Modeling the heating and atomic kinetics of a photoionized neon plasma experiment

Transcription

1 University of Nevada, Reno Modeling the heating and atomic kinetics of a photoionized neon plasma experiment A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics. by Thomas E. Lockard Dr. Roberto C. Mancini, Dissertation Advisor December 25

2 Copyright c 25 - Thomas E. Lockard All Rights Reserved

3 THE GRADUATE SCHOOL We recommend that the dissertation prepared under our supervision by THOMAS E. LOCKARD Entitled Modeling The Heating And Atomic Kinetics Of A Photoionized Neon Plasma Experiment be accepted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Dr. Roberto Mancini, Advisor Dr. Radu Presura, Committee Member Dr. Yasuhiko Sentoku, Committee Member Dr. Paul Neill, Committee Member Dr. Hyung Shin, Graduate School Representative David W. Zeh, Ph. D., Dean, Graduate School December, 25

4 i Abstract Motivated by gas cell photoionized plasma experiments performed by our group at the Z facility of Sandia National Laboratories, we discuss in this dissertation a modeling study of the heating and ionization of the plasma for conditions characteristic of these experiments. Photoionized plasmas are non-equilibrium systems driven by a broadband x-ray radiation flux. They are commonly found in astrophysics but rarely seen in the laboratory. Several modeling tools have been employed: () a view-factor computer code constrained with side x-ray power and gated monochromatic image measurements of the z-pinch radiation, to model the time-history of the photon-energy resolved x-ray flux driving the photoionized plasma, (2) a Boltzmann self-consistent electron and atomic kinetics model to simulate the electron distribution function and configuration-averaged atomic kinetics, (3) a radiation-hydrodynamics code with inline non-equilibrium atomic kinetics and radiation transport to perform a comprehensive numerical simulation of the experiment and plasma heating, and (4) steady-state and time-dependent collisional-radiative atomic kinetics calculations with fine-structure energy level description and non-equilibrium radiation transport to assess transient effects in the ionization and charge state distribution of the plasma. The results indicate that the photon-energy resolved x-ray flux impinging on the front window of the gas cell is very well approximated by a linear combination of three geometrically-diluted Planckian distributions that represent the contribution of the z-pinch radiation as well as re-radiation of the hardware surrounding it. Knowledge of the spectral details of the x-ray drive turned out to be important for computing the heating and ionization of the plasma. The free electrons in the plasma thermalize quickly relative to the timescales associated with the time-history of the x-ray drive

5 ii and that of the plasma atomic kinetics. Hence, plasma electrons are well described by a Maxwellian energy distribution of a single temperature. This finding is important to support the application of a radiation-hydrodynamic model to perform multi-physics simulations of the experiment, since this model assumes that the electrons are in equilibrium and thus characterized by a single temperature. It is found that the computed plasma heating compares well with experimental observation when the effects of spectral distribution of x-ray drive, window transmission and hydrodynamics, and non-equilbirium collisional-radiative neon emissivity and opacity are employed. The atomic kinetics shows significant time-dependent effects because the timescale of the x-ray drive is too short compared to that of the photoionization process. These modeling results are important to test theory assumptions and approximations, and also to provide guidance on data interpretation and analysis.

6 iii Dedication To my loving family and friends

7 iv Acknowledgments I would like to thank Professor Roberto C. Mancini, my advisor, for his guidance and support throughout my education, and I would like to acknowledge the useful suggestions and efforts by my dissertation committee members: Drs. R. Presura, Y. Sentoku, H.K. Shin, and P. Neill. I would also like to thank Drs. J. Bailey, G. Rochau, G. Loisel, and R. Falcon (Sandia National Laboratories, SNL) for providing the experimental data, training, and guidance needed to prepare the data for analysis; Dr. J. Abdallah Jr. (Los Alamos National Laboratory, LANL) for his assistance, guidance, and support with data from their Botlzmann model code. I want to thank Drs. Tirtha Joshi, Taisuke Nagayama and Iain Hall, former group members of Dr. Mancini s research group for their always helpful thoughts and advice. I would like to thank Dr. Tunay Durmaz and Ms. Sam Nasewicz for their discussions and support with physics and life, smoothing out even the rockiest of times. I want to thank the other members of the group Mr. Dan Mayes and Mr. Kyle Carpenter for their insight and friendly conversations. My especial gratitude goes to my parents and family members for their patience, help and support during this work, especially, to Bailey and Emma who supported and lifted me up everyday. This work is sponsored in part by the National Nuclear Security Administration under the High Energy Density Laboratory Plasmas grant program through DOE grant DE-FG52-9NA2955, the Z Facility Fundamental Science Program, and Sandia National Laboratories.

8 v Contents Abstract i Dedication iii Acknowledgments iv Introduction. States of matter Spectroscopy of plasmas Z-pinch plasma Z-machine facility Laboratory plasmas relevant to astrophysics Experimental requirements and conditions Summary of thesis Chapters Author s role in experiment and code development Atomic and radiation physics 8 2. Introduction Atomic processes in plasma Bremsstrahlung & inverse bremsstrahlung Collisional excitation & de-excitation Photoionization & radiative recombination Electron capture & Autoionization Dielectronic recombination Electron impact ionization & 3-body recombination Photoexcitation, spontaneous radiative decay, & stimulated emission Understanding rates Equilibrium conditions and atomic population models Thermodynamic equilibrium

9 vi Local thermodynamic equilibrium Non-local thermodynamic equilibrium Coronal Equilibrium Collisional-Radiative equilibrium Photoionization Equilibrium Equations of radiative transfer Escape factor Z-machine photoionization experiment Purpose and scope of photoionized plasma experiment Experimental design and diagnostics Z-machine Gas cell specifications Gas cell testing station Gas cell fill station Gas cell window transmissions Transducer diagnostic and pressure measurements X-ray spectrometer characteristics Elliptical crystal spectrometer formulae Review of experimental data X-ray measurements Modeling of x-ray radiation drives Introduction Z-pinch radiation and re-radiation effects Planckian fit algorithm IMH closer position radiation drive fits IMH further position radiation drive fits Extrapolation of GPL closer position radiation drive Extrapolation of GPL further position radiation drive GPLe closer position radiation drive fits GPL further position radiation drive extrapolation Flux comparisons Power comparisons Brightness temperature comparisons Summary Electron kinetics simulations Introduction Overview of Boltzmann electron kinetics model Electron energy distribution function Two temperature model approximation Electron-electron equilibration times Population distributions

10 vii 6 Heating simulations Introduction HELIOS-CR radiation-hydrodynamic model Motivation for the simulations Simulation parameter description Radiation drive characterization Electron temperature and average charge state Simulation parameter explanation Impact of hydrodynamics and atomic models on plasma heating Atomic kinetics simulations Introduction Steady-state results Cloudy results PrismSPECT SS results Steady-state Li- through H-like ions Time-dependent results Boltzmann results Steady-state & time-dependent comparisons Calculation of ionization parameter X-ray drive intensity profiles Electron density profiles Time evolution of ionization parameter H-like neon and the ionization parameter Conclusions 254 A Atomic Kinetics 26 A. Distribution equations A.. The Boltzmann Equation A..2 The Saha Equation A..3 Planck Function B Spectroscopy notation primer 272 B. Basic definitions B.. Configuration B..2 Term B..3 Level B..4 Selection rules B..5 KLM notation

11 viii C Miscellaneous 28 C. Electron density calculation Bibliography 282

12 ix List of Tables. Neon Ionization potentials Approximate parameter values of different plasmas Window material characteristics Maxwell-Botlzmann EEDF fits summary Equilibration times between hot and cold temperatures of Maxwell- Boltzmann distributions Atomic processes Region spacing Region parameters Neon densities Region zoning Frequency groups Output times Average electron temperatures for each fill pressure Helios-CR run parameters Closer LTE peak drive electron temperatures Closer NLTE peak drive electron temperatures Further LTE peak drive electron temperatures Further NLTE peak drive electron temperatures B. Quantum numbers B.2 Possible values B.3 Orbital configurations breakdown B.4 LS term splitting B.5 Fine structure splitting B.6 Selection rules

13 x List of Figures. Representation of the shell structure of the helium-like neon atom (a) Schematic of the z-pinch nested wire array with inner foam used to create a z-pinch dynamic hohlraum shown in the inset above the wire arrays, (b) A VISRAD schematic model showing emission power around the pinch [] The multiple lines of sight of the Z-machine and the multiple experiments they support [] An example of slab or plane-parallel geometry, as it relates to the version of the radiation transport equation Where here L is the size of the plasma and the generalized τ is the optical depth of the plasma The transmission probability contributions for the two different line shapes listed in equations For example, at an optical depth of τ = 3 gives T L T G. Also note the asymptotic behavior of the natural logarithm in the Gaussian term explaining the discontinuity The arcs and sparks photo taken during a shot on the Z-machine where the extent of electricity used can be seen. Courtesy, Sandia National Laboratories Schematic of the experimental setup The side on view of a experimental setup that used an older version of the gas cell and gas cell base. This is also a pre-transducer experiment. Courtesy, Sandia National Laboratories Looking top down on the experimental setup the tungsten wires of the z-pinch aremore clearly visible. The neon gas cell seen on the right has the transducer lines and gas fill lines leaving off to the right. Courtesy, Sandia National Laboratories Another side view of the experiment showing the gas cell in relation to the surrounding hardware with the rear aperture in place. Courtesy, Sandia National Laboratories An older design of the gas cell hardware. The dimensions of the gas cell itself remain unchanged while mounting (baseplate) and viewing (windows) hardware are continually updated for reliability and reproducibility

14 xi 3.7 A zoomed in picture of the gas cell with aperture. The arrow shows boththeline ofsight to the spectrometer which isthe paththe photons take traveling from the radiating z-pinch through the neon gas. Courtesy, Sandia National Laboratories An extreme close-up of the gas cell body and rear aperture, with the body using the silicon nitride windows Transmission of Mylar window, computed with LASNEX, shown in red, when subjected to radiation from the Z-machine whose power as a function of time is shown in blue. Courtesy of Dr. Tom Nash Transmission of 5 nm Silicon Nitride windows, computed with LAS- NEX, shown in red, when subjected to radiation from Z-machine whose power, as a function of time, is shown in blue. Courtesy of Dr. Tom Nash Transmission calculations made for.5 (green) and 2. (blue) micron thick mylar and also for 5 nanometers of silicone nitride all at a constant solid density [2] The behavior of the transducer signal when powered by different excitation or bias voltages is extremely linear (red). The blue vertical line in the Figure are the values quoted by the manufacturer. Courtesy, Dr. Ross Falcon Figure showing the reproducibility of different transducer using the same bias voltage. Courtesy, Dr. Ross Falcon Illustration of how the baselines are calculated for finding the fill pressure of the gas cell. This also shows the filling and purging process of the gas cell. Courtesy, Dr. Ross Falcon A schematic of the TREX spectrometer showing different possible slit positions along with the positions of crystals and where the detector (X-ray film, MCP) are placed. Courtesy, Sandia National Laboratories A real image of the TREX spectrometer being calibrated by a manson x-ray source. Courtesy, Sandia National Laboratories Diagram of the geometry setup of the x-rays traveling from the source reflecting of the crystal and then traveling to the detector Diagram showing the vector leaving the crystal surface as they would fall onto the detector Diagram of the geometry setup of the x-rays traveling from the source reflecting of the crystal and then traveling to the detector An example of the x-ray film developed and then densitometered for use of processing and analysis Inspection of the developed film shows the evidence of spectral features sought after in the experiment A finalized spectra showing different signatures of line transitions from different ion species An example of fitting the convolution of three Planckian distributions, each of a given radiation temperature and scaling constant, to an arbitrary radiation drive; with the arbitrary radiation drive in red and the fit in green with crosses The percent difference between the fit value and the original value, shown as the absolute value of the difference over the mean

15 4.3 The first time steps (-5 ns) of the IMH radiation drive and its corresponding Planckian drives that were fit, where the pink, blue, and cyan colored traces are the three Planckian fit distributions and the green and red traces are the IMH drive and the convolved fit Progressing in time (6-9,9-92 ns) it can be seen how the two cooler Planckian s contribution to the lower energies of the photon distribution increase Leading up to the peak of the radiation drive (93-98 ns), even though the cooler Planckian peak intensity level approaches that of the hotter Planckian, the overall distribution is still dominated by the high energy tail of the hotter Planckian distribution Steps leading up to and after the peak radiation drive (99-4 ns) Showing the last six time steps (5- ns) of the radiation drive a steady cooling trend can be seen by the lowering of intensities at each time step The first time steps (-5 ns) of the IMH radiation drive and its corresponding Planckian drives that were fit, where the pink, blue, and cyan colored traces are the three Planckian fit distributions and the green and red traces are the IMH drive and the convolved fit As with the closer position the same trend of the two cooler Planckians in photon energy slowly increase to be comparable intensities to the hotter Planckian (6-9, 9-92 ns) Though the peak intensity of the blue cooler Planckian surpasses that of the hotter Planckian, the hotter Planckian is still the dominating factor in terms of higher energy photons (93-98 ns) The time steps around the peak of the radiation drives (99-4 ns) Showing the last time steps (5- ns) of the further radiation drive a steady drop in peak intensity can be seen Matching trendsofimhdriveandaseparategpldrive, GPL2, togive the extrapolated values of GPL the correct trend at the peak turnover of the drive Matching the trend a few nano seconds before the drive is important because it is a region where the temperature experiences sharp increasing rate of change Projecting the decreasing trend of IMH onto GPL Fitting the ramping stage of the second radiation temperature Planckian Continuing fitting the ramping stage of GPL and beginning fitting the ending points Finishing projecting the ending slope onto GPL For the final and coolest temperature profile the fitting is very similar to that of IMH only with a steeper peak Fitting the late in time points again here the temperature levels are very similar simplifying the fit IMH and GPL closer position a coefficient values and fits to values The further away position fitting the temperature profiles follow similar trends as was done for the closer position, using the second GPL drive and IMH around the peak and then find the slope of the ramping up stage xii

16 4.23 Fitting the late in time points of GPL with the polynomial fit of IMH The second radiation temperature profile of the further away position fit using trends of second GPL drive and IMH in the further position Polynomial fit of IMH further position used to extrapolate the late in time points of GPL The trends of the third and coolest temperature profile follow similar trends of a slow linear increase followed by a quick ramping up until the peak is reach and almost symmetric decrease Matching ending point trends of IMH further to GPL further IMH and GPL further position a coefficient values and fits to values GPLe closer extrapolated radiation drive points with their three Planckian contribution breakdown for time steps GPLe closer extrapolated radiation drive points with their three Planckian contribution breakdown for time steps GPLe closer extrapolated radiation drive points with their three Planckian contribution breakdown for time steps GPLe closer extrapolated radiation drive points with their three Planckian contribution breakdown for time steps GPLe closer extrapolated radiation drive points with their three Planckian contribution breakdown for time steps The total integrated power per unit time and the breakdown planckian power contributions to the total power The brightness temperature profile as a function of time with individual contributions GPLe further extrapolated radiation drive points with their three Planckian contribution breakdown for time steps to 7 ns GPLe further extrapolated radiation drive points with their three Planckian contribution breakdown for time steps 8 to 93 ns GPLe further extrapolated radiation drive points with their three Planckian contribution breakdown for time steps 94 to 99 ns GPLe further extrapolated radiation drive points with their three Planckian contribution breakdown for time steps to 5 ns IMH C vs IMH F flux distributions: to 7 ns IMH C vs IMH F flux distributions: 8 to 94 ns IMH C vs IMH F flux distributions: 95 to ns IMH C vs IMH F flux distributions: to 6 ns GPLe C vs GPLe F flux distributions: to 7 ns GPLe C vs GPLe F flux distributions: 8 to 94 ns GPLe C vs GPLe F flux distributions: 95 to ns GPLe C vs GPLe F flux distributions: to 6 ns IMH C vs GPLe C flux distributions: to 7 ns IMH C vs GPLe C flux distributions: 8 to 94 ns IMH C vs GPLe C flux distributions: 95 to ns IMH C vs GPLe C flux distributions: to 6 ns IMH F vs GPLe F flux distributions: to 7 ns IMH F vs GPLe F flux distributions: 8 to 94 ns IMH F vs GPLe F flux distributions: 95 to ns IMH F vs GPLe F flux distributions: to 6 ns IMH C vs IMH F Power as a function of time xiii

