Hydrodynamic stability and Ti-tracer distribution in low-adiabat OMEGA direct-drive implosions

Transcription

1 University of Nevada, Reno Hydrodynamic stability and Ti-tracer distribution in low-adiabat OMEGA direct-drive implosions A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics by Tirtha R. Joshi Dr. Roberto C. Mancini/Dissertation Advisor August 2015

2 c Tirtha R. Joshi All rights reserved.

3 THE GRADUATE SCHOOL We recommend that the dissertation prepared under our supervision by TIRTHA R. JOSHI Entitled Hydrodynamic Stability and Ti-Tracer Distribution in Low-Adiabat OMEGA Direct-drive Implosions be accepted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Dr. Roberto C. Mancini, Advisor Dr. Yasuhiko Sentoku, Committee Member Dr. Peter Hakel, Committee Member Dr. Nathalie Le Galloudec, Committee Member Dr. Sean M. Casey, Graduate School Representative David W. Zeh, Ph. D., Dean, Graduate School August, 2015

4 Abstract i We discuss the hydrodynamic stability of low-adiabat OMEGA direct-drive implosions based on results obtained from simultaneous emission and absorption spectroscopy of a titanium tracer added to the target. The targets were deuterium filled, warm plastic shells of varying thicknesses and filling gas pressures with a submicron Ti-doped tracer layer initially located on the inner surface of the shell. The spectral features from the titanium tracer are observed during the deceleration and stagnation phases of the implosion, and recorded with a time integrated spectrometer (XRS1), streaked crystal spectrometer (SSCA) and three gated, multi-monochromatic X-ray imager (MMI) instruments fielded along quasi-orthogonal lines-of-sight. The time-integrated, streaked and gated data show simultaneous emission and absorption spectral features associated with titanium K-shell line transitions but only the MMI data provides spatially resolved information. The arrays of gated spectrally resolved images recorded with MMI were processed to obtain spatially resolved spectra characteristic of annular contour regions on the image. A multi-zone spectroscopic analysis of the annular spatially resolved spectra permits the extraction of plasma conditions in the core as well as the spatial distribution of tracer atoms. In turn, the titanium atom distribution provides direct evidence of tracer penetration into the core and thus of the hydrodynamic stability of the shell. The observations, timing and analysis indicate that during fuel burning the titanium atoms have migrated deep into the core and thus shell material mixing is likely to impact the rate of nuclear fusion reactions, i.e. burning rate, and the neutron yield of the implosion. We have found that the Ti atom number density decreases towards the center in early deceleration phase, but later in time the trend is just opposite, i.e., it increases towards the center of the implosion core. This is in part a consequence of the convergent effect of

5 ii spherical geometry. The spatial profiles of Ti areal densities in the implosion core are extracted from space-resolved spectra and also evaluated using 1D spherical scaling. The trends are similar to the Ti number density spatial profiles. The areal densities extracted from data and 1D spherical scaling are very comparable in the outer spherical zones of the implosion core but significantly deviate in the innermost zone. We have observed that approximately 85% of the Ti atoms migrate into the hot core, while 15% of the atoms are still on the shell-fuel interface and contributing to the absorption. In addition, a method to extract the hot spot size based on the formation of the absorption feature in a sequence of annular spectra will be discussed. Results and trends are discussed as a function of target shell thickness and filling pressure, and laser pulse shape.

6 Acknowledgements iii Firstly, I would like to express my sincere gratitude to my advisor Dr. Roberto C. Mancini for the continuous support of my Ph.D study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing of this dissertation. Besides my advisor, I would like to thank Dr. Yasuhiko Sentoku, Dr. Peter Hakel and Dr. Nathalie Le Galloudec and Dr. Sean Casey for serving on my committee. I would like to thank physics department, UNR for providing me a teaching assistantship for the first two years in my graduate study. I thank Dr. Taisuke Nagayama, Dr. Tunay Durmaz, Dr. Heather Johns former group members of Dr. Mancini s research group for their help. I also thank Tom Lockard, Danial Mayes and Kyle Carpenter for their support and friendly conversations. I express my special thanks to Danial Mayes for his help in OMEGA experiments and IDL codes once again. Most importantly, none of this would have been possible without the love and patience of my family. My deepest gratitude goes to my parents and family members for their endless love, support and encouragement. My special gratitude goes to my beloved wife Hemu, little children Sarthak and Shambhavi, and sisters Rekha and Lalita. This work was supported by DOE/NLUF Grants DE-NA and DE-NA , Los Alamos National Laboratory C10 academic collaboration grant and Lawrence Livermore National Laboratory..

7 iv Contents Abstract Acknowledgements Contents List of Tables List of Figures i iii iv vi vii 1 Introduction 1 2 Experiments at OMEGA and MMI data Nuclear fusion Conceptual discussion of direct-drive implosion experiments The importance of spectroscopic tracers for implosion diagnosis Target and laser parameters of specific experiments Instruments and spectroscopic measurements MMI data processing Titanium emission and absorption K-shell spectrum Atomic and radiation physics of titanium ions Atomic processes and plasma atomic kinetics Spectral line profiles Radiation transport Ti K-shell emission spectrum from an implosion core Titanium K-shell absorption in a dense plasma Hot-spot evolution in the implosion core Transition from emission-dominated spectrum to absorption-dominated spectrum Extraction of hot-spot size from annular spatially resolved spectra.. 45

8 v 4.3 Results and trends for hot spot size Multiple lines-of-sight view Temporal evolution Effect of changing parameters Methods for spectroscopic analysis and Ti spatial distribution Multi-zone model and geometry reduction Extraction of electron temperature and density Determination of the titanium atom number density Results for Titanium penetration in the core Time in the implosion Annular-type spatially resolved spectra Testing of the multi-layer spectroscopic model Ti penetration in the core Spectroscopic data analysis Ti number and areal density distributions in the core Multi-view spatial profile Temporal evolution of Ti mixing Summary 88 A Solution of the one-dimensional radiation transport equation through a slab of uniform plasma conditions 91 B Opacity correction for upper level population 94 C Calculation for one-dimensional spherical scaling of Ti-areal density 98 Bibliography 101

9 vi List of Tables 6.1 Overview of the target parameters and laser pulse shapes for the shots used for the analysis Comparison of spatial profiles of plasma conditions used to generate the synthetic spectra with the best fit solutions obtained by using multizone spectroscopic model Percentage distributions of the Ti atoms contributing to emission and absorption in the implosion cores of OMEGA direct-drive shots Summary of the results for the emission zones in the implosion core of OMEGA shot TIM 3 Frame 2. Where the column Data and 1D Sp. Sc. represent experimental areal density and 1D spherical scaling areal density respectively C.1 Experimental areal densities and percentage distributions in different spherical zones in implosion core

10 vii List of Figures 2.1 Mean binding energy per nucleon of stable nuclei as a function of mass number. (Credit: S. Pfalzner, An introduction to inertial confinement fusion, Taylor & Francis group) The stages of direct-drive inertial confinement fusion. (Credit: An assessment for the prospect of the inertial fusion energy. National Academies Press, 500 Fifth Street, NW, Washington DC 20001) a) OMEGA shot TIM 4-MMI data without Ti tracer, b) OMEGA shot TIM 4-MMI data with Ti tracer Schematic of a direct-drive target pie-slice Laser pulse shapes to drive high and low-adiabat implosions a) highadiabat 1ns, b) low-adiabat α 2, c) low-adiabat α a) MMI instrument built in the Machine shop at UNR and b) Schematic of MMI Three TIMs (Ten Inch Manipulators in the OMEGA target chamber) Processed MMI data from shot TIM4. The target parameters were, core size: 400µm, shell thickness: 15µm and laser pulse shape: α Annular region in implosion core and its projection on detector plane a)processed MMI data, OMEGA shot TIM 4 Frame 2, b) MMI mask image associated with spatial region of interest, c) MMI data in a) after the application of mask, d) space-resolved spectrum associated with spatial region of interest Area normalized absorption line shapes for Be-like Ti n=1 to n=2 transitions, for several electron temperature and density conditions [49] Plasma electron temperature and density dependence of the titanium emission K-shell spectrum

11 viii 3.3 Optical depth due to n=1 n=2 transitions in titanium ions for several electron temperatures and densities. The titanium areal density is cm 2 [49] Titanium transmission spectrum for the plasma conditions displayed in figure 3.3b, not including instrumental broadening effect (a), and including instrumental broadening effect (b) Transition from an emission dominated spectrum to an absorption dominated spectrum with increasing backlight radiation and keeping plasma conditions constant Fitting of the continuum level: OMEGA shot TIM 3 Frame Space-resolved spectra (SRS) from different annular regions on the image plane (outermost to central): OMEGA shot TIM 3 Frame Annular region corresponding to the boundary of hot-spot: OMEGA shot TIM 3 Frame Temporal evolution of continuum intensity along three quasi-orthogonal lines-of-sight: OMEGA shot (TIM3, TIM4 and TIM5) and (TIM3 and TIM5) Temporal evolution of hot-spot size: OMEGA shot a) TIM 4, b) TIM 4, TIM 3, and TIM Change of hot-spot size with changing shell thickness of the target Change of hot-spot size with changing filling pressure Change of hot-spot size with changing laser pulse shapes Spatially integrated, time-resolved spectrum recorded with the SSCA streaked spectrometer displaying simultaneous line emission in He- and H-like Ti ions and n=1-2 line absorption transitions in F- through Lilike ions Schematic illustration of annular image regions on the image (detector) plane and spherical shell zones in object space (core, plasma source) Illustration of the fitting of the absorption feature of I nu (R3) Multi-layer slabs for fitting spectrum I ν (R3) Illustration of the emission/absorption fitting of spectrum I nu (R3) Multi-layer slabs for fitting spectrum I ν (R2) Multi-layer slabs for fitting spectrum I ν (R1)

12 ix 5.8 Continuum subtracted total line intensities. For each line, total intensities are found from all three space resolved spectra. Integrated regions are shown as shaded a) Processed X-ray image data recorded by streaked crystal spectrometer (SSCA) from the OMEGA shot b) temporal evolution of laser pulse (low-adiabat α 2, SSCA and NTD data Area normalized and rescaled space-resolved spectra from annular regions on the image plane: OMEGA shot TIM 3 Frame Synthetic space-resolved spectra extracted from three annular regions on the synthetic implosion core image a) Analysis of synthetic spectra from region 3, b) region 2, c) region 1, and d) region 3 but fitting only the absorption spectra Analysis of space resolved spectra for OMEGA shot TIM 3 Frame 2. Plots a) and b) are for region 3, c) and d) are for region 2, e) and f) are for region 1. Plots a), c) and e) are the fittings corresponding to the absorption spectral range ( ) ev. Plots b), d) and f) are the simultaneous fittings of absorption and emission corresponding to the spectral range ( ) ev Temperature and density spatial profiles of the emission region in the implosion core of OMEGA shot TIM 3 Frame 2. Plots a) and b) show temperature and density gradient respectively a) Spatial profile of excited level population density in the implosion core of OMEGA shot TIM3 Frame 2, b) spatial profile of Ti number density, c) spatial profile of areal density (data and 1D spherical scaling), d) spatial profiles of Ti atom number distributions Multi-view spatial profiles of Ti atom number areal density distributions in the implosion core of OMEGA shot (along TIM 4 and TIM 5) Multi-view spatial profiles of Ti atom number density distributions in the implosion core of OMEGA shot (along TIM 4 and TIM 5) Temporal evolution of spatial profiles of Ti atom number areal density distributions in the implosion core for OMEGA shot TIM Temporal evolution of spatial profiles of Ti atom number density distributions in the implosion core for OMEGA shot TIM

13 A.1 Schematic of the slab geometry used in the solution of the radiation transport equation x

14 1 Chapter 1 Introduction Inertial confinement fusion is an approach to initiate nuclear fusion reactions by the heating and compressing of a fuel target, typically in the form of a spherical plastic shell filled, for example, with deuterium. The deuterium is compressed to very high densities and temperatures by applying laser or x-rays. The confinement times are extremely short ( s) but the plasma density is very high [1]. There are mainly two techniques to achieve inertial confinement fusion: indirect-drive and direct drive. In the indirect-drive approach, the target is placed inside a hohlraum and laser beams irradiate the interior walls of the hohlraum from both sides. The laser beam interaction with the walls produces soft x-rays, which drive the implosion of the capsule by ablating the outer surface material [1, 2, 3]. In the direct-drive, the surface of the plastic shell target is irradiated directly by the intense laser beams or ion beams. The experiments for this dissertation project were direct-drive, and laser beams were used for the target s surface irradiation. The irradiation ablates the outer surface material, which in turn generates shock waves and compress the remaining fuel and shell as a result of the rocket like blow-off of the hot surface material. The inner shell surface moving inward is decelerated by the pressure build up due to the compression and shock waves reflected at the center of the implosion. At some point

15 2 during the implosion, the plasma enclosed by the inner shell surface achieves a fairly uniform pressure profile, and the central region of the core with the uniform pressure profile is called hot-spot. At the peak compression, the temperatures and densities of the plasmas in the hot-spot become very high and may begin the burn of the fusion material in the central area. A thermonuclear burn front then propagates rapidly outward into the main fuel region. In ICF experiments, the diagnosis of plasma temperature and density conditions, and the mixing of shell material with the fuel are critical to understand the performance of the implosion s hydrodynamics. The mixing occurs primarily due to the Rayleigh-Taylor instability in the deceleration phase of an implosion and is seeded by the target imperfection and laser non-uniformity [4, 5, 6]. It increases the radiative losses from the hot-spot, and decreases the peak pressure of the fuel and the fusion yield [7, 8, 9, 10, 11]. The X-ray spectroscopy has been used as a powerful, non-intrusive tool in the field of inertial confinement fusion [10, 12, 13, 14, 15, 16, 17, 18, 19]. An extensive amount of work has been done to diagnose plasma conditions in ICF implosion cores by using X-ray spectroscopy [10, 12, 13, 14, 20]. After its first use in implosion experiments in 1977 [21], there have been continuous improvements because of the advances in instrumentation and in the theory and modeling calculations required to interpret the data [22]. Mid-Z tracer elements like Ar or Ti are partially ionized and emit or absorb photons from K-shell line transitions, which carry information about local material conditions. The direct-drive ICF experiments were performed at the OMEGA laser facility of the Laboratory for Laser Energetics at the University of Rochester, NY [23]. The targets were deuterium filled, spherical plastic shells of varying thicknesses and gas pressures with a thin Ti-doped tracer layer at the fuel-shell interface. The spectral features from the titanium tracer are primarily observed during the deceleration phase

