 45 Three-Body Recombination Being the inverse process of collisional ionization means that the three-body recombina-t ion rate can be calculated by the method of detailed balance. Thus, where g(Z, 0) is the mult ipl ici ty of the ground state and g(Z — is the multiplici ty of the state formed by the recombination process. (OSZB)Z O 1 j n a s units of [cm 6 /s]. Radiative Recombination The radiative recombination rate, in [cm 3/s], is calculated using the method of Seaton [66], (<XR)zfld = 5 - 2 x l ( T 1 4 Z ( £ 1 / 2 ( 0 . 4 3 + 0.5 In 0 + O . 4 7 0 _ 1 / 3 ) , (3.30) where <f> = xf /(kTe) . Dielectronic Recombination The dielectronic recombination rate, in [cm 3/s], is calculated with the Burgess and Merts [67] prescription 2.4 x 1 0 - 9 _ ^ , (-E* where («C/)K''°= ' ^,/2 W E W J l - P ^ ) (3.31) B { z ) = fiff? z^20 P-32) E* Xij 1 + 0 . 0 1 5 Z 3 / ( Z + 1)^ (3.33) and A(Z, j) = xxl2l{\ + 0.105a; + 0.015a:2) A n = 0 (3.34) = 0 . 5 z 1 / 2 / ( l + 0.21a: + 0.30a;2) A n ^ 0 Chapter 3. Numerical Simulations 46 where the definition x = (1 + Z)*f- has been used. In these equations, Xij a n d hj a r e the excitation energy and oscillator strength respectively of the i —> j transition of the recombining ion of charge Z. S p o n t a n e o u s E m i s s i o n The spontaneous emission rate is derived from the oscillator strength. For transition ^% 1.5 xV' (3'35) where Qj and <?, are the statistical weights of the upper and lower levels, respectively, is the oscillator strength of the transition, and A is the wavelength of the emitted photon. C o l l i s i o n a l E x c i t a t i o n a n d D e - e x c i t a t i o n The collisional excitation rate is calculated with the semi-classical method of impact parameters [68]. Specifically, Xn = 1.578 x 10~*FII < y exp ( ^ ) (3.36) XijTe' v 7 e y where is the oscillator strength. < ^ • > is the four-parameter thermally averaged Gaunt factor taken from the work of Mewe [69]. Detailed balance is used to calculate the collisional de-excitation rate "^=IHi)^  •  (3-37> B o t h X^ and X^1 have units of [cm 3/s]. C o n t i n u u m L o w e r i n g In addition to the atomic processes listed above, the N L T E model also accounts for continuum lowering. This effect reduces the ionization potential of each state xf by an Chapter 3. Numerical Simulations 47 (3.38) Z' = <z2> <z> (3.39) z'(z' + i)jv; T ± e -,1/2 K = 1.937 x 10~ (3.40) 3.3.3 Photoexcitation Optical pumping of ion populations in the plasma is incorporated into the model through the concept of photon escape probability. The region under consideration is divided into n cells. Photons emitted in cell j have a finite probability Cji of being absorbed in cell i. If Nf is the total upper level population of cell i, and Au\ is the spontaneous transition rate between the levels of interest, then the upper level population of cell % is governed by a rate equation of the form The first two terms on the right hand side describe mechanisms responsible for populating and depopulating the level N™, and hence include al l of the atomic processes previously described. The third term accounts for the photoexcitation of the level by line radiation emitted from al l cells j into cell i. The occurrence of a photon being emitted and re-absorbed in the same region is implici t in this final term. As before, the steady state approximation dN^/dt = 0 is used to simplify calculations. A line photon emitted in cell j has a probability (3.41) Pji = Pe(Tji) ~ Pe(Tji + A T ; ) (3.42) Chapter 3. Numerical Simulations 48 of being absorbed in cell i. The angle-averaged probability Pe{r) that a photon traverses an optical depth r without being absorbed or scattered is defined by [70] 1 r°° P"(T) = 2J0 ^MrtJdy (3.43) where ipu and </>„ are the normalized emission and absorption line profiles and E2 is the second order exponential integral. The total line center optical depth A T , is related to the line center absorption coefficient KQ and the cell width Axi by Ar , ; = KoAo-i . . (3.44) The following approximate expressions for Pe, adopted for the model, Fe = (1 + 1.861607r0 + 0.817393r 0 2 ) - 1 r 0 < 3 ' (3.45) = 0 .286ro V / ( ln(1.95r 0 ) ) - 1 r 0 > 3 have been calculated assuming a Doppler line profile. The probability Cji is then calcu-lated with where the factor of 1/2 accounts for the assumed isotropic forward and backward emission of radiation. 3 .3 .4 Line Transport Non-resonant absorption of line photons is accounted for in the following manner. The optical depth of cell i at photon energy h u 0 is calculated wi th • A ^ W = [«//("<>)+ «f>o.)] AXi (3.47) where the bremsstrahlung absorption coefficient /«// is given by 3 4 x 10 6 n «// = ' ^ 1/2 ,  £ I 1 ~ exp(-/w/0/Te)] x Y.9ffZ2NZ (3.48) Chapter 3. Numerical Simulations 49 and.K^(^o) is the total inner shell photoabsorption coefficient as described in [56]. Neglecting photoexcitation, the number of photons N? absorbed in cell j due to line emission from cell i is evaluated wi th N[i=lNrAUL[E2(T^ + Ar^/2)-E2(T^ + ^ / 2 + Arr)}. (3.49) B y defining the continuum escape probability Pj%l(r) as P^(T) = E2(T™ + ATZ/2 + T), (3.50) the coupling coefficient Cji used to couple resonant absorption of line radiation to the ion population distribution can be approximated wi th A ppe C» = NH— — — (3.51) which includes al l of the absorption processes. Equation 3.51 employs the definitions AP** = P% (0) - Pg (Ar . ) (3.52) and APcn = P j f (0) - P^(Ari) . (3.53) Consequently the number of photons absorbed due to all other processes is b P APcn Nji - Nji ^ppe + p^cn • (3-54) which represents an absorbed energy of E% = hv0N$' (3.55) in the calculation of Xrad. Chapter 3. Numerical Simulations 50 3.3-5 Continuum Transport The method of multi-group diffusion [23] is used to transport bremsstrahlung and recom-bination radiation. This allows the radiation within each energy bin to be treated in a manner similar to that used for line transport. Specifically, for bin k, extending from uk to vk+u E%{k) = \mAxi [E2(T™ + A r f / 2 ) - E2(r% + + Ar™)] (3.56) where Ii(k) is given by Ti(k)= r+\lff{v) + Ifb{v))dv . (3.57) The spectral intensity per unit frequency interval per unit volume for bremsstrahlung and recombination radiation is calculated with IfM = J/2 "« E Z2Nzgff exp(-hu/Te) (3.58) and 6 8 x 1 fl~47 P v- 2 - 1 W = ' "« E Z2NZ e x p ( - ^ / T e ) £ ^ r - 9 f b expfe*"1/^ ) (3.59) respectively. 3.4 Modifications for Present Work The original implementation of the N L T E calculation was specific to homogeneous alu-minum targets. A number of changes were necessary to enable H Y R A D to handle the multi-material targets used for the present work. The C H layer on the laser deposition side is expected to dissociate completely into carbon and hydrogen ions. For this reason, the photoabsorption edge energies required Chapter 3. Numerical Simulations 51 Ionization Stage K-Edge (eV) Reference C+° 283.84 [20] C + 1 310.00 [71] C + 2 320.13 [71] • C + 3 354.31 [72] C + 4 390.67 [71] G + 5 489.99 [71] Table 3.1: Photoabsorption edge energies for the different ionization stages of carbon. are those of the various ionization stages of carbon atoms alone. Hydrogen ions do not contain any photoabsorption edges. Table 3.1 summarizes the K-edge energies used in calculating the inner shell optical depth in the C H layer. Opacities needed for the radiation transport are calculated by fitting fourth order polynomials to existing opacity data [20] in energy regions between absorption edges, as suggested by Biggs and Ligh th i l l [73]. In the case of C H (Z=3.5), the opacity is weighted by atomic mass, i.e. 86% for carbon and 14% for hydrogen. The N L T E calculation described previously made no reference to the specifics of the target material. Mater ia l dependence is introduced through the atomic input data. Atomic data for only one species could be included in the C R E calculation, and therefore the hydrogen ions in the C H plasma have been ignored. The carbon atomic data includes al l ionization states plus four excited states in each of the C + 4 and C + 5 excited states. Table 3.2 summarizes the electronic configurations considered by the present calculation. Electronic configurations in the lower ionization stages of carbon have been ignored since the relatively small ionization potentials coupled wi th moderate-to-high laser irradiances means that the majority of carbon atoms wi l l be fully ionized. Table 3.3 lists the chosen line transitions with corresponding oscillator strengths. Oscillator strengths have been obtained from the work of Wiese [74]. Chapter 3. Numerical Simulations 52 Ionization Stage Outer Electron Ionization Potential (eV) c +o 2p 2 11.260 C + 1 2p 24.384 c + 2 . 2s 2 47.888 C + 3 2s 64.494 C+4 I s 2 392.09 C + 4 2p 84.128 C + 4 3p 37.502 C + 4 4p 21.088 C + 4 5p 13.433 C + 5 l s 489.99 C + 5 2p 122.36 C + 5 3p 54.255 C + 5 4p 30.425 C + 5 5p 19.468 C+6 — ' 0.00 Table 3.2: Ionization potentials for the ground and excited states included in the N L T E model. Chapter 3. Numerical Simulations 53 Transition Oscillator Strength l s 2 - l s 5 p 0.024 l s 2 - l s 4 p 0.051 l s 2 - l s 3 p 0.141 l s 2 - l s 2 p 0.647 l s -5p 0.014 l s - 4 p 0.029 l s -3p 0.079 l s - 2 p 0.415 Table 3.3: Transitions and oscillator strengths used in the N L T E calculation. 3.5 C o m p a r i s o n o f N L T E C a l c u l a t i o n w i t h R A T I O N C o d e To test the validity of the present N L T E calculation, average ionizations produced by our calculation for a carbon plasma were compared to those produced wi th the widely used [75] atomic physics code R A T I O N [76]. A brief description of the physics included in R A T I O N is presented here, wi th an emphasis on how R A T I O N differs from our N L T E calculation. A more detailed description of the level structure and coupling considered by the R A T I O N code can be found in References [76] and [106]. 