17 xiv 4.57 GPLe C vs GPLe F Power as a function of time IMH C vs GPLe C Power as a function of time IMH F vs GPLe F Power as a function of time IMH C vs IMH F Brightness temperature as a function of time GPLe C vs GPLe F Brightness temperature as a function of time IMH C vs GPLe C Brightness temperature as a function of time IMH F vs GPLe F Brightness temperature as a function of time The series of EEDF s at different time steps for a filling pressure of 3 Torr in the closer position, the EEDF at 8 ns is highlighted in green while the blue is the EEDF at the peak of the radiation drive, ns The average charge state of the neon gas as a function of time for different filling pressures in the closer position The EEDF distribution calculated in the blue, along with the red line of the Maxwell-Boltzmann fit The thermal electron temperature as a function of time for the closer position radiation drive The thermal electron temperature as a function of time for the further position radiation drive Comparing the thermal electron temperatures as functions of time for the closer and further positions each of a different fill pressure: (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr Fitting the EEDF generated by the closer radiation drive at 95 ns with a two temperature Maxwell-Boltzmann model where the colder Maxwell-Boltzmann distribution is the red trace and the hotter distribution is the green trace. Where the individual plots correspond to (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr Fitting the EEDF generated by the further radiation drive at 95 ns with a two temperature Maxwell-Boltzmann model where the colder Maxwell-Boltzmann distribution is the red trace and the hotter distribution is the green trace. Where the individual plots correspond to (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr Fitting the EEDF generated by the closer radiation drive at ns with a two temperature Maxwell-Boltzmann model where the colder Maxwell-Boltzmann distribution is the red trace and the hotter distribution is the green trace. Where the individual plots correspond to (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr Fitting the EEDF generated by the further radiation drive at ns with a two temperature Maxwell-Boltzmann model where the colder Maxwell-Boltzmann distribution is the red trace and the hotter distribution is the green trace. Where the individual plots correspond to (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr Isoelectronic sequence distributions as functions of time for different filling pressures at the closer position: (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr Isoelectronic sequence distributions as functions of time for different filling pressures at the further position: (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr

18 xv 6. Schematic of geometry used for the multi-angle diffusion approximation The brightness temperature extracted from two different radiation drives, closer position (red), further position (green) Attenuation of radiation through a planar slab of mylar Average charge state distribution vs time for an LTE vs NLTE neon plasma Average electron temperature in unshocked neon as a function of time with LTE (red) and NLTE (green) treatement for a mylar-neon-mylar plasma Average electron temperature of unshocked neon gas in mylar-neonmylar (solid) and neon only system (dashed) as a function of time with LTE (red) and NLTE (green) treatment Electron temperature as a function of filling pressure. The shot number associated with each data point is also shown Heating from the GPLe (red) and IMH (green) drives using LTE (a) and NLTE (b) atomic kinetics Plots each containing the same four filling pressures with the four variations of simulations using the closer radiation drive (a) LTE of mylarneon-mylar, (b) LTE of mylar-neon-mylar without hydro motion, (c) LTE with only a neon slab, (d) LTE with only a neon slab and no hydro motion Plots each containing the same four filling pressures with the four variations of simulations using the closer radiation drive (a) NLTE of mylar-neon-mylar, (b) NLTE of mylar-neon-mylar without hydro motion, (c) NLTE with only a neon slab, (d) NLTE with only a neon slab and no hydro motion Plots each containing the same four filling pressures with the four variations of simulations using the further radiation drive(a) LTE of mylarneon-mylar, (b) LTE of mylar-neon-mylar without hydro motion, (c) LTE with only a neon slab, (d) LTE with only a neon slab and no hydro motion Plots each containing the same four filling pressures with the four variations of simulations using the further radiation drive (a) NLTE of mylar-neon-mylar, (b) NLTE of mylar-neon-mylar without hydro motion, (c) NLTE with only a neon slab, (d) NLTE with only a neon slab and no hydro motion The hydrogen-like neon s from the two different drives IMH (red) and GPLe (green) using both the steady-state (dashed) and time-dependent (solid) treatment Charge state populations as a function of time as calculated by Cloudy using the radiation drive in the closer position Charge state populations as a function of time as calculated by Cloudy using the radiation drive in the further position Charge state populations as a function of time as calculated by PrismSPECT using the radiation drive in the closer position Charge state populations as a function of time as calculated by PrismSPECT using the radiation drive in the further position

19 7.6 Lithium-, Helium-, and Hydrogen-like neon fractions as a function of time for each code calculation in the closer position: Cloudy(dash), PrismSPECT(dot); Li-like(red), He-like(green), H-like(blue) Lithium-, Helium-, and Hydrogen-like neon fractions as a function of time for each code calculation in the further position: Cloudy(dash), PrismSPECT(dot); Li-like(red), He-like(green), H-like(blue) Charge state populations as a function of time as calculated by the Boltzmann model and using the radiation drive in the closer position Charge state populations as a function of time as calculated by the Boltzmann model and using the radiation drive in the further position Lithium-, Helium-, and Hydrogen-like neon fractions as a function of time for each code calculation in the closer position: PrismSPECT(dash), Boltzmann(dot); Li-like(red), He-like(green), H-like(blue) Lithium-, Helium-, and Hydrogen-like neon fractions as a function of time for each code calculation in the further position: PrismSPECT(dash), Boltzmann(dot); Li-like(red), He-like(green), H-like(blue) PrismSPECT time-depedent calculations using the closer position radiation drive and filling pressures in Torr: 3.5(red), 7.5(green), 5(blue), 3(pink). Zbar, Neutral through Carbon-like neon PrismSPECT time-dependent calculations using the closer position radiation drive and filling pressures in Torr: 3.5(red), 7.5(green), 5(blue), 3(pink). Boron-like through fully stripped neon PrismSPECT time-dependent calculations using the further position radiation drive and filling pressures in Torr: 3.5(red), 7.5(green), 5(blue), 3(pink). Zbar, Neutral through Carbon-like neon PrismSPECT time-dependent calculations using the further position radiation drive and filling pressures in Torr: 3.5(red), 7.5(green), 5(blue), 3(pink). Boron-like through fully stripped neon Steady-state and time-dependent calculations using closer position radiation drive, Zbar Steady-state and time-dependent calculations using closer position radiation drive, neutral neon Steady-state and time-dependent calculations using closer position radiation drive, F-like Steady-state and time-dependent calculations using closer position radiation drive, O-like Steady-state and time-dependent calculations using closer position radiation drive, N-like Steady-state and time-dependent calculations using closer position radiation drive, C-like Steady-state and time-dependent calculations using closer position radiation drive, B-like Steady-state and time-dependent calculations using closer position radiation drive, Be-like Steady-state and time-dependent calculations using closer position radiation drive, Li-like Steady-state and time-dependent calculations using closer position radiation drive, He-like xvi

20 xvii 7.26 Steady-state and time-dependent calculations using closer position radiation drive, H-like Steady-state and time-dependent calculations using closer position radiation drive, fully stripped neon Steady-state and time-dependent calculations using further position radiation drive, Zbar Steady-state and time-dependent calculations using further position radiation drive, neutral neon Steady-state and time-dependent calculations using further position radiation drive, F-like Steady-state and time-dependent calculations using further position radiation drive, O-like Steady-state and time-dependent calculations using further position radiation drive, N-like Steady-state and time-dependent calculations using further position radiation drive, C-like Steady-state and time-dependent calculations using further position radiation drive, B-like Steady-state and time-dependent calculations using further position radiation drive, Be-like Steady-state and time-dependent calculations using further position radiation drive, Li-like Steady-state and time-dependent calculations using further position radiation drive, He-like Steady-state and time-dependent calculations using further position radiation drive, H-like Steady-state and time-dependent calculations using further position radiation drive, fully stripped neon Closer and further position photon-energy integrated flux profiles: closer (red), and further (green) Electron density profiles for each filling pressure in Torr: 3.5 (pink), 7.5 (blue), 5 (green), 3 (red) Boltzmann model ionization parameter results for each filling pressure in the closer (solid) and further (dashed) positions. Where 3.5 Torr (pink), 7.5 Torr (blue), 5 Torr (green), and 3 (Red) Cloudy ionization parameter results for each filling pressure in the closer(solid) and further(dashed) positions. Where 3.5 Torr (pink), 7.5 Torr (blue), 5 Torr (green), and 3 (Red) PrismSPECT ionization parameter results for each filling pressure in the closer(solid) and further(dashed) positions. Where 3.5 Torr (pink), 7.5 Torr (blue), 5 Torr (green), and 3 (Red) The ionization parameter as a function of time for each filling pressure from each of the atomic kinetics calculations: Boltzmann (B), Cloudy (C), PrismSPECT (P), in the closer (Cl) and further (F) positions Steady-state calculation data (black) using PrismSPECT along with a polynomial fit to the calculation data (red) Steady-state calculation data (black) using Cloudy along with a polynomial fit to the calculation data (green)

21 xviii 8.9 The fits to the calculations, PrismSPECT (red) and Cloudy (green), along with the experimental data points and the fit to those experiment data points (black-dashed) [3] Time-dependent (green) and steady-state (red) PrismSPECT calculations along with the experimental data fit (black-dashed) [3] B. Atomic structure model showing a conceptual model of the KLM notation [4]

22 Chapter Introduction The purpose of this Chapter is to provide the reader with an introduction and background to the basic subject matter of this dissertation. The main theme of this manuscript is the discussion of laboratory photoionized plasmas. Specific experiments and model simulations were carried out in order to understand the behavior and characteristics of these plasmas, mainly with neon gas. Before results are delved into a review of complementary material will be covered. This will start with a basic treatment given to a few key concepts of plasma physics in general, followed by the more specific experimental plasmas produced and how they are necessary for this research. Next, atomic physics and spectroscopy is discussed, motivated by the necessity to diagnose the temperature and density conditions within these laboratory plasmas. The bridge is then made relating laboratory astrophysical plasma experiments to the astrophysical conditions they are designed to shed light onto. It is then discussed how a large effort has been made to understand these systems from a modeling stand point. This modeling, though, has yet to undergo a large scale benchmarking control. The Chapters following will then discuss a large group of simulations addressing the hydrodynamics and the atomic kinetics of the neon plasma. The results of these calculations along with a study of the intensity of the source radiation lead to a discussion

23 2 about the ionization parameter, an established parameter used in the astrophysics community for characterizing photoionized plasmas.. States of matter Matter is commonly referred to by three main states: solids, liquids, and gases. Expandingthisdescriptionfromthreetofourstatesofmatterallowsustoaddanionized gas or plasma. Generally addressed these states of matter relate to the temperature and the density of the material, though as shown in statistical mechanics there are times when these defining classifications are not as easily made (e.g. triple point [5]). Also, as science continues the progression of research into new extreme limits of temperature and density new states of matter are observed (e.g. Bose-Einstein condensates). A simplistic explanation of these phase changes starts with a material in solid form at a constant density, heat (energy) is applied and the solid sublimates or transitions to a liquid. The liquid is then heated to higher temperatures (more energy) and the liquid transitions into a gas. As expected when even higher temperatures are achieved by the gas the electrons bound to the individual atoms gain enough energy to free themselves from the bonds generated by the atomic nucleus. When these negatively charged electrons leave, they leave the atom with a net positive charge, thus ionizing the atom in the gas. This collection of ionized atoms then is what we define as our plasma. Of specific interest and application in this thesis is the neon atom. Neon at equilibrium has ten bound electrons in the configuration: s 2 2s 2 2p 6 As energy in the form of light hits these atoms the light can ionize and excite the electrons. The energies required to free each subsequent electron from its shell are

24 3 Figure.: Representation of the shell structure of the helium-like neon atom called ionization potentials. A point of clarification, regarding K-shell ionization potentials (see the appendix B), specifically the Kα (α referring to a transition between principle quantum number n=2 to n=, see Figure.) is important. The term K-shell refers to the inner most electron shell or electrons with principal quantum number equal to one. The K- shell ionization potential refers to removing the inner most K-shell electrons with or without any number of the outer shell electrons present to adjust this K-shell ionization potential due to screening effects. When a cold Kα is discussed an element with all the outer shell electrons are still in place. Of interest to our research are the higher charge states of a neon plasma, specifically, Be-like neon up to H-like neon. As previously discussed, a plasma can be thought of as a very high temperature gas, one which has been heated sufficiently so that it is a mixture of free electrons and ions with bound electrons. The position and velocity of these particles are used to describe a plasma at any point in time. This

25 4 Ionization stage Charge Table.: Neon Ionization potentials Isoelectronic state Ionization potential [ev] I Ne II F III 2 O IV 3 N V 4 C VI 5 B VII 6 Be VIII 7 Li IX 8 He X 9 H XI FS K-shell ionization potential [ev] behavior is governed by the particles interaction with all the varying electromagnetic fields [6]. As the plasma continues to get hotter, enough electrons will eventually be freed so that the electrons surrounding the now charged ion will create a screen, these screen potentials are shown in table.. This influence of the ion beyond a certain distance is then cancelled out [7, 8, 9]. This distance is known as the Debye length. The sphere surrounding an ion with characteristic radius being the Debye length is known as the Debye sphere. In other words this screening effect says that any electrostatic fields originating outside of the Debye sphere are blocked and do not contribute to the electric field at the center of this sphere. Using these ideas we will now define a classical plasma with the following conditions [7, 9, 6]:. Number of electrons within the Debye sphere 2. Ion-Ion spacing than the Debye length 3. Plasma physical length interparticle spacing There are many plasmas where all of these conditions are not met. Though not

26 5 explicitly, we have been speaking of weakly coupled plasmas. On the contrary, a plasma whose density is so large that the spacing between ions is on the order of the Debye length, then all charged ions are affected by the potential energies of their neighboring particles. This is referred to as a strongly coupled plasma. For a weakly coupled plasma the Debye length is: λ D = kte +kt i 4πe 2 Zn (.) Here, k is Botlzmann s constant, T e and T i are the electron and ion temperatures respectively, Z and n are the charge and density of the test particles, respectively. Considering the neon plasmas discussed in this manuscript the length is on the order of tens of nanometers. Another metric used is the plasma parameter, which is roughly defined as the number of particles in the Debye sphere. This is important because as mentioned earlier the electrostatic fields of every particle interact with one another. The complexity of this many body problem quickly becomes overwhelming. Using the definition of the Debye sphere this simplifies the problem because each particle of the plasma only interacts collectively with the other charges inside its Debye sphere. Λ p = Zn ( kte 4πe 2 ) 3/2 (.2) This dimensionless parameter is also useful when considering the collisionality of a plasma, which is very important for understanding photoionized plasmas. For a plasma to be photoioinized the majority of ionization events need to be caused by the interaction of an ion and a photon. In opposition are ionization events caused by a collision interaction either with an ion or free electron. The ionization parameter specifically relates these competing criterion and will be addressed and given special treatment in a later Chapter.