16 3 of the implosion and recorded with a time integrated spectrometer (XRS1), a streaked spectrometer (SSCA) and three identical gated, multi-monochromatic x-ray imager (MMI) instruments fielded along quasi orthogonal lines-of-sight. The MMI instrument consists of a pinhole array, a multi-layered mirror and an X-ray framing camera with four gated strips. X-rays from the implosion core pass through the pinholes to image the implosion core and are dispersed by reflecting them off a multi-layer mirror. The result is a large collection of spectrally-resolved images of the core. The MMI data has spectral, spatial and temporal resolution, whereas the streaked data has only spectral and temporal resolution. The XRS1 gives space and time integrated data. All the time integrated, streaked and gated data have shown simultaneous emission and absorption features associated with titanium K-shell line transitions. We have extracted spatially- resolved spectra characteristics of the annular regions in the image by processing the array of spectrally-resolved images recorded with MMI [12, 13]. The K-shell emission and absorption lines originated from the imploding core are very sensitive to the plasma conditions. The sensitivity comes from the fact that the line emission and absorption strengths and shapes depend on the atomic level populations, which in turn depend on the plasma conditions in the implosion core [12, 17, 19]. The effect of these dependencies can be seen in the line intensity ratio of the similar line transitions in adjacent ionization stages. For example, if the ratio of intensity of Lyβ (n=3 to n=1 in H-like Ti) to Heβ (n=3 to n=1 in He-like Ti) increases, it indicates that the implosion core is becoming hotter and the ionization balance of the source plasma is shifting from He-like to H-like Ti ions, and vice versa. Detailed modeling and analysis of absorption and emission lines can yield core electron temperature, density, areal density, and mixing of shell mass into the core. This work aims to study the spatial distribution of the dopant material (Ti for our targets), initially located on the inner surface of the target, into the OMEGA direct-

17 4 drive implosion core in the deceleration phase of the implosion by using space-resolved spectra characteristics of annular regions on the implosion core image. Several works have been done at OMEGA and NIF previously to understand the mixing of the shell material into the hot core [6, 11, 24, 25, 26] but none of them have used space resolved spectra to report the mixing of tracer into the deep core. We use the spatially resolved spectra (characteristics of annular regions in the image) extracted from the MMI data for the first time to diagnose the plasma conditions and study the spatial distributions of the tracer in the OMEGA implosion core. We also use the spatially-resolved spectra to estimate the size of the hot-spot formed in an implosion. This dissertation manuscript is organized as follows. Chapter 2 presents details about our OMEGA direct-drive inertial confinement fusion experiments, laser and target parameters, and diagnostic instruments (mainly about the MMI). It also describes about the MMI data processing and extraction of space-resolved spectra characteristics of the annular regions on the implosion core image. In chapter 3, we will discuss the formation of Ti K-shell emission and absorption spectrum in the OMEGA implosion core. We will start with the details about the atomic processes and plasma atomic kinetics, line broadening of spectral line shapes and radiation transport. In chapter 4, space-resolved spectra characteristics of annular regions in the implosion core image will be used to estimate the hot-spot size. The key concept here will be the gradual transition of an emission dominated spectra to an absorption dominated spectra as we increase the backlight radiation to the plasma system while keeping the plasma condition constant. We will also discuss the change of hot spot size with varying target parameters and pulse drives. Chapter 5 will describe the multi-layer spectroscopic model to extract plasma conditions in the implosion core. We have developed a new spectroscopic method to obtain the tracer spatial distribution in OMEGA implosion core. In chapter 6, we will apply the methods developed in chap-

18 5 ter 5 to obtain the spatial distribution of the Ti tracer initially located on the inner surface of our targets. Results and trends of Ti mixing will be also discussed. Finally, we will summarize the dissertation and talk about the ideas to advance the methods developed during this dissertation further.

19 6 Chapter 2 Experiments at OMEGA and MMI data 2.1 Nuclear fusion Nuclear fusion is a reaction in which two or more light nuclei fuse together to form a heavier nucleus with the release of energy. The mass of the combined nucleus will be less than the sum of the masses of the individual nuclei which took part in the fusion process. In a nucleus, the difference in mass between the mass of the nucleus and the sum of the masses of it s nucleons (protons and neutrons) is called the mass defect( m), and is given by m = Zm p + Nm n M, (2.1) Where m n and m p are masses of each neutron and proton respectively, M is the mass of the combined nucleus, N is the number of neutrons, Z is the atomic number. The binding energy of the nucleus is the product of the mass defect times the square of the speed of light. This is the total energy required to hold all the nucleons in the

20 7 Figure 2.1: Mean binding energy per nucleon of stable nuclei as a function of mass number. (Credit: S. Pfalzner, An introduction to inertial confinement fusion, Taylor & Francis group) nucleus together. In other words, this is the energy needed to break up the nucleus and to separate all the nucleons to distances where they no longer interact via nuclear force. Iron with atomic weight A = 56 is the natural limit for fusion processes to deliver energy because it has the highest binding energy per nucleus. For nuclei much lighter or much heavier than iron, the binding energy per nucleon is considerably smaller [1]. The fusion of two nuclei with smaller masses than iron (which, along with nickel, has the largest binding energy per nucleon) generally releases energy, while the fusion of nuclei heavier than iron absorbs energy. The mean binding energy per nucleon of stable nuclei as a function of mass number is shown in figure 2.1. Positively charged nuclei strongly repel each other due to the mutual repulsive electrostatic force i.e., Coulomb repulsive force. Fusion is possible when the distance

21 8 between nuclei becomes smaller than m. At this point, the attractive nuclear force dominates the electrostatic repulsive force. However, under normal conditions, nuclei are separated by distances far larger than the nuclear force range. Therefore, a nuclear fusion reaction is quite unlikely. There are many scenarios to achieve nuclear fusion reactions; such as gravitational confinement, magnetic confinement, and inertial confinement fusion. The gravitational forces in the stars compress matter, mostly hydrogen, up to very high densities and temperatures at the star centers, triggering the fusion reaction. The primary process of fusion in the sun is known as the proton-proton chain [1]. The first step is the fusion of two pairs of hydrogen (protons) and formation of a diproton. Then, the diproton undergoes beta plus decay, which forms a deuterium, positron and a neutrino. This first step is extremely slow, because the beta-plus decay of the diproton to deuterium is extremely rare. In the second step, the deuterium fuses with an additional proton and forms helium-3. Then, mainly two helium-3 nuclei fuse together and form a helium-4 and two protons with the release of MeV energy or the one helium-3 nuclei fuses with one proton and form a helium-4, positron and a neutrino. Most of the energy streaming from the sun come from these fusion reactions. To a lesser degree, other fusion processes using different reaction cycles leading to the formation of helium take place at the same time. All these processes are unsuitable for terrestrial fusion in the laboratory due to their extremely slow reaction rates [1]. There are currently two techniques employed to achieve nuclear fusion in the laboratory: magnetic confinement fusion and inertial confinement fusion. Magnetic confinement fusion uses magnetic fields to confine the hot plasma and creates the conditions needed for fusion. It confines a low density plasma for the relatively long time of several seconds [1]. The two major projects based on magnetic confinement for

22 9 nuclear fusion are International Thermonuclear Experimental Reactor (ITER) under construction in France, and Tokamak Fusion Test Reactor (TFTR) at Princeton. The latter was completed in 1997 [1]. The inertial confinement fusion and the conceptual discussion of direct-drive implosion experiments are explained in the next section. 2.2 Conceptual discussion of direct-drive implosion experiments Inertial confinement fusion is an approach to initiate nuclear fusion reactions by the heating and compressing of a fuel target, often in the form of a spherical plastic shell filled with deuterium. The deuterium is compressed to very high densities and temperatures by applying lasers or x-rays on the target s surface. In contrast to magnetic confinement fusion, the confinement times are extremely short ( 10 9 s) but the plasma density is very high [1]. There are two techniques to achieve inertial confinement fusion: indirect-drive and direct drive. In the indirect-drive approach, the target is placed inside a gold-coated hohlraum cavity and laser beams irradiate the interior walls of the hohlraum from both sides. The laser beam interaction with the walls produces soft x-rays, which drive the implosion of the capsule by ablating the outer surface material [1, 2, 3]. In direct-drive, the surface of the plastic shell target is irradiated directly by intense laser or ion beams. The experiments for this dissertation project were directdrive, and laser beams were used for the target s surface irradiation. The irradiation heats up and breaks down the outer surface material immediately turning it into a coronal plasma i.e., a low density plasma that expands outward from the surface. Figure 2.2a) shows the target irradiation by the laser beams and energy delivered to the target. The laser energy propagates to the critical density surface, i.e., the region

23 10 Figure 2.2: The stages of direct-drive inertial confinement fusion. (Credit: An assessment for the prospect of the inertial fusion energy. National Academies Press, 500 Fifth Street, NW, Washington DC 20001) where the laser frequency is equal to plasma frequency. The critical density is given by [1, 27], n c = (10 21 /λ 2 )cm 3, (2.2) where λ is the wavelength in micrometers. The location of the critical density surface is characteristic of the wavelength, intensity, and pulse length of the laser beam [1]. At the critical density surface, most of the laser energy is absorbed mainly by inverse bremsstrahlung, and also by resonance absorption and parametric processes. The absorbed energy is transported to the ablation surface via electron thermal conduction [1, 28]. The ablated shell mass surrounds the target and accelerates the remaining shell material inward via rocket effect. Figure 2.2b) shows the ablation of the outer surface material and generation of shock waves as a result of the rocket like blow-off of the hot surface material, which compresses the remaining fuel and shell. The orange arrows represent the ablation of the outer surface material and blue arrows indicate the inward compression by the shock waves. As a reaction of the ablated shell material, multiple shock waves may be launched inward, which compress the core fuel and unablated shell. These shocks merge together and become a singular strong shock before reaching the center of the capsule

24 11 [29]. The shock moving inward is reflected at the center and subsequently reflected off the incoming inner shell surface multiple times. The interaction of the incoming inner shell surface with the reflected shock wave from the center decelerates the inner shell surface. Typically, the reflected shock becomes weak after the first or second reflection off the shell [30]. Successive shock waves, generated due to multiple reflections off the center and the shell inner surface, increase the pressure of the core, and slow down the incoming shell in a gradual manner. At some point the plasma enclosed by the inner shell surface develops a fairly uniform pressure profile, which is large enough to stagnate the system. The core region with the uniform pressure profile is called hot-spot. Figure 2.2c) shows the stagnation phase of the implosion. The peak compression occurs at the stagnation phase. During the stagnation phase, the temperature and density of the hot spot becomes very high and may begin the burn of the fusion material in the central area. A thermonuclear burn front then propagates rapidly outward into the main fuel region. Figure 2.2d) shows the burning phase of the fuel. During acceleration and deceleration of the implosion, the target is subject to Rayleigh-Taylor instability (RTI) which makes the implosion unstable. There are many factors for seeding the perturbations early in the implosion: laser imprint, beam to beam power imbalance, outer/inner shell surface roughnesses which seeds the RTI and determine the final target performance. The RTI in the deceleration phase causes mixing of shell material with the hot-spot [28].

25 2.3 The importance of spectroscopic tracers for implosion diagnosis 12 In our targets for direct-drive inertial confinement fusion experiments, we use Ti and Ar as tracer elements. At the temperature and density conditions achieved in the implosion core, deuterium is fully ionized and emits continuum radiation due to Bremsstrahlung. However, the mid-z elements like Ti or Ar become partially or fully ionized, and change into either fully striped or H-, He-, Li-like and lower ionization stage ions. These ion species can exhibit emission and absorption lines as well as continuum radiation from the imploding core during the period leading upto and including peak compression. Only the Ti K-shell line emissions and absorption spectra have been used for the analysis of the data for this dissertation work. X-ray spectroscopy has been used as a powerful, non-intrusive tool in the field of inertial confinement fusion to diagnose plasma conditions. Detailed modeling and analysis of absorption and emission lines can yield core electron temperature, density, areal density, ionization state of the dopant and mixing of shell material into the core. The temperature and density sensitivity of k-shell emission and absorption lines comes through the dependence of atomic level populations on plasma electron temperature and density. Also, the Stark broadening of lines depend on electron density. The sensitivity of line emission and absorption shapes on temperature and density can be used as a diagnostic tool for extracting implosion core and shell conditions in ICF implosions. The spectral range ( ) ev is sensitive to plasma electron temperature and density, which consists of Ti K shell line emissions Heα, Lyα, Heβ, Heγ, Lyβ and Lyγ, and their associated Li- and He-like satellites. The line intensity ratio of similar line transitions in adjacent ionization stages reflects the ionization balance of

26 13 the source plasma. For example, if the ratio of intensity of Lyβ(n=3 to n=1 in H-like Ti) to Heβ(n=3 to n=1 in He-like Ti) increases, it indicates that the implosion core is becoming hotter and the ionization balance of the source plasma is shifting from He-like to H-like Ti plasma, and vice versa. When Ti-ions in the cold shell are backlit by X-ray continuum due to Bremsstrahlung emission from hot implosion core, the Ti K-shell absorption spectra is produced. The absorption lines that we observe in our Ti spectra are mainly due to n=1 2 transitions in Ti L-shell ions in the doped region. The spectroscopic analysis of absorption features is very useful to determine the state of the compressed shell in terms of electron temperature, density and areal density [14]. The amount of tracer must be chosen properly so that it does not alter the hydrodynamics of the implosion due to the radiation cooling effect while at the same time produce enough spectroscopic signatures for data analysis [9, 10]. Figure 2.3a) and b) show the processed MMI data with and without Ti tracer doped plastic layer respectively. Figure 2.3a) is from OMEGA shot 65692, TIM4, the target in this shot does not contain Ti-doped plastic layer in the shell. Figure 2.3b) is from OMEGA shot 65695, TIM4. The target in this shot contains a Ti-doped plastic layer on the inner surface of the plastic shell. The vertical line features in b) arise from line emissions from partially ionized Ti ions. The dark vertical features are due to the absorption of Bremsstrahlung radiation by the Ti ions in the colder region. 2.4 Target and laser parameters of specific experiments The direct-drive ICF experiments were performed at the OMEGA laser facility of the Laboratory for laser energetics at the University of Rochester, NY. The 60 laser