3.5.1 E n e r g y L e v e l s a n d L e v e l C o u p l i n g In the calculation of ion level populations by R A T I O N , a distinct emphasis is placed on the Li - l ike through fully-stripped ionization stages. Ion species of neutral through Be-like are represented simply as a single ground state and its associated ionization potential. There are a total of 88 energy levels included in the R A T I O N calculation for carbon, and the ionization stages of Li- l ike through fully stripped comprise 84 of these levels. Li- l ike levels include detailed nl configurations for all / levels up to n = 4, wi th level energies calculated from a method due to Edlen [77]. Energies for the n — 5 level are Chapter 3. Numerical Simulations 54 calculated with the formula of Vainstein and Safronova [78]. Energies of levels with principal quantum number, n = 6-10 are calculated with another method of Edlen [79] He-like energy levels in states wi th principal quantum number n = 2-7 are taken from the data compilation of Scofield [80]. For levels between n — 7 and n = 10, the hydrogenic formula n . (3.60) ' E(n,Z) = Ip(Z) is used, where IP(Z) is the ionization potential of the He-like species, also obtained from Scofield [80]. Equation 3.60 is used for H-like energy levels with n = 2-10, where Ip(Z) is again taken from the work of Scofield [80]. The energy levels used in the N L T E calculation of this work were taken from the data tables of K e l l y [81]. Similar to the R A T I O N code, our calculation contains an expanded level structure in higher ionization stages, specifically 4 excited levels in each of the C + 4 and C + 5 ionization stages (refer to Table 3.2). However, the N L T E calculation contains a total of only 15 states, far fewer than considered by R A T I O N . The 88 energy levels considered by R A T I O N represent specific atomic transitions. Many of these transitions occur between excited levels of adjacent and non-adjacent ionization states, in contrast to our N L T E calculation which couples excited states to ground states of the next ionization stage (refer to Figure 3.1). The coupling scheme in R A T I O N thus represents a more complete description of the atomic processes in the plasma. 3.5.2 Rate Coefficients A wide variety of rate coefficients are employed by R A T I O N , typically with the choice of equation wi thin a particular process dependent upon the ion species. This represents a level of complexity not included in our calculation. Chapter 3. Numerical Simulations 55 The collisional ionization rate is calculated with the prescription of Lotz [82], and detailed balance with this rate is used to determine the three-body recombination rate. The spontaneous radiative recombination rate for all ion stages except H-like is calculated with the formula of Spitzer [83]. For H-like ions, this rate is determined by the method of Seaton [66]. The stimulated radiative recombination, photoionization, and photoex-citation rates, for al l ion species, are calculated according to the methods outlined in the work of Mihalas [84]. The collisional excitation rate for H-like species employs the formula found in the work of V a n Regemorter [85], whereas the methods of Cochrane and McWhi r t e r [86] are used for ionization stages between neutral and He-like. Autoion-ization rates for He-like ions are calculated from the work of Vainstein and Safronova [87], whereas this rate for Li - l ike ions is taken from the work of Gabriel [88]. Inverse autoionization rates for the two ion species are determined by the principal of detailed balance. The R A T I O N code contains al l of the atomic processes considered by our N L T E cal-culations, plus many that are not, namely, autoionization, inverse autoionization and stimulated radiative recombination. The only rate coefficient formula common to R A -T I O N and our N L T E calculation is the spontaneous radiative, recombination rate of Seaton [66]. 3.5.3 Results of Comparison Figures 3.2 to 3.6 illustrate the difference in average ionizations produced by the two calculations for a carbon plasma over the temperature range 50 to 550 eV and electron density range 10 1 9 to 10 2 3 c m - 3 . The greatest difference between the two calculations occurs for high density and low temperature plasma conditions. For example, at an electron density of 10 2 3 c m - 3 and plasma temperature of 50 eV, R A T I O N predicts an average ionization of 3.60, whereas the N L T E calculation predicts a value of 3.31. The Chapter 3. Numerical Simulations 56 6.00 5.95 h •I 5.90 CO N 'c o CD O) CO S 5.85 5.80 -5.75 100.0 200.0 300.0 400.0 Temperature (eV) 500.0 600.0 Figure 3.2: Ionizations produced by the present N L T E calculation and R A T I O N for a carbon plasma with an electron density of 10 1 9 c m - 3 . The dashed lines with circles correspond to R A T I O N calculations, whereas the solid lines wi th squares correspond to N L T E calculations. Chapter 3. Numerical Simulations 57 6.00 | — - n 1 • r < 5.80 -5 75 I , I lL I i _ l i I , L i I . 0.0 100.0 200.0 * 300.0 400.0 500.0 600.0 Temperature (eV) Figure 3.3: Ionizations produced by the present N L T E calculation and R A T I O N for a carbon plasma with an electron density of 1020 c m - 3 . The dashed lines wi th circles correspond to R A T I O N calculations, whereas the solid lines with squares correspond to N L T E calculations. Chapter 3. Numerical Simulations 58 6.00 Temperature (eV) Figure 3.4: Ionizations produced by the present N L T E calculation and R A T I O N for a carbon plasma wi th an electron density of 1021 c m - 3 . The dashed lines with circles correspond to R A T I O N calculations, whereas the solid lines wi th squares correspond to N L T E calculations. Chapter 3. Numerical Simulations 59 6.5 5.5 c o CO _N ' c o CD U) CO i _ CD > < 4.5 - . - - - . - 0 - T - - - 0 3.5 1 1 1 • 1 • 1 • 1 i • I 0.0 100.0 200.0 300.0 400.0 500.0 600.0 Temperature (eV) Figure 3.5: Ionizations produced by the present NLTE calculation and RATION for a carbon plasma with an electron density of 1022 cm"3. The dashed lines with circles correspond to RATION calculations, whereas the solid lines with squares correspond to NLTE calculations. Chapter 3. Numerical Simulations 60 2 0 I i i i J i i i i J i i ' 0.0 100.0 200.0 300.0 400.0 500.0 600.0 Temperature (eV) Figure 3.6: Ionizations produced by the present N L T E calculation and R A T I O N for a carbon plasma with an electron density of 10 2 3 cm"" 3. The dashed lines wi th circles correspond to R A T I O N calculations, whereas the solid lines wi th squares correspond to N L T E calculations. Chapter 3. Numerical Simulations 61 difference between the two values is always less than 10%, and decreases as electron density and temperature increase. A t near-solid density and high temperature conditions the two calculations give very similar results (refer to Figures 3.5 and 3.6). Thus, even though the level coupling scheme used in our N L T E calculation is theoretically only valid for ne < 10 2 2 c m - 3 [64], the fact that it predicts ionizations very close to those predicted by R A T I O N (which considers a much broader coupling scheme) allows the N L T E model to be used for the near solid-density plasmas encountered in these simulations. 3.6 Survey of Simulation Parameters In al l simulations, the numerical mesh is divided into five regions consisting of 150-300 cells (refer to Figure 3.7). The number of cells used varies with target thickness, and is chosen to simultaneously maximize both numerical resolution and computational speed. The first three regions on the laser deposition side consist of the C H compound. The first of these three regions is uniformly zoned with a thickness per cell of ~ . 0.05 A. Cel l width in this region must be less than 10% of the laser wavelength to achieve sufficient resolution in the ablation process. The second region is geometrically zoned to provide a smooth transition between the finely zoned front side region and the more coarsely zoned third region. The bulk of the C H lies in the th i rd region, constructed with a width per cell of 0.2 u.m. Simulations with a more finely zoned target did not produce quantitatively different results, but required more computing time. The fourth region contains the required 3.0 pm of aluminum. The code more easily handles constant volume per cell across dissimilar material interfaces (as opposed to constant mass per cell), and hence the aluminum layer spans 15 cells. Following the convention used at the Chapter 3. Numerical Simulations LASER C H CH CH Al Si Figure 3.7: Five-zone scheme used in target design. Chapter 3. Numerical Simulations 63 CH-A1 interface, the cell width in the silicon layer is also set to 0.2 //m per cell. The thickness and hence number of cells in this fifth region depends upon the specific target, and therefore no single set of values was used for all of the simulations. The S E S A M E [89] equation of state table 7590 describes the thermodynamics of the C H layer. Electr ical and thermal conductivities are calculated with the Spitzer formulae as tabular data is currently unavailable in our laboratory. The ionization state and opacity of each C H cell is calculated wi th the N L T E model. The Q E O S [90] equation of state is used to generate E O S data along with electrical and thermal conductivities for the aluminum. Ionization state was also obtained with Q E O S , but opacity values were calculated using the N L T E model. Equat ion of state data, electrical and thermal conductivities and ionization state for the silicon layer is interpolated from the Q E O S tables. Opacity calculations follow from the Biggs and Lighth i l l approach used in the N L T E model. A l l simulations considered the following processes in the C H layer : collisional ion-ization, radiative, three-body, and dielectronic recombination, collisional excitation , col-lisional de-excitation, spontaneous de-excitation and finally, continuum lowering. The full radiation transport package, consisting of continuum transport, line transport, and photoexcitation, was used for the C H , aluminum, and silicon layers. Results of the sim-ulations are discussed in the following chapter. Chapter 4 Results of Radiation-Hydrodynamic Simulations This chapter presents a survey of the results of the radiation-hydrodynamic computer code simulations. F ina l parameters of target and laser pulse design are summarized, along wi th density and temperature profiles corresponding to each of the shock compression processes. 4.1 General Results 4.1.1 Hydrodynamics Figure 4.1 shows a snapshot of the hydrodynamic profiles wi th in the target during a simulation of the laser ablation process. Tables 4.1 and 4.2 present a summary of the pressures and shock speeds produced in the C H and aluminum layers from single shock wave simulations. Many features of the hydrodynamic evolution of the shock wave agree with previous work [38],[91]-[97]. 