27 6 Plasma type Plasma size [cm] Atom density [#/cm 3 ] Temperature [ev] Solar wind 7 [] < Space plasma >> 3 Flourescent tube 2 Lightning Inertial fusion 22 3 Table.2: Approximate parameter values of different plasmas. To understand the collisionality of a plasma an important idea to talk about is the frequency at which collisions occur within the plasma. If we take, for example, a simple model that assumes electrons are being scattered by fixed ions we can ascertain an estimation of the electron momentum relaxation collisional frequency: ν ei = Λ C = 8πZ 2 e 4 n i 3 3/2 m /2 e (kt e ) 3/2lnΛ C (.3) (kt) 3/2 2Ze 2 4πe 2 n i (.4) Where Λ C is the Coulomb logarithm [6] and is representative of all the Coulomb collisions within the Debye sphere and is equal to the plasma parameter to within a factor of unity. For neon plasmas this electron-ion collisional frequency is on the order of 2 Hz. Getting back to the simple picture that a plasma is just an ionized gas, these plasmas exist in a wide range of conditions of composition, temperature, density, and size. Table.2 gives a few examples of these different plasmas and their characteristic parameters..2 Spectroscopy of plasmas In this section, we discuss the roll of spectroscopy and how plasma conditions may be inferred from the measured and simulated plasma spectral features. This method is

28 7 necessary due to the extreme conditions at which these plasmas can occur. For everyday living temperatures the thermal expansion of a column of liquid can be employed to measure temperatures or using a manometer to measure pressure. For the neon plasmas of interest in this manuscript this is impractical. The Z-machine environment, discussed in the next section, is very tumultuous making the use of Langmuir probes, which could measure the conditions directly, and other such devices impractical, hence the need for spectroscopic measurements. A widespread, accepted method for inferring these plasma conditions indirectly is to use the emission and absorption features produced from within the plasma. This indirect method also keeps any detectors from affecting and changing the properties of the plasma, though using this method to infer these conditions indirectly is not without its own difficulties, as these plasmas are a conflux of many complicated factors including thermodynamic properties, physical sizes and compositions, and neighboring materials and their compositions. The purpose of spectroscopy is to understand the atomic and radiation kinetics of the plasma, where atomic kinetics here refers to the details of the movements of the individual electrons in a given atomic level structure. This allows for the inference of this spectroscopic information and other such plasma properties [, 2]. It is also important to note that in the case of classical thermometers a single temperature is given for the environment; this is not usually true for high energy plasmas. Even if this assumption is made these plasmas are, in the strictest cases, characterized by multiple temperatures within the system. The electrons will have a kinetic temperature (or multiple) as do the varieties of ions. If the plasma radiation is relatively Planckian in nature there is also a characteristic brightness and color temperature. In the laboratory, the differences between these temperatures often arise when the duration of the relaxation and ionization time scales are very different compared to

29 8 the duration of the environmental conditions of the experiment. Using x-ray emission and absorption spectroscopy does not represent a different method for inferring plasma properties, it is merely an extension of well-established techniques that are more applicable to these plasmas [3]. The movements of the individual electrons, or the excitations and de-excitations of electrons, are the main mechanisms for the production of x-ray spectra of a plasma. These excitations and de-excitations occur within ion states and the continuum and are referred to as either bound-bound, bound-free, or free-free transitions. Boundbound transitions occur due to ions being built of discrete energy levels, when an electron transitions from one bound state to another bound state within the ion, either by gaining or losing the discretized energy difference between the levels. This behavior results in the observation of either absorption or emission line spectra. Bound-free transitions can be triggered either by the absorption of a photon or by the capture of a free electron. In the case of photon absorption the energy absorbed by the photon eitherexcitesorionizesaboundelectron, andinthereversecaseofacapturedelectron a photon is emitted. The two cases of bound-bound and bound-free absorptions are important for our discussion later because they are proportional to the number of absorbing atoms per unit volume of gas [4]. The third category of transitions are free-free or bremsstrahlung transitions. In this case a free electron traveling through the electric field of an ion can either emit a photon without losing all of its kinetic energy (bremsstrahlung) or it absorbs a photon and gains additional kinetic energy (inverse bremsstrahlung). The total spectral signature measured by a detector or calculated is a conglomerate with each of the aforementioned possible transitions contributing to the whole. This spectra therefore contains information about both the atomic structure of the ions and the atomic kinetics of the ions and electrons from

30 9 which information about the temperature and density can be inferred..3 Z-pinch plasma In order to photoionize a gas a large flux of high energy photons are needed. Creating this large flux of photons in the laboratory is accomplished by using the Z-machine at Sandia National Laboratories. This Z-machine uses z-pinch physics to create an emitting plasma. This plasma is produced by a collection of wires colliding on a centralized z-axis (hence the determination of the name) and begins to heat up as a large current passes through the wires in a very short amount of time. Stated simply a current passes through wires and the internal resistance of the wires cause them to heat up. When a current passes through a wire a magnetic field is contemporaneously generated perpendicular to the direction of the traveling current. When multiple wires are parallel to one another the application of the magnetic fields cause the wires to attract to one another. This force is called the Lorentz force and in simplified form is F = q v c B (.5) F = I L B (.6) Where q is the charge, I is the current, L is the inductance, and B is magnetic field. When mega-amps of current are traveling through the wires, on the order of nanosecond timescales, these wires begin to ablate and form plasma. This plasma surrounding the wires is subject to the J B Lorentz force (J being the current density) and is also pulled towards the central z-axis, together with the more solid density wire. This intense heating of the wires causes them to radiate a distribution of x-rays that grows in intensity until the wires in the system collide, at which point

31 this distribution peaks both in total power and intensity. The x-rays emitted, off the z-axis, by this z-pinch plasma are those used to drive the inert Neon gas in our experiments. The behavior of this neon gas due to the interaction with the z-pinch x-ray radiation is the main subject matter of this manuscript..4 Z-machine facility Inorder toproduceaphotoionizedplasma avery largeamountofhighenergyphotons needs to be produced. Creating such a source in a controlled, safe laboratory environment is no trivial task and isprecluded by many factors. The pulsed power Z-machine [5] facility as Sandia National Laboratories is capable of fulfilling all these needs. Earlier attempts of neon photoionized plasmas were performed on the Z-machine prior to a refurbishment that took place in which the energy capable of being stored was doubled [6]. Now this large scale facility is capable of storing 2 MJ of energy in 36 Marx bank capacitors each able to achieve up to 85 kv of charge. This allows over 26 MA of current to be coupled into the wire of array of the z-pinch []. In terms of x-ray yield and power this translates into an accurately measurable output of 2 MJ and 33 TW respectively [7]. For the neon photoionized plasma experiment the Z-machine load is composed of two nested cylindrical tungsten wire arrays of 24.4 µm diameter wires in the outer array and 2.4 µm diameter wires in the inner array. Figures.2a and.2b show the layout of the nested wire arrays and the hardware surrounding the wire array load. Another important point to note is that the Z-machine has the capability of utilizing the z-pinch radiation for multiple experiments at once. Due to the large scale of the facility the hardware real estate is a

32 (a) (b) Figure.2: (a) Schematic of the z-pinch nested wire array with inner foam used to create a z-pinch dynamic hohlraum shown in the inset above the wire arrays, (b) A VISRAD schematic model showing emission power around the pinch [] precious commodity and highly sought after. This leads to large scale collaborative efforts spanning universities, national laboratories, and private industries. One such collaboration is the ZAPP collaboration [], of which the neon photoionized plasma experiment is a part. As Figure.3 shows, the Z-facility can host a large collaborative effort like the ZAPP collaboration capable of executing multiple laboratory astrophysics experiments simultaneously [8, 9, 2, 2, 22, 23]. The experimental work presented in this manuscript will be from data collected and simulations carried out on the experiment from the line of sight labelled Photoionized Ne-Gas Cell in Figure.3..5 Laboratory plasmas relevant to astrophysics Why is studying plasmas important to our understanding of the universe? 99.9 percent of the known universe is made up of plasma [24]. Photoionized plasmas in

33 2 Figure.3: The multiple lines of sight of the Z-machine and the multiple experiments they support []. particular occur over a wide range of astrophysical objects from black hole accretion disks to x-ray binary stars and active galactic nuclei (e.g., quasars) [25, 26, 27]. These different objects occur between drastically differing conditions from cold ( 4 5 K) and dense (> cm 3 ) to hot ( 7 8 K) and sparse ( 3 cm 3 ) [28]. Historically, sensitivity and spectral resolution of available instrumentation have limited the amount of detailed spectroscopic information available from these astrophysical objects. This situation has been greatly improved since 999 when orbiting x-ray observatories CXO (Chandra X-ray Observatory, previously known (pre-998) in literature as AXAF or Advanced X-ray Astrophysics Facility) and XMM-Newton were launched into Earth s orbit. Data from these observatories with the new high level of accuracy and resolution has brought into question the accuracy of the spectral models that are used to interpret these astrophysical photoionized plasmas. The ability to test and benchmark these spectral models against laboratory experiments is one of the aims of this study. The plasmas we are interested in can be split into two gener-

34 3 alized categories: collisionally dominated or photoionization dominated. This is why the photoionization parameter was introduced by astronomers and astrophysics as a means of understanding the collisionality of space plasmas. The different classes of astrophysical objects mentioned here contain photoionized plasmas due to the strong source of ionizing radiation from neighboring sources. This is where one difficulty in creating a laboratory photoionized plasma arises. Creating a highly energetic source of ionizing x-rays in a controlled laboratory environment, like those created by the Sandia Z-machine, is a non-trivial task. Another difficulty producing these laboratory plasmas involves the target gas that is heated by the source photons. The density of this gas needs to be low enough so that collisions do not dominate over the photoionization processes. This becomes difficult in the laboratory due to the lack of signal received from the low density samples of gas used in experiments. As mentioned earlier, in cosmic environments gas densities approach levels in space that are considered vacuum for many laboratory experiments. The difference arises in the sheer size of cosmic gases. Therefore, a compromising middle ground is found where densities are low enough to confidently acquire data while still allowing photoionization events to be a significant contributing factor..6 Experimental requirements and conditions Forming a static -dimensional plasma in the laboratory is a non-trivial task. This can be explained by breaking down the experiment into its individual components. First consider the photons, they originate from the source and travel to the detector. These photons need an undeterred path to the neon gas and then continue to a

35 4 detector where the photons can be recorded. Next we consider the neon gas that is to be photoionized. This gas needs to be accessible to the photons in a static, well characterized way that does not interact with the neighboring environment of the Z-machine. This all needs to be setup in such a way that it is visible to the x-ray spectrometers while keeping the spectrometer free from any debris from the z-pinch. One of the goals of this experiment is to benchmark atomic kinetics codes. In order to do this, it is crucial for us not only to have well characterized experimental data but also to have accurate simulations of the experiment. The design of the targets and the experiment have undergone many iterations in order to simplify the design robustness and reproducibility of the experiment. Specific details of affecting the state of the neon gas include fluid hydrodynamics, heating, and atomic kinetics. The hydrodynamic behavior of the neon gas and the windows enclosing the gas to keep it from leaking out of the cell can also have major effects and need to be accounted for. This leads to a discussion on the windows effects and the different window materials used. Radiation from the z-pinch source deposits energy into gas, continually heating it during the lifetime of the source. This highly energetic plasma then undergoes a series of atomic processes that try to force the gas to a state of equilibrium. The time scales of all these competing conditions also play a role in how and to what state the plasma relaxes to..7 Summary of thesis Chapters This work is made up of many inter-connected pieces and as such the results of one Chapter are often used in many other Chapters of this manuscript. This interdependence between Chapters is most easily shown using the example of fitting an

36 5 arbitrary radiation source in Chapter 4. The radiation drives created from the method in Chapter 4 is then taken and used as input in both the hydrodynamic simulations and atomic kinetics simulations later on. As an aside, the method for fitting an arbitrary radiation drive are not limited to the z-pinch plasmas and are also not restricted to act only upon neon. Chapter 2 This Chapter provides a description of the theory used in the rest of this thesis. Atomic kinetics and atomic rates are discussed along with population distributions of bound and free electrons. A description of different equilibrium conditions and the radiative models are given in the appendix. Radiation transfer and its relation to atomic kinetics spectroscopy is also briefly discussed. Chapter 3 A more in depth detailing of the experiment is given including schematics of target hardware and testing diagnostics, both those produced by Sandia National Laboratories and those by our group, also, a review of experimental data that has been processed. This also gives a detailed description of the x-ray spectrometer used in the experiment. Chapter 4 This Chapter, will discuss the method of fitting any arbitrary radiation source with multiple Planckian distributions. The accuracy of the fit is discussed and also the extrapolation method. A detailed review of the photon energy distributions of these drives as a function of time is also reviewed. Chapter 5 This Chapter in the Boltzmann model. This model is used to selfconsistently solve the electron and atomic kinetics in order to simulate the free electron distribution function and configuration-averaged atomic kinetics.

37 6 Chapter 6 This Chapter focuses on the hydrodynamic modeling of the neon plasma using the Helios-CR code [29]. These results are analyzed to understand the heating of the complex system of windows and gas in the experiment and the way they are all driven by a distribution of photons. This will be introduced with an initial review of the first principles of thermodynamics. Chapter 7 Here, the atomic kinetics are studied using the well established codes PrismSPECT [3, 3] and Cloudy [32]. The level populations are investigated as a function of time using the different radiation drives. The different audiences each of the two codes are designed for are discussed and how this experiment can help bridge the gap between the two communities. Chapter 8 The ionization parameter, which is extensively used in the astrophysical community, is given a more extensive review for the conditions of a photoionized plasma. Here we study the ionization parameter as calculated using experimental results and the results of the codes PrismSPECT and Cloudy..8 Author s role in experiment and code development In the Z-machine experiment, the author s main roles were to: ) aid in the calibration, alignment, and fielding of x-ray diagnostics; 2) to assemble, test, and field the neon gas cell. I also supported the setup and development of x-ray film on x-ray spectrometers, not only on this experiment, but on the experiments that were running concurrently on the z-machine as part of the ZAPP collaboration. The fitting routine

38 7 presented in Chapter 4 to fit an arbitrary radiation drive by multiple Planckian distributions was written, and, unless otherwise stated, the simulation work was carried out by the author. The majority of the simulations were carried out on established codes that will be discussed in the following Chapters.

39 8 Chapter 2 Atomic and radiation physics 2. Introduction This Chapter discusses the basic theories and ideas that are used to describe the state of the neon plasma. Primary focus, here, is given to ideas that lend to a conceptual understanding of the physics involved. In a plasma, electrons undergo bound-bound, bound-free, and free-free transitions. These transitions are the primary sources of x- ray emission and absorption. Each of these transitions have a contribution to the total x-ray emission and/or absorption of the plasma. Each relative contribution depends on the atomic structure of the ions and the distribution of the electrons within the bound and free energy levels. In order to understand the spectrum given by the system one has to understand both the atomic level structure of the system and the transitions that are taking place within the system, where the system is classified as the interaction of the radiating x-ray source incident upon the neon gas. In a photoionized plasma it is assumed that collisional processes are not the dominating transitional processes, otherwise the assumption of having a photoionized plasma is rendered invalid. For these photoionized plasmas photoexcitation and photoionization are the dominant upward processes. It is also important to note that for plasmas of

40 9 non-negligible thickness spatial variations can occur. This means that, any photons emitted within the plasma must propagate through the rest of the plasma and escape its boundary in order to be observed. Emission and re-absorption of local radiation can also influence the distribution of the bound state electrons, i.e. atomic level populations. This Chapter begins by discussing the individual atomic processes that occur within the plasma and their associated rates. It then goes into a review of the distribution and equilibrium conditions that are often used to model a distribution of either electrons or photons. Discussion on the derivations of these are given in appendix A. Next, a discussion will be given on the basic ideas of radiative transfer and the escape factor approximation. How these ideas relate to spectroscopic models will then be briefly discussed. 2.2 Atomic processes in plasma In the following subsections the letter Z refers to the number of bound electrons, hν or ω refers to photon energy, e refers to free electrons, and A is the ion species where an * denotes an excited state. There are more atomic processes that occur in a plasma than those listed here, but for brevity only those relevant to our application are considered. These processes can be looked at in terms of bound-bound processes and bound-free processes. A bound-free process is a process that occurs between the bound levels of an ion and continuum levels of the ion. A bound-bound processes is a processes that occurs entirely within the bound levels of an ion. Bremsstrahlung radiation, for example, is free-free transition that occurs when an electron is slowed by the interaction with an ion and the loss of energy is emitted as a photon into the continuum. Dielectronic recombination is an example of a bound-free transition