27 14 Figure 2.3: a) OMEGA shot TIM 4-MMI data without Ti tracer, b) OMEGA shot TIM 4-MMI data with Ti tracer Figure 2.4: Schematic of a direct-drive target pie-slice

28 15 a) b) c) Figure 2.5: Laser pulse shapes to drive high and low-adiabat implosions a) highadiabat 1ns, b) low-adiabat α 2, c) low-adiabat α 3 beams (λ = 350 nm, total energy 22 KJ) were used to irradiate the surface of deuterium filled plastic shell targets. The targets had varying shell thicknesses and gas pressures with a half micrometer tracer layer at the fuel-shell interface. The Ti atomic concentration in the layer was either 3% or 6%. The filling pressures were 5, 10 and 20 atm of deuterium gas. The targets had an internal radius approximately 400 µm, the wall thicknesses were 15, 20 and 27 µm, and an outer aluminum coating of 0.1 µm for sealing purposes. Some of the targets had Ar (0.072 atm) in the core. The laser pulse shapes used for the target irradiation were high-adiabat, and lowadiabat (α 2 and α 3 ). The adiabat is defined as the maximum local ratio of the plasma pressure to the Fermi pressure of a degenerate electron gas at the same density. The

29 16 Figure 2.6: a) MMI instrument built in the Machine shop at UNR and b) Schematic of MMI value of α is 2 and 3 for α 2 and α 3 respectively, and higher than 3 for high-adiabat pulse. The high-adiabat delivers energy to the target in a single step, resulting in a higher degree of heating. Conversely, a low-adiabat drive has a low intensity foot, followed by a high intensity main peak. The low-adiabat laser pulse shape is expected to minimize any preheating of the fuel [31]. The lower α maintains the compressibility of the target and maximizes the yield [28]. The laser beam smoothing was obtained by distributed phase plates (DPPs), 2D spectral dispersion (SSD), and distributed polarization rotators [32, 33, 34]. 2.5 Instruments and spectroscopic measurements The X-ray signal from the titanium tracer is primarily observed at the collapse of the implosion and recorded with a streaked crystal spectrometer (SSCA), time integrated spectrometer (XRS1) and the gated, multi-monochromatic x-ray imager (MMI) instrument. The SSCA records image data that is temporally and spectrally resolved but spatially integrated. The sweep speed of the camera was 50ps/mm, with resolving power of (E/ E) = 350 and temporal resolution of 30ps [14]. The XRS1 spectrometer records a time- and space-integrated spectrum of X-ray radiation emitted during the

30 17 Figure 2.7: Three TIMs (Ten Inch Manipulators in the OMEGA target chamber) implosion. The spectral resolution of XRS1 is 1200 [14]. The MMI instrument consists of a pinhole array, a multi-layered mirror, and an X-ray framing camera with four micro-channel plates [35, 36, 37]. Each pinhole array has about a thousand pinholes made by laser drilling on a tantalum substrate in a hexagonal pattern, each 10 µm in diameter and an average separation of 115 µm between pinholes. The multi-layered mirror consists of 300 bilayers of boron carbide and tungsten with an average thickness of 15 Å each. X-rays from the implosion core enter the nose tip of the MMI, and pass through the pinholes, which in turn create a large number of implosion core images. Then, the core images fall on the mirror which reflects the images towards the micro-channel plate according to Bragg s law yielding a spectral resolution along the axis parallel to the plane of the incidence [12]. The spectral resolving power (E/ E) of MMI is 150, the spatial resolution is 10 µm, and the magnification is 8.1 [35, 36]. Each micro-channel plate is triggered at different times with an interstrip delay of 100 ps to record four snapshots of spectrally resolved implosion core images during the implosion. Figure 2.6a) shows the MMI instrument fabricated in our Machine shop at UNR and 2.6b) shows the schematic of MMI [12].

31 Figure 2.7 shows the quasi-orthogonal lines of sight from three TIMs i.e., ten-inch 18 manipulators, in the OMEGA target chamber. Three identical MMI instruments were fielded along these three quasi-orthogonal lines of sight by using the OMEGA target chamber diagnostic ports labeled by TIM3, TIM4, and TIM5. The spherical coordinates for location of the TIM3, TIM4 and TIM5 are given (in degrees) by (θ: ; φ: ), (θ: 63.44; φ: ), (θ: ; φ: ) respectively. The angles between TIM3 and TIM5, TIM3 and TIM4, TIM4 and TIM5 are 70.5deg, 79.2deg, and 79.2deg respectively [12]. The spectrally-resolved implosion core images are recorded on the film of gated micro channel plates in an X-ray framing camera detector. The film is then digitized. Figure 2.8 shows the processed MMI data, OMEGA shot recorded along TIM4. There are four frames, the top frame, i.e., frame 1 was recorded at the earliest time. The second, third and the fourth frames were recorded approximately 100, 200 and 300 ps after the recording of the first frame. The horizontal axis represents the spectral resolution axis increasing from left to right measured in units of ev. In addition to the spectral resolution, there is also a time difference between the left and right end of each frame due to the finite sweep speed of the gating voltage applied across each MCP. The left end is earlier in time than the right end. The spectral features from the titanium tracer are primarily observed at the collapse of the implosion. The data shows simultaneous emission and absorption features associated with titanium K-shell line transitions. The spectral range includes Ti K- shell absorption (principal quantum number n = 1 2 transitions in Ti L-shell ions, spectral range: ev, Ti Heα (1s 2-1s 1 2p 1, line center: 4750 ev), Ti Lyα (1s 1-2p 1, line center: 4973 ev), Heβ (1s 2-1s 1 3p 1, line center: 5580 ev), and X-ray continuum due to the Bremsstrahlung emission and radiative recombination in the implosion core.

32 19 Figure 2.8: Processed MMI data from shot TIM4. The target parameters were, core size: 400µm, shell thickness: 15µm and laser pulse shape: α 2 We observe that early in time, i.e., in frame 1, the Ti-absorption is barely visible, while in later frames it is clearly observed. The line features are better in the earliest frame than in the later frames. In frames 2, 3 and 4, implosion core images are larger and close to each other on left sides, while they are smaller and better separated on right sides. There are no images on the higher energy side of frame 4.

33 2.6 MMI data processing 20 The data recorded by MMI are X-rays in the form of gated, spectrally resolved implosion core images, each of which is formed by photons in a slightly different energy range. The data are very rich in information, and therefore, a good processing tool is essential to extract information useful for spectroscopic diagnosis. Our processing codes are written in IDL language and were developed by T. Nagayama et al. [12, 13]. The important steps of processing the raw MMI data include conversion of film density to intensity using film calibration, artifact removal and flat-fielding of the data, center determination of monochromatic core images, photon energy dependent intensity corrections associated with beryllium filter transmission, reflectivity of the multilayered Bragg mirror, and spectral response of the MCP [12, 38]. After the corrections and center determination, MMI data can further be processed to extract narrowband/broadband images and space-integrated/space-resolved spectra. The algorithm implemented in the processing code simultaneously determines the centers of all core sub-images by finding parallel line grids, which fall into the dark patterns in between the core subimages. To find the parallel line grids, it uses three core adjacent subimages as a starting point, which are not on the same straight line and whose centers are approximated using intensity weighted averages. The centers of these three sub-images provide three basis vectors. Then, the parallel lines are recursively constructed starting from the origin. These grid lines sometime go over some of the core sub-images located far from the origin, which is improved by optimizing the initial parameters i.e., centers of the three core subimages using Powel s method [12]. After center determination, narrow-band or broadband images can be constructed.

34 21 Figure 2.9: Annular region in implosion core and its projection on detector plane The processing code [12, 13] utilizes the following equation [39] i(x, y ) = N s(x,y ) 1 j=0 h(x + x j, y + y j )/N s (x, y ) (2.3) Where the two-dimensional arrays h and i represent the raw and the reconstructed images, (x j, y j ) is the center position of the j th sub-image, and N s (x, y) is the number of pixels summed for the reconstruction of the raw image h. An important criterion for reliable MMI diagnostics is appropriate tilting of pinhole array, since it has huge impact on image reconstructions [40]. A properly tilted pinhole array efficiently provides spatial information of the object at each photon energy, however; an improperly tilted pinhole arrays may not provide spatial information of the object at each photon energy and hence the spatial sampling is biased. In order to reconstruct narrow-band images, one has to select the energy band width greater or equal to reconstruction or sampling width [40]. The sampling or reconstruction width is defined as the minimum width required to reconstruct a narrow-band image completely. If we choose an energy band smaller than the reconstruction width, the image reconstruction will be incomplete, which means that it can not construct the whole image. If we choose a wider energy band than the reconstruction width, contributions from different core images overlap, image reconstruction becomes complete and signal-to-noise ratio improves [12].

35 22 We can also extract space-integrated and space-resolved spectra from MMI data. To extract space integrated spectrum, intensities are summed up vertically at each photon energy across the MMI data. Spatially resolved spectra can be extracted from the MMI data by constructing an MMI mask image associated with the spatial region of interest on the image plane. Application of the mask on the MMI data will allow contributions only from the spatial region of interest and block the contributions from outside of the mask. Then, summing up intensities vertically at each photon energy across the MMI data will give a space-resolved spectra from the spatial region of interest. For this dissertation work, spatially resolved spectra characteristics of annular regions on the image plane are extracted. Each annular region represents the projection of the domain of integration in the core on the detector plane. MMI data can spatially resolve the core in many ways i.e., it depends on how the mask is defined. The minimum width of the annular region must be equal to or greater than the spatial resolution of the MMI i.e., 10µm. Figure 2.9 shows the spatial region of interest in the implosion core and its projection on the image plane. Figure 2.10 shows the steps to obtain space-resolved spectra. Figure 2.10a) shows the processed MMI data from OMEGA shot TIM 4 Frame 2, and 2.10b) shows the MMI mask image associated with the spatial region of interest. The red annular region on the image represents the mask associated with the spatial region of interest. Similarly, figure 2.10c) represents the MMI data after the application of MMI mask image in figure b), and figure 2.10d) shows the space-resolved spectrum from the spatial region of interest.

36 Figure 2.10: a)processed MMI data, OMEGA shot TIM 4 Frame 2, b) MMI mask image associated with spatial region of interest, c) MMI data in a) after the application of mask, d) space-resolved spectrum associated with spatial region of interest 23

37 24 Chapter 3 Titanium emission and absorption K-shell spectrum The measurement of plasma conditions during the deceleration and burning phases of an ICF implosion experiment relies on observations of nuclear reaction products and/or radiation emission and absorption. In particular, x-ray spectroscopy is the method used in this dissertation. It employs the analysis of the line radiation signal emitted (or absorbed) by a tracer element added to the core fill (or located in the shell) in order to extract the ionization, areal-density, temperature and density conditions in the plasma core or the dense shell confining the core. Indeed, over the years x- ray line spectroscopy has proved to be a powerful, non-intrusive diagnostic of ICF implosions [22, 41]. To this end, a tracer amount of a low- or mid-z element is added to the target. Typically, noble gases have been used for the core, e.g. neon, argon, krypton and xenon, and chlorine, titanium or vanadium in the form of a thin, doped tracer layer for the shell. The selection of the tracer element depends on the plasma conditions expected and the type of line transition used in the analysis, e.g. usually K-shell or L-shell transitions. The amount of dopant has to be sufficient so

38 25 that a quality signal is recorded suitable for detailed quantitative analysis but, at the same time, the radiation emitted (or absorbed) by the tracer does not have to perturb the hydrodynamics of the implosion, i.e. the difference between the implosion hydrodynamics with and without tracer must be negligible. In addition, the photonenergy dependent optical depth has to be moderate so that the information content of the emerging line intensity distribution is characteristic of spatial regions near the surface as well as deep inside the plasma source. In the ICF implosion experiments discussed here the tracer element is a titanium-doped (Z=22) plastic layer with 3% or 6% atomic concentration, 0.5 microns thick, initially located on the inner surface of the plastic shell, i.e. the ablator. The spectroscopic analysis considers K-shell line emission from n=2 1, n=3 1 and n=4 1 in He- and H-like Ti ions, and their associated satellite transitions in Li- and He-like Ti ions. Also, K-shell n=1 2 line absorption in F- through Li-like Ti ions. In this chapter, we discuss the plasma dependence of K-shell line emission and absorption in highly-charged titanium ions inside the core and in the outer region near the shell, respectively. We take into account the atomic processes that determine the atomic kinetics, ionization and level population distribution, the broadening effects of the spectral line shapes, and the emergent line intensity distribution that results from the radiation transport through the plasma source as well as the transmission of backlit radiation through a Ti-doped warm, dense plasma. We illustrate the sensitivity of the spectrum to the plasma temperature and density conditions. It is the dependence of the titanium emission and absorption spectrum on plasma environmental conditions that is at the core of the analysis methods discussed in Chapter 5.