4.1.2 Radiation Transport The most obvious artifact of radiation transport is the radiatively heated zone ( R H Z ) . This is the optically thick region of nearly constant density and temperature occurring immediately behind the shock compressed material. The explanation for the formation of this region involves analysis of the mean free path of x-rays emitted from the front side plasma. As the shock propagates through the target, the width of the compressed 64 Chapter 4. Results of Radiation-Hydrodynamic Simulations 65 1000.00 100.00 10.00 1.00 0.10 0.01 7 1 — i — l — i 1—l—l—r -CORONA - | — l — l — l — l — I — l — i — i — i — i — i — i — r -0.0 \ Pressure (MBar) Density (x100kg/m3j| Temperature (eV) SHOCK / RHZ Al CH Si 10.0 20.0 30.0 Position (microns) 40.0 Figure 4.1: Snapshot of hydrodynamic profiles within the target at 200 picoseconds of simulation time. The shock wave is shown .propagating through the C H layer, imminent upon the CH-A1 boundary. This particular target consists of 27 C H - 3 pmA\ - 6 /j,m S i . The laser irradiance is 3.75 x 10 1 4 W / c m 2 . Chapter 4. Results of Radiation-Hydrodynamic Simulations 66 Irradiance(10 1 3 W / c m 2 ) Pressure(MBar) Speed(10 4 m/s) 1.25 2.36 1.75 2.50 3.93 2.43 6.25 9.29 3.02 37.5 36.2 6.75 Table 4.1: Pressures and speeds characteristic of single shock waves in the C H layer. Irradiance(10 1 3 W / c m 2 ) Pressure(MBar) Speed(10 4 m/s) 1.25 3.90 1.50 2.50 6.49 1.88 6.25 15.8 2.72 37.5 61.5 6.02 Table 4.2: Pressures and speeds characteristic of single shock waves in the A l layer. region becomes larger than the mean free path for some subset of wavelengths emitted from the front side plasma. Hence, a larger portion of x-ray deposition occurs behind the shock front, rather than ahead of it. In uniform aluminum targets, this effect can actually eliminate a significant fraction of preheat [38]. This is clearly not the case for C H , arid so x-ray preheat is an important factor for al l times during the hydrodynamic evolution. The width of the R H Z grows linearly in time, similar to the results obtained by Salzmann et al. [38] in homogeneous aluminum targets. The radiation front associated with the R H Z , propagating behind the shock front, is responsible for the second, delayed heating process observed in rear surface emission studies [96, 98, 99]. For moderate irradiances, the shock wave outruns the radiation front. However, for irradiances of ~ 1 0 1 6 W / c m 2 and higher, the radiation front w i l l run ahead of the shock front [97], greatly heating and decompressing the target material so that the shock process is relatively ineffective. The levels of radiation emission and transport are consistent with previous studies Chapter 4. Results of Radiation-Hydrodynamic Simulations 67 using laser heated carbon foils [40]. For laser irradiances of 1.25, 2.50, and 6 . 2 5 x l 0 1 3 W / c m 2 , the percentages emitted by the various processes breaks down approximately as : 6% free-free, 87% bound-free, and 7% bound-bound. For the .3 .75xlO 1 4 W / c m 2 case, the percentages are : 7% free-free, 89% bound-free, and 4% bound-bound. The fraction of bremsstrahlung and recombination (recombination is pr imari ly into K shell levels [40]) increases with increasing laser irradiance, and the level of bound-bound radi-ation decreases accordingly. This can be attributed to the higher average ionization and hence fewer available electrons to make bound-bound transitions within the ions. The bound-free intensity greatly exceeds the bound-bound intensity due to,electron collisional quenching of the line radiation [40]. The population density contributing to continuum radiation lies below the collisional quenching density, and hence is unaffected. The small difference in the amount of line radiation between this work and that of Ref. [40] can be attributed to the number of bound transitions included in the model. However, since continuum radiation accounts for over 90% of the emitted radiation, a difference of a few percent in the level of line radiation cannot be expected to have any significant effect on target hydrodynamics. Final ly, the ionization burn wave [96] which induces transparency to certain emission lines in aluminum targets is not expected to have a significant effect on radiation transport through the C H layer. Most of the emission lies above the 490 eV K-shel l energy of C 6 + [40], and hence wi l l be attenuated by inner shell photoionization processes regardless of the ionization state. 4.1.3 Laser Absorption For certain sets of simulation parameters, the two methods of calculating the deposition of laser energy within the target (refer to Section 3.2.1) predict different hydrodynamic conditions. The question then arises as to the validity of each of the absorption routines. Chapter 4. Results of Radiation-Hydrodynamic Simulations 68 Routine % Dump at n c r C H . Pressure(MBar) A l Pressure(MBar) E M Wave Solver — 15.3 26.5 Inv. Brem. 100 26.7 45.5 Inv. Brem. 10 19.3 32.7 Inv. Brem. 1 15.6 27.1 Inv. Brem. 0.1 9.6 16.0 Table 4.3: Pressures in the C H and aluminum layers when different laser absorption routines are used. The inverse bremsstrahlung absorption routine contains a free parameter which dic-tates the amount of laser energy deposited at the crit ical density surface. This energy dump is crucial in the ini t ia l formation of the plasma, but can be expected to become less important for laser pulses with nanosecond risetimes since a sizable fraction of laser energy wi l l reach the crit ical density layer only at very early times during the pulse. O n the other hand, for laser pulses with sub-nanosecond risetimes, the absorption process wi l l show a significant dependence on the value of this parameter. The calculation which solves the Helmholtz equations, on the other hand, contains no such free parameter. To study the effect of the energy dump parameter contained in the inverse bremsstrahlung absorption routine, a series of simulations were performed. A single trapezoidal pulse of intensity 2 x l 0 1 4 W / c m 2 was assumed'incident on a 10.9/xm C H - 2/xm A l - 10pm Si target. The two absorption calculations were employed, and the switch to control the energy dump in the inverse bremsstrahlung routine was varied from 100% to 0.1%. Normal incidence with s-polarized light (i.e. E parallel to the plane of incidence) was used in the electromagnetic wave solver calculation. The flux limiter, another free pa-rameter, was fixed at 0.1, a value suggested by detailed Fokker-Plank calculations [100]. Table 4.3 presents a summary of the shock wave pressures in the C H and aluminum layers corresponding to the different simulation parameters. Chapter 4. Results of Racliation-Hydrodynamic Simulations 69 The inverse bremsstrahlung absorption routine with 1% dump at the critical density layer predicts roughly the same hydrodynamic conditions as the electromagnetic wave solver. The reason for this is evident in Figure 4.2, where the reflectivity of the absorbing plasma is plotted against time. Dur ing the first ~100 picoseconds, the two calculations predict, very different absorptions. However, at ~200 picoseconds, the absorption levels are roughly the same, and by 300 picoseconds, they are nearly identical. The spikes occuring in the reflectivity level of the inverse bremmstrahlung absorption routine are a result of the finite number of cells used to model the plasma density gradient in the absorbing region. Hence, for few-hundred picosecond time-scale and longer simulations, the two absorp-t ion routines wi l l lead to the same hydrodynamic conditions. However, for simulations on time-scales shorter than ~300 picoseconds, care must be taken to choose the correct absorption mechanism. 4.1.4 Flux Limiter One of the largest uncertainties in computer simulations of this type concerns value of the flux limiter, / , to be used. A value of / = 0.6, corresponding to a free-streaming flux for a Maxwell ian electron velocity distribution, is almost certainly too large for the present laser irradiances. Lower values of / l imit the amount of electronic heat conduction from the crit ical density surface to the cold solid, thus producing a much hotter coronal plasma. As less heat is transported to the ablation front, the the shock pressure within the target is reduced and the shock arrival at the aluminum layer is delayed. To obtain the general trend of shock characteristics on flux l imiter value, three simu-lations have been performed with the following parameters : 2.0 x 10 1 4 W / c m 2 on 10.9/xm C H - 2yum A l - 10/xm Si . Absorption of laser energy is calculated with the electromag-netic wave solver routine. The value for / is varied, with al l other simulation parameters oo 1— 1— 1—•— 1— 1—•— 1— 1— J— 1— 1—•—>— 1 1 • • • 1 • • •—• I 0.0 100.0 200.0 300.0 400.0 500.0 Time (ps) Figure 4.2: Plot of reflectivity as a function of time for both the electromagnetic wave solver and inverse bremsstrahlung absorption routines. Chapter 4'. Results of Radiation-Hydrodynamic Simulations 71 F l u x Limi ter Peak Pressure (MBar) Shock Ar r iva l T ime (ps) 0.60 36.9 214 0.10 26.5 235 0.03 18.5 247 Table 4.4: Dependence of peak pressure within and shock arrival time at the aluminum layer on the value of the flux limiter used. Simulation parameters are described in the text. held constant. The results for conditions wi thin the aluminum layer are summarized in Table 4.4. The different values of the flux limiter have significant effects on target hydrody-namics, with shock speed and pressure showing a larger variation than compression or temperature. The variation in shock speed wi l l shift the temporal position of the steady state conditions and so this factor must be carefully considered during experimental de-sign. Furthermore, the flux l imiter wi l l also play a cri t ical role in schemes where the t iming between multiple shock waves is important. B y experimentally characterizing shock transit through aluminum and C H targets, and comparing with numerical simula-tions, the correct value for the flux limiter can be deduced. 4.1.5 F ina l Choice of Absorption Routine and Flux Limiter The absorption routine which solves the Helmholtz equations, assuming plane-polarized light, was chosen for the final calculations. This routine, coupled with a flux l imiter of 0.1, predicts hydrodynamic conditions similar to those of more well established hydrocodes [101]. Chapter 4. Results of Radiation-Hydrodynamic Simulations 72 4.2 Optimized Shock Compression Schemes 4.2.1 Laser Pulse and Target Design The laser pulse consists of one or two intensity plateaus, wi th linear rise and fall times between each level. The rise and fall times are fixed at 100 picoseconds, in accordance with current shaped laser pulses. Moreover, this short, linear rise time is capable of generating shock waves at intensities below which shock wave formation is not possible with Gaussian laser pulses. The individual intensities 7; and plateau lengths Tj of each laser pulse are varied to achieve the appropriate hydrodynamic conditions. Figure 4.3 presents a visual aid to the relevant laser parameters. Al though the essential target characteristics were described in the introduction, a more detailed description would prove useful. The C H layer on the laser deposition side serves three purposes. Firs t ly , it provides an absorption medium to minimize the effects of radiation preheat on the aluminum layer. Secondly, by acting as a material of low shock impedance, the impedance mismatch between this layer and the aluminum layer allows enhancement of the shock pressure as the shock wave passes through the CH-a luminum boundary. Final ly , the C H provides support to the shock compressed aluminum layer, preventing release out the front side before steady state is obtained for the necessary time. B y varying the C H thickness, the proper delay between multiple shocks, rarefactions and release waves can be achieved. Whi le increasing the thickness of this