41 2 that begins with an initially free electron that gets captured by an ion in a doubly excited state and from there decays to a less excited state emitting a photon. In a bound-bound transition, like that of collisional excitation or collisional de-excitation, the initial and final states of the electron remain within the levels of the ion Bremsstrahlung & inverse bremsstrahlung e +A(Z) e +A(Z)+hν (2.) For an ion that starts off neutral and is then continually heated so that the bound electrons begin to burn off the pool of free electrons grows larger and larger. This causes the level of Bremsstrahlung radiation to increase. Inversely the absorption of a photon by a free electron makes that electron s energy increase by energy of the photon absorbed. hν e e k i Of particular interest to our case is inverse Bremsstrahlung, investigated by looking at the absorption coefficient κ. κ IB = 32π3 3 e 6 ( c) 2 n (3) m e c 2 (2πm e c 2 T e ) /2(Z)2 i n e ( ω) 3g ib( ω) (2.2)

42 2 Where here is the Dirac constant, also known as Planck s constant divided by 2π, m e is the mass of the electron, c is the speed of light, T e is the electron temperature, n e,n i are the electron and ion number densities respectively, Z is the average charge state distribution, ω is the energy of the incoming photon, and g IB is the degeneracy of energy ω Collisional excitation & de-excitation e +A (Z) e +A (Z) (2.3) In collisional excitation and de-excitation, a free electron interacts with an electron in an atomic level. In the upward process of excitation, the free electron loses energy to the bound electron promoting it to a higher energy level within the ion. In the downward de-excitation case, the bound electron loses energy and drops to a lower level within the ion. The same idea follows for collisional ionization and recombination, except that instead of transitions between levels there are transitions between charge states. e j i cont. R CE,R CdE

43 22 With the associated rates: ( ) 3/2 ( 2 R CE,CI = n e kt e πm e ) /2 E σ(hν)(hν)e E kte de (2.4) R CD = g l e E kter CE (2.5) g u Here, h is Planck s constant, k is Boltzmann s constant, n e is the electron density, T e is the electron temperature, m e is the mass of the electron, E is the transition energy between levels, and g l g u is the statistical weight of lower/upper level Photoionization & radiative recombination A(Z)+hν e +A(Z ) (2.6) An ion becomes photoionized when an incoming photon is absorbed and its energy used to free a bound electron. The inverse of this process is when a free electron is captured by the ion and in turn emits a photon of energy hν, this is called radiative recombination. j i R PI,R RR k With the associated rates: R PI = 8π h 3 c 2 E (hν) 2 σ(hν) dhν (2.7) e hν/kte g l 2 R RR = N e g u c 2 π(m e kt e ) 3/2e E/kTe E σ(hν)(hν) 2 e hν/kte dhν (2.8)

44 23 T e is the electron temperature of the free electrons, n e is the electron number density, E is the energy spacing between levels, hν is the energy of the incoming photons, and σ is the absorption cross section Electron capture & Autoionization A (Z) e +A(Z +) (2.9) In the case of electron capture, as the name would suggest, a free electron is captured within the levels of a neighboring ion. For autoionization an electron is in a highly (doubly,multiply) excited state. In this case these highly excited states are above the normal ionization limit of the respective ion. The electron is then spontaneous emitted changing the state from charge Z to Z+. e j R AI,R EC k i Dielectronic recombination e +A (Z +) A (Z) A (Z +)+hν (2.)

45 24 Dielectronic recombination is the name given to a two step process consisting of electron capture followed up by radiative decay or autoionization. Here an incoming electron is resonantly captured into a double excited state, j. This state will then either autoionize or decay to a lower state. Refer to the previous discussion on autoionization and electron capture. e j hν R EC,R RD k i Electron impact ionization & 3-body recombination e +A(Z) e +A(Z )+e (2.) In this process an electron collides with an ion with charge Z, this colliding electron then loses energy to the ion, ionizes a bound electron increasing the ion s charge to Z+, freeing a second electron. In the reverse process, two free electrons collide with an ion, and one of the electrons recombines and reduces the ion s charge state from Z+ to Z, leaving one free electron.

46 25 e j R AI,R EC k i Photoexcitation, spontaneous radiative decay, & stimulated emission A (Z) A (Z)+hν When a photonis absorbed by an ionthe energy is deposited to a boundelectron, this causes the electron to be promoted to a higher energy level. Once the electron is in this higher level, it can undergo one of two processes. It can either A) spontaneously emit a photon and drop to a lower energy level or B) it can be stimulated by a photon causing the electron to then emit a photon and drop back down to a lower energy level.

47 Understanding rates i R AI,R EC k R CE,R CdE j R PI,R RR Shown graphically, in the simplified example above, is that the populations of levels within an ion are dependent upon the combination of all level populations and associated rates populating and depopulating each individual level. The energy level structure can be found in many ways, one such way involves using atomic level structure codes like HULLAC, FAC or other [33, 34, 35]. This atomic level structure and rate coefficient information is then used in atomic kinetic rate equations, like the three level system shown below, i : dn i dt j : dn j dt k : dn k dt = N j R CDE +N k R RR N i R CE N i R PI = N j R CDE +N k ()+N i R CE +N j () = N j () N k R RR +N i R PI +N j () (2.2) Here N i,j,k are population densities for level i,j,k and the R terms represent the collisional or radiative rates from the respective levels. It is possible here to convert population number densities to fractional ionic populations, f j,k with the relationship f i = N i N a (2.3)

48 27 Where in this equation it is assumed that particle number, equation 2.4, and charge neutrality, equation 2.5, have been conserved. N i = N a (2.4) i q i N i = N e (2.5) i Where here q i is the charge of ion i. Summing over all the s will result of unity. f i = (2.6) i Summing over all the s multiplied by their respective charge will give the average charge state, or average ionization of the plasma, Z, q i f i = Z (2.7) i The rate matrix equations, shown in equations 2.2, are explicitly the collection of individual level time-dependent rate equations. In order to find the steady-state solutions to these equations they can be simplified by setting dn =. Taking timedependent effects into accout we set the level population solutions to dn, as was dt dt done in equations Equilibrium conditions and atomic population models For a system to be in an ionization equilibrium, the system must have relaxed to a state where the ionization processes are balanced by the recombination processes. The balance between competing effects is where the term equilibrium originates. Due

49 28 to the complexity of the many body processes in plasmas, both in the laboratory and in astrophysical systems, assumptions need to be made so that specific problems and questions can be treated. Each of the following equilibrium systems contain assumptions specific the regimes they address (e.g. low density vs. high density systems). It is important to note the descriptions below are only very simple, cursory treatments. For a thorough and detailed explanation of the more subtle intricacies of these models, see references [2, 36, 37, 38, 39]. The relevance of discussing these models relates to the differences between laboratory astrophysical plasmas and astrophysical plasmas in space. Designing and executing a controlled laboratory experiment to understand these space plasmas is a non-trivial task. Therefore, in order to understand a plasma that is unable to be probed directly, assumptions are made so that observations can be modeled, and laboratory experiments can mimick astrophysical situations Thermodynamic equilibrium A very simplified method of describing a plasma, in terms of using the most assumptions to constrain a plasma, goes back to the first laws and principles of thermodynamics [4, 4, 42]. This is done by looking at a distribution of electrons and how they are spread across various electron states. For more information on how this is done, refer to the references or section A.. that discusses the Boltzmann equation. Taking thermodynamic equilibrium as being the most general case for our intents and purposes the qualifications for a state to be in a strict thermodynamic equilibrium are as follows: The radiation field is characterized by a Planckian distribution. The free electron distribution follows either the Maxwell-Boltzmann distribution

50 29 or the Fermi-Dirac distribution. The level and ionic populations follow the Boltzmann and Saha equations, respectively. The principle of detailed balance is satisfied, meaning that the rate every atomic process is directly balanced by the rate of its inverse process Local thermodynamic equilibrium Local thermodynamic equilibrium (LTE) can be defined by following the conditions of thermodynamic equilibrium (TE) being held over a localized section of the plasma in time and space. For the strict purposes discussed here, LTE will be defined similarly to the condition of thermodynamic equilibrium, with the exception that the radiation field cannot be characterized by a Planckian distribution Non-local thermodynamic equilibrium The next equilibrium state in the progression from TE to LTE is non-local thermodynamic equilibrium (NLTE), or sometimes called kinetic equilibrium [39]. This is the most general, in this list, in terms of using the least assumptions to describe the plasma. In this case, the radiation field is described by the solution to the radiative transfer equation by calculating the source function. Ionic and level populations are found by solving the rate matrix, see section The only assumption that these different models share is that the free electron energy distribution falls into a Maxwell-Boltzmann distribution.

51 Coronal Equilibrium This is a special case that was introduced, historically, to explain the solar corona. Here the plasma needs to be at a low enough density so that the collisional processes are very weak compared to the radiative processes. This is contrary to the models like TE and LTE where collisional processes are balanced by their inverse processes. This sets the environment so that depopulations of atomic levels are dominated by radiative decay rather than any collisional processes. Another important condition of this model is that the radiation is assumed to escape the plasma without interacting much with the plasma; this is known as the optically thin approximation. This is important because, for astrophysical plasmas, this is often the case due to the extremely low densities of space, whereas laboratory densities are much higher. It is also assumed that the free electrons follow a Maxwellian distribution. Another assumption is that the majority of level populations are held mainly in the ground states of the ions. The populations of the excited levels are determined by a balance between the rate of electron collisional ionization and the rate of radiative recombination [2]. Similarily, the equilibrium condition for a collisional plasma is n A +xn e σ CI f v vdv = n A +(x+)n e σ RD f v vdv (2.8) Since by definition collisions are negligible and the plasma is optically thin, it holds that the majority of the level populations lay in the ground states Collisional-Radiative equilibrium The coronal model significantly restricts collisional processes. One attempt to build upon this and put more weight upon these collisional processes is called the collisional-

52 3 radiative (CR) model. The CR model expands the coronal model by including electron collision processes that cause transitions between upper levels and three-body recombination. The assumptions held by this model are that, like the coronal approximation, the free electrons follow a Maxwellian distribution. Unlike the coronal approximation, ionization and level populations are dependent electron collisions. Like in the coronal model, the inclusion of the collisional processes also takes into account radiation emitted when an electron makes a transition to a lower level, either from an upper bound state or from the continuum. Models have been expanded to make this either optically thin or optically thick. In the case of an optically thick plasma, any photon emitted is reabsorbed within the plasma. The model calculation results shown in later Chapters for Helios-CR [29] employ the collisional radiative model, as it is assumed to be the most complete atomic kinetics model and includes the coronal and LTE model in the limits of low and high densities, respectively Photoionization Equilibrium The specific upward and downward processes of concern here are the bound-free photoionization and photorecombination (also known radiative recombination) transitions. hν +A +x A +(x+) +e [PI & RR] (2.9) e+a +x A +(x+) +2e [CI & 3BR] (2.2) The photoionization equilibrium equation for any two successive stages of ionization Z and Z+ of any element A is: n A +Z ν σ PI N ν dν = n A+(Z+)n e σ R f v vdv (2.2)

53 32 In this equation, n A +Z and n A +(Z+) are the number densities of the two successive stages of ionization, σ PI is the photoionization cross section and σ R is representative of the radiative recombination and dielectronic cross sections, N ν is the number of photons of frequency ν, n e is the electron density and f v is the electron velocity, v, distribution. At photoionization equilibrium the photoionization rate is balanced by the rate of radiative recombination and dielectronic recombination. 2.4 Equations of radiative transfer The equation of radiative transfer are well established and documented by many authors and apply to any situation involving the transfer of photons [38, 43, 44, 45]. They are also important when considering the influence of radiation in a plasma, whether generated internally or externally, in relation to the plasma. Photons become absorbed or re-absorbed within a plasma when the opacity of the plasma is such that a photon is trapped within the plasma before it can escape the plasma. This is often called radiation trapping. The radiative transfer equation describes the absorption and emission of these photons (assuming no scattering). where. di ν ds = α νi ν +j ν (2.22) Specific intensity (I ν ) - Defined as the energy passing through an area in a given time, solid angle and frequency, with units of: I ν = energy (time)(area)(solid angle)(frequency) Absorption coefficient (α ν ) - Defined asthe energy emitted inagiven time, solid

54 33 angle, volume and frequency, with units of: α ν = (distance) Emissivity (j ν ) - A beam traveling a differential distance ds loses intensity representative of the absorbers that the beam passes through. j ν = energy (time)(solid angle)(frequency)(volume) Path length ds - a differential length element along the path of the photon. The absorption coefficient is related to the opacity of the plasma. The opacity is a measure of a plasma s optical depth, τ ν. The optical depth is defined as how far a photon can travel into a plasma before being absorbed. dτ ν = α ν ds (2.23) s τ ν (s) = α ν (s )ds (2.24) s The opacity of the plasma is photon energy dependent, so the photon energy distribution must be considered. To conceptually understand equation 2.22 for a group of photons traveling through a plasma, the amount of energy removed from the group is equal to α ν I ν, and the amount of energy added to the group is equal to j ν, thus describing the rate of change for photon energy, hν. If we now look at the specific case of a plasma of given length L, we can rewrite the radiation transport equation discretized as position z and angle cos(θ) = µ. cos(θ) di ν(z,θ) dz = α ν (z)i ν (z,θ)+j ν (z) (2.25) Using the optical depth, shown in radiative transfer equation version 2.25 and some elaboration we can change the dependence from z to a dependence on τ, with the relation shown in Figure 2.. First, looking at the total optical depth along the

55 34 L= L=L max n } z = max = Figure 2.: An example of slab or plane-parallel geometry, as it relates to the version of the radiation transport equation Where here L is the size of the plasma and the generalized τ is the optical depth of the plasma. beam, through the plasma by integrating the length of the plasma L, along the spatial variable z. z max τ ν z dz = τ ν (2.26) Applying this to equation 2.25 we obtain the radiation transfer equation in terms of optical depth. µα ν di ν dτ ν = α ν I ν +j ν (2.27) To simplify things even more we can introduce the source function S ν which is simply the ratio of the emission coefficient over the absorption coefficient. S ν j ν α ν (2.28)

56 35 This allows us to write the radiative transfer equation as di ν dτ ν = S ν I ν (2.29) Solving this equation we can obtain a solution for the intensity in the plasma by multiplying with the integration factor e τν and integrating. This eventually leads us to I ν (τ ν ) = I ν ()e τν +S ν ( e τν ) (2.3) This tells us that if an initial source of photons I ν () is incident upon a plasma the intensity of these photons will be reduced exponentially by the optical depth of the plasma. The additional source term accounts for any self-emission within the plasma Escape factor Models discussed previously generally make the assumption of being either in an optically thin or an optically thick regime. This is done in order to trap the photons (optically thick) or not to trap (optically thin). This radiation trapping effect can be approximated by an escape factor. Conceptually stated, the escape factor can be defined as the ratio of photons leaving a plasma by the photons generated from within the plasma [46]. This is a way of describing whether or not a photon generated from within a plasma will escape the plasma before being absorbed by the plasma. This becomes very important when in the region between optically thin, where a photon has a high probability of escaping, and optically thick plasmas, where the photon has a high probability of being absorbed. These factors are dependent upon characteristics of the line shape of the emitted photon and upon the geometry of the plasma. To go into more depth on this topic, we can begin by revisiting the solution

57 36 to the radiative transfer equation we found previously and simplify this treatment by ignoring any self emission from within the plasma. We can then rewrite the equation in the form of a transmission probability, T(τ ν ), which describes the probability that a photon will travel a given optical depth before being absorbed. I ν (τ ν ) = I ν ()e τν I ν (τ ν ) I ν () = eτν = T ν Remembering that the optical depth describes the amount of absorption a group of photons has undergone while traveling through a plasma, we can now investigate the more intricate mechanisms of the optical depth term. Stated in general terms the optical depth is dependent on the atomic physics of the plasma, the number of particles N within the plasma and the physical length R of the plasma. We now define the optical depth in a more formal manner. ( τ ν = πe2 f i,j φ ν (i,j) F i )N R (2.3) mc i,j The factor on the left side of the parenthesis are fundamental constants, e being the charge of the electron, m the mass of the electron and c is the speed of light in vacuum. The terms controlled by the summation relate to the atomic structure and line transitions, f i,j is the absorption oscillator strength of a transition from lower level itoupper level j. φ (i,j) ν is the area-normalized line shape of the absorption feature centered at frequency ν of the transition from level i to j. Finally F i is the fractional population. The last terms on the right, outside of the parenthesis are the same as mentioned previously and relate to the environment conditions of the plasma. It is also important to note that the line shape φ is normalized such that φ ν dν = (2.32)