39 3.1 Atomic and radiation physics of titanium ions Atomic processes and plasma atomic kinetics During the deceleration and burning phases of OMEGA direct-drive inertial confinement fusion implosion cores, the electron temperatures and densities are very high and in the ranges of 300 ev to 3000 ev and 1e +23 cm 3 to 1e +25 cm 3, respectively. For these plasma conditions, titanium is highly ionized and the ionization distribution is dominated by the population of Be-, Li-, He-, and H-like Ti ions and the fully-stripped ion. The population distribution among atomic levels and ionization states in a given ion is normally in a state of non-local thermodynamic equilibrium (NLTE). In NLTE the ionization and population distribution are determined by both collisional and radiative atomic processes. For an accurate calculation of the atomic kinetics of highly-charged titanium ions in implosion experiments, following atomic processes are very important and need to be considered. Spontaneous radiative decay Photo-excitation Radiative recombination Photo-ionization Spontaneous autoionization Resonant electron capture Electron collisional excitation Electron collisional de-excitation Electron collisional ionization

40 27 Electron collisional recombination The objective of the atomic kinetics calculation is to provide the ionization and distribution of level populations of the titanium ions in the plasma. In other words, it provides total population of each ion species and, for each ion species, total population in the ground state and in each excited state. The method of calculation is based on collisional-radiative (CR) model. It sets up a system of atomic kinetic rate equations, and solves them for the level populations. Thus, in a CR model the distribution of population among energy levels is determined by the competition between multiple collisional and radiative atomic processes that connect energy levels within a given ion and across adjacent ions. Of the atomic processes listed above, spontaneous radiative decay, photo-excitation, and electron collisional excitation and de-excitation connect the populations of energy levels in a given ion. Radiative recombination, photoionization, spontaneous autoionization, resonant electron capture, electron collisional ionization and electron collisional recombination link the populations of energy levels across adjacent ionization stages. There are two limiting cases that the general CR atomic kinetics model satisfies i.e., the low-density or corona model limit and high density or local thermodynamic equilibrium model (LTE) limit. In the low density limit, the population of each ion is dominated by the population of the ground state. In this limit, the level populations are determined by the competition between electron collisional excitation and radiative and dielectronic recombination between ground states in adjacent ions. The density cancels out in the atomic kinetic rate equations and the population distribution is only dependent on the electron temperature because the rates of all these binary processes are proportional to the electron number density. After computing the ground state populations for all ions in the model, the populations of the excited states are determined (in a perturbation approximation) by a balance between

41 28 electron collisional excitation from the ground state and spontaneous radiative decay to the ground state. In LTE limit, the atomic kinetics is dominated by electron collisions. Thus, the populations are obtained only by collisional processes, namely electron collisional excitation and de-excitation to connect energy levels in a given ion, and electron collisional ionization and recombination across ionization stages. The Boltzmann factors and Saha equations can be used to connect energy levels in the same and across ionization states, respectively. In this dissertation, atomic kinetics calculations for titanium ions in implosion core plasmas have been performed with the ATOKIN CR model and code [42]. The ATOKIN model developed for titanium considers all ionization states from Ne-like Ti to the fully-stripped ion and includes 143 energy levels, with the following maximum number of energy levels per ion: 1 fully stripped, 16 H-like, 22 He-like, 29 Li-like, 14 Be-like, 17 B-like, 17 C-like, 14 N-like, 9 O-like, 4 F-like and 1 Ne-like Ti. Energy levels and spontaneous radiative decay and autoionization rates, and scattering cross sections were computed using the atomic structure and scattering code FAC [43] including configuration interaction. All non-autoionizing and autoionizing states characterized by principal quantum numbers consistent with the continuum lowering criterium [42] are taken into account. Therfore, the actual final number of levels included in the calculation depends on the plasma conditions of each case. We have considered the H-like and He-like Ti with principle quantum number n up to 4, the Li-like Ti with n up to 3, and rest of the Ti ions mentioned above with n up to 2 in the calculation. After obtaining level populations from the solution of the CR atomic kinetics model, the local values of the emissivity and opacity can be computed. We have included the effects of all high-order satellite transitions with spectator electrons of principal quantum n consistent with the continuum lowering criterium. When the principal quantum number of spectator electron increases, the difference between the

42 29 photon energy of satellite transition and the parent line decreases (photon energy of satellite shifts towards the parent line). As a result of the shifting, the satellite transition overlaps and blends with the parent line, and thus affects both the emissivity and opacity of the composite spectral feature defined by parent and satellite lines [44] Spectral line profiles There are several line broadening mechanisms which affect the spectral line shapes of Ti line transitions, for example, natural line broadening, Doppler line broadening, and Stark line broadening [22]. The natural line broadening arises from the uncertainty (Heisenberg uncertainty) in the finite lifetime of the states involved in the transition, e.g. spontaneous radiative decay and spontaneous autoionization. The spectral line profile of this effect is defined by a Lorentzian function. The full-width-half-maximum (FWHM) is given by ha/(2π) where h is Plancks constant and A is the total decay rate, including contributions from spontaneous radiative decay and autoionization of the initial and final states involved in the transition. For the K-shell lines of titanium considered here this FWHM is about or less than 1 ev. Doppler broadening of spectral lines is due to the Doppler effect caused by a distribution of velocities of radiator (or absorber) ions in the plasma. Different velocities of the radiating (or absorbing) ions result in different Doppler shifts, the cumulative effect of which is the line broadening. The distribution of velocities of the ions can be obtained from Boltzmann distribution, and the line profile is given by a Gaussian function. For temperatures in the range of 300 ev to 3000 ev and the wavelengths of the titanium K-shell lines, the Doppler FWHM is of the order of a few ev. In high energy density plasma, ions and electrons produce electric fields and line transitions are perturbed by the fields. Stark broadening of a line transition is due to

43 30 a) b) c) Figure 3.1: Area normalized absorption line shapes for Be-like Ti n=1 to n=2 transitions, for several electron temperature and density conditions [49].

44 31 the plasma electric microfields produced by the ions and electrons of the plasma, and is strongly dependent on the plasma density. It is the dominant line broadening effect for our cases of application. For the titanium lines this effect results in an overall FWHM of about and larger than 10 ev for the plasma conditions of ICF implosion experiments. In the standard Stark broadening theory approximation the (heavier) ions are considered static during the time of the line transition [22], and the (lighter) electrons are considered dynamic. The static microfield (produced by ions) splits and shifts the energy levels of the radiators, and this effect is characterized by the ions microfield distribution function. The microfield produced by the electrons further broadens the line transitions between the field-dependent energy levels. However, in our OMEGA implosion core plasmas the majority of the perturbing ions are actually deuterium ions that are much lighter than titanium ions and thus the effect due to the motion of the ions has to be taken into account as well [45]. For this dissertation, a database of emission and absorption Stark broadened line shapes including the effects of both plasma ions and electrons was computed using the formalism and codes discussed in references [46, 47, 48, 49]. Figure 3.1 shows area normalized absorption line shapes for the case of Be-like Ti as an illustration of the line profile calculations [49]. The configurations for the initial levels were 1s 2 2s 2, 1s 2 2s 1 2p 1 and 1s 2 2p 2, and for the final levels were 1s 1 2s 2 2p 1, 1s 1 2s 1 2p 2 and 1s 1 2p 3. This group of initial and final configurations gives rise to a problem with 10 initial and 30 final J-energy levels, and 84 fine structure line transitions. Line shapes for several values of the electron density n e have been displayed in figure 3.1a. The line shapes show little change and become independent of electron density for values of n e below cm 3. This clearly indicates that line broadening is dominated by natural and thermal Doppler broadening effects. The line shape feature changes significantly when the electron density increases from to , suggesting that the Stark broadening effect

45 32 is gradually becoming more important. For electron densities above cm 3 the Stark effect clearly dominates the shape and broadening of the line profile, making them dependent on n e. Figure 3.1b displays the n e dependence of the line profiles for several electron densities between cm 3 and cm 3. Figure 3.1c) shows the temperature sensitivity of the line shapes at n e = cm 3 for several temperatures. We observe that the temperature sensitivity of the line shapes at these high electron densities is negligible. Similar results are obtained for other electron densities larger than cm Radiation transport The system of CR atomic kinetics rate equations and the radiation transport equation are coupled and have to be solved self-consistently and simultaneously. On one hand, we need radiation field intensity to compute photoexcitation and photoionization rates for the atomic kinetic rate equations. On the other hand, the level populations are needed to compute the emissivity and opacity that determine the radiation field intensity [44]. To solve the radiation transport equation self-consistently with the set of atomic kinetics rate equations, ATOKIN accounts for photoexcitation and photoionization effects using an escape factor approximation [50]. In this approximation the spontaneous radiative decay and radiative recombination rates are multiplied by escape factors that approximately take into account the trapping effect of bound-bound (i.e. line) and bound-free transition photons in the plasma source by effectively reducing the values of the spontaneous radiative decay and radiative recombination rates [50]. The escape factors were calculated for the case of a spherical plasma source, i.e. a spherical geometry was used to approximate the radiation transport effect on the atomic kinetics in the implosion plasma. This radiation transport effect on the atomic kinetics is particularly important for

46 33 the resonance n=1 n=2 line transition in the He-like titanium ion (Heα line) since, in spite of the low titanium concentration used in our implosion experiments, the Heα line can still have a substantial optical depth. Indeed, it is known that neglecting the photoexcitation effect driven by the Heα photons in the atomic kinetics of a tracer element can result in an underestimation of the titanium ionization state [51]. In turn, this effect results in an overestimation of the plasma temperature. 3.2 Ti K-shell emission spectrum from an implosion core After computing the level populations and line profiles, the local emissivity and opacity of the plasma can be calculated. We include the contributions from all boundbound (line) transitions, as well as additional contributions from bound-free and freefree transitions. Once the emissivity and opacity database is available, the radiation transport equation is integrated for each photon energy along chords parallel to the line-of-sight, and then all individual chord contributions are further integrated to obtain the spectrum integrated over the domain of integration inside the plasma source suitable for comparison with the experimental spectrum. The domain of integration depends on types of measurements, space-integrated or space-resolved spectra. To account for the finite spectral resolution of the instrument, the synthetic spectrum is convolved with the instrumental function before comparing with data. In general, these spatial integrations are performed numerically. However, for the case of a sphere of uniform conditions of temperature and density the integral over the whole sphere that yields the spatially-integrated spectra is analytical [52]. The spectroscopic analysis relies on the electron temperature and density sensitivity of the titanium line emission through the density and temperature dependence of the

47 34 atomic level populations and the density sensitivity of the Stark broadening effect of the line shapes. This dependency of K-shell emissions on plasma conditions makes the K-shell emission spectrum useful for diagnostic applications of ICF experiments. Figure 3.2 displays a matrix arrangement of spatially-integrated spectra for a uniform sphere computed for four combinations of temperature and density. The changes in the intensity distribution reflect the dependencies with temperature and density. In particular, the increase in line broadening correlates with the increase in density, and the increase in temperature results in a Lyβ line that becomes stronger compared to the Heβ line. The overall level of intensity of radiative recombination radiation emission also increases with density. 3.3 Titanium K-shell absorption in a dense plasma We now turn our attention to the transmission of backlit radiation through a Tidoped dense plasma. The following equation gives photon energy dependent optical depth due to n=1 n=2 line transitions in Ti ions, in a slab of plasma with physical thickness R, τ ν = πe2 mc f ij φ ij (ν)f i N R, (3.1) ij where e, m and c stand for the electron charge and mass, and the speed of light, respectively. The sum goes over all line transitions i j in all ions included in the model, with i and j labeling the initial and final states of the transitions, f ij is the absorption oscillator strength, φ ν is the area-normalized line shape, F i is the fractional population of the initial state, and N is the total Ti atom number density. To compute the optical depth using the equation 3.1, we need the detailed area-normalized absorption line shapes, fractional populations F i of the initial states (i.e. lower energy

48 Figure 3.2: Plasma electron temperature and density dependence of the titanium emission K-shell spectrum 35

49 36 states) of the transitions. The initial states of the transitions are calculated with the CR atomic kinetics model. The transmission through a uniform absorbing layer can be modeled according to, I ν = Iν 0 e τν, (3.2) where the incident and transmitted intensities are represented by Iν 0 and I ν, respectively. The effect of self-emission of the line is neglected. Equation 3.2 gives the solution of the radiation transport equation for the case of uniform opacity and negligible emissivity. Figure 3.3 shows calculations of τ ν for several densities and temperatures [49]. The optical depth τ ν is dependent on the electron density n e and temperature T e of the plasma through the temperature and density dependence of the fractional level populations F i and the density dependence of the Stark-broadened, area-normalized absorption line profiles [49]. Figure 3.3a displays τ ν for an electron density of cm 3 and three electron temperatures: 270 ev, 340 ev and 420 ev. The optical depth changes as the temperature increases, and the change in the optical depth reflects the shift in the ionization balance. For the lowest temperature, the optical depth is dominated by F-, O- and N-like Ti ions, while for the highest one C-, B- and Be-like Ti ions are more important. We note that, for the latter case, the maximum values of τ ν for absorption by the dominant ions are larger than in the former case. This is due to the increase in absorption oscillator strength with ionization stage [49]. Figure 3.3b illustrates the Stark broadening effect on the absorption lines due to the increase of electron density. It shows τ ν for several combinations of T e and n e that keep the ionization balance approximately constant [49]. Therefore, in each case τ ν is dominated by the same ions. When the density increases from cm 3 to cm 3, the optical depth becomes less structured and the width of the absorption peaks

50 Figure 3.3: Optical depth due to n=1 n=2 transitions in titanium ions for several electron temperatures and densities. The titanium areal density is cm 2 [49]. 37

51 38 become broader due to the density dependence of the line shapes. In fact, as density increases there is a significant decrease of the optical depth of the absorption peaks. The change in n e affects both the ionization balance and the Stark broadening of the absorption line profiles. When T e is constant, increase in n e will shift the ionization balance downward due to the increase in three-body recombination. An increase in n e will also increase the Stark broadening of the absorption line shapes. In figure 3.3b, several combinations of temperature and density have produced approximately the same ionization distribution, but the corresponding optical depths are different due to the density dependence of the Stark-broadened absorption line shapes. Figure 3.4 shows Ti transmissivity spectrum (I ν /Iν 0 in equation 3.2) for the plasma conditions displayed in figure 3.3b [49]. Figure 3.4a shows that the Stark broadening effect of the absorption line shapes is also observed in the transmissivity, while figure 3.4b displays the same results as figure 3.4a, but the transmissivity is now convolved with an area-normalized Gaussian of FWHM=7 ev.