layer would certainly reduce the level of radiative preheat, excessive thickness would degrade the absorption measurement. The C H thicknesses presented as final target designs represent a trade off between preheat and absorption measurement factors. The sole restriction in target design concerns the thickness of the aluminum layer, which is preset to 3.0 yum. This value represents a compromise between transmission level of the absorption signal (from the backlighter) and observable effect. If the layer is too Chapter 4. Results of Radiation-Hydrodynamic Simulations 73 100 X1 ioo x 2 1 0 0 Time (ps) Fi gure 4.3: Visual aid to the important parameters in laser pulse design. The rise and fall times between levels is fixed at 100 ps. Chapter 4. Results of Radiation-Hydrodynamic Simulations 74 thick, the intensity of the transmitted x-rays may be too faint to detect wi th the desired accuracy. Conversely, if the aluminum is too thin, the amount of material displaying the photoabsorption edge shift may be too litt le for a definitive measurement. The silicon layer serves only to prevent premature shock breakout and subsequent release of the aluminum layer. This layer must be made as thin as possible to minimize the effects of absorption and hence signal loss, and yet must maintain steady state in the aluminum layer for the required time. The similar atomic properties between this layer and the aluminum make silicon the natural choice for target design. Moreover, the use of silicon could aid in target fabrication ; bulk silicon could easily be etched to the desired thickness, and the aluminum and polystyrene layers deposited on top. 4.2.2 Density and Temperature Conditions Table 4.5 and Figure 4.4 summarize the density and temperature conditions achieved in the aluminum layer with the different shock compression processes. The quoted values represent the average value of five of the Lagrangean mesh points (regularly spaced across the aluminum layer) at a time half way through the steady state plateau. Steady state is considered to have been reached after the rear aluminum mesh has been shocked up and the rarefaction generated at the aluminum-silicon interface (described later) has traversed the entire aluminum layer. The uncertainties are the difference between the highest and lowest values achieved by any one of the Lagrangean mesh points in the layer (during the steady state time) and the average value described above. 4.2.3 Discussion of Individual Compression Schemes To aid in future target design, a brief discussion of the important processes in each of the shock compressions of Figure 4.4 is given. Emphasis is placed on the description of the shock and rarefaction waves propagating inside the target. Chapter 4. Results of Radiation-Hydrodynamic Simulations 75 100 > CD "co 10 zs •I—• CO I — CD CL E CD ~1 1 1 r - ~| 1 T 1 f~ Reference d x ^ High Z* 5 Isochore t x ^ 1.5 2.0 2.5 3.0 Compression H 1 I l-Isotherm J 1 i i i I i i i i I _l I L_ 3.5 4.0 Figure 4.4: Illustration of the densities and temperatures achieved in the shock compres-sion schemes. Chapter 4. Results of Radiation-Hydrodynamic Simulations 76 State P/PAI T ( e V ) 1 C H - A l - S i nm r l , 2 (P'S) Ii,2 10 1 3 W / c m 2 Reference. 2 .4±0.2 8 .5±0.5 18-3-4 700 , — 6.25 , — Isochore 2 .4±0 .3 3 .3±0 .3 22-3-5 900 , 600 1.25 , 2.25 Ptx, Ttx 2 .0±0 .1 2 .5±0 .1 22-3-5 900 , 600 1.25 , 2.25 Isotherm 3 .7±0 .4 8 .5±0 .8 27-3-7 900 , 600 2.5 , 10.0 Pdx > Tfa 2.2±0 .1 4 .9±0 .3 27-3-7 900 , 600 2.5 , 10.0 High Z* 3 .1±0 .3 29 .0±2 .0 27-3-6 700 , — 37.5 , — Table 4.5: Compression and temperature conditions obtained in the shock compression schemes, along wi th final target and laser pulse parameters! Chapter 4. Results of Radiation-Hydrodynamic Simulations 77 The most important factor to consider when choosing the compression and temper-ature conditions of the reference state, and hence the isothermal and isochoric paths, is radiative preheat. The relatively low compressions and temperatures chosen to map out the compression-temperature space can be obtained with sufficiently low laser irradiances that radiative preheat does not seriously degrade steady-state conditions. O n the other hand, the plasma conditions generated should st i l l produce a measurable K-edge shift [16, 17, 56]. The graphs in Figures 4.5 to 4.17 represent the time evolution of 5 Lagrangean mesh cells wi thin the aluminum layer. The first and last cells of the aluminum layer are included, as well as three others which are equally spaced among the remaining cells. T ime zero on the graph corresponds to the start of the laser pulse. The time interval over which quasi-steady state conditions are assumed to he present is indicated by a horizontal line above the plot of the Lagrangean mesh. In a l l of the figures, the cells are labeled as follows : front side (first cell) —, fourth cell • - -, eigth cell , twelfth cell -— , and rear side (fifteenth cell) - • - . Reference State Plots of compression, temperature and pressure wi thin the aluminum layer during the shock compression process associated wi th the reference state appear in Figures 4.5 to 4.7. As stated previously, the term 'reference state' is merely a label for the state with which conditions on the three branches are compared in order to deduce the effects of varying hydrodynamic conditions on the K-edge energy. Much of the analysis' of this state is common to the other shock produced states, and is only described once. Radiat ion transport is a noticeable factor in this compression scheme, manifesting itself in a variety of ways. First , there are the 0.8 eV and ~10% heated and decompressed conditions in the aluminum prior to shock arrival. However, these conditions are not Figure 4.5: Compression (i.e. density) as a function of time within the aluminum layer for the reference state. The decompression prior to shock arrival is due to radiative preheat. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. Figure 4.6: Temperature as a function of time within the aluminum layer for the reference state. Note the temperature increase prior to shock arrival due to radiation preheat. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. Chapter 4. Results of Radiation-Hydrodynamic Simulations 80 Figure 4.7: Pressure as a function of time within the aluminum layer for the reference state. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. Chapter 4. Results of Radiation-Hydrodynamic Simulations 81 uniform across the layer, and as expected, the higher temperatures and decompressions experienced by the cells closer to the laser deposition side of the target result in lower peak shock compression. Connected to the non-uniformity in radiation preheat is the spread of densities and temperatures observed across the aluminum layer at any time during the steady state plateau. This range arises as a consequence of the cells starting at different in i t ia l conditions in the shock compression process. Final ly , radiation emitted from the front side plasma causes continual decompression of the aluminum layer after the shock compression process is complete. The heating due to radiation absorption is offset by the cooling associated with the expansion process, and so even though the density drops, the temperature remains nearly constant. Single shock processes are the easiest to describe simply because there are fewer counter- and co-propagating waves within the target. The decompression (from rear to front) visible immediately after the rear-most aluminum cell is shock compressed can be attributed to a rarefaction wave generated at the aluminum-silicon interface. This effect is apparent in all of the simulations, resulting from an impedance mismatch between layers common to al l the target designs considered in this work. After this rarefaction has traversed the entire aluminum layer, the reference state (po,T0) forms, existing for ~150 picoseconds. The lifetime of this state is l imited by shock breakout. The rarefaction wave associated with release from the target rear surface propagates back into the aluminum, illustrated as the rear mesh being the first to decompress. Theoretically, a thinner silicon layer could be used to force the steady state to a 100 picosecond lifetime, and improve the absorption measurement by l imi t ing signal loss through the silicon. However, simulations with 3//m of silicon show that not enough support is provided for the shock compressed aluminum, and it begins to decompress almost immediately. Four microns of silicon appears to be the min imum useful thickness. Chapter 4. Results of Radiation-Hydrodynamic Simulations 82 High Average Ionization Density, temperature and pressure conditions necessary to achieve a high average ioniza-tion (relative to those achieved in the other shock compression schemes) are presented in Figures 4.8 to 4.10. Radiation transport is clearly an important factor for this irradiance, as the aluminum layer is heated to nearly 4 eV and decompressed by ~20% prior to shock arrival. The purpose of producing such a high shock pressure is to raise the average ionization of the aluminum layer to ~3.5. Although Figure 4.11 suggests that the average ionization should be greater than 4 (given the shock pressure of this process), QEOS is known to overestimate the average ionization, and a value of ~3.5 is more likely [41]. Figure 4.11 serves to demonstrate the large shock pressure increases necessary to generate small increases in the average ionization. This shock compression process allows a study of ionization effects on the K-edge position. Since single shocks produce higher temperatures for a given compression than multiple shocks, the single shock used to form this state will produce a density not too different from that of the Reference state, aiding in isolating the ionization effects. Once the temperature effects have been studied (from the isochoric branch), these can be unfolded from the observed K-edge dependence in this scheme to arrive at the true ionization effect. The irradiation conditions used to generate this state represent the limit of practicality for obtaining any form of steady state conditions in experiments of this type. The 10% variation between highest and lowest densities achieved over the 100 picosecond plateau pushes the limits of acceptability. Originally, a state containing mostly A l + 4 ions was to be produced, but the irradiance needed was almost 6 x l 0 1 4 W / c m 2 , and the final shock compressed state displayed 30% variation over a 100 picosecond plateau. Chapter 4. Results of Radiation-Hydrodynamic Simulations 83 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 Time (ps) Figure 4.8: Compression (i.e. density) as a function of time within the aluminum layer for the high average ionization state. Note the much higher level of decompression caused by radiative preheat in this case as opposed to other simulation results. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. 0.0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I l l " 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 Time (ps) Figure 4.9: Temperature as a function of time within the aluminum layer for the high average ionization state. Radiation preheat is clearly an important effect for this laser irradiance. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. Chapter 4. Results of Radiation-Hydrodynamic Simulations 85 : I | ' ; Q l~i i i I i i i I i i i i i i i i i i i i i i i i , , i I , , , "100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 Time (ps) Figure 4.10: Pressure as a function of time within the aluminum layer for the high average ionization state. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. Chapter 4. Results of Radiation-Hydrodynamic Simulations 86 14.0 12.0 10.0 c o "CO N 'c o CD O) CO \ CD > < 8.0 6.0 4.0 2.0 0.0 0.1 T r r ] 1 1—i—r~i-ri-T | 1 1—i i i i i i j _ i i i _ J 1 L I I I I I I I - 1 u_d_ _ l I I I I I I I 1.0 10.0 100.0 1000.0 10000.0 Shock Pressure (MBar) Figure 4.11: Average ionization as a function of shock pressure for aluminum along the single shock Hugoniot. These values are obtained from the QEOS model. Chapter 4. Results of Radiation-Hydrodynamic Simulations 87 Isochoric State Figures 4.12 to 4.14 present hydrodynamic conditions within the aluminum layer during the double shock process used to obtain the point on the isochore of Figure 4.4. The compression associated with each plateau of the laser pulse is easily recognizable, and remains uncomplicated by the presence of radiation effects. One obvious benefit of using a low laser irradiance is the lack of significant radiation preheat. This is clearly illustrated in Figure 4.13. Neither of the two shock compressions are complicated by the decompressed and heated ini t ia l conditions, and hence these results are closely approximated by simulations which do not include any form of radiation transport. The relative t iming of the two shocks used to create the hydrodynamic conditions (p t, Tt) leads to the formation of an 80 picosecond steady state (ptx,Ttx) prior to the arrival of the second shock. This state is formed just after the rarefaction wave generated by the first shock at the aluminum-silicon interface has propagated back through the aluminum, decompressing it. Al though the density and temperature conditions created at this first plateau do not lie on either the isotherm or isochore of the reference state, they do provide an opportunity to obtain K-edge data without using a separate pulse-target scheme. The 200 picosecond wide plateau at (pt,Tt) provided by the second shock is ultimately l imited by the thickness of the silicon layer. The rarefaction wave associated with shock breakout decompresses the aluminum, demonstrated in Figure 4.12 as the density drops in order of rear to front. Isothermal State Compression, temperature and pressure conditions associated with the state isothermal to the reference state appear in Figures 4.15 to 4.17. The analysis of the shock and Chapter 4. Results of Radiation-Hydrodynamic Simulations 88 3.0 2.0 c o CO CO CD Q. E o O 1.0 I | 1 I T'l" I II I I j I I I I I I I I [ [• I I 1 I | | | | | | | | | I I I l 1 1 I I 1 1 1 I 1 i l l 1 ''I II I I I 0.0 1000.0 1200.0 1400.0 1600.0 1800.0 2000.0 2200.0 Time (ps) Figure 4.12: Compression (i.e. density) as a function of time within the aluminum layer for the isochoric state. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. Chapter 4. Results of Radiation-Hydrodynamic Simulations 89 0.0 1 1 1 1 1 1 1 1 1 1 1 , 1 1 1 1 1 1 1 ' 1000.0 1200.0 1400.0 1600.0 1800.0 2000.0 2200.0 Time (ps) Figure 4.13: Temperature as a function of time within the aluminum layer for the iso-choric state. The low laser irradiance used for the first pulse leads to a situation where radiation preheat is not a significant factor. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. Chapter 4. Results of Radiation-Hydrodynamic Simulations 90 10.0 CO CD ZJ CO CO CD 8.0 -6.0 -Ql 4.0 h 2.0 \-0.0 1000.0 1200.0 1400.0 1600.0 1800.0 2000.0 2200.0 Time (ps) Figure 4.14: Pressure as a function of time within the aluminum layer for the isochoric state. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. Chapter 4. Results of Radiation-Hydrodynamic Simulations 91 rarefaction waves generated in this shock process is identical to that of the isochoric state. Again , radiation effects do not play a significant role. Two quasi-steady state plateaus are produced which can generate K-edge data. The first plateau, occurring between 1300 and 1400 picoseconds, contains hydrodynamic con-ditions which lie on the isochoric branch, and provide a convenient median point to the reference state and the previous shock compression scheme. The second plateau, occur-ring between 1490 and 1600 picoseconds, is l imited by the thickness of the aluminum layer. Final Comment The six distinct hydrodynamic conditions generated by the four shock compression schemes described here should provide enough data to unravel the density, tempera-ture, and ionization state dependencies of the K-edge position. Three of the main points, namely the reference, isochoric, and isothermal, lie along nearly exact isochoric and isothermal branches from the reference point. The point labeled (pdxiTdx) in Figure.4.4 is a rather fortuitous by-product of the isothermal state, and serves to fill out the iso-choric branch. Another point along the isothermal branch would be ideal, but one could not be found which did not suffer greatly from radiation preheat. Chapter 4. Results of Radiation-Hydrodynamic Simulations 92 900.0 1100.0 1300.0 1500.0 1700.0 1900.0 Time (ps) Figure 4.15: Compression (i.e. density) as a function of time within the aluminum layer for the isothermal state. The time interval over which quasi-steady state conditions are achieved is.indicated by the horizontal line. Chapter 4. Results of Radiation-Hydrodynamic Simulations 93 Q I i i i i i i i i i I i ' i i iii i i i i i i , 1 900.0 1100.0 1300.0 1500.0 1700.0 1900.0 Time (ps) Figure 4.16: Temperature as a function of time within the aluminum layer for the isother-mal state. The low laser irradiance used for the first pulse leads to a situation where radiation preheat is not a significant factor. The time interval over which quasi-steady " state conditions are achieved is indicated by the horizontal line. Chapter 4. Results of Radiation-Hydrodynamic Simulations 94 900.0 1100.0 1300.0 1500.0 1700.0 1900.0 Time (ps) Figure 4.17: Pressure as a function of t ime within the aluminum layer for the isothermal state. The time interval over which quasi-steady state conditions are achieved is indicated by the horizontal line. Chapter 5 Experimental Facility This chapter presents a description of the laser facility and analysis techniques used in obtaining x-ray spectra from various elements. 5.1 Laser Facility A schematic of the laser facility appears in Figure 5.1. The laser chain consists of a Quantel N d : Y A G oscillator and pre-amplifier followed by four Quantel Nd:Glass am-plifiers. The oscillator is passively Q-switched with a saturable absorber to produce a single 2.3 ns (full width at half maximum) temporally and spatially Gaussian pulse at 1.064 /xm. A second saturable absorber is placed between the oscillator and the rest of the chain to prevent premature seeding of the oscillator by spontaneous emission from the amplifiers. To reduce beam nonuniformities incurred in the amplification process, a vacuum spatial filter follows each amplifier. The horizontal polarization of the beam is preserved by pairs of thin film polarizers situated between each amplifier and the fol-lowing spatial filter. The amplified 1.064 p:m pulse then passes through a deuterated potassium dihydrogen phosphate ( K D P type II) crystal where approximately 50% of the beam energy is converted to 0.532 / im light. Beam energies of up to 15 J at 0.532 u,m can be obtained from this system. The remaining infrared beam is dumped, and the 0.532 //m light is steered to the target chamber and beam diagnostic equipment by a series of dichroic mirrors. The pulse shape and beam energy of every laser shot is monitored by measuring the 95 Chapter 5. Experimental Facility B S R / _ 96 BSR BSR B S R \ LENS h ' G E N T E K LENS ND I -| | PHOTODIODE IRM SF 5 A4 T P7, P8 SF 6 SHG BSR F3SR A3 SF 4 P4'P3 A2 IRM SF 3 P2 P1 A 1 7 ^ S / I R DUMP SF 2 PA v H = h \ X-SF 1 AP I R M \ - 0 — I I OSC SA Figure 5.1: Schematic of the laser facility. O S C is the oscillator, P A is the preamplifier and A l - 4 are the amplifiers. I R M is an infrared mirror and SF1-6 are the vacuum spatial filters. A P is the hard aperture and S A is the saturable absorber. P l - 8 are polarizers, B S R is a beam splitter-reflector, N D is a neutral density optical absorber, IF is an interference filter, and G E N T E K is the piezoelectric beam energy monitor. S H G is the second harmonic generation crystal. Chapter 5. Experimental Facility 97 leakage through a beam splitter-reflector. A Gentek [102] piezoelectric detector is used to obtain the energy measurement. The absolute energy calibration of the Gentek is accomplished by firing the full beam into a Scientech [103] calorimeter and comparing the calorimeter reading wi th the piezoelectric reading. The pulse shape is obtained with a Hammamatsu [104] photodiode coupled to a Tektronix 7104 oscilloscope. A sample laser pulse appears in Figure 5.2. Shot to shot variations in laser power are less than ten percent. 5.2 Focal Conditions In order to achieve known irradiation conditions, it is first necessary to map out the focal distr ibution of the lens. The following procedure is used. A surface deposition graticule is placed in the target plane, and using an incandescent lamp in conjunction wi th an f/1.4 camera lens, an image of the graticule is obtained on Polaroid film. A 5300A interference filter (100A bandwidth) is placed in front of the Polaroid camera to help avoid chromatic aberrations (see Figure 5.3). The image of the graticule is used to determine the magnification of the image relay system as well as focus the camera lens on the target plane. The graticule is removed and a high reflectance mirror is placed at the entrance of the target chamber to divert nearly all of the laser light into the Scientech calorimeter. This is to avoid damaging the image relay system, as the full laser chain must be fired in order to obtain the most accurate focal distribution. Effects such as thermal lensing, which can alter the focal position by several millimeters, appear only when the full chain is fired. A n image of the focussed beam at the target plane is then obtained as the^ focussing lens position is varied. Different neutral density filters placed in front of the Polaroid camera allow images of the focal spot corresponding to different intensity levels on film, thus expanding the dynamic range of the measurement. Chapter 5. Experimental Facility Figure 5.2: Sample laser pulse. The horizontal scale is 1 nanosecond per division. Chapter 5. Experimental Facility 99 Figure 5.4 shows the smallest focal spot obtained (best focus) at different intensity levels. The calculation of the irradiance on target is somewhat arbitrary, as there is considerable freedom in choosing the size of the spot radius to use. The present work uses the spot radius corresponding to the 60% level, as has been used in previous work [56, 55]. W i t h this definition the irradiance of the beam on target is calculated with 0.6 xE 1 = — , (5.1) where E is the beam energy, r is the full width half maximum duration of the laser pulse, and r 6 0 is the radius of the spot which contains 60% of the beam energy, in this case 30/im. This yields an irradiance on target of 1.5 x 10 1 4 W / c m 2 . Originally, three different irradiances were to be used to investigate how plasma condi-tions affected spectral emission. However, the lower irradiances achieved by attenuating the beam wi th neutral density filters, and enlarging the spot size on target did not pro-duce sufficient x-ray emission levels for analysis. The only irradiation condition which could produce sufficient emission was that corresponding to best focus. 