58 37 If we now look back at the transmission probability and average the optical depth at line center(τ ) by the spread of all frequencies of the line profile we come up with T(τ ) = e τ φ νφ φ ν dν (2.33) From here we can treat the photons whose frequencies deviate from the center frequency of the line shape. If a photon has a frequency matching that of the center of the line profile it will have a higher chance of being absorbed than photons whose frequencies lie in the wings of the line shape. This conversely means that the photons with frequencies from the wings of the line profile have a small optical depth and can travel longer through the plasma without being absorbed. This leads to the discussion of two different standardized line shapes: Gaussian and Lorentzian. These two line shapes represent two extreme cases. In a Gaussian profile the area around line center holds most of the weight where the profile from a Lorentz holds the weight in the wings. In terms of photons, if a photon originates with frequencies from around line center they are readily absorbed and the transmission for this profile is low, while photons more characteristic of a Lorentz profile have a higher level of transmission. For large optical depths (e.g. τ 3), and using equation 2.33 yields the following expressions for transmission with purely Gaussian (T G (τ )) or Lorenztian line shapes (T L (τ )) [47, 48]. T G (τ ) = τ (πln(τ )) /2 (2.34) T L (τ ) = (πτ ) /2 (2.35) These factors show that the Lorentz profile has a factor of (τ ln(τ )) /2 greater than that of the Gaussian profile. Many photons in a laboratory astrophysics experiment have a wide array of

59 38. Transmission probability. e-6 e-8 e- e-2 e-4 e τ Figure 2.2: The transmission probability contributions for the two different line shapes listed in equations For example, at an optical depth of τ = 3 gives T L T G. Also note the asymptotic behavior of the natural logarithm in the Gaussian term explaining the discontinuity. T G TL optical depths due to different conditions where neither Gaussian nor Lorentz profiles are accurate. In these cases the Voigt profile, which is the convolution of the Gaussian and Lorentzian line shapes, becomes appropriate [49]. Another point of note when considering line shapes in experiments that generate strong magnetic fields is the possibility of line broadening due to Zeeman splitting. This energy level shifting due to the magnetic field is known as the Zeeman effect [5, 5]. In order for this broadening mechanism to be considered it would need to broaden the experimental line shapes on an order comparable to the Doppler broadening. In order for this to occur the magnetic field in the gas cell would need to be greater than 6 Gauss, which is not reached in our experiments.

60 39 Chapter 3 Z-machine photoionization experiment 3. Purpose and scope of photoionized plasma experiment The primary aim of the Z-machine photoionized plasma experiment was to form a uniform plasma column and then use x-ray spectroscopic measurements to diagnose the plasma s conditions. The gas cell target s design was made to allow the largest amount of flux possible while maintaining spatial uniformity. The selection of the gas cell location was determined as to maximize the intensity of the flux of photons from the source; placing the gas cell as close as possible to the wire array load and return current canister while not interfering with the load hardware. The diagnostics used were chosen to provide spectroscopic information so that the plasma could be analyzed and experimental variables could be extracted and tested against simulation work. Time-resolved and time-integrated x-ray spectroscopy was used to interrogate the wavelengths of the plasma that could be used to accurately diagnose the temper-

61 4 ature and density of the plasma where spectral features were believed to occur. The generation of these spectral features rely on the interaction of the gas held within the gas cell and incoming photons generated by the z-pinch source. These spectral features, from which one can infer level populations and charge state distributions, can then be used to test the accuracy of atomic kinetic simulations. By testing the accuracy of these simulations, we can begin to build a physics model of the plasma which reflects the experimental conditions as closely as possible. 3.2 Experimental design and diagnostics 3.2. Z-machine The experiment was performed using the Z-Machine pulsed power driver at the Sandia National Laboratory. This is a large scale facility based on Marx-bank capacitor technology. Marx-banks are characterized by charging a capacitor bank in parallel and then discharging the bank in series and was originally proposed by Erwin Otto Marx in 924. These ideas relate back to fundamental principles of electrical circuits. The facility, commonly referred to its moniker, the Z-machine, has undergone many changes and upgrades since its initial inception in 98. Initially, the facility was named PBFA-I (Particle Beam Fusion Accelerator-I) [52]. This was then upgraded to PBFA-II [53] in 985. Eleven years later in 996 PBFA-II made the transition and was upgrade into PBFA-Z or initial Z-machine [54]. The Z-machine then went through its most up to date refurbishment, (ZR) [55, 56]. This latest refurbishment is the configuration of the machine, in which, our experiments took place. With such large amounts of energy stored by the Z-machine (as discussed in

62 4 Figure 3.: The arcs and sparks photo taken during a shot on the Z-machine where the extent of electricity used can be seen. Courtesy, Sandia National Laboratories. Chapter ) impressive images like that in Figure 3. can be taken. To visualize the amount of energy released in these experiments, this picture shows only the static bleed off. This break down is not contained by the insulated transmission lines guiding the electricity to the circuit load. The Z-machine, however, is only used as a source of energetic x-rays that are used to drive the experiment. The gas cell containing the neon gas to be photoionized is placed close to the source of the x-rays. The diagram in Figure 3.2 shows an example of the basic geometry of the setup. Also shown in Figure 3.2 the return current can slots are effective apertures ( x 3 mm) along the lines of site of the diagnostics. The line of site to the spectrometer viewing the gas cell is offset by a twelve degree angle from the bottom center of the z-pinch axis. The gas cell viewing windows are then centered along this line of site axis. Figures3.3and3.4arepicturesshowingtheplacementofthegascellinrelationto the load hardware. The slots of the return current cannister can be seen in Figure 3.3.

63 42 Figure 3.2: Schematic of the experimental setup These pictures also show an evolution of design hardware of the gas cells. One example of this can be seen by looking at the baseplate in Figure 3.3. It can be seen how the width expands behind the gas cell where this width expansion in Figure 3.4 has been removed. Figure 3.5 shows a more recent side view of the experimental setup, where directly behind the gas cell, attached to the base plate, is a rear horizontal aperture. This aperture is meant to limit and mitigate the view of any spatial variations from within the neon plasma in the gas cell. The transducer can be seen on the top of the gas cell in Figures 3.4 and 3.5 with the conformable RG-45 coaxial cables taped with copper tape to the gas cell fill lines. The bundles of copper cylinders grouped together are the weights attached to the ends of the tungsten wires that are used for the wire array z-pinch driver. As stated before, this uses a double nested tungsten cylindrical wire array of 2 wires in the inner cylinder and 24 wires in the outer cylinder. With these wire array configurations, mounting all external experimental hardware is a very sensitive and painstaking process that is completed with delicacy and patience. Figures 3.3, 3.4, and 3.5 also show the other diagnostics and ride along experiments that can be fielded on a shot. Directly to the left and right of the gas

64 43 cell are the b-dot diagnostics; and opposite of the neon gas cell, other hardware can be seen directed towards the central axis. Figure 3.3: The side on view of a experimental setup that used an older version of the gas cell and gas cell base. This is also a pre-transducer experiment. Courtesy, Sandia National Laboratories

65 44 Figure 3.4: Looking top down on the experimental setup the tungsten wires of the z-pinch are more clearly visible. The neon gas cell seen on the right has the transducer lines and gas fill lines leaving off to the right. Courtesy, Sandia National Laboratories Figure 3.5: Another side view of the experiment showing the gas cell in relation to the surrounding hardware with the rear aperture in place. Courtesy, Sandia National Laboratories

66 Gas cell specifications The gas design has undergone several iterations of improvements and modifications, from interfacing with the load hardware of the Z-machine to individual components of the gas cell. The gas cell consists of a main stainless steel rectangular body with dimensions of. in (±.5 in) x.8 in (±.5 in) x.453 in (±.5 in) with a rectangular hollowed out portion with dimensions.8 in (±.5 in) x.43 in (±.5 in) to allow for filling with a neon gas. The gas cell body, on each open face, has an encircling O-ring groove at the edge of the hollowed out region for use to seal the gas cell window frame plate to the main gas cell body. These O-rings are Viton.8 ID x.87 OD, where here ID is the inner diameter and OD is the outer diameter of the O-ring. All O-rings were ordered from Precision Associates Inc. On these same faces of the gas cell and parallel to the outside edges and.9 in ±.5 in from this same outside edge are screw holes for mounting the the window frame plate. On the bottom of the main gas cell body are grooved tabs, one on each side, used for attaching to the baseplate fixture. On the bottom of the gas cell are two holes surrounded by an oval shaped o-ring groove. These two holes connect to two holes on the base plate that are at the end of the gas fill line. These holes allow for gas to enter into the gas cell. The O-ring groove fits a.246 ID x.382 OD Viton O-ring that acts to seal the interface of the baseplate and the main gas cell body. The top of the gas cell has a hole that allows a Swagelok fitting to be welded. This fitting accommodates a nut that allows the pressure transducer to be attached. The pressure sensor used comes from the PX72 series sold by Omega Engineering Inc. The sensors from the PX72 series come in ranges of sensitivities. The most frequently used model was the PX72-.5GV, which is a differential pressure transducer with a specific range

67 46 of to.5 PSI which converts to approximately to 8 Torr of fill pressure. The window frame plates of the gas cell have also undergone many iterations to improve the reliability and reproducibility of the experiment. These plates originally held a.5 micron mylar (C H 8 O 4 ) a sheet in place over the hollowed out opening of the main gas cell body. This was then modified for the use of Silicon Nitride (Si 3 N 4 ) windows. The Silicon Nitride windows only had a 5 x 5 mm opening for use in transmission measurements. This made it so that a rectangular stainless plate can be used to cover the hollowed out region and only have a 5 x 5 mm square hole machined out where the window would then be attached. These different windows will be discussed in more detail in section Figure 3.6: An older design of the gas cell hardware. The dimensions of the gas cell itself remain unchanged while mounting (baseplate) and viewing (windows) hardware are continually updated for reliability and reproducibility. If Figure 3.5 is inspected more closely more detail can be seen as in Figure 3.7. This gas cell used large mylar windows encompassing the height and width of the gas fill area. This led to the need for a rear aperture to reduce any spatial variation of the neon gas. The transducer can be seen on top of the Swagelok fitting that is on top of

68 47 the gas cell. This small cylinder on top of the fitting has the electrical circuit pins and are connected to the three coaxial cables seen approaching the cylindrical transducer. Directly below the red line of sight arrow are the two gas fill lines that feed directly to the baseplate allowing gas to flow in through one line traveling through the baseplate into the gas cell and then another line leaving the gas cell through the baseplate to the second gas fill line. Figure 3.7: A zoomed in picture of the gas cell with aperture. The arrow shows both the line of sight to the spectrometer which is the path the photons take traveling from the radiating z-pinch through the neon gas. Courtesy, Sandia National Laboratories Finally, infigure3.8we canseethe most updatedversion ofthegascell thatuses 5 nanometer thick silicon nitride windows. The updated window plates are designed to accommodate the delicate windows. These thicker windows were a necessity when screwing the window plate to the gas cell. With a thin window of these dimensions just the act of screwing the window plate to the gas cell body would induce stress through the window plate that would be transferred to the thin windows. With thicknesses of only 5 nm this made the windows very susceptible to breaking from

69 48 this stress. Using thicker windows plates does not completely reduce the risk for the breakage of windows due to assembly stress but does reduce the risk greatly. The white seen ontheedges ofthewindow ishysol R loctitetwo partepoxy used toadhere the windows to the metal window plate. Figure 3.8: An extreme close-up of the gas cell body and rear aperture, with the body using the silicon nitride windows Gas cell testing station Due to the expense and limited availability of experimental shots that are available, assembled gas cells need to be tested before shots to minimize the risk of faulty hardware. This was accomplished with the setup of a testing station. Originally set up by Dr. Iain M. Hall, a former group member, for the singular purpose of testing the neon gas cell, this station has been expanded to test similar gas cell designs of other experiments. The testing is a simple vacuum chamber evacuated with the gas cell inside and all lines of the gas cell also open to vacuum. The chamber was pumped down to the limit of the turbo pumps (approximately 2x 5 Torr). Since the gas

70 49 cell inside was also open to the vacuum, the inside of the gas cell pumped down at the same rate as the bulk of the chamber volume. This was crucial so that stress on the delicate windows could be minimized. Once the chamber and gas cell were at the zero level of vacuum, the gas cell lines were sealed off from the vacuum chamber. This then allowed for the gas cell to be filled with gas. Testing whether or not the gas cell had a leak happened in two ways. The first method was maintaining the vacuum of the chamber; if there was a leak in the cell that gas would immediately leave the cell and enter the vacuum chamber. This would be recognized by the cold ion vacuum gauges used to monitor the 5 Torr vacuum levels that would immediately be compromised by the leaking gas cell. The second case is needed when the gas cell does not have an immediate leak or if the leak is small of enough to not be noticed immediately. The gas cell can be filled to the target capacity of the shot or in some cases over the target capacity of the shot in order to over test the gas cell. To fill the gas cell to the appropriate level, the pressure transducer was used to monitor and attain the appropriate fill pressure. Once this pressure had been met the pressurized (pressurized due to the gas fill nature of the cell but still under standard atmospheric pressure ( 76 Torr) level it would be left and monitored. Any drop in pressure could be monitored and recorded by the associated drop in voltage produced by the transducer. This also let us obtain a leak rate of the gas cell. If another gas cell was not available for the shot this leak rate could then be used to determine the amount of gas in the gas cell if it needed to be used during a shot with a longer pre-shot wait time.

71 Gas cell fill station The purpose of the fill station is to fill the gas cell with the requested gas prior to the shot and maintain the fill pressure up until the time of the shot. This pressure station is maintained and operated by SNL technologists. The station uses feed lines to attach to the gas cell baseplate fill lines. The gas cell is filled after the vacuum chamber of the Z-machine has reached vacuum. Once appropriate shot vacuum levels have been reached, the gas fill station on the Z-machine can begin its filling procedure. The gas cell has been tested on a system that is very similar, only different in scaling (Z-machine vacuum chamber 6 m 3, testing vacuum chamber.3 m 3 ), as such the probability for leaks compromising the Z-machine s vacuum are low. The gas fill station is then used to fill to the requested fill pressure. This is monitored in real time by the pressure transducer atop the gas cell and by the diagnostics of the fill station. The fill station diagnostics are not monitored at the moment right before the shot, but the transducer attached to the gas cell is. This gives two complementary measurement diagnostics that add reliability to the fill pressure measurement.