52 Figure 3.4: Titanium transmission spectrum for the plasma conditions displayed in figure 3.3b, not including instrumental broadening effect (a), and including instrumental broadening effect (b). 39

53 40 Chapter 4 Hot-spot evolution in the implosion core In direct-drive inertial confinement fusion a spherical plastic shell filled with deuterium is irradiated by laser beams and the irradiation is designed to drive multiple shocks through the shell while minimizing entropy [1, 29]. These shocks merge into a single shock before reaching the center of the capsule [2, 3, 29]. The shock moving inward is reflected at the center of the core and is subsequently reflected off of the incoming inner shell surface, and as a consequence of this, the incoming shell is impulsively decelerated. The reflected shock off the inner surface again reflects at the center and off the incoming inner surface a second time. Typically, the reflected shock becomes weak after the first or second reflection off the shell [3]. Successive shock waves, generated due to the multiple reflections off the center and the shell inner surface, increase the pressure in the core each time, and slows down the shell in a continuous manner [3]. The plasma enclosed by the inner shell surface develops a fairly uniform pressure profile. The core region with a uniform pressure profile is known as the hot-spot in the implosion core.

54 41 Hot-spot formation in the central ignition technique of inertial confinement fusion requires substantial implosion symmetry. It is hindered by the RayleighTaylor instability (RTI) developed at the fuel shell interface during the deceleration phase of an implosion [31, 53]. The RTI occurs when a lighter fluid accelerates another fluid of higher density and may dramatically reduce the performance of ICF experiments by degrading the symmetry of implosion or even by breaking the shell. The RTI perturbations reduce the size of the central hot-spot and delay ignition [53]. Hence, it is important to know the size and symmetry of a hot-spot in an implosion core along different lines-of-sights. By comparing the sizes with each other, one can also extract qualitative and quantitative information about the relative RTI that have occurred along different directions in an implosion core. A large body of theoretical and simulation work have been done previously to study the hot-spot dynamics in deceleration phase [29, 30, 54, 55]. However, these efforts do not discuss the hot-spot size along different lines-of-sight. In this work, for the first time, the space-resolved spectra extracted from a direct-drive ICF implosion have been used to estimate the hot-spot size in an implosion. The temporal evolution and dependency on target parameters have been also studied. The details about experiments, data processing and the extraction of space-resolved spectra are discussed in chapter 2. We have estimated hot-spot sizes along multiple lines of sight. The variations in hot-spot size along different lines of sight could be associated with the different perturbations arising due to Rayleigh-Taylor instability or laser power imbalance along different directions in an implosion [53]. In section 4.1, we discuss the formation of absorption features in a titanium-doped direct drive OMEGA implosion. In section 4.2, we talk about extraction of hot-spot size from annular spatially resolved spectrum, and in section 4.3 we discuss the results and trends of the hot-spot size.

55 4.1 Transition from emission-dominated spectrum to absorption-dominated spectrum 42 In this section we illustrate gradual transition of emission-dominated spectrum to absorption-dominated spectrum with respect to the increase of the backlighting photons while we keep the Ti plasma conditions constant. Let us consider a slab of uniform plasma conditions as discussed in the section 3.3 chapter 3. Suppose the width of the slab is L, and Ti emissivity and opacity at photon energy ν are ε ν and κ ν, respectively. Iν 0 is the total backlighting radiation obtained to the slab. The formal solution of the radiation transport equation for the slab (equation A.10 in appendix A) is given by, I ν = I 0 ν e κνl + ε ν κ ν (1 e κνl ) (4.1) The first term represents transmission of the backlight radiation and the second term represents Ti self-emission from the slab. In section 3.3 chapter 3, we discussed about the transmission of backlight radiation through a slab of uniform plasma conditions and negligible Ti self-emission. The sensitivity of absorption line shapes and optical depth on plasma conditions were also discussed. We illustrated that Stark broadening dominates Natural and Doppler broadening at high densities (around and above). We had also discussed the dependency of opacity (and optical depth) on both temperature and density via the level populations atomic kinetics and stark broadening of line shapes. In this section, we consider Ti self-emission as well as transmission of backlight radiation. Figure 4.1 shows the plots for transmission of backlight radiation (first term in the equation), Ti self-emission (second term in the equation) and the total emergent intensity. The emissivity and opacity database used for these calculations was same

56 43 as discussed in chapter 3 section The Ti ions included in the calculations for these plots were He-like, Li-like, Be-like, B-like, C-like, N-like, O-like, F-like and Nelike Ti ions. The backlighting radiation was gradually increased (plot 4.1a to 4.1f) while keeping the plasma conditions in the slab constant. The electron temperature and density were 400 ev and 3e +24 cm 3, respectively. In each plot, the blue trace represents the self-emission of Ti, green represents transmitted backlighting radiation, and the red represents the total intensity due to Ti self-emission. In figure 4.1a) the backlight radiation was 1.0e +7 erg/(sr s cm +2 ev). The transmission of the backlight radiation is negligible. The total intensity is approximately equal to the Ti selfemission from the slab. In figure 4.1b) the backlight radiation is increased to 5.0e +8 erg/(sr s cm +2 ev. The backlight radiation used in the plot 4.1c), 4.1d), 4.1e) and 4.1f) were 1.0e +9 erg/(sr s cm +2 ev), 1.5e +9 erg/(sr s cm +2 ev), 3.0e +9 erg/(sr s cm +2 ev) and 8.0e +9 erg/(sr s cm +2 ev), respectively. The Ti self-emission was always same since the plasma conditions in the slab were kept constant. The total intensity showed the absorption spectral features only when the backlight radiation was equal or higher than 3.0e +9 erg/(sr s cm +2 ev). In other words, when the Ti selfemission is dominated by the transmitted backlight radiation, the resulting spectra showed absorption feature instead of the emission features. The key factor here is the quantity of the backlighting radiation received by the plasma system. The MMI data obtained from the titanium doped implosions consist of K-shell emission and absorption lines, and continuum emission. The absorption feature is produced in the colder shell region when the Ti ions (F-like, O-like, N-like, C-like, B- like, Be-like, Li-like, He-like and H-like Ti) in the region are backlit by the continuum radiation due to Bremsstrahlung emission coming from the central hot spot of the implosion core [14].

57 44 a) b) c) d) e) f) Figure 4.1: Transition from an emission dominated spectrum to an absorption dominated spectrum with increasing backlight radiation and keeping plasma conditions constant.

58 45 Figure 4.2: Fitting of the continuum level: OMEGA shot TIM 3 Frame Extraction of hot-spot size from annular spatially resolved spectra The space-resolved spectra (SRS) extracted from annular regions on the implosion core image have been used to estimate the hot-spot size in an implosion core. The first step towards estimating hot-spot size is to extract SRS from the outermost annular region on the implosion core image with region width approximately equal to the spatial resolution of the MMI i.e., 11 µm. Then, we extract space-resolved spectra from all the possible annular regions inward from the outermost region by keeping the width of each region constant. Each annular region is one pixel inward from the adjacent outer region. Next, the zero-level of absorption for each space-resolved spectrum is determined from a linear fit to the un-attenuated continuum observed on both sides of the absorption feature, as shown by the left-right arrows in figure 4.2. The spectral range of the continuum used for estimating the zero level of the absorption shown in the figure 4.2 (Shot TIM 3 Frame 3) was ev

59 and ev on the left and right sides of the absorption feature respectively, 46 and these same ranges were used for all other SRS shown in the figure 4.3. The chi-square (χ 2 ) calculation is done between each SRS and its corresponding zero-level absorption. The spectral range for χ 2 calculation for all the cases shown in the figure 4.3 is ev. The χ 2 calculation is defined by, n χ 2 i = (Idata,k i Ifit,k) i 2, (4.2) k=1 Where I data,k and I fit,k are the intensities corresponding to the absorption feature of the experimental data and zero-level absorption respectively. The index i (i=1 for the outermost region and increases inward) represents the location of the annular region on the image plane. The summation is taken over the total number of intensity points included in the chi-square calculation (same range for each SRS). Figure 4.3 shows the annular regions on the image plane and corresponding space-resolved spectra with linear fitting of continuum level to find zero-level absorption. Some of the annular regions which lie between region 5 and the central region are not shown in the figure (all of those regions, which are not shown in the figure, also show simultaneous emission and absorption of Ti spectral features). The lowest chi-square value (Fig. 4.3b) is associated with the annular region which shows emission only..it separates the absorption region from the no-absorption region in the implosion core. All the other inward annular regions show simultaneous emission and absorption, and all the other outward regions show emission only. The spherical zones (see figure 5.2, Chapter 5) in the implosion core contributing to this annular region and all the other outward regions beyond it receive very little continuum radiation coming from the hotter regions in the core which may not be enough to produce absorption lines. Therefore, the outer boundary of this annular region defines the hot-spot size in the implosion core. Figure 4.4 shows the average outer boundary of the hot-spot in

60 47 Figure 4.3: Space-resolved spectra (SRS) from different annular regions on the image plane (outermost to central): OMEGA shot TIM 3 Frame 3 the implosion core. The distance of each pixel on the implosion core image from its center can be obtained from the MMI processing codes [12, 13] discussed in chapter 2. The approximate diameter of the implosion core image (OMEGA shot TIM 3 Frame 3) obtained from the post-processing is: d = 120µm. The average distance of the annular region corresponding to the lowest chi-square value from the outermost boundary of the implosion core is represented by dx (which is 12µm for the given case). The total width of the region, which is not backlit by the continuum radiation, is 2dx i.e., 24µm. Therefore, the hot-spot size estimation is d - 2dx = = 96µm. The uncertainty in the estimation of hot spot size is approximately 5µm.

61 48 Figure 4.4: Annular region corresponding to the boundary of hot-spot: OMEGA shot TIM 3 Frame Results and trends for hot spot size Multiple lines-of-sight view In order to compare the hot-spot size of an implosion core estimated along different lines of sight, we need to be sure that we are comparing results from the same time during the evolution of implosion. In principle, the three MMI instruments should start recording x-rays reflected from the multi-layered mirror simultaneously; however, due to the jittering in the fast electronics, the start of recording for each strip is not precisely at the same time. The time correlations between MMI data recorded along different lines of sight were performed by comparing time histories of continuum intensity, since the peak Bremsstrahlung emission must have occurred at the same time. The time history of continuum intensity was obtained in two different ways. Firstly, by extracting the narrow-band continuum image and obtaining the average

62 49 Figure 4.5: Temporal evolution of continuum intensity along three quasi-orthogonal lines-of-sight: OMEGA shot (TIM3, TIM4 and TIM5) and (TIM3 and TIM5) intensity associated with the narrow-band extracted from all the frames in a TIM. Secondly, by finding the space integrated spectrum from MMI data and then, obtaining the integrated continuum intensity from a narrow-band spectral range in it for all the frames in a TIM. The same spectral range has been used in both of the methods for all the frames and TIMs in MMI data. Figure 4.5 shows the temporal evolution of continuum intensity for shot and 62086, where Image represents the temporal evolution of the continuum intensity obtained from the first method, whereas Line out represents that obtained from the second method. The spectral range 5100 to 5300 ev was used in all cases for both of the methods. In shot 62085, the peak continuum emission recorded in TIM 3 was at frame 3, and in TIM 5 was at frame 2. However, in TIM 4, the peak emission was recorded in between frame 2 and frame 3, therefore, to find the hot-spot size in the implosion, we find the average of hot-spot sizes estimated at frame 2 and frame 3. The hot-spot size at the peak continuum emission in TIM 3, 4 and 5 were 77, 93,

63 50 Figure 4.6: Temporal evolution of hot-spot size: OMEGA shot a) TIM 4, b) TIM 4, TIM 3, and TIM 4 and 98µm respectively. TIM 3 showed the smallest hot-spot size. For shot 62086, there was MMI data of good spectroscopic quality only in two TIMs (TIM 3 and TIM 5). The peak continuum emission recorded in TIM 3 was at Frame 3, and in TIM 5, it was at F2. The hot-spot sizes at the peak continuum emission in TIM 3, and 5 were 77, and 89µm respectively. Again, TIM 3 showed a smaller hot-spot size in the implosion. Different hot-spot sizes along different lines-of-sight in an implosion core indicate that an unequal Rayleigh-Taylor instability occurred during the implosion.

64 4.3.2 Temporal evolution 51 Figure 4.6 shows the temporal evolution of the hot-spot size. The upper two plots (OMEGA shots: TIM 4, TIM4) show the decrease of the hot-spot size with time. However, the lower two plots (OMEGA shot (TIM3 and TIM4) show first a decrease and then become flat. Hot-spot size decreases with respect to time because of the compression. From the early deceleration phase to the stage of maximum compression, hot-spot size will decrease. The timings for the lower plots could be very close to the stagnation phase as a result we don t see change in the hot spot size. Based on the timing sheet provided by the OMEGA lab, the frame 2 and 3 are recorded 80 and 160 ps later than frame 1 respectively in OMEGA shot TIM 4. The frame 2 and 3 were recorded 100 and 160 ps later than frame 1 respectively in OMEGA shot TIM 4. In OMEGA shot TIM 4, frame 2 and frame 3 are 40 and 120 ps later than frame 1 respectively. In OMEGA shot TIM 3, frame 2 and frame 3 are 80 and 180 ps later than frame 1 respectively Effect of changing parameters We have found that the hot-spot size decreases with increasing shell thickness of the target. In Figure 4.7, the red curve represents the variation of the hot-spot size with changing shell thickness keeping the filling pressure at 10 atm and laser pulse shape α 3 constant. While the green curve represents the change of the hot-spot size when changing shell thickness keeping the filling pressure at 10 atm and laser pulse shape α 2 constant. Similarly, the blue curve represents the change of the hot-spot size when changing shell thickness and keeping the filling pressure at 20 atm and laser pulse shape α 2 constant. All three cases show decrease of hot-spot size with respect to the increase of shell thickness. When the same total driver (laser) energy is used for the irradiation of the thicker and thinner target shell, the shock speed

65 52 Figure 4.7: Change of hot-spot size with changing shell thickness of the target is higher for the thinner-shell implosion than that of the thicker-shell. This means that the thinner-shell implosion achieves a uniform pressure profile sooner and at a larger implosion radius than in the thicker-shell implosion. Therefore, the thinnershell shows larger hot-spot size. One can also interpret this in the following way: the reflected shock wave off the center in the thinner-shell implosion will decelerate the inward moving shell-fuel interface more than that of the thicker-shell. Hence, the thinner- shell implosion will possess larger size than thicker one. Figure 4.8 shows the increase of the hot-spot size with increasing filling pressure. Higher filling pressure implies a lower convergence ratio, which in turn implies larger hot-spot size. In Figure 4.9 we see that the hot spot size decreases with changing laser pulse shape from α 2 to α 3. In an ICF implosion, it is desired that the compression be as close as possible to adiabatic and the fuel entropy should be kept as low as possible for the high gain in the implosion [53]. Any preheating of the target, i.e. heating pre-

66 53 Figure 4.8: Change of hot-spot size with changing filling pressure ceding compression, should be minimized. Preheating is measured by using isentrope parameter α, defined as the ratio of the actual pressure to the Fermi pressure. For better compression α is desired as close as possible to unity [53]. The α 2 has value equal to 2 and α 3 has value equal to 3. Therefore, laser pulse shape α 3 produces higher fuel entropy and a higher Rayleigh-Taylor instability on the inner surface of the shell than that of the α 2 [53], which in turn reduces the hot-spot size.