5.3 X-Ray Spectrometers Figure 5.5 shows the layout of the target chamber. A single flat pentaerythritol ( P E T ) Bragg diffraction crystal is used to record emission between 7.3A and 8.3k . B-10 foil [105].. is used to protect the fragile crystal from plasma debris, and provides an absorption edge on the exposure which is used to verify the wavelength calibration. Spurious film exposure due to secondary fluorescence is prevented by i) inserting a plastic shield between the plasma and the spectrometer crystal, ii) fabricating the equipment from a low atomic number material such as plastic, and i i i) covering aluminum parts wi th thin layers of plastic. The emission spectra are recorded on Kodak PF469 film, which consists of Kodak emulsion number 2530 on an acetate base. The film is cut into strips approximately 3 Chapter 5. Experimental Facility 100 Figure 5.3: Experimental arrangement for scanning the focal distribution. N D is a neutral density optical absorber, and IF is the interference filter described in the text. Chapter 5. Experimental Facility 101 Figure 5.4: Images of the laser spot at best focus. Attenuation of the top picture is 100 times greater than the attenuation of the bottom picture, and 10 times greater than the attenuation of the center picture. The magnification of these images is 165 times. Chapter 5. Experimental Facility 102 cm long by 1 cm wide and housed inside a portable, light tight film cassette. The film cassette is machined from a solid block of aluminum and the front surface (excluding the entrance window) covered wi th plexiglass (refer to Figure 5.6). The entrance window of the cassette is 0.6 cm tal l by 2.0 cm wide and is covered with a strip of beryl l ium, allowing the sealed unit to be light tight for normal room light operation, while st i l l allowing x-ray emission to reach the emulsion. A removable mask allows the aluminum reference spectrum to be recorded on one half of the emulsion, and then by flipping the mask, the sample spectrum can be recorded on the other half of the emulsion. The film cassette is not moved between reference shot and sample shot, and so wi th both exposures on the same piece of film, an accurate wavelength calibration can be obtained. Each sample element spectrum is obtained wi th two laser shots in an effort to improve spectral quality. 5.4 Targets Target thickness is compensated for by translating the target mount axially along the beam line. This ensures that the different spectra can be overlaid on a single exposure while st i l l obtaining an accurate wavelength calibration. Table 5.1 presents a summary of the different targets used in this work. 5.5 Processing of Data Each piece of exposed film is carefully developed and dried, and the resulting negative is mounted between glass slides to avoid damage. The spectra are then measured using a microdensitometer. The microdensitometer uses a photomultiplier tube to measure the film exposure of a 1 mm tal l by 8 pm wide region, and the output, in the form of a current, is recorded Figure 5.5: Target chamber and P E T crystal spectrometer. The diameter of the target chamber is approximately 90 cm. Chapter 5. Experimental Facility 104 CASSETTE BODY ACCESS CAP P L A S T I C C O V E R I N G BERYLLIUM WINDOW /////. / / / / / BERYLLIUM WINDOW FILM PLANE Figure 5.6: F i l m cassette. The film plane is indicated by the dotted line. Chapter 5. Experimental Facility 105 Element Atomic Number Sample Type Bismuth 83 50/im on 125/xm plastic Lead 82 100/xm foil Go ld 79 25u,m foil Tungsten 74 25/xm foil Tantalum 73 25/xm foil Samarium 62 30/im foil T i n 50 50/xm foil Silver 47 50//m foil Niobium 41 100/xm foil Y t t r i u m 39 100/xm foil Germanium 32 500//m wafer A luminum 13 25/jm foil Magnesium 12 25/xm foil Table 5.1: Summary of sample elements used in this work in order of decreasing atomic number. Chapter 5. Experimental Facility 106 for every 5 pm step of the scanning stage: The dynamic range of the microdensitometer has been determined to be approximately 100 [106] whereas the dynamic range of the film is only 30, and hence there is no loss of information wi th this technique. The data from the microdensitometer scan represents the film exposure as a function of distance along the scan. The first step in the calibration procedure consists of con-verting this distance measurement into a wavelength scale. This spectral calibration is derived using three lines whose wavelengths and spatial relationship on the exposure are accurately known. A list of the calibration lines used in this work appears in Table 5.2. The calibration lines should span as large a portion of the recorded spectrum as possible. Figure 5.7 shows the spectrometer arrangement used to derive the spectral dispersion at the film plane. For photons of wavelength A, the Bragg diffraction angle 0 is given by where d is the lattice spacing of the crystal (4.37.1 A for P E T ) . Using this equation, and three calibration lines, one can derive a and R and the following relation to convert the where X is the spatial film coordinate. The angle QQ'IS the Bragg diffraction angle of the longest wavelength reference line. The variable R is the path length from the plasma to the film plane for this reference line, and a is the angle of intersection this line makes wi th the film plane. The choice of calibration lines is somewhat arbitrary, although more consistent results are obtained when satellite lines are used instead of resonance lines.. Resonance lines have high opacity and tend to overlap lines from other, more highly excited atomic states. Thus, while the resonance lines are dominant, it is more difficult to accurately locate A = 2ds in0 (5.2) spatial coordinate on film to a wavelength : (5.3) Chapter 5. Experimental Facility 107 PLASMA o CRYSTAL Figure 5.7: Arrangement of the x-ray spectrometer. The path length R = a + b of the longest wavelength reference line is shown. Chapter 5. Experimental Facility 108 Transition Wavelength (A) Reference #12 7.2710 [107] #11 7.3635 [107] He a 7.7571 [108] abed 7.8574 [107], k 7.8683 [107] lj 7.8717 [107] op 8.0713 [107] Table 5.2: A luminum wavelengths used in deriving the spectral dispersion relation. their center position. This inaccuracy can mean the difference between values for R, a and #0 which are consistent wi th the experimental set up, and those which are not. This sensitivity is due almost entirely to the small overall size of the spectrometer. The longest path length is only a few times the entire exposure width on fi lm. As well, the sphericity of the emitting plasma introduces curvature to the spectral lines on film. This curvature translates into extra line width when the exposure is scanned, hence leading to uncertainty in determining the true line position. The second step in the analysis is to convert the film exposure obtained from the microdensitometer scan into relative intensities. This is done using a best-fit polynomial obtained from previous work [106], and is expected to be accurate to within 20% in the relative intensity range of interest. Specifically, the relative intensity is obtained from the film exposure using 1 = 0.02819/ 4 - 0 .2562/ 3 + 0.8369/ 2 - 0.2298/ - 3.455 (5.4) where X is the relative intensity and / is the film exposure. The final step is to correct the relative intensity for the transmission through the beryl l ium window on the film cassette and through the B-1'0 foil used to protect the crys-tal . The beryl l ium window correction follows exponential attenuation, where the opacity Chapter 5. Experimental Facility 109 is calculated from a best fit polynomial in the energy region of interest. Unfortunately, the exact composition of the B-10 was unavailable, and so the attenuation properties had to be determined experimentally. Relative intensities of lead spectra taken with and without the B-10 foil are compared to derive an effective attenuation as a function of wavelength, which is applied to the data. The following chapter presents the x-ray spectra obtained by the methods outline above. Chapter 6 Results and Discussion of X - R a y Spectra This chapter presents the results of the x-ray spectra obtained from various elements by the methods outlined in the previous chapter. Comparison with previously published spectra .is used to determine select transitions, but many other transitions remain un-classified. Suitabil i ty of elements as backlighters is also discussed. 6.1 X - R a y Spectra A print of a sample aluminum and tantalum exposure appears in Figure 6.1. Some of the known lines for both spectra are labeled (Refer to Figures 6.2, 6.3, and 6.8). The most important point to note about the following spectra is that they are neither time nor space resolved. Rather, they represent the average spectral emission of the plasma over the entire emission volume. For example, any density or temperature diagnostics made from the aluminum satellite line ratios represent only average conditions. Typica l aluminum reference spectra appear in Figures 6.2 and 6.3. Figure 6.3 shows that the spectrometer was able to resolve the k from the l,j satellite lines of Li- l ike aluminum. This suggests a resolving power A / A A of 2300. Higher resolving power could not be demonstrated as no other closely spaced lines appeared in this region. Using the wavelength calibration equation (5.3) and the range of values obtained for a and R during processing, the derived uncertainty o f the spectrometer is ± 1 eV in the region of the K-edge. This estimate is supported by two pieces of evidence. The wavelength calibration with typical values of a and R is applied to a random aluminum 110 Chapter 6. Results and Discussion of X-Ray Spectra 111 Aluminum Tantalum Figure 6.1: A print of a sample aluminum and tantalum exposure. Some of the known lines are labeled (refer to Figures 6.2, 6.3, and 6.8). Chapter 6. Results and Discussion of X-Ray Spectra 112 Figure 6.2: Sample aluminum reference spectrum. Some common lines are labeled. Chapter 6. Results and Discussion of X-Ray Spectra 113 Figure 6.3: Enlarged aluminum reference spectrum showing the clearly resolved satellite lines qr, a-d, and klj of A l XI. Chapter 6. Results and Discussion of X-Ray Spectra 114 spectrum and agreement is seen to be excellent (see Table 6.1). The second piece of evidence is the consistent appearance of the K-edge on the sample spectra between 1559.2 and 1600.8 eV, visible prior to correction for the B-10 attenuation. Each microdensitometer scan has had the same film calibration curve applied in con-verting film exposure to relative intensity. Although the accuracy of this curve is only 20% for the intensities seen here, comparisons can st i l l be made between the different spectra. The spectra are discussed in order of decreasing atomic number. 