72 Gas cell window transmissions In order for the x-rays from the radiating z-pinch to reach the neon gas in the gas cell they need a path to travel that isn t impeded by the stainless steel cell encasing the neon. This was accomplished by using x-ray transmissive windows whose attenuation of the photons can be minimized. Initial experiments used.5 micron Mylar windows. The transmission for these windows in a z-pinch environment were simulated as a function of time using the modelling code LASNEX by Dr. Tom Nash, of SNL. This Mylar response is shown in Figure 3.9. Here it can be seen that the Mylar remains very opaque to the incident radiation for the first half of the radiation drive. Only then does it begin to allow radiation to reach the neon. The peak of the radiation drive is denoted by when the total integrated power of the radiation drive peaks. Seeing the possibility to improve upon the transmission allowed by the Mylar windows, the change was made to use Silicon Nitride windows (Si 3 N 4 ). These windows are produced and sold by Norcada, Inc. of Edmonton, Alberta, Canada. These windows are available in varying thicknesses and sizes. For our experiment we used windows that are 5 x 5 mm square. Thickness vary depending on the fill pressure of the gas cell. For fill pressures of 5 Torr and below 5 nm windows were used. While for a fill pressure of 3 Torr 75 nm windows were used. LASNEX simulations were also done for the transmission response for these windows also and are shown in Figure 3.. As can be seen in the following Figures, these windows are transmissive from the earliest times of the simulation, reaching 8 % transmission before the radiation drive has reached its halfway point. It then fluctuates around this 8 percent level through the rest of the drive. This is important because the main x-rays that drive the inner shell transitions of hydrogen-, helium- and lithium-like ions occur ten to

73 52 Table 3.: Window material characteristics Mylar(.5 µm) Silicon Nitride(5 nm) Density [g/cm 3 ] Areal density [g/cm 2 ] 2e-4 2e-5 Total mass [µg] twenty nanoseconds before the peak of the radiation drive. While the transmission is simulated by the model calculations the input for the power seen in the plots are taken from experimental measurements using x-ray diodes (XRD) and bolometer measurements [57, 58, 59, 6, 6, 62, 63, 64]. These XRD and bolometers signals can be combined in such a way that the power can be ascertained. V XRD Power = Yield[J] XRD Vdt (3.) In this simple equation the yield is a function of the bolometer signal and the characteristics of the bolometer like the thickness, density, and view factor to name a few, where the V s in the equation refer to voltages output by the devices. The bolometer is used to find a total amount of energy deposited on it. The XRD is sensitive to a specific bandwidth of x-rays and returns a signal that is sensitive to these specific responses. Combining all these factors and the power as a function of time, the power of the source can be measured andthen used in model calculations in order to connect experiments to simulations. This lessens the disconnect seen between experiment and theory and gives validity to using the information found from these simulations. As seen in the following Figures we can estimate, with some confidence, the transmission through our windows heated by the same driver we use in our experiments.

74 53 Figure 3.9: Transmission of Mylar window, computed with LASNEX, shown in red, when subjected to radiation from the Z-machine whose power as a function of time is shown in blue. Courtesy of Dr. Tom Nash. Figure 3.: Transmission of 5 nm Silicon Nitride windows, computed with LASNEX, shown in red, when subjected to radiation from Z-machine whose power, as a function of time, is shown in blue. Courtesy of Dr. Tom Nash.

75 54 Taking a step back and looking at the materials in room conditions we can look at the response of the material if they were in standard atmospheric solid density conditions. This can be done using the Center for X-Ray Optics databases (CXRO [2]) as shown in Figure 3.. If first we look at two different thicknesses of Mylar we can see that these seemingly small changes in thickness show a large sensitivity to the transmission of the material. If, for example, we look at the lowest photon energy of 8 ev we can see that increasing the thickness of the mylar by half a micron, or 5 nanometers, drops the transmission by approximately 5 percent. Whereas with the silicon nitride the transmission is a nice steady level above 9 percent for the whole range of photons energies. Transmission Mylar 2 um.2 Silicon Nitride. Mylar.5 um hv (ev) Figure 3.: Transmission calculations made for.5 (green) and 2. (blue) micron thick mylar and also for 5 nanometers of silicone nitride all at a constant solid density [2].

76 Transducer diagnostic and pressure measurements Using a pressure transducer in situ on the neon gas cell to make real time monitoring possible was established, previously by Drs. Iain M. Hall and Roberto C. Mancini [2, 65, 66, 22, 67] and was then expanded upon by Dr. Ross Falcon [68]. The cm neon gas cell front window is located at 4.3 or 5.9 cm away from central axis of the z-pinch. Due to the destructive nature of the experiments, the gas cell and pressure transducer are destroyed during each shot. This led to the need of a small, low cost transducer that could be replaced every shot while still giving accurate measurements. The piezoresistive pressure sensor model discussed in the gas cell specifications works by measuring a change in resistivity due to the deformation of a diaphragm that is having pressure applied to it. The diaphragm in this sensor is a thin silicon membrane withawheatstone bridgemadeofgoldwires etched ontoitssurface. Whenthese thin wires are stressed they deform changing the cross section of the wires which changes the physical properties and hence the resistivity. This leads directly to a change in the output voltage given by the sensor [69, 7]. The pressure transducer is initially attachedandtested during thegasfill testing phaseof construction. Itisat thisphase that the calibrations of transducer are also carried out. The aim of the transducer is to get a measure of how the voltage output by the sensor changes with an applied pressure. When going through all the transducer tests, windows were not used on the gas cells. Instead, blank stainless steel plates were screwed onto the gas cell. Once the gas cell and the vacuum chamber were both at vacuum levels just as during the filling procedure, the gas cell was valved off and sealed from the vacuum chamber. The cell is then filled with the neon gas in steps, recording (with a multimeter) the voltage from the pressure sensor and the pressure from a separate transducer independent

77 56 of the gas cell transducer. The voltages were recorded using a Fluke 89 Digital Multimeter and the pressure by an Omegadyne, Inc. model PXC-2A5T that measures the absolute pressure instead of a differential pressure. The fit lines were done using a Levenberg-Marquardt least-squares minimization [7, 72]. To calibrate these transducers, a few considerations must be made. Firstly, the transducer, in order to operate, needs to be powered by an excitation voltage. Figure 3.2 shows the V P as a function of excitation voltages. The vertical blue line and point in this Figure is the V P given by the manufacturer for a constant excitation voltage of 5 Volts. Many tests were made using different excitation voltages to find the variation in scaling from the linear trend shown in Figure 3.2. Of the different excitation voltages tested the range of standard deviations was. -.4 percent. This allows a large confidence in the deviation based on any variations in excitation voltage. Figure 3.2: The behavior of the transducer signal when powered by different excitation or bias voltages is extremely linear (red). The blue vertical line in the Figure are the values quoted by the manufacturer. Courtesy, Dr. Ross Falcon.

78 57 The next step in calibrating the transducers is to test for differentiations between different individual sensors themselves. Figure 3.3 shows the calibration curves of a single sensor performed 5 times. For each calibration the same excitation voltage of 5 Volts was used. It is shown that the baseline initial voltage changes between calibration but the slopes are all very similar, to the effect of V =.678 mv P Torr, with a standard deviation of σ =.5 mv Torr. Again, even with the visible variations seen, the reproducibility of the slope of the curves is of direct importance and significance to our experiment. Figure 3.3: Figure showing the reproducibility of different transducer using the same bias voltage. Courtesy, Dr. Ross Falcon. With the calibration of the transducers complete they are now ready to be used on an actual Z-machine shot using the mechanisms in section The fill station connects to the gas cell by a long fill line, 9 meters. This introduces the possibility of obstructions and contaminates. Because of this the fill station follows a fill and purge procedure. The gas cell is systematically filled with gas and then the gas cell

79 58 Figure 3.4: Illustration of how the baselines are calculated for finding the fill pressure of the gas cell. This also shows the filling and purging process of the gas cell. Courtesy, Dr. Ross Falcon. hasthefillgasevacuated. Thisallowsanycontaminantstobeflushed fromthesystem and any obstructions to be identified. As shown in Figure 3.4, the graph is split into two frames. The left frame shows the time of the initial fill and purge. The voltage level before the first fill is taken as the initial baseline voltage level V. After all of the fill and purges are complete the gas cell outputs a voltage of the fill level. This is shown in Figure 3.4 in the right frame. This information along with the calibration response of the transducer V P is then combined to ascertain the pressure of gas in the cell at the time of the shot. Fill pressure = V fill V V P (3.2) If a leak should occur or if the gas cell burst at a time close to the time of the shot this real time monitoring of the pressure make this information known. In the case of the leak using the calibration information the exact amount of gas that has left the cell can be calculated and the shot can still yield valuable data even if the target fill pressure was not accessible.

80 X-ray spectrometer characteristics The TREX [73] elliptically bent crystal spectrometer used to record the x-ray measurements on the photoionized plasma experiments was located on line of sight (LOS) 33 on the Sandia Z-machine. This designation is built on the concept of the degrees units of a circle, 33 relating to a 33 degree rotation from a reference line of sight. The line of site that it is placed at is also twelve degrees off axis, see Figure 3.2 and 4 meters away from the gas cell. Leaving the gas cell along the line of sight towards the crystals the photons follow a path encountering the following devices. First, leaving the gas cell, the photons encounter the first horizontal aperture on the baseplate. Beyond this is a large scale debri aperture to protect the line of site of the spectrometer from major debri. After this aperture, the blast shield covering the z-pinch experimental platform. Beyond the blast shield, and beginning at the nose of the spectrometer, are the baffle plates and imaging slits. These slits project a group of spectrally resolved target images onto the detector. This spectrometer uses six 5 micron slits that can image different parts of the pinch either the pinch radius or the pinch axis. Also at this point two of the slits are covered 3.6 microns of Kimfol in order to study the second order photon contributions to the total signal. Beyond these baffles is a large.5 micron window that is used a debri check. After this window are the crystals. The crystals used are potassium acid phthalate (KAP) and, in the present configuration, cover a range of 83 to 2 ev of photon energy (or in wavelength.33 to 5.25 Å). Resolution for the TREX spectrometer can be as low as 5 µm and achieve a spectral resolution of λ/dλ 8. X-rays reflecting from the crystal are then focused through the second focal point of the ellipse and onto the detector. This crossover point is filtered by. microns of aluminized Lexan.

81 6 This gives the. micron Lexan a.2 micron aluminum layer, which serves to prevent undesired x-rays from scattering onto the detector. Figure 3.5: A schematic of the TREX spectrometer showing different possible slit positions along with the positions of crystals and where the detector (X-ray film, MCP) are placed. Courtesy, Sandia National Laboratories Figure 3.6: A real image of the TREX spectrometer being calibrated by a manson x-ray source. Courtesy, Sandia National Laboratories

82 Elliptical crystal spectrometer formulae The TREX spectrometer is built on the equations of diffraction from an elliptically bent crystal [74, 75, 76, 77, 78]. The spectrometer views photons in the line of sight traveling from the source through the neon gas and other filtering mechanisms on then reflecting off two elliptically bent crystals onto a detector. In our cases x-ray film for time-integrated measurements, or to micro-channel plates for time-resolved measurements. If you first consider the positions of rays striking the crystal, we Figure 3.7: Diagram of the geometry setup of the x-rays traveling from the source reflecting of the crystal and then traveling to the detector. can define a parameterized description of positions on the crystal by vector p c. p c (θ) = acosθ î+bsinθ ĵ (3.3) As can be scene from Figure 3.7 this ignores source broadening and both crystal and detector extensions in the z-direction. We then want to define the position of the source given by, s = S x î+s y ĵ (3.4)

83 62 With the position of the source identified and positions on the crystal parameterized we can define the vector from the source to a point on the crystal by r in, r in = p c (θ) s (3.5) = (acosθ S x )î+(bsinθ S y )ĵ (3.6) Taking the dot product between the vector r in and the normalized surface normal vector (n c ) we can find the angle φ n c = cosθ î+sinθ ĵ (3.7) cosφ = r in n c r in n c (3.8) Working through the algebra and making the appropriate substitutions for the various vectors in both the dot product of the numerator and the magnitudes of the denominator we find an equation relating the Bragg angle to the source position and to the position on the crystal surface, cosφ = acosθ(acosθ S x )+bsinθ(bsinθ S y ) (acosθ Sx ) 2 +(bsinθ S y ) 2 a 2 cos 2 θ+b 2 sin 2 θ (3.9) Nowthatwehaveanexpression relatingthesourcetothecrystal we needaexpression relating the crystal to the detector surface. If we consider the geometry set up in Figure 3.8 we can relate the outgoing vector (r out ) by the incoming vector and the surface normal (n ref ). Due to the bisection of this angle, the diffracted ray is then, r out = r in +2n ref (3.) Since we have already found cosφ and r in we can compute n ref and then use these relations to calculate the vector position on the detector r out. cosφ = n ref r in (3.)

84 63 Figure 3.8: Diagram showing the vector leaving the crystal surface as they would fall onto the detector. Without computing the large algebraic form of this equation we can write it in simplified form as, r out = r in +2 p c(θ) p c (θ) r in cosφ (3.2) The final position of the outgoing ray onto the detector (r det ) is given by adding ǫ multiples of r out to the position of the ray leaving the crystal surface. It is also important to remember that ǫ needs to be chosen such that the ray still falls on the finite length of the detector. r det = p c (θ)+ǫr out (3.3) If we take the simplest case of a flat planed detector, which is the case in our photoionized neon experiments, then the vector from the point of intersection on the detector to the origin of the plane (p det ) will be perpendicular to the surface normal

85 64 of the plane (n det ), this gives of the ǫ multiple values of interest. (r det p det ) n det = (3.4) (p c +ǫr out p det ) n det = (3.5) ((p c n det +ǫr out n det )p det n det = (3.6) (p det p c ) n det r out n det = ǫ (3.7) This description of the characteristic rays and vectors is not always the most obvious and conceptual description of how elliptical geometry is useful when imaging onto a detector. Still using Figure 3.7 and parameterizing the equations in a more familiar form: x = asin(t) y = bcos(t) This in terms of the complete ellipse is, (x) 2 + ( y) 2 = (3.8) a b It is important to note here that the parameter t, above, is not the same as the angle θ. They are related through, θ = arctan(y/x) tanθ = (b/a)cot(t) The eccentricity, ǫ of an ellipse is a description of how far away its shape deviates from a circle and is defined as ǫ = ( b 2 /a 2 ) /2 (3.9)

86 65 If the dot product between the vector from the focus to the crystal and the normal vector of the crystal surface as functions of θ. We can find a relation between the ellipse parameter (t) and the Bragg angle (θ B ), sinθ B = ( ǫ2 ) /2 ( ǫ 2 sin 2 t) /2 (3.2) We can now look at a more conceptual version of the detector plane in relation to the crystal surface. Figure 3.9: Diagram of the geometry setup of the x-rays traveling from the source reflecting of the crystal and then traveling to the detector. Equation 3.9 can now be written in a much simpler manner as functions of the eccentricity. cosφ = (sinθ ǫ)(sinθ ǫ)+( ǫ 2 )cosθcosθ ( ǫsinθ)( ǫsinθ ) (3.2) Here, θ is the ray traveling the center of the crystal. The distance then of a ray off from this central ray on the detector denotes by x is related to the detector-focus distance, as shown in Figure 3.9 by, ( ) /2 x = cos 2 φ (3.22)

87 Review of experimental data The processing and analysis of the x-ray film acquired from the shots was initially carried out by Dr. Iain M. Hall and then continually improved upon by Mr. Dan C. Mayes [3]. A short summary of the basic steps and ideas used in processing the x-ray film will be discussed below. This experimental data is crucial to the understanding of these experiments and is the baseline for the carrying out the modeling effort in the following chapters. It is this experimental data with which we want to develop a confidence and comparison with the simulations X-ray measurements Data from the x-ray film is seen in Figure 3.2. The six dark lines seen are spectrally resolved images of the neon photoionized plasma backlit by the z-pinch radiation. In order to make this raw densitometered image useful for analysis a few steps need to be taken. First each of the dark strips is an image from an individual slit from the spectrometer. Each of these images, though ideally would be perfectly straight, are not. When taking average of sections, it is important that the same positions are being averaged over; this is the cause of the straightening. After this an approximate calibration needs to be made using the spectrometer dispersion. This sets the spectral axis of the data. Next, each piece of film has an associate fog due to the chemical nature of the film processing. This background film density needs to be subtracted. From here, all the points on the film are still in units of film density but we are interested in the level of intensity on the film; so this conversion is carried out. As seen in Figure 3.2 spectral features and/or crystal imperfections and defects can easily be seen. If a well known spectral feature is identified but the initial spectral

88 67 Figure 3.2: An example of the x-ray film developed and then densitometered for use of processing and analysis. dispersion did not line up the features exactly, fine tuning these positions can be made. This can also be a good time to remove any artificial artifacts in the spectra that can be mistaken for real spectral line transitions. From here spectra taken and processed from each of these slits can then be taken and averaged to improve the signal to noise qualities of the final spectra, an example is shown in 3.22.

89 68 Figure 3.2: Inspection of the developed film shows the evidence of spectral features sought after in the experiment. Figure 3.22: A finalized spectra showing different signatures of line transitions from different ion species.