67 Figure 4.9: Change of hot-spot size with changing laser pulse shapes 54

68 55 Chapter 5 Methods for spectroscopic analysis and Ti spatial distribution In this chapter we discuss the method developed to analyze the spatially resolved spectra recorded with the MMI instrument as well as the extraction of the titanium atom number density. The former produces plasma electron temperature and density in the implosion core, and the latter determines the penetration or migration of the titanium tracer into the core. In the ideal one-dimensional spherical implosion without asymmetries and hydrodynamic instabilities, the titanium tracer initially located on the inner surface of the shell would remain there and produce a characteristic absorption feature backlit by the continuum emission from the core. However, as seen in figure 5.1, the experimental observation shows a spectrum that displays simultaneous emission and absorption features. The data displayed in figure 5.1 was recorded with the SSCA streaked spectrometer. We emphasize that gated data recorded with the MMI instrument also shows simultaneous emission and absorption (see Fig. 2.10). There are two conclusions that can be drawn from this observation. First, titanium tracer located in the outer

69 56 Figure 5.1: Spatially integrated, time-resolved spectrum recorded with the SSCA streaked spectrometer displaying simultaneous line emission in He- and H-like Ti ions and n=1-2 line absorption transitions in F- through Li-like ions. portion of the core close to the unablated shell confining the core is expected to produce the absorption as it is backlighted by the continuum emission from the core, as discussed in Chapter 4. Hence, the presence of simultaneous emission and absorption in the data shows evidence of deviation from a stable spherical implosion and suggests inwards migration of tracer into the core where temperatures are higher and can excite line emission. Second, the analysis of the spectrum will require a model that includes several plasma regions of different temperatures and densities in order to accommodate the emission and absorption features. The latter motivates the development of a multi-zone spectroscopic model to analyze the data. 5.1 Multi-zone model and geometry reduction The set of space-resolved spectra for the analysis is based on data recorded along a single line of sight (LOS) and extracted from annular regions of the image plus an

70 57 Figure 5.2: Schematic illustration of annular image regions on the image (detector) plane and spherical shell zones in object space (core, plasma source). outermost region to model the absorption spectral feature. Figure 5.2 schematically illustrates the annular regions on the image plane. The selection of the annular regions must be consistent with three requirements: (1) size larger than spatial resolution of the MMI instrument (around 10um), (2) good signal-to-noise ratio so that the spectrum is suitable for quantitative analysis, and (3) a gradual change and stable trend in the collection of photon-energy resolved intensity distribution (i.e. spectrum) across regions. We have found that, for the experiments discussed in this dissertation, the image can be divided into three annular regions that are consistent with these requirements. To model the set of annular spaceresolved spectra the volume of the core is discretized into three concentric spherical shell zones that project into the regions selected on the image. The sizes of these zones are given by the details of the annular regions selected on the image. Each one of these shells is characterized by an electron temperature T e and density n e. The emergent intensity distribution to model and compare with the experimental

71 58 measurement is then computed by transporting the radiation along chords inside the plasma source parallel to the lines of sight. In general, this calculation requires a numerical integration of the radiation transport equation. However, we have found in previous work that the result of the numerical integration can be well approximated by a system of effective slabs [18, 56]. Hence, the spherical onion shell geometry is further reduced to a system of five slabs plus an additional slab to model the absorption feature. The physical thickness of each slab is estimated from the average chord lengths of the spherical shells using the size of the annular contour regions defined on the image plane. 5.2 Extraction of electron temperature and density Given the set of three annular spatially resolved spectra I ν (R1), I ν (R2), I ν (R3) obtained from the MMI data and the corresponding modeling spectra, the analysis proceeds in three steps motivated by the idea that as we move from the outside to the inside of the core only one of the spectrum makes a new contribution to a previously analyzed case. Each of the steps consists of two sub-steps. The label R 3 denotes the outermost spectrum, R 2 the middle one, and R 1 the innermost spectrum. The very first step is to determine the zero level of absorption and fit the absorption spectral feature of I ν (R3) to obtain the electron temperature and density parameters of the absorption region of this spectrum. Transmission through a uniform absorbing layer is discussed in equation 3.2, chapter 3. This step is illustrated in Fig The parameters of the absorption spectrum are obtained from the first step. Next step is to fit the emission/absorption spectrum simultaneously searching in parameter space for the electron temperature and density parameters of the equivalent slab that

72 59 Figure 5.3: Illustration of the fitting of the absorption feature of I nu (R3). Figure 5.4: Multi-layer slabs for fitting spectrum I ν (R3). characterizes region R3. The emergent intensity distribution is given by the equation 5.1 I ν = ε(3) ν κ (3) ν (1 e κ(3) ν L 3 )e τ abs + Iν 0 e (κ(3) ν L 3 +τ abs ) (5.1) The schematic of the slabs used to model the spectrum from the region 3 (I ν (R3)) in the implosion core image is shown in the figure 5.4. Since the spectrum from annular region 3 has contribution from spherical zone 3 and absorption region, we define two slabs i.e., one for the spherical zone 3 and the other for absorption region. In the equation 5.1, the first term represents radiation emitted by and transported through the spherical zone 3 and further attenuated by the transmission through

73 60 Figure 5.5: Illustration of the emission/absorption fitting of spectrum I nu (R3). the absorption region. The second term represents the transmission of backlighter (coming from central hotter part of the implosion core) through the spherical zone 3 and absorption region. In the equation, ε (3) ν and κ (3) ν stand for the temperature, density and photon energy dependent emissivity and opacity of Ti in the spherical zone 3. L 3 is the width of the slab equivalent to the chord length in spherical zone 3, and I 0 ν is the backlight radiation (continuum radiation) obtained from the zero level absorption fitting. This step is illustrated in Fig To model the spectrum from the annular region 2 in the image plane, we first determine the zero level of absorption and then fit the absorption spectral feature of I ν (R2) to obtain the electron temperature and density parameters of the absorption region of this spectrum. Given the parameters obtained for the analysis of I ν (R3), and for the absorption spectrum, we simultaneously fit the emission/absorption spectrum searching in parameter space for the electron temperature and density parameters of the equivalent slab that characterizes region R2 in the image. The schematic of the

74 61 Figure 5.6: Multi-layer slabs for fitting spectrum I ν (R2). slabs used to model the spectrum from the region 2 (I ν (R2)) in the implosion core image is shown in the figure 5.6. It consists of four slabs i.e., one for the spherical zone 2, two for spherical zone 3 (front and rear sides of implosion) and the other for absorption region, this is because the spectrum from annular region 2 gets contribution from spherical zone 2, 3 and absorption region. The emergent intensity distribution for annular region 2 is I ν = ε(3) ν κ (3) ν (1 e κ(3) ν L I 3 )e (κ (2) + ε(2) ν κ (2) ν (1 e κ(2) ν ν L 2 +κ (3) ν L I 3 +τ abs) L 2 )e (κ(3) ν L I 3 +τ abs) + ε(3) ν κ (3) ν +Iν 0 e (κ(3) ν L I 3 +κ(2) (1 e κ(3) ν L I 3 )e τ abs ν L 2 +κ (3) ν L I 3 +τ abs) (5.2) In the equation 5.2, the first term represents radiation emitted by the rear side of the spherical zone 3 (see figure 5.2, equivalent to section d but on the rear side) and its transmission through spherical zone 2 and 3 (front side) and absorption region. The second term represents the self-emission by spherical zone 2 and its transmission through section spherical zone 3 and absorption region. The third term represents the self emission by spherical zone 3 and its transmission through the absorption region. The last term represents the transmission of the backlight radiation through the spherical zone 2, 3(rear and front) and absorption region. In equation 5.2, ε (2) ν

75 62 Figure 5.7: Multi-layer slabs for fitting spectrum I ν (R1). and κ (2) ν stand for the temperature, density and photon energy dependent emissivity and opacity of Ti in the spherical zone 2. L 2 is the width of the slab equivalent to the chord length in spherical zone 2, while L I 3 is the width of the slab equivalent to the chord length in spherical zone 3. Iν 0 is the backlight radiation obtained from the zero-level absorption fitting. To model the spectrum from the annular region 1 in the image plane, we first determine the zero level of absorption and then fit the absorption spectral feature of I ν (R1) to obtain the electron temperature and density conditions of the absorption region of this spectrum. Given the parameters obtained for the analysis of I ν (R2), I ν (R3) and for the absorption spectrum, we simultaneously fit the emission/absorption spectrum searching in parameter space for the electron temperature and density parameters of the equivalent slab that characterizes region R1 in the image. The schematic of the slabs used to model the spectrum from the region 1 (I ν (R1)) in the implosion core image is shown in the figure 5.7. It consists of six slabs i.e., two for the spherical zone 3 (rear and front sides), two for spherical zone 2 (front and rear sides), one for spherical zone 1 and the last for the absorption region, this is because the spectrum from annular region 1 gets contributions from sections in spherical zone 1, 2, 3 and absorption region. The emergent intensity distribution for annular region 1 is,

76 63 I ν = ε(3) ν κ (3) ν (1 e κ(3) ν L II + ε(2) ν κ (2) ν 3 )e (κ (2) ν L I 2 +κ(1) ν L 1 +κ (2) ν L I 2 +κ(3) ν L II 3 +τ abs) (1 e κ(2) ν L I 2 )e (κ (1) + ε(1) ν κ (1) ν (1 e κ(1) ν + ε(2) ν κ (2) ν ν L 1 +κ (2) ν L I 2 +κ(3) ν L II 3 +τ abs) L 1 )e (κ(2) (1 e κ(2) ν L I 2 +κ(3) ν L II 3 +τ abs) ν L I 2 )e (κ (3) ν L II 3 +τ abs) +Iν 0 e (κ(3) ν L II + ε(3) ν κ (3) ν (1 e κ(3) ν L I 3 )e τ abs 3 +κ(2) ν L I 2 +κ(1) ν L 1 +κ (2) ν L I 2 +κ(3) ν L II 3 +τ abs) (5.3) In the equation 5.3, the first term represents radiation emitted by the rear side of the spherical zone 3 (see figure 5.2, equivalent to section d II but on the rear side) and its transmission through spherical zone 2 (rear and front sides), 1, 3 (front side) and absorption region. The second term represents the self-emission by spherical zone 2 (rear side) and its transmission through spherical zones 1, 2, 3 and absorption region. The third term represents the self emission by spherical zone 1 and its transmission through the spherical zones 2, 3 and absorption region. The fourth term represents the self emission by spherical zone 2 (front side) and its transmission through the spherical zones 3 and absorption region. The fifth term represents the self emission by spherical zone 3 (front side) and its transmission through the absorption region. The last term represents the transmission of the backlighter through the spherical zone 2 (front and rear sides), 3(rear and front) and spherical zone 1 and absorption region. In the equation 5.3, ε (1) ν and κ (1) ν stand for the temperature, density and photon energy dependent emissivity and opacity of Ti in the spherical zone 1. L 1 is the width of the slab equivalent to the chord length in spherical zone 1, while L I 2 is the width of the slab equivalent to the chord length in spherical zone 2, L II 3 is the

77 width of the slab equivalent to the chord length in spherical zone 3. I 0 ν is the backlight radiation (continuum radiation) obtained from the zero level absorption fitting Determination of the titanium atom number density Once the spectroscopic analysis is completed, the atom number density can be determined from the total (i.e. photon energy integrated) line intensity. To this end, total line intensities are obtained from the experimental annular spectra by estimating the background (continuum emission) and numerically integrating the area under the line relative to the continuum level as shown in the figure 5.8. For each line, total intensities are found from all three space resolved spectra. The narrow-band spectral range is always same for a chosen spectral line. Figure 5.8 shows an example to illustrate the method to obtain continuum subtracted total intensity associated with lines. Integrated regions are shown as shaded. In the optically thin approximation, the total line intensity is proportional to the population number density of the upper energy level of the line emission, the photon energy of the line transition, the spontaneous radiative decay, and the volume of emission. We note that the only unknown is the upper level number density. The total line intensity in the optically thin approximation (area normalized line shape) is given by, I = 1 4π n ua ul hν ul V (5.4) Where n u is the upper level (or excited-level) population number density, A ul is the spontaneous radiative decay, and hν ul is the photon energy of the line transition and V is the volume of emission. To work out the upper level population (excited-level)

78 65 Figure 5.8: Continuum subtracted total line intensities. For each line, total intensities are found from all three space resolved spectra. Integrated regions are shown as shaded. number densities, a system of equations is set up for the total line intensities associated with each of the transitions observed in the annular spectra, and solved for each region starting with outer region R3, next region R2, and finally region R3. Just as in the case of the spectroscopic analysis, this strategy produces a sequence of equations with only one unknown but dependent on previously worked out populations. The solution of all the equations produces the set of upper level number densities. I 3 = 1 4π n udv d A ul hν ul (5.5) I 2 = 1 4π (n ucv c + V d In ud )A ul hν ul (5.6) I 1 = 1 4π (n ubv b + n uc V c I + n ud V d II)A ul hν ul (5.7) The equation 5.5 will produce the upper-level population density (in arbitrary units) in the spherical zone 3. This will be used as one of the input in the equation 5.6 which