6.1.1 Bismuth, Lead and Go ld The spectra of bismuth, lead and gold appear in Figures 6.4 to 6.6. The observed emissions are remarkably similar given the range of atomic numbers that these elements span, and therefore the spectra are discussed together. The most prominent feature of the three spectra is the appearance of transition arrays between 1545 and 1635 eV. It is interesting to note that these arrays appear at nearly the same center energies and wi th almost identical widths. This would suggest that the arrays come from transitions which are relatively insensitive to the nuclear structure and number of shielding electrons, and hence are derived from inner subshell transitions. Previous work [109, 110, 111, 120] in analysing the spectra from these elements in different wavelength regions allows certain transitions to be ruled out. Candidates for the observed arrays are the N i I like ion transitions [111] of 3 d 1 0 - 3 d 9 / 2 n f 5 / 2 , 3 d 1 0 - 3 d | / 2 n f 7 / 2 , and 3 p 6 3 d 1 0 - 3 p i j / 2 3 d 1 0 n d 5 / 2 wi th principal quantum number n equal to 4 or 5. The average intensity of the bismuth emission in the ± 3 0 eV region around the K -edge is nearly 1.5 times that of lead and gold, although al l three display roughly the same amount of fractional variation. The prominent line structure of bismuth in the region centered at 1555 eV, which is not seen in lead or gold, make the latter two more attractive as backlighters. In fact, bismuth has been used previously as a backlighter [11, 12] in the Chapter 6. Results and Discussion of X-Ray Spectra 115 Transition Experiment (eV) Theory (eV) #12 1706.2 1705.5 #11 1684.1 1684.2 H e a 1598.3 1598.7 y 1589.3 1588.5 s,t 1588.5 1588.0 q,r 1580.1 1580.0 • a-d 1578.1 1578.3 k 1575.5 1576.1 1J 1574.5 1575.3 o,P 1536.3 1536.4 Table 6.1: Experimental and theoretical energies of lines of an aluminum spectrum when the spectral calibration equation is applied. Chapter 6. Results and Discussion of X-Ray Spectra 116 Bismuth 5.0 4.0 CO -4—' c Z) -e 3.0 < 0 CD 2.0 > •4—' _CJJ CD DC 1.0 ~ i — i — i — F — i — i — i — i — I — r T 1 1 1 1 I 1 1 1 J-Q Q I—i—"—i—i—i— i—i—i—i— l—i—i— i—J i i i i i l t i i i i i i i -t i i i I i i i i i i i i i i i 725.0 1675.0 1625.0 1575.0 1525.0 Energy (eV) Figure 6.4: Sample bismuth spectrum. The position of the aluminum K-edge is shown by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra 117 Lead 5.0 4.0 CO -e 3.0 < CO c CD CD 2.0 > CD DC 1.0 ~ i — i — i — i — i — i — i — i — I — i — r J—i—i—I—I—I—i—i—i I I I i j 0.0 1725.0 1675.0 1625.0 1575.0 1525.0 Energy (eV) Figure 6.5: Sample lead spectrum. The position of the aluminum K-edge is shown by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra 118 K-edge region, but the sample bismuth spectrum obtained by these researchers had less spectral resolution. It is unknown how the spectral structure seen in Figure 6.4 would affect the results of this previous work. On the basis of observed line structure, lead and gold provide more suitable backlighter characteristics than bismuth. This structure, however, will be enhanced or degraded depending on irradiance, and therefore none or all of these elements may turn out to be useful at other irradiances. 6.1.2 Tungsten The observed x-ray emission spectrum of tungsten appears in Figure 6.7. Analyses exist for this element in the ranges of 2.6-3.3 keV [109] and 1.6-1.9 keV [118]. Four of the spectral lines seen in this work have been identified previously, and have been labeled according to Kef. [118]. The g,h and i transitions originate from the Cu I like 3d104s-3d94s4p array. The (3 line group belongs to the 3d 1 04s 2-3d 94s 24p Zn I like transition. The lower energy lines remain unclassified, but most likely belong to inner subshell transition arrays. As with previous elements, the line structure around the K -edge makes tungsten unsuitable as a backlighter, even though it is one of the more intense emitters studied in this work. 6.1.3 Tantalum Tantalum, whose x-ray spectrum appears in Figure 6.8, is one of the more actively studied elements in the classification of unresolved transition arrays [111, 117, 118, 119]. Again, the lines are labeled in accordance with Ref. [118]. The Cu I like transition, 3d104s-3d94s4p, labeled with lowercase letters, gives rise to the majority of lines in this energy region. The two Ni I like resonance lines are the most prominent emission lines, in agreement with previous work [109]—[121]. A few lines escape Chapter 6. Results and Discussion of X-Ray Spectra 119 Figure 6.6: Sample gold spectrum. The position of the aluminum K-edge is shown by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra Figure 6.7: Sample tungsten spectrum. The position of the aluminum K-edge is by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra 121 Tantalum Z) JQ < CO c CD 0 > 16.0 14.0 12.0 10.0 8.0 > 6.0 05 CD rr 4.0 2.0 h 0.0 ~i—r—\—r i—i—i—r-J — i — i — i — i — i — i — I — i — i — i — i — i — i — i i i I i i i i i i i t_ i I i i I 1725.0 1675.0 1625.0 1575.0 1525.0 Energy (eV) Figure 6.8: Sample tantalum spectrum. The position of the aluminum K-edge is shown by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra 122 classification, but like tungsten, probably originate from inner subshell transition arrays. Unfortunately, tantalum displays more line structure near 1560 eV than is acceptable for a backlighter element. 6.1.4 Samarium A sample samarium spectrum appears in Figure 6.9. Burkhalter et al. [116] have obtained spectra of the rare earth elements Sm through Y b , but have published results only for G d . Comparison of the Sm spectrum wi th the published G d spectrum does not lead to any line identifications. The intense transition array structure over the observed energy range makes the Sm spectrum ideal as data for transition array modeling, but useless as a backlighter. 6.1.5 Tin Spectra of laser produced t in plasmas below 1240 eV have been published previously [113, 114, 115] but Figure 6.10 shows the first spectrum obtained above 1500 eV. Transition arrays appear superimposed on a sloped background of bremsstrahlung radiation. Reference [113] allows certain transitions to be ruled out, so that candidates for the observed emission are transition arrays of the types 3d f e -3d f e _ 1 5f and 3d f e -3d f c _ 1 5p in Sn X X X I I I to Sn X X I I I ions. Whi le the extensive transition array structures around the K-edge render Sn unsuitable as a backlighter, they do provide a testbed for further transition array modeling. 6.1.6 Silver Previous spectral analyses for this element have concentrated on a much lower energy regime [123, 126, 124], whereas Figure 6.IT shows the spectrum in the region 1 5 0 0 - 1725 Chapter 6. Results and Discussion of X-Ray Spectra 123 Figure 6.9: Sample samarium spectrum. The position of the aluminum K-edge is shown by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra Tin I Q I i ' ' ' i i i ' i i i i i i i i i i i , i ; i • • • i 1725.0 1675.0 1625.0 1575.0 1525.0 Energy (eV) Figure 6.10: Sample t in spectrum. The position of the aluminum K-edge is shown the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra 125 eV. Reference [126] suggests that the observed spectrum may originate from 2s-3p, 2p-3d, and 2p-3s transitions in A g + 3 9 . However, detailed line classifications do not exist, and none are offered here. Theoretically, the temporal evolution of the intensity of the 1560 eV line of silver could be used to monitor the aluminum K-edge shift. As the K-edge shifts towards the red, the lower energy components experience less attenuation, and the line would appear more broad and with increased intensity. Conversely, as the K-edge shifts towards the blue, the line would suffer greater attenuation, and would appear more narrow and with less intensity. This method, however, suffers from several serious drawbacks. First of al l , the width of the emission line is much less than the magnitude of the expected edge shift, making this technique useful only during the ini t ia l moments of the experiment, and not over the entire measurement period. Furthermore, the K-edge measurement is of the time-resolved form, and thus the temporal dependence of the emission of the 1560 eV line of silver would be required, adding yet another level of complexity to the measurement. Final ly , the attenuation of this line only provides K-edge information right at 1560 eV, whereas a spectrally flat backlighter would provide a K-edge measurement over a much broader spectral range. However, silver may find a use as a spectral calibration source. Many intense, well defined emission lines surround the K-edge, and the resonance at 1560 eV may serve as a convenient reference point by which to calibrate spectral measurements. 6.1.7 Niob ium Spectra of highly ionized N b have been obtained from vacuum spark measurements, [123] and like those of silver, these measurements have focussed on a much lower energy region of the spectrum. Figure 6.12 presents the N b spectrum in the region of 1500 -Chapter 6. Results and Discussion of X-Ray Spectra 126 5.0 4.0 cn 'E -Q 3.0 < CO c CD CD 2.0 as CD rr 1.0 Silver ~~| i i i—i—i—i—i—i—i—]—i—i—i—i—i—i—i—i—i—|—r~ -i—i—i—r~ 0.0 1725.0 1675.0 1625.0 1575.0 1525.0 Energy (eV) Figure 6.11: Sample silver spectrum. The position of the aluminum K-edge is shown by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra 127 1725 eV. No line identifications can be made, although the existence of transition arrays seems likely, as is the case with medium to high atomic number elements. As atomic number decreases, however, there should be a trend toward non-overlapping resonance line structures and transition array structures should become less frequent. 