90 69 Chapter 4 Modeling of x-ray radiation drives 4. Introduction The neon photoionized plasma discussed in this manuscript is driven by an external radiation source. In order to accurately model the dynamics of the plasma an external radiation source representative of the drive used in the experiments needs to be well modeled. With a dynamic radiation system, like that produced with the Z- machine at Sandia National Laboratories, modeling this drive is very difficult and so a collection of experimental and simulated data has been combined in order to create a radiation drive that is most representative of the produced by the Z-machine and impinging on the front window of the neon gas. Having an accurate understanding of the radiation drive is important because the conditions of a plasma heated by an external source cannot be understood without it. This is because the distribution of photons produced by the radiation source determine how the atoms in the plasma will emit and/or absorb the light. The level of excitation of these atoms caused by the absorption of the incoming light can be directly linked to a temperature characterization of the plasma. One of the most simplistic characterizations is to assume the system is in a state of of local thermodynamic equilibrium (LTE), though the

91 7 requirements governing what LTE is can change between different communities one common accepted rule is that the processes that populate and depopulate a given energy state occur at equal rates and that the system can be characterized by a single radiation temperature T r. In LTE the number of electrons in adjacent excited states is expressed by the Boltzmann equation (see Appendix A ). The trouble comes when considering an interstellar medium where the conditions of LTE do not usually hold and the relative populations of excited states cannot be described by the Boltzmann equation. The Boltzmann equation can also be expanded to derive the Saha equation (see Appendix A), which can give populations of neighboring ionization states. Though, again, like with the Boltzmann equation, the conditions of an interstellar medium are not compatible with the Saha equation. This makes it very clear that a single temperature may be inappropriate to describe an interstellar gas. The kinetic energy of the particles in the gas (namely free electrons) can be thought to resemble a thermal distribution, namely given by the Maxwell-Boltzmann distribution, which is defined by a temperature of the electrons T e. These temperature descriptors for an LTE plasma can be described as a single temperature where T = T r = T e. This means the electron and photon distributions are in equilibrium. The LTE assumption is powerful when considering the details of upward and downward processes because there is no need to consider the fundamental atomic cross-sections. The difficulties arise when the system is non-lte. This can happen if the system is changing rapidly enough that the electron and/or photon energy distributions do not reach thermal equilibrium (T r T e, or when the plasma is optically thin so the radiation escapes and is not available balancing through re-absorptions of atomic processes or if noncollisional atomic processes become important. When this is the cause, the level populations cannot be found through simple analytic formulas and must be found

92 7 by solving detailed kinetic rate equations. Another temperature parameter to be considered when characterizing an x-ray drive is its brightness temperature T b. This is the temperature characterizing a spectrum described by the Planck function (see Appendix A). The radiation emitted by the Z-machine is not a strictly Planckian distribution and so cannot be described by a single temperature, so a method was needed to characterize the radiation from the pinch in a way that could be applicable to other calculations [79], namely atomic kinetics and radiation hydrodynamics. This led to the idea of fitting the radiation extracted from the Z-machine with three Planckians. Using three well defined Planckian distributions gives a six parameter workspace in which the radiation drive can be analytically characterized for inputing model calculations. 4.2 Z-pinch radiation and re-radiation effects In order to extract a radiation drive from the Z-machine, an effort is needed to be made due to the complexity of the hardware system and the diagnostic available for use. A large contributor to this work is the 3-D view factor code VISRAD [8] from Prism Computational Sciences Inc. (PCS) and the large amount contributions from collaborators [65, 66, 22, 67, 8] VISRAD was developed to model radiation environments like those on the Z-machine. This made it ideal for our purposes. One characteristic of the code that makes it very convenient is its simplicity. It calculates fluxes from a coupled set of steady-state power balance equations. These coupled equations relate the emission and absorption from a surface to the source that is

93 72 heating the surface. B i α i F ij B j = Q i (4.) j For a given surface element i the equation of balance is implemented as following, where B i is the total radiation leaving a surface area i, α i is surface area i s albedo (ratio of the amount of reflected light to the incident light), Q i is the radiation coming from the source and F ij is the configuration factor of the environment. This configuration term is powerful in the fact that the VISRAD code can handle anywhere from a few to several thousand surface elements. The solution of finding this fractional energy leaving a surface i versus that incident on the surface j is, F ij = cosθ i cosθ j da 2πA i A i A j πs 2 i da j (4.2) Here A i and A j are the area of surfaces i and j. θ i and θ j are the angles between the surface and the normal vector of the surface area element da i,j and S is the distance between the two surface elements. Using the parameters of the experiment, as inputs into this code, the configuration environment and the time history of the radiation coming from the pinch was used to find the radiation and re-radiation from all the elements of the hardware system. This gives a more complete understanding of the radiation field than that of just using the hot radiation from the pinch. Since cooler re-radiating hardware still contributes to the x-ray drive in a significant manner. 4.3 Planckian fit algorithm Using the Planck s law for a description of the electromagnetic radiation, B ν, we approximate the radiation impinging on the front surface of the gas cell, F ν, by combining three Planckian energy distributions each with a geometric dilution scaling

94 73 coefficient, a, a 2, and a 3. Three Planckian distribution were because two did not appropriately fit the radiation drive and any more than three did not significantly better the fit. F ν = a B ν (Tr )+a 2 B ν (Tr 2 )+a 3 B ν (Tr 3 ) (4.3) Where Planck s law in terms of frequency and temperature is, B ν (T) = 2hν3 c 2 e hν kt Here h is Planck s constant, and k is Boltzmann s constant. Given Tr, Tr 2, and Tr 3 combine the three Planckian equations and minimize the square of the differences with F νi χ 2 = N [F νi a B ν (Tr )+a 2 B ν (Tr 2 )+a 3 B ν (Tr 3 )] 2 i= In order to minimize the χ 2 with respect to the a, a 2, and a 3 we set, χ 2 a = χ 2 a 2 = χ 2 a 3 = Expressing the partial derivative with respect to a and similarly for a 2 and a 3 requires, χ 2 a = N 2[F νi (a B ν (Tr )+a 2 B ν (Tr 2 )+a 3 B ν (Tr 3 ))] ( B νi (Tr )) i= = 2 N [F νi (a B ν (Tr )+a 2 B ν (Tr 2 )+a 3 B ν (Tr 3 ))] B νi (Tr ) = i=

95 74 Working through the algebra leads to, = N N N N F νi B νi (Tr ) a Bνi 2 a 2 B νi (Tr 2 )B νi (Tr ) a 3 B νi (Tr 3 )B νi (Tr ) i= i= re-writing in a way that will make substitution into matrix form we find, a +a 2 +a 2 N Bνi(Tr 2 ) i= i= N Bνi 2 (Tr 2)Bνi 2 (Tr ) i= N Bνi(Tr 2 3 )Bνi(Tr 2 ) = i= N F νi B νi (Tr ) After applying the same process for a 2 and a 3 we find the analytic solution by solving a 3x3 matrix. Where the elements of the array are defined as follows, i= b d d a e b 2 c 2 d 2 a 2 = e 2 (4.4) b 3 c 3 d 3 a 3 e 3 b = N Bνi(Tr 2 ) c = N B νi (Tr 2 )B νi (Tr ) d = N B νi (Tr 3 )B νi (Tr ) i= i= b 2 = N B νi (Tr )B νi (Tr 2 ) c 2 = N Bνi 2 (Tr 2) d 2 = N B νi (Tr 3 )B νi (Tr 2 ) i= i= b 3 = N B νi (Tr )B νi (Tr 3 ) c 3 = N B νi (Tr 2 )B νi (Tr 3 ) d 3 = N Bνi(Tr 2 3 ) i= i= e = N F νi B νi (Tr ) i= e 2 = N F νi B νi (Tr 2 ) i= e 3 = N F νi B νi (Tr 3 ) i= Solving the 3x3 matrix is now a simple process available by various algebraic methods, giving the analytic solutions for a, a 2, and a 3. As shown in the Figure below, the three Planckian distributions are created with the analytically solved values for a, a 2, and a 3 while the temperature space is searched in order to find the minimum difference of squares or the minimum χ 2 values. i= i= i= i=

96 Planckian Planckian 2 Planckian 3 fit to F ν F ν 6 intensity [J/cm2/s/eV] Figure 4.: An example of fitting the convolution of three Planckian distributions, each of a given radiation temperature and scaling constant, to an arbitrary radiation drive; with the arbitrary radiation drive in red and the fit in green with crosses. 6 % difference Figure 4.2: The percent difference between the fit value and the original value, shown as the absolute value of the difference over the mean.

97 76 As can be seen from the Figure, the fit to the arbitrary F ν using the combination of three Planckian distritubion is very good. Even with a closer inspection at the peak of the distribution at approximately ev the difference between the fit and real values is less than.%. Now that there is a method to fit an arbitrary radiation drive with three Planckian distributinos of a given temperature and photon energy distribution this method can be used to first fit the convolution of the three Planckians to an existing radiation drive and second using the three Planckian parameters to extrapolate or interpolate data points extending these radiation drives in time based on shape trends of existing similar radiation drives.

98 IMH closer position radiation drive fits The first radiation drive that the three Planckian fitting routine was applied to was createdbydr. IainHallforthecaseofaz-pinchimplosionandislabeledasIMH.This radiation drive was modeled using the code VISRAD, simulating the environment on the Z-machine at Sandia National Laboratories. The closer position as listed here relates to a position of 4.3cm from the center z-axis of the pinch and is modeled with radiation and re-radiation from simulated load hardware. This drive also contains experimental information about the power radiated off axis from the center of the pinch and the source size as a function of time of the pinch a few nanoseconds around stagnation time. With the gas cell containing the neon being closer to the pinch, a more limited view of the pinch and surrounding hardware is created when compared to the radiation drive as seen by the gas cell at a farther away position as will be shown after. The plots of the IMH drive in the closer position, figures 4.3, 4.4, 4.5, 4.6, and 4.7 show the three Planckian distributions that when convoluted fit the IMH closer positionradiationdrive ateachstep intime. The changes intime oftherelative intensities of the three drives and the distribution of photon energies of each can be seen from the plots giving an idea of the possible source of each Planckian. Where the cooler Planckian distributions are usually representative of the re-radiating hardware and the hotter Planckian drives are expected to be generated by the much hotter z-pinch. Where hotter and cooler relate to the location of the peak of the photon energy distribution and not the relative intensities or total integrated flux of each drive.

99 78 8 IMH C VISRAD ns Fit to VISRAD 2.5e+6 IMH C VISRAD ns Fit to VISRAD 7 2e+6 6 Intensity [J/cm2/s/eV] Intensity [J/cm2/s/eV].5e+6 e e+6 IMH C VISRAD 2ns Fit to VISRAD 6e+6 IMH C VISRAD 3ns Fit to VISRAD 4.5e+6 4e+6 5e+6 Intensity [J/cm2/s/eV] 3.5e+6 3e+6 2.5e+6 2e+6.5e+6 Intensity [J/cm2/s/eV] 4e+6 3e+6 2e+6 e+6 5 e e+6 IMH C VISRAD 4ns Fit to VISRAD 2e+7 IMH C VISRAD 5ns Fit to VISRAD 8e+6.8e+7 7e+6.6e+7 Intensity [J/cm2/s/eV] 6e+6 5e+6 4e+6 3e+6 2e+6 Intensity [J/cm2/s/eV].4e+7.2e+7 e+7 8e+6 6e+6 4e+6 e+6 2e Figure 4.3: The first time steps (-5 ns) of the IMH radiation drive and its corresponding Planckian drives that were fit, where the pink, blue, and cyan colored traces are the three Planckian fit distributions and the green and red traces are the IMH drive and the convolved fit.

100 79 3e+7 IMH C VISRAD 6ns Fit to VISRAD 6e+7 IMH C VISRAD 7ns Fit to VISRAD 2.5e+7 5e+7 Intensity [J/cm2/s/eV] 2e+7.5e+7 e+7 Intensity [J/cm2/s/eV] 4e+7 3e+7 2e+7 5e+6 e e+7 IMH C VISRAD 8ns Fit to VISRAD e+9 IMH C VISRAD 9ns Fit to VISRAD 6e+7 8e+8 Intensity [J/cm2/s/eV] 5e+7 4e+7 3e+7 2e+7 Intensity [J/cm2/s/eV] 6e+8 4e+8 e+7 2e e+9 IMH C VISRAD 9ns Fit to VISRAD e+9 IMH C VISRAD 92ns Fit to VISRAD 8e+8 8e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e Figure 4.4: Progressing in time (6-9,9-92 ns) it can be seen how the two cooler Planckian s contribution to the lower energies of the photon distribution increase.

101 8 e+9 IMH C VISRAD 93ns Fit to VISRAD e+9 IMH C VISRAD 94ns Fit to VISRAD 8e+8 8e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e e+9 IMH C VISRAD 95ns Fit to VISRAD e+9 IMH C VISRAD 96ns Fit to VISRAD 8e+8 8e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e e+9 IMH C VISRAD 97ns Fit to VISRAD e+9 IMH C VISRAD 98ns Fit to VISRAD 8e+8 8e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e Figure 4.5: Leading up to the peak of the radiation drive (93-98 ns), even though the cooler Planckian peak intensity level approaches that of the hotter Planckian, the overall distribution is still dominated by the high energy tail of the hotter Planckian distribution.

102 8 e+9 IMH C VISRAD 99ns Fit to VISRAD e+9 IMH C VISRAD ns Fit to VISRAD 8e+8 8e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e e+9 IMH C VISRAD ns Fit to VISRAD e+9 IMH C VISRAD 2ns Fit to VISRAD 8e+8 8e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e e+9 IMH C VISRAD 3ns Fit to VISRAD e+9 IMH C VISRAD 4ns Fit to VISRAD 8e+8 8e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e Figure 4.6: Steps leading up to and after the peak radiation drive (99-4 ns).

103 82 e+9 IMH C VISRAD 5ns Fit to VISRAD e+9 IMH C VISRAD 6ns Fit to VISRAD 8e+8 8e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e e+9 IMH C VISRAD 7ns Fit to VISRAD e+9 IMH C VISRAD 8ns Fit to VISRAD 8e+8 8e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e e+9 IMH C VISRAD 9ns Fit to VISRAD e+9 IMH C VISRAD ns Fit to VISRAD 8e+8 8e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 Intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e Figure 4.7: Showing the last six time steps (5- ns) of the radiation drive a steady cooling trend can be seen by the lowering of intensities at each time step.

104 IMH further position radiation drive fits The next series of plots, figures 4.8, 4.9, 4., 4., and 4.2, show the IMH drive calculated at the further position located 5.9cm from the center z-axis of the pinch. The same types of information were used to constrain this calculation as were used on the closer position. The main changed created by the difference in positions of the gas cell, at which the radiation is calculated, is the viewfactor that the window (effectively an aperture) of the gas cell allows. Being located further away from the pinch permits a smaller solid angle of radiation to pass into the gas cell. Both gas cell positions give a full view of the radius of the pinch, so moving the cell further away gives the gas cell a broader view of the re-radiating hardware around the pinch. This is seen in the radiation drives by a shift in the peak of the distribution, moving to the left, towards lower photon energies. This is because the re-radiating hardware emits at a cooler temperature, lending more weight to the lower photon energies of the distribution. The following set of figures show the time evolution of this further away radiation drive and the three Planckian distributions that were used to fit them. With the radiation peaking at ns in time, this smaller solid angle seen in the further position creates a geometrical dilution effect. This can be seen in the smaller intensity scales of this further position.