79 66 will output the upper-level population density in the spherical zone 2. Similarly, the upper-level population densities produced by equations 5.5 and 5.6 will be used as the input in the equation 5.7 which produces the upper-level population density in the spherical zone 1 i.e., innermost zone in the implosion core. The volumes (spherical sections) used in these equations are shown in the figure 5.2 in section 5.1. Finally, to calculate the titanium atom number density, we note that the upperlevel number density is given by the product of the atom number density and the upper-level fractional population. The latter can be evaluated with the electron temperature and density from the spectroscopic analysis. Hence, the atom number density can be computed from the ratio of the upper-level number density divided by the upper-level fractional population. To account for the opacity effect (i.e. radiation transport through the plasma source) on the line transition the optically thick formula for the emergent intensity of a slab can be integrated with respect to photon energy to obtain an optically thick approximation to the total line intensity [see Appendix B]. This optically thick line intensity is smaller than that obtained in the optically thin approximation. For the cases considered here the net result of this effect can be approximated by considering an effective volume of integration in the optically thin formula of about one half the actual value. Titanium atom number densities are determined in this way from as many line emission features as observed in the data and compared to each for consistency. We also find the total number of Ti atoms in spherical zones by multiplying their Ti atom number densities with respected volumes of spherical shell zones. Also, the areal densities can be also obtained by multiplying the Ti number densities with the width of the spherical shell zones. These Ti number densities are related to emissions. To calculate the Ti density associated with the absorption region during the implosion [14], we have used the relation between the

80 67 electron density and total ionization balance of the Ti ions. n T i = Z av CH δ n e + Z av T i (5.8) In the equation 5.8, n e is the free electron density obtained for absorption region using the multi-zone spectroscopic model discussed in 5.1 and 5.2, ZCH av is the average ionization state of C (6) and H (1) total of 7. The C and H will be fully stripped at the time in the implosion when the Ti absorption becomes strong [14]. Z av T i is the average ionization state of Ti atoms. We have used 20 for Ti, charge state for He-like Ti. δ is given as the ratio of total number of Ti atoms in our plasma system to 2 times the total number of carbon (or hydrogen) in the doped layer. We assume that total number of Ti atoms remains conserved in our implosion. Our targets had shell thickness of at least 15 µm. The laser irradiation ablates approximately 12 µm. So, our assumption of conservation of Ti atoms is reasonable. The Ti areal density for the absorption region can be obtained by multiplying it with the width of the absorption region (obtained from the spectroscopic analysis). Similarly, the total number of atoms in the absorption region can be obtained by multiplying it with the volume of the region (since we know the width of absorption region from spectroscopic analysis, we can find volume). Once we know the total number of atoms in the absorption region, we can find the number of Ti atoms in the emission regions by subtracting it from the total number of Ti atoms known from initial target designs.

81 68 Chapter 6 Results for Titanium penetration in the core In this chapter we apply the methods developed in chapter 5 to MMI data from OMEGA implosion experiments discussed in chapter 2. For each OMEGA implosion, we have the MMI data recorded along multiple lines-of-sight, and for each lines-ofsight we have snapshots of the data recorded at different times during the deceleration and burning phase of the implosion. We also apply the methods to study the data with different drive pulse shapes and target designs. In ICF experiments, mixing of shell material into the core has remained the main reason for the degradation of neutron yield and lack of achieving ignition. It is critical to know the cause and quantity of mixing for improving implosion performance towards achieving ignition. This work aims to study the spatial distribution of dopant material (Ti for our targets), initially located on the inner surface of the target, into the OMEGA directdrive implosion core in the deceleration and burning phase by using the space-resolved spectra characteristics of annular regions on the implosion core image. Extensive amount work has been done at OMEGA and NIF previously to understand mixing

82 69 of shell material into the hot core [6, 11, 24, 25, 26]. P.B. Radha et al. [6] reports the direct-drive ICF experiments with targets using 20 µm CH shells and 15 atm DD fuel, and other shots using CH shells with embedded CD layers filled with 3 He fuel. They have found approximately 20% of the compressed shell areal density and 30% of the fuel density is in the mixed region from experiments and model. S. P. Regan et al. [24] have estimated the density of shell material mixed into the outer implosion core as 3.4g/cc using time resolved X-ray spectroscopic measurements, nuclear measurements of fuel areal density, and core X-ray images with the OMEGA direct-drive implosion data having shell thickness 20 µm and 15 atm DD fuel pressure. Similarly, C.K. Li et al. [25] have studied mixing of shell-fuel by using charged-particle spectrometry data from an experimentally constrained model. They have found the size of the mix region decreasing with increasing filling pressure. In the current work, spatial distributions of the number density of the Ti mixed into the implosion core has been estimated for the first time using space-resolved X-ray data extracted from OMEGA direct-drive implosions. We have also estimated spatial distributions of areal density and total number of Ti atoms in the OMEGA implosion core. In section 6.1, we discuss the timing in the implosion at which the MMI data were recorded. We use the time-resolved streaked camera (SSCA) data and neutron temperature diagnostic (NTD) data for correlating the snapshot times in MMI. In section 6.2, we talk about the annular space-resolved spectra extracted from the implosion core image. We test our multi-layer spectroscopic model discussed in chapter 5 by fitting with synthetic spectra and compare the best fit solution obtained from fitting with the input parameters used to generate the synthetic spectra. Section 6.3 covers the spatial distribution of Ti in the implosion core. This section includes the estimation of Ti atom number density, areal density distribution and total Ti atom

83 70 number in different locations in the hot implosion core. Similarly, in this sections we will discuss the multi-view spatial profiles, temporal evolution and dependency on target designs and drive pulses respectively. Table 6.1 gives an overview of the data used for this chapter. Shot Number Drive Pulse (ns) Shell thickness Fill (µm) pressure (atm) α α α α α α α α α Table 6.1: Overview of the target parameters and laser pulse shapes for the shots used for the analysis 6.1 Time in the implosion In our OMEGA direct-drive experiments, we had used three different instruments (MMI, SSCA and XRS1) for recording X-ray data from the imploding core. The details of the instruments are discussed in chapter 2. Figure 6.1a shows the processed image data recorded by SSCA from OMEGA shot 65694, and 6.2b shows the temporal evolution of the laser pulse (low-adiabat α 2 ) used for the target surface irradiation, neutron yield recorded by neutron temporal diagnostic (NTD), continuum

84 71 a) b) Figure 6.1: a) Processed X-ray image data recorded by streaked crystal spectrometer (SSCA) from the OMEGA shot b) temporal evolution of laser pulse (lowadiabat α 2, SSCA and NTD data

85 72 compression (CC) and continuum subtracted line emissions (Lyα and Heβ) extracted from the processed SSCA image data. The OMEGA shots and are used for the spectroscopic data analysis and mixing distributions in coming sections and these shots are nominally identical with shot The horizontal axis represents time in ps, whereas vertical axis is in normalized units. The lines peak before the neutron yield and continuum compression. The yield peaks slightly earlier than compression. Both the lines (Lyα and Heβ) show asymmetric trends at the time close to the maximum neutron yields and maximum continuum compression. The MMI data is primarily observed during the deceleration and burning phases of the implosion which is represented by the red arrow in the figure 6.1b. 6.2 Annular-type spatially resolved spectra Figure 6.2 shows the annular regions in the implosion core image of the OMEGA shot TIM 3 frame 2 on its image plane and corresponding space resolved spectra extracted from the regions. In figure 6.2 on the image of the implosion core, the regions are represented by green color. We define three annular regions in the implosion core image: region-1, region-2 and region-3. The region-1 is the innermost and the region- 3 is the outermost. The radius of these three regions are represented by R 1, R 2 and R 3 respectively. On the other hand, the implosion core in the object space can also be sectioned into three concentric spherical regions. The region between each concentric sphere is characteristics of its own temperature and density. We call the central spherical region the spherical zone-1, the region between central sphere and second sphere the spherical zone-2 and the space between the second sphere and third sphere is the spherical zone-3. The radius of the innermost sphere is equal to the radius of the region 1 on the image. Similarly, the radius of middle sphere is equal to the radius of the image region 2 and the radius of the outermost sphere is equal

86 73 to the radius of the image region 3. The number of spherical zones in the implosion core in object space are equal to the number of regions on the image plane. The space-resolved spectra extracted from the outermost region (Region-3) on the image has the contributions only from part of the outermost spherical zone in the implosion core. The spectra extracted from the region 2 on the image has the contributions from parts of the spherical zone 2 and 3. Similarly, the spectra extracted from the region 1 on the image has the contributions from parts of all spherical zones in the implosion core. The details about the extraction of space-resolved spectra is discussed in chapter 2. The space resolved spectra shown in the figure 6.2 have area normalized and scaled intensities. For the area normalization, we divide the intensities by the area of the corresponding region. We use the area normalized space-resolved spectra for the spectroscopic analysis of the data. The area normalization will make the spectra independent of the area of the region and the spectrum becomes intrinsic to the corresponding region. The characteristics of the spectra reflect plasma conditions in different parts of the core. The Ti-absorption feature and all line emissions become stronger in the spectra extracted from the inner regions. They suggest penetration of the Ti tracer deep inside the core Testing of the multi-layer spectroscopic model Before starting to use the multi-layer spectroscopic model mentioned in chapter 5 to analyze the actual experimental space-resolved spectra, we compute the synthetic space-resolved spectra from three annular regions on the synthetic implosion core image similar to the OMEGA experimental implosion core image on its image plane. The information about the radial coordinates and corresponding plasma conditions is obtained from LILAC hydrodynamic simulation [57]. We calculate the spatially averaged plasma conditions associated with the width of each region to generate

87 74 Figure 6.2: Area normalized and rescaled space-resolved spectra from annular regions on the image plane: OMEGA shot TIM 3 Frame 2 the synthetic spectra. The atomic kinetics model used for creating the synthetic spectra is same as used in the multi-layer spectroscopic model. Then we use our multi-layer spectroscopic model to obtain the best fit to the synthetic spectra. If the values for the plasma conditions from the best fit compare well to the LILAC parameters used to generate the synthetic spectra in the first place, we can verify the reliability of the model. The LILAC simulation of the example case (nominally identical with our OMEGA shots) discussed here included a total of more than 500 fluid elements or zones, of which about 150 comprise the core of the plasma. We convolve the synthetic spectra with a Gaussian of 30 ev at FWHM to approximate the instrumental broadening in the experiment. In table 6.2, R c and dr represent the average radius and width of each spherical zone in the implosion core respectively. Similarly, T e and n e represent the free electron temperature and density, respectively. We have found that the best fit values (represented by Bestfit in the table) for free electron temperatures and densities in different spherical zones in the implosion core are close with the actual values (rep-

88 75 Figure 6.3: Synthetic space-resolved spectra extracted from three annular regions on the synthetic implosion core image a) b) c) d) Figure 6.4: a) Analysis of synthetic spectra from region 3, b) region 2, c) region 1, and d) region 3 but fitting only the absorption spectra

89 76 Spherical zones T e (ev) n e (cm 3 ) no. R c (µm) R (µm) LILAC Best fit LILAC Best fit e e e e e e +24 absorp e e +24 Table 6.2: Comparison of spatial profiles of plasma conditions used to generate the synthetic spectra with the best fit solutions obtained by using multi-zone spectroscopic model resented by LILAC in the table) used to generate the synthetic spectra. Figure 6.3 shows the synthetic space-resolved spectra computed from region 1, 2 and 3 respectively from the synthetic implosion core image. Similarly, figure 6.4 shows fittings of synthetic space-resolved spectra using multi-layer spectroscopic model. Figures labeled a), b), c) show the comparison between the synthetic spectra and model best fits for the region-3, region-2 and region-1 respectively. As seen in the figure, good agreement was found for all cases. For each of the case, the χ 2 minimization was done for the spectral range (4400 to 6500) ev. Figure 6.4d) is the fitting for the region-3 when we use the spectral range (4500 to 4700) ev for the χ 2 minimization. This gives the plasma conditions for the compressed shell region which is relatively colder than the fuel region [14]. For all cases, the searches in parameter in space (exhaustive search) yielded a single global minimum. Some discrepancies between synthetic spectra and best model fits are observed in the continuum levels on lower energy side of the absorption feature and higher energy side after the Lyβ, we attribute these differences to the fact that the synthetic spectra is obtained by using spherical geometry where the model is 1D calculation.