6.1.8 Yttrium Previous analyses of Y t from vacuum spark and laser produced plasma spectra [123, 126] have been done in the 120 - 1200 eV range. Figure 6.13 shows the spectrum in a much higher energy region. Comparison with previous work does not lead directly to any line classifications. The relatively small amount of line structure coupled with small fractional intensity variations make Y t a good candidate for a backlighter if the low emission levels seen here can be increased by increasing the laser irradiance. 6.1.9 Germanium Figure 6.14 presents the spectrum of Ge in the range 1500 - 1725 eV. No previous analy-ses exist for the spectrum of this element in this energy range, and no line classifications are offered here. The spectrum most likely consists of unresolved transition arrays su-perimposed upon a sloped background of bremsstrahlung radiation. The intense spectral emission over the entire range makes Ge unusable as a backlighter source, which is un-fortunate given its unparalleled level of brightness. Interestingly, Ge has been used as a backlighter for shock waves studies where spectral structure is not inhibi t ing [127]. This spectrum provides a rich source of data to be modeled for further transition array studies. Chapter 6. Results and Discussion of X-Ray Spectra 128 5.0 - i — i — i — i — i — r -Niobium i — i — i — i — i — i — i — i — i — j — i — i — i — i — i — i — i — i — i — j — i — i — i — i — 4.0 'c 3 -E 3.0 < CO c CD CD 2.0 > 03 CD DC 1.0 | Q I—I 1 1—I—I 1—I—I 1—I—I I—I—I I I I I I I I I ' ' I i i i i i i i i i I i i i . 1725.0 1675.0 1625.0 1575.0 1525.0 Energy (eV) Figure 6.12: Sample niobium spectrum. The position of the aluminum K-edge is shown by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra 129 Yttrium 5.0 4.0 CO 'c -a 3.0 < CO c CD CD 2.0 03 CD rr 1.0 • • • . i J— i—i—i— I—i—i—i—i—i i i i i i i i } i • i i i i i 0.0 1725.0 1675.0 1625.0 1575.0 1525.0 Energy (eV) Figure 6.13: Sample yt t r ium spectrum. The position of the aluminum K-edge is shown by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra 130 Germanium 30.0 25.0 CO E 20.0 JD < % 15.0 c o c: 0 > 03 CD DC •g io .o 5.0 —I—I—I—I—I—I—I—I 1 I I —1—I—I I } I 1 I I l_ 0.0 1725.0 1675.0 1625.0 1575.0 1525.0 Energy (eV) Figure 6.14: Sample germanium spectrum. The position of the aluminum K-edge is shown by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra 131 6.1.10 Magnesium A typical M g spectrum appears in F i g . 6.15. The emission lines at 1700, 1660 and 1580 eV are the He^, H e 7 , and He^ resonance lines of M g X I , respectively. The structure at the low energy edge of the recorded spectrum consists of the 1-6 satellite lines of M g X . Aside from the line structure wi thin 30 eV on either side of the K-edge, the lack of intensity of the observed spectrum makes magnesium unusable as a backlighter. 6.2 Discussion The aim of this work was to find a suitably flat and bright spectral source for use as a backlighter. Classification of spectral features for each element was never a priority. The computational resources needed to model the unresolved transition arrays present in these spectra were not available, and hence the primary means by which lines are identified for high-Z elements could not be used. Moreover, much of the previous work done in classifying these spectra has been concentrated in different spectral regions than the one studied in this work. The multitude of possible transitions originating from the many coexisting ionization stages in the plasma makes line identifications a labourious task. Certain trends can be employed to aid in the identification process, but these are often of l imited use. One method employs the systematic Z2 dependence of atomic features on atomic number by means of a Moseley plot [116], as illustrated in F i g . 6.16 (taken from Reference [116]). The square root of the transition energy for a select transition is plotted against atomic number. Interpolation from the resulting nearly linear fit is used to determine the energy of the same transition for different elements. O f course, this method is only approximate, and relies heavily on previous calculations. Another technique used to unravel atomic spectra involves the systematic addition Chapter 6. Results and Discussion of X-Ray Spectra 14.0 12.0 — 10.0 CO "c Z) -Q < 8 0 'co c CD I 6.0 CD .> "•+—< _C0 CD DC 4.0 2.0 ~ i — — i — i — i — i — i — r Magnesium i i i i i I—i—i—i—i—i—i—i—i—i—i—i—i—m—i—i—i—r~ i i i i i i 0.0 1725.0 1675.0 1625.0 1575.0/ 1525.0 Energy (eV) Figure 6.15: Sample magnesium spectrum. The position of the aluminum K-edge shown by the dotted line. Chapter 6. Results and Discussion of X-Ray Spectra 133 45 -I 1 : I I i i 28 30 40 50 60 70 ATOMIC NUMBER Figure 6.16: Moseley plot used to aid in line classification. Chapter 6. Results and Discussion of X-Ray Spectra 134 and subtraction of spectator electrons [113, 116, 121]. The energy of a specific transition for a single ionization state is calculated. Spectator electrons added to or subtracted from various shells w i l l shift the transition energy by constant amount, allowing identification of the transition in different ionization states. For example, removal of successive 3d elec-trons in G d + 3 6 shifts the 3d-4f transition up by +39 eV. This has been used to identify 3d-4f lines in G d + 3 T and G d + 3 8 [116]. This method, however, suffers from a form of de-generacy. Different transitions originating in different ionization stages can lead to nearly identical energies [109]. As well, line shifting due to asymmetric Doppler broadening [113] can shift spectral lines, making the spectator electron technique inaccurate. It should be noted that the spectral classifications which are offered in the previous sections are by no means final. Instead, they have been determined by extrapolation and comparison wi th previous analyses, and simply serve as a starting point for future work. One of the most important aspects of these spectra, as was alluded to earlier, was the dependence of line structure and continuum emission strength on laser irradiance. Increasing the irradiance should lead to greater ion populations in the higher ionization states. Unfortunately, much of the spectra seen in this work remains unclassified as to specific line transitions and even from which ionization species they originate. Hence, increasing the laser power could have one of two effects. The line structure already observed could become enhanced, rendering al l of the elements studied here unusable, or, the relative populations of the different ionization stages could shift so that the line emission would appear in a different spectral region. The spectra observed here would then seem to flatten out, as the plasma becomes more like a continuum emitter. Likely candidates to be used as a backlighter source are narrowed to three. Lead and gold show reasonable levels of brightness, wi th lit t le observed line structure. However, the intensity variation displayed by these elements around 1560 eV could prove to be a problem. Y t t r i u m , on the other hand, shows excellent spectral flatness, and would be Chapter 6. Results and Discussion of X-Ray Spectra 135 suitable provided it is sufficiently bright. Finally, although not usable as a backlighter, silver seems promising as an emission source by which to calibrate spectrograph^ instru-ments in an experimental situation. Chapter 7 Summary and Conclusions This final chapter presents a summary of the work undertaken in this thesis project and the conclusions which can be drawn from it. Suggestions for future work are also offered. 7.1 Summary of Present Work The temporal evolution of the K photoabsorption edge of aluminum under shock com-pressed conditions allows the study of the atomic physics of hot, dense matter. This thesis work has concentrated on providing the necessary background information for the design of an efficient experiment to probe this physics regime. In order to decouple to effects of density, temperature and ionization state on the position of the K-edge, four separate shock compression schemes are used. Each scheme couples a tailored laser pulse to a specific C H - A l - S i target in order to produce a well characterized plasma state within the aluminum layer. B y comparing three of these states back to a reference state, density, temperature and ionization state effects can be individually studied. The shock compression schemes are designed with a one dimensional hydrodynamic code incorporating non-local thermodynamic equilibrium physics. The N L T E model con-siders the atomic physics of the C H layer only, but line and continuum radiation, as well as inner-shell photoionization processes are considered for a l l cells in the simulation. The target designs represent the most efficient thicknesses in terms of absorption measurement signal, radiation preheat, and laser pulse length. The laser pulse is modeled following 136 Chapter 7. Summary and Conclusions 137 presently available shaped pulses. The second aspect of a K-edge experiment involves the x-ray emission used for the photoabsorption measurement. In search of a suitable source, the x-ray emission between 1525 and 1725 eV of 12 elements between M g and B i has been obtained wi th a P E T crystal spectrometer. Accuracy of the spectroscopic measurement is ± 1 eV. Suitable backlighter sources are identified, and a wealth of new data for unresolved transition array studies is produced. 7.2 Conclusions The pulse-target schemes summarized in Chapter 4 present a viable scheme for study of density, temperature and ionization effects on K-edge position. A l l steady state plasma conditions used to generate K-edge data have sufficiently long lifetimes and small varia-tions in hydrodynamic conditions to generate well characterized plasma states. Candidates for suitable x-ray emission sources are narrowed to three. Lead and gold show reasonable levels of brightness, wi th l i t t le observed line structure. Y t t r i u m , on the other hand, shows excellent spectral flatness, but displays rather low emission levels. Final ly , although not usable as a backlighter, silver seems promising as an emission source by which to calibrate spectrographic instruments in an experimental situation. 7.3 Future Work Ideally, the N L T E calculation would include a l l of the layers of the target, as opposed to just the C H layer as considered here. This would result in a far more accurate calculation of the ionization in the aluminum layer than is currently provided by the Q E O S or S E S A M E tables. As well, some improvement would be expected on the absolute levels of radiation emission, as now line radiation could originate from the aluminum and silicon Chapter 7. Summary and Conclusions 138 layers. To gain further understanding of target preheat, fast electron generation could also be incorporated into the code. This effect may prove significant for higher irradiances, and eventually needs to be considered. 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