90 6.3 Ti penetration in the core Spectroscopic data analysis In this section we analyze the experimental space resolved spectra extracted from the three annular regions on the implosion core image as shown in the figure 6.2 for OMEGA shot TIM 3 Frame 2. The filling pressure of the target was 10 atm of DD, the shell thickness was 15 µm and pulse drive was α 2. The result is shown in the figure 6.5, which shows the comparison between experimental and best fit model spectra. Plots a) and b) are from region 3, c) and d) are from region 2, e) and f) are from region 1. plots a), c) and e) are the fittings corresponding to the absorption spectral range ( ) ev, whereas plots b), d) and f) are the simultaneous fittings of the absorption and emission corresponding to the spectral range ( ) ev. The uncertainties in the best fit parameters come from the uncertainties associated with the χ 2 calculations and uncertainties in the spectrum data points [58]. We had applied the three criteria discussed in chapter 5 to choose the MMI data for the extraction of space-resolved spectra from annular regions in the implosion core image. The spectral quality of the space-resolved spectra (signal-to-noise ratio) for the shots used in this dissertation were also comparable. The uncertainties in the best fit values of electron densities and temperatures for different shots (used for this dissertation) were also comparable. The uncertainties for electron temperatures and densities were approximately 10% and 15%, respectively. We have found the best fit values for the temperature and density for the absorption region from three space resolved spectra (for case TIM 3 Frame 2) very comparable to each other. The average value of temperature and density for the absorption region is (417±42) ev and (8.5±1.3)e +24 cm 3 respectively. The temperature and density spatial profiles in the hot core are shown in the figures 6.6 a) and b) respectively. The best fit

91 78 a) b) c) d) e) f) Figure 6.5: Analysis of space resolved spectra for OMEGA shot TIM 3 Frame 2. Plots a) and b) are for region 3, c) and d) are for region 2, e) and f) are for region 1. Plots a), c) and e) are the fittings corresponding to the absorption spectral range ( ) ev. Plots b), d) and f) are the simultaneous fittings of absorption and emission corresponding to the spectral range ( ) ev

92 79 a) b) Figure 6.6: Temperature and density spatial profiles of the emission region in the implosion core of OMEGA shot TIM 3 Frame 2. Plots a) and b) show temperature and density gradient respectively. values extracted from the analysis for the temperature and density in the innermost spherical zone are (2025±202) ev and (1.3±0.3)e +23 cm 3 respectively. Similarly, the temperature and density in the spherical zone 2 are extracted as (1825±182) ev and (3.8±0.6)e +23 cm 3 respectively. For the spherical zone 3, the temperature and density are (1125±112) ev and (5.3±0.8)e +23 cm 3 respectively Ti number and areal density distributions in the core To find the spatial profiles of excited level population densities in the implosion core, first of all we extract continuum subtracted integrated intensity spatial profiles of line emissions present in the space-resolved spectra. Figure 6.7a shows the spatial profile of excited level population density (relative to the outermost spherical zone and in arbitrary units) in the implosion core of OMEGA shot TIM 3 Frame 2. The spatial profile in the figure was obtained by averaging the spatial profiles of excited level populations corresponding to the narrow-band spectral ranges of Lyα, Heβ, and Lyβ and Heγ lines. Trends shown by all the lines are consistent with the averaged one as shown in the figure. The energy difference between the Lyβ and Heγ is only 20 ev which falls below the spectral resolution of the instrument

93 80 MMI. Therefore, when we find the excited level populations by using the narrowband spectral range corresponding to Lyβ and Heγ lines, we are actually finding the excited level populations contributed by Lyβ and Heγ. The vertical axis represents the upper level population density distributions relative to the outermost region in arbitrary units, and the horizontal axis represents the radius of the implosion core in µm. The excited level population densities obtained by using all the lines present in the space resolved spectra show the trends of increasing towards the center of the implosion. As an initial attempt to take into account the opacity correction, we have excluded the rear side volume of the implosion core in the excited level populations extraction method. As explained in chapter 5, to obtain the Ti atom number density (n T i ), we need fractional level populations F u (T e, n e) ) and excited level population density n u. To this end, we have extracted plasma electron temperatures T e and densities n e in different spherical zones in the implosion core from the spectroscopic data analysis, and excited level population densities which is discussed in the previous paragraph. Now using the plasma conditions information, we can find the F u (T e, n e) ) from ATOKIN [42] calculations. The ratio of the excited level populations density to the fractional level populations will give the Ti atom number density. Figure 6.7b shows the average spatial profile of (n T i ) in the implosion core of OMEGA shot TIM 3 Frame 2, which is obtained by averaging the spatial profiles of n T i due to excited level populations associated with lines Lyα, Heβ, and Lyβ and Heγ. Here, the vertical axis represents the average Ti atom number density (cm 3 ), the horizontal axis represents the radial coordinates in the implosion core. Similarly, the figure 6.7d shows the spatial profile of Ti atom number distributions in the implosion core. This was also obtained by averaging the spatial profiles of Ti atom number distributions using Lyα, Heβ, and Lyβ and Heγ lines. The vertical axis represents the Ti atom number and

94 81 the horizontal axis represents the radial coordinates in the implosion core. Using the method to find the Ti number density in the compressed shell region discussed in chapter 5, we have found that approximately 11% of the atoms are still located on the outer colder compressed shell region and contributed to the Ti K shell n = 1 2 absorption, and 89% of the Ti atoms have migrated to the hot core i.e in the emission region. Using the initial conditions of the target provided, the total number of Ti atoms in our plasma system is 5.5e +15, and the total number of Ti atoms in the hot core i.e in the emission region is 89% of 5.5e +15 which is approximately 4.9e +15. The total number of Ti atoms in the absorption region is approximately 11 % of 5.5e +15 which is approximately 0.6e +15. Table 6.3 displays the percentage distributions of Ti atoms contributing to emission and absorption. The uncertainty in the percentage of Ti-atoms in the absorption region is approximately 15%, equal to the uncertainty in the best fit electron density as discussed above. The reason is that the Ti atom number density for the absorption reason is obtained by using equation 5.8, which in turn depends on the best fit electron density obtained from spectroscopic analysis. Figure 6.7c shows the comparison of the areal densities obtained using spaceresolved data and the 1D spherical scaling [see Appendix C for 1D spherical scaling calculation of Ti areal density]. Again the experimental areal density spatial profile is the average of the spatial profiles of Ti atom areal densities obtained by using Lyα, Heβ, and Lyβ and Heγ lines. The vertical axis represents the Ti atom areal density (cm 2 ) and the horizontal axis represent the radial coordinates in the implosion core. The details of the procedures to obtain Ti atom areal density distribution is explained in chapter 5. Ti atom number density and areal density increase towards the center of the implosion core. This is in part a consequence of the convergent effect of spherical geometry. We have found that significant deviations of experimental areal density from the areal density obtained from 1D spherical scaling are found in the innermost

95 82 a) b) c) d) Figure 6.7: a) Spatial profile of excited level population density in the implosion core of OMEGA shot TIM3 Frame 2, b) spatial profile of Ti number density, c) spatial profile of areal density (data and 1D spherical scaling), d) spatial profiles of Ti atom number distributions.

96 83 Shot Number Emission (%) Absorption (%) T4F T4F T3F T4F T4F T5F T4F T4F T5F T5F T3F T3F T4F Table 6.3: Percentage distributions of the Ti atoms contributing to emission and absorption in the implosion cores of OMEGA direct-drive shots. zone of the implosion core, while areal densities at other two outer zones are very comparable with each other. Table 6.4 summarizes the results for the spatial profiles of Ti number densities, Ti atom areal densities and the total Ti atoms in the hot implosion core of the OMEGA shot TIM 3 Frame 2. The uncertainties in the Ti atom number density come from the uncertainties associated with the spectroscopic data analysis, sensitivity of the experimental space-resolved spectra on width of the annular region in the image and uncertainties associated with the excited level populations. By taking into account all these factors, the uncertainties inferred were 24%, 18% and 12% in the values obtained from innermost, middle and outermost zones in the implosion core respectively.

97 Spherical zones n T i R (cm 2 ) no. R c (µm) R (µm) n T i (cm 3 ) N T Data 1D Sp. Sc e e e e e e e e e e e e +19 Table 6.4: Summary of the results for the emission zones in the implosion core of OMEGA shot TIM 3 Frame 2. Where the column Data and 1D Sp. Sc. represent experimental areal density and 1D spherical scaling areal density respectively. 84 Figure 6.8: Multi-view spatial profiles of Ti atom number areal density distributions in the implosion core of OMEGA shot (along TIM 4 and TIM 5) Multi-view spatial profile Figure 6.8 shows the multi-view spatial profiles of Ti atom number areal density in the implosion core of OMEGA shot along TIM 4 and TIM 5. The solid green line represents experimental areal density spatial profile extracted from TIM 5 frame 2, and the dashed green line represents areal density spatial profile obtained from 1D spherical scaling for the same TIM and frame. Similarly for TIM 4 frame 2, the solid red line represents the experimental spatial profile of areal density, while the dashed one is obtained from 1D spherical scaling. Similarly, figure 6.9 shows the multi-view

98 85 Figure 6.9: Multi-view spatial profiles of Ti atom number density distributions in the implosion core of OMEGA shot (along TIM 4 and TIM 5) spatial profiles of Ti atom number density in the implosion core of OMEGA shot extracted from TIM 4 frame 2 and TIM 5 frame 2. The green line is for TIM 5 frame 2 and red line for TIM 4 frame 2. Based on the size, shape and quality of the implosion core images recorded by MMI, and MMI timing sheet provided by the OMEGA lab, we found these two frames in two different lines of sight of shot to be very comparable in time with each other. The results show that the areal densities and number densities are very comparable in the outer two zones in the implosion core, but significantly different in the innermost zone. The deviation in the results along TIM 4 and TIM 5 in the innermost zone show the non-uniformity in the propagation of Ti into the deep core along different lines-of-sight Temporal evolution of Ti mixing In this section we talk about the temporal evolution of the Ti atom number and areal densities. Figure 6.10 shows the temporal evolution of spatial profiles of Ti atom number areal density in the implosion core of OMEGA shot extracted from TIM 4 frame 1 and 2. The time difference between these two frames were 80 ps. Frame

99 86 Figure 6.10: Temporal evolution of spatial profiles of Ti atom number areal density distributions in the implosion core for OMEGA shot TIM 4 Figure 6.11: Temporal evolution of spatial profiles of Ti atom number density distributions in the implosion core for OMEGA shot TIM 4

100 87 2 was recorded 80 ps later than frame 1. The solid green line represents experimental areal density spatial profile from frame 2, while the dashed green line represents areal density spatial profile obtained from 1D spherical scaling for frame 2. For frame 1, the solid red line represents the experimental spatial profile of areal density, while the dashed green is obtained from 1D spherical scaling for frame 1. Figure 6.11 shows the temporal evolution of spatial profiles of Ti atom number density in the implosion core of OMEGA shot extracted from frames 1 and 2. The green line is for frame 2 and red line for frame 1. Early in time i.e., in frame 1, the Ti areal density has the highest value for the outermost zone and the values are decreased for the other two zones in the implosion core. This means that early in time a significant fraction of Ti atoms is still located on the compressed shell-fuel interface or not very far into the hot core. For later in time i.e., in frame 2, the trend is just the opposite. The higher values are now in the inner zones and decrease outwards. The large amount of Ti atoms migrated into the deep core in the interval of 80 ps during the implosion.

101 88 Chapter 7 Summary In ICF experiments, the mixing of shell material with the fuel are critical to understand the performance of the implosion hydrodynamics. It has remained the main reason for degrading target efficiency and lack of achieving ignition. It increases the radiative losses from the hot spot, and decreases the peak pressure of the fuel and the fusion yield [7, 8, 9, 10, 11]. This dissertation has contributed to understand hydrodynamic stability and the mixing of tracer material (initially located on the shell-fuel interface) into the implosion core and the plasma conditions in the core by exploring the application of X-ray line emission and absorption spectroscopy. In addition, it has provided the information about the implosion s hot spot size. The direct-drive ICF experiments were performed at the OMEGA laser facility of the Laboratory for laser energetics at the University of Rochester, NY [23]. The targets were deuterium filled, spherical plastic shells of different thicknesses and filling gas(dd) pressures with a 0.5µm Ti-doped plastic layer on the inner surface of the shell. The spectral features from the titanium tracer were observed during the deceleration and stagnation phases of the implosion, and recorded with a time integrated spectrometer (XRS1), streaked crystal spectrometer (SSCA) and three gated, multi-monochromatic X-ray imager (MMI) instruments fielded along quasi-orthogonal

102 89 lines-of-sight. The time-integrated data, streaked and gated data show simultaneous emission and absorption spectral features associated with titanium K-shell line transitions but only the MMI data provides spatially resolved information. The arrays of gated spectrally resolved images recorded with MMI were processed to obtain spatially resolved spectra characteristic of annular contour regions on the image. Simultaneous observation of emission and absorption from OMEGA implosion core indicated that the plasma source is not uniform, which motivated us to consider a multi-zone spectroscopic model comprised of several slab regions characteristics of different plasma conditions. The model was used for the spectroscopic analysis of the annular spatially resolved spectra to extract core electron temperatures and densities. The atomic data was computed using Flexible Atomic Code (FAC) [43]. The emissivity and opacity data base for the Ti was computed using ATOKIN [42]. We extracted space-resolved spectra (SRS) from three annular regions on the implosion core image. The size of the implosion core was obtained by using MMI data processing codes [12, 13] for the extraction of space-resolved spectra. The spectroscopic analysis of SRS using the multi-layer spectroscopic model provided the plasma conditions in different spherical zones in the implosion core. A new method was developed to determine the spatial distribution of total Ti atom number density and number of atoms in the core. The titanium atom distribution provided direct evidence of tracer penetration into the core and thus of the hydrodynamic stability of the shell. The observations, timing and analysis indicated that during fuel burning the titanium atoms have migrated deep into the core and thus shell material mixing is likely to impact the rate of nuclear fusion reactions, i.e. burning rate, and the neutron yield of the implosion. We have found that the Ti atom number density decreases towards the center in early deceleration phase, but later in time the trend is just opposite i.e., it increases towards the center of the implosion core. The spatial profiles of Ti

103 areal densities were extracted from space-resolved spectra characteristics of annular region in the implosion core image and evaluated by using 1D spherical scaling. The 90 trends were similar to the Ti number density spatial profiles. The areal densities obtained by using two methods were very comparable in the outer spherical zones in the implosion core but significantly deviated in the innermost zone in the implosion core. This is in part a consequence of the convergent effect of spherical geometry. We have also observed that approximately 85% of the Ti atoms migrated into the hot core, while 15% of the atoms were still on the shell-fuel interface and contributed to the absorption. In addition, a method to extract the hot spot size based on the formation of the absorption feature in a sequence of annular spectra was discussed. Results and trends were presented as a function of target shell thickness and filling pressure, and laser pulse shape. In this dissertation, spatially resolved spectra from the implosion core along single line-of-sight have been used for finding spatial distributions of a tracer initially located in the shell. My future work includes use of genetic algorithm and Levenberg- Marquardt methods with the multi-layer spectroscopic model to build an automated search and optimization technique to extract 3-D profiles of electron temperatures, densities, and tracer spatial distributions in the core.

104 91 Appendix A Solution of the one-dimensional radiation transport equation through a slab of uniform plasma conditions This appendix presents the solution of radiation transport equation for a uniform slab geometry in the steady state approximation. Let s consider the opacity and emissivity of the plasma at the photon energy hν in the slab are K ν and ε ν, respectively. Figure A.1 shows the schematic of the slab geometry used in the solution of the radiation transport equation. θ is the angle between the axes z and z. The radiation transport equation in the presence of a radiation field of intensity I νµ at the photon energy hν along z is given by, Figure A.1: Schematic of the slab geometry used in the solution of the radiation transport equation.