105 84 35 IMH F VISRAD ns Fit to VISRAD.2e+6 IMH F VISRAD ns Fit to VISRAD 3 e+6 Intensity[J/cm2/s/eV] Intensity[J/cm2/s/eV] hv[ev] hv[ev].8e+6 IMH F VISRAD 2ns Fit to VISRAD 2.5e+6 IMH F VISRAD 3ns Fit to VISRAD.6e+6.4e+6 2e+6 Intensity[J/cm2/s/eV].2e+6 e Intensity[J/cm2/s/eV].5e+6 e hv[ev] hv[ev] 3e+6 IMH F VISRAD 4ns Fit to VISRAD 7e+6 IMH F VISRAD 5ns Fit to VISRAD 2.5e+6 6e+6 Intensity[J/cm2/s/eV] 2e+6.5e+6 e+6 Intensity[J/cm2/s/eV] 5e+6 4e+6 3e+6 2e+6 5 e hv[ev] hv[ev] Figure 4.8: The first time steps (-5 ns) of the IMH radiation drive and its corresponding Planckian drives that were fit, where the pink, blue, and cyan colored traces are the three Planckian fit distributions and the green and red traces are the IMH drive and the convolved fit.

106 85.2e+7 IMH F VISRAD 6ns Fit to VISRAD 3e+7 IMH F VISRAD 7ns Fit to VISRAD e+7 2.5e+7 Intensity[J/cm2/s/eV] 8e+6 6e+6 4e+6 Intensity[J/cm2/s/eV] 2e+7.5e+7 e+7 2e+6 5e hv[ev] hv[ev] 3e+7 IMH F VISRAD 8ns Fit to VISRAD 5e+7 IMH F VISRAD 9ns Fit to VISRAD 4.5e+7 2.5e+7 4e+7 Intensity[J/cm2/s/eV] 2e+7.5e+7 e+7 Intensity[J/cm2/s/eV] 3.5e+7 3e+7 2.5e+7 2e+7.5e+7 5e+6 e+7 5e hv[ev] hv[ev] 6e+7 IMH F VISRAD 9ns Fit to VISRAD 7e+7 IMH F VISRAD 92ns Fit to VISRAD 5e+7 6e+7 Intensity[J/cm2/s/eV] 4e+7 3e+7 2e+7 Intensity[J/cm2/s/eV] 5e+7 4e+7 3e+7 2e+7 e+7 e hv[ev] hv[ev] Figure 4.9: As with the closer position the same trend of the two cooler Planckians in photon energy slowly increase to be comparable intensities to the hotter Planckian (6-9, 9-92 ns).

107 86.2e+8 IMH F VISRAD 93ns Fit to VISRAD.6e+8 IMH F VISRAD 94ns Fit to VISRAD e+8.4e+8.2e+8 Intensity[J/cm2/s/eV] 8e+7 6e+7 4e+7 Intensity[J/cm2/s/eV] e+8 8e+7 6e+7 4e+7 2e+7 2e hv[ev] hv[ev] 2.5e+8 IMH F VISRAD 95ns Fit to VISRAD 3e+8 IMH F VISRAD 96ns Fit to VISRAD 2e+8 2.5e+8 Intensity[J/cm2/s/eV].5e+8 e+8 Intensity[J/cm2/s/eV] 2e+8.5e+8 e+8 5e+7 5e hv[ev] hv[ev] 3.5e+8 IMH F VISRAD 97ns Fit to VISRAD 4e+8 IMH F VISRAD 98ns Fit to VISRAD 3e+8 3.5e+8 2.5e+8 3e+8 Intensity[J/cm2/s/eV] 2e+8.5e+8 e+8 Intensity[J/cm2/s/eV] 2.5e+8 2e+8.5e+8 e+8 5e+7 5e hv[ev] hv[ev] Figure 4.: Though the peak intensity of the blue cooler Planckian surpasses that of the hotter Planckian, the hotter Planckian is still the dominating factor in terms of higher energy photons (93-98 ns).

108 87 4.5e+8 IMH F VISRAD 99ns Fit to VISRAD 4.5e+8 IMH F VISRAD ns Fit to VISRAD 4e+8 4e+8 3.5e+8 3.5e+8 Intensity[J/cm2/s/eV] 3e+8 2.5e+8 2e+8.5e+8 Intensity[J/cm2/s/eV] 3e+8 2.5e+8 2e+8.5e+8 e+8 e+8 5e+7 5e hv[ev] hv[ev] 4.5e+8 IMH F VISRAD ns Fit to VISRAD 4e+8 IMH F VISRAD 2ns Fit to VISRAD 4e+8 3.5e+8 3.5e+8 3e+8 Intensity[J/cm2/s/eV] 3e+8 2.5e+8 2e+8.5e+8 Intensity[J/cm2/s/eV] 2.5e+8 2e+8.5e+8 e+8 e+8 5e+7 5e hv[ev] hv[ev] 3e+8 IMH F VISRAD 3ns Fit to VISRAD 2.5e+8 IMH F VISRAD 4ns Fit to VISRAD 2.5e+8 2e+8 Intensity[J/cm2/s/eV] 2e+8.5e+8 e+8 Intensity[J/cm2/s/eV].5e+8 e+8 5e+7 5e hv[ev] hv[ev] Figure 4.: The time steps around the peak of the radiation drives (99-4 ns).

109 88.8e+8 IMH F VISRAD 5ns Fit to VISRAD.6e+8 IMH F VISRAD 6ns Fit to VISRAD.6e+8.4e+8.4e+8.2e+8 Intensity[J/cm2/s/eV].2e+8 e+8 8e+7 6e+7 Intensity[J/cm2/s/eV] e+8 8e+7 6e+7 4e+7 4e+7 2e+7 2e hv[ev] hv[ev].2e+8 IMH F VISRAD 7ns Fit to VISRAD e+8 IMH F VISRAD 8ns Fit to VISRAD 9e+7 e+8 8e+7 Intensity[J/cm2/s/eV] 8e+7 6e+7 4e+7 Intensity[J/cm2/s/eV] 7e+7 6e+7 5e+7 4e+7 3e+7 2e+7 2e+7 e hv[ev] hv[ev] 9e+7 IMH F VISRAD 9ns Fit to VISRAD 9e+7 IMH F VISRAD ns Fit to VISRAD 8e+7 8e+7 7e+7 7e+7 Intensity[J/cm2/s/eV] 6e+7 5e+7 4e+7 3e+7 Intensity[J/cm2/s/eV] 6e+7 5e+7 4e+7 3e+7 2e+7 2e+7 e+7 e hv[ev] hv[ev] Figure 4.2: Showing the last time steps (5- ns) of the further radiation drive a steady drop in peak intensity can be seen.

110 4.3.3 Extrapolation of GPL closer position radiation drive 89 The methods used by Dr. Hall to model the radiation drive was expanded by Dr. Guillaume Loisel of SNL. In addition to side power and radius measurements, Dr. Loisel included the intensity coming from the pinch as a function of time. These radiation drives use the notation of GPL, GPL2, signifying different shots. These intensity measurements were made on the Sandia Z-machine using a multilayer mirror assembly (MLM) [82]. This extra constraint, characterizing the pinch, add confidence in understanding the radiation used in model calculations. The drawback to this added level of information is that the MLM diagnostic can only take a select number of data points per z-shot/experiment. This leaves, for the data we have available, five and six data points to work with. With a method to characterize arbitrary radiation fluxes by a collection of Planckian distributions, now the points calculated by Dr. Loisel need to be expanded so that full time-dependent calculations can be made. Using the capability of fitting a radiation drive of a given photon energy distribution with three Planckian distributions each of a given temperature and dilution coefficient, each of those temperatures and coefficients can be tracked in time as the radiation drive, overall, either grows or shrinks in time. The following series of figures each take the coefficients from a single Planckian distribution, and, using the information from the IMH drive, extrapolate around the GPL points. To explain this processes I will use, as an example, the processes taken to extrapolate the radiation temperature used by a single Planckian as a function of time. Figure 4.3 starts with plot (a) containing the IMH closer drive and GPL2. Since the points for GPL2 are available around the peak of the drive, this matched against IMH at the peak of the drive. The pinch hardware driving both these shots are different, but it is assumed the z-pinch

111 9 quantitative behaviors will be similar. This means there is a slow ramping stage as the wires of the pinch begin to heat up and travel toward the center axis. Then once the wires become close enough together or they collide with a central axis foam, there will be a ramping up convergence effect where the power increases dramatically. This is one of the motivations for extrapolating information from GPL and GPL2 data points from other experimental shots. Returning to plot (a), there is a dashed and a dotted line. These lines fit the slope of the data over different ranges. The dotted lined fits the point just before the peak and the peak of the radiation drive and the dashed line fits a linear slope to the ramping up stage of the pinch. This slope information is then compared to the same available information on GPL2. Once comparisons are made and all the information is determined copacetic the GPL drive to be extrapolate and the appropriate line is projected onto the first extrapolation reference point. We now move on to Figure 4.4. In plot (a) we see the line from the peak of the drive to the point immediately before passes through the point that is determined as the peak for GPL. Next, the slope of the ramp phase of IMH is projected onto GPL both without the first line slope determined, (b), and within (c). Now that these two line features have been projected onto GPL, the first extrapolation point and those of the ramp stage for GPL can be determined. The time before the ramping stage of the drive is assumed to be very similar in all cases. This is a reasonable assumption due to the colder nature of the radiation emitted at this point of the drive. So once the ramping up stage of the GPL drive has been determined, the next step is to make a smooth transition into the cooler slow rise of the early time radiation drive. This is done, again, by matching slopes so that the drive behaviors are as similar as possible. This is shown in plots (e) and (f). In the Figure 4.5 we take away the slope information and add the newly created points and lines connecting them. The final step is to

112 9 extrapolate the end of the drive behavior. The end behavior of IMH most closely resembles a second order polynomial curve. This was used to fit the end points of the IMH drive, plot (c), and was then projected onto the last two point of the GPL drive. Since no new energy is being fed into the pinch, it is unlikely it will begin to heat up again, as the polynomial would suggest. For this reason, the minimum of the polynomial fit was used as the end point of the late time GPL extrapolation, finalized in plots (e) and (f). With this method there is now a complete time history for the radiation temperature used to fit a single Planckian distribution. The steps used in this method are the same as those used in each of the following Planckian distribution radiation temperatures, both for the remaining closer position and for the further position values. Following the radiation temperature extrapolated value plots of both the closer and further positions fit the a parameters. These are the same a values that are listed in 4.3 and for the closer position are seen in Figure 4.2. In this figure plots (a) and (b) are the a coefficients of known values for IMH and GPL, respectively. Plots (c), (d), and (e) then use a linear fit to the known IMH a coefficients and project this onto the center averaged value of the GPL a coefficients. In like manner are the a coefficients dnd temperature values determined for the GPL drives in Figures

113 92 Blackbody radiation temperature 25 IMH C GPL2 IMH C (99-) IMH C (93-99) 25 IMH C GPL2 GPL2(98-99) 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] (a) (b) 25 IMH C GPL2 IMH C (99-) GPL2(98-99) 25 IMH C GPL C GPL2 GPL2(98-99) GPL C (99) 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] (c) (d) Figure 4.3: Matching trends of IMH drive and a separate GPL drive, GPL2, to give the extrapolated values of GPL the correct trend at the peak turnover of the drive.

114 93 25 IMH C GPL C GPL2 GPL C (99) 25 IMH C GPL C GPL2 IMH C (93-99) GPL C (98) 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] (a) (b) 25 IMH C GPL C GPL2 GPL2(99) GPL C (98) 25 IMH C GPL C GPL2 GPL C 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] (c) (d) 25 IMH C GPL C GPL C IMH C (85)-GPL C (94.5) 25 IMH C GPL C GPL C 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] (e) (f) Figure 4.4: Matching the trend a few nano seconds before the drive is important because it is a region where the temperature experiences sharp increasing rate of change.

115 94 25 IMH C GPL C 25 IMH C GPL C GPL C GPL C 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] (a) (b) 25 IMH C GPL C 25 IMH C GPL C GPL C polynomial fit GPL C polynomial fit 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] (c) (d) 25 IMH C GPL C GPL C polynomial fit 25 IMH C GPL C GPL C 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] (e) (f) Figure 4.5: Projecting the decreasing trend of IMH onto GPL

116 95 Blackbody radiation temperature 2 IMH C GPL2 IMH C (93-98) GPL2(98-99) IMH C GPL C GPL2 GPL2(98-99) GPL C (99) temp [ev] 4 temp [ev] time [ns] time [ns] IMH C GPL C GPL2 IMH C (93-98) GPL C (98) GPL C (99) IMH C GPL C GPL2 GPL C (99) GPL C (98) temp [ev] 4 temp [ev] time [ns] time [ns] IMH C GPL C GPL2 GPL C IMH C GPL C GPL2 GPL C temp [ev] 4 temp [ev] time [ns] time [ns] Figure 4.6: Fitting the ramping stage of the second radiation temperature Planckian

117 96 IMH C GPL C GPL C IMH C (85)-GPL C (94.5) IMH C GPL C GPL C temp [ev] 4 temp [ev] time [ns] time [ns] IMH C GPL C IMH C GPL C GPL C GPL C temp [ev] 4 temp [ev] time [ns] time [ns] IMH C GPL C IMH C GPL C GPL C polynomial fit GPL C polynomial fit temp [ev] 4 temp [ev] time [ns] time [ns] Figure 4.7: Continuing fitting the ramping stage of GPL and beginning fitting the ending points

118 97 IMH C GPL C IMH C GPL C GPL C polynomial fit GPL C polynomial fit temp [ev] 4 temp [ev] time [ns] time [ns] IMH C GPL C IMH C GPL C GPL C temp [ev] 4 temp [ev] time [ns] time [ns] Figure 4.8: Finishing projecting the ending slope onto GPL

119 98 Blackbody radiation temperature IMH C GPL C GPL IMH C GPL C GPL2 IMH C (93-98) GPL C (99-) temp [ev] 25 2 temp [ev] time [ns] time [ns] 45 4 IMH C GPL C GPL2 IMH C (93-98) GPL C (98) 45 4 IMH C GPL C GPL C temp [ev] 25 2 temp [ev] time [ns] time [ns] 45 IMH C GPL C 45 IMH C GPL C GPL C GPL C temp [ev] 25 2 temp [ev] time [ns] time [ns] Figure 4.9: For the final and coolest temperature profile the fitting is very similar to that of IMH only with a steeper peak.

120 IMH C GPL C GPL C 45 4 IMH C GPL C GPL C polynomial fit temp [ev] 25 2 temp [ev] time [ns] time [ns] 45 IMH C GPL C 45 IMH C GPL C 4 GPL C polynomial fit 4 GPL C polynomial fit temp [ev] 25 2 temp [ev] time [ns] time [ns] 45 4 IMH C GPL C GPL C GPL C polynomial fit 45 4 IMH C GPL C GPL C temp [ev] 25 2 temp [ev] time [ns] time [ns] Figure 4.2: Fitting the late in time points again here the temperature levels are very similar simplifying the fit.

121 Closer position a coefficients.25 a a2 a3.6 a a2 a3.2.5 IMH C a coefficient.5. GPL C a coefficient time [ns] time [ns] (a) (b).5 IMH a GPL a f(x) f(x).4 IMH a2 GPL a2 f(x) f(x).4.2 GPL C a coefficient.3.2 GPL C a coefficient time [ns] time [ns] (c) (d).6.5 GPL C a coefficient IMH a3 GPL a3 f(x) f(x) time [ns] (e) Figure 4.2: IMH and GPL closer position a coefficient values and fits to values.

122 4.3.4 Extrapolation of GPL further position radiation drive Blackbody radiation temperature 25 IMH F GPL2 GPL2(98-99) IMH F (94-99) 25 IMH F GPL F GPL2 GPL2(98-99) IMH F (94-99) 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] 25 IMH F GPL F GPL2 GPL2(98-99) IMH F (94-99) (GPL2-GPL(99)) GPL F (98) 25 IMH F GPL F GPL F (98) IMH F (94-99) GPL F (98) 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] 25 IMH F GPL F GPL F (98) IMH F (94-99) GPL F (98) (IMH F -GPL F (98)) 25 IMH F GPL F GPL F (IMH F -GPL F (98)) 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] Figure 4.22: The further away position fitting the temperature profiles follow similar trends as was done for the closer position, using the second GPL drive and IMH around the peak and then find the slope of the ramping up stage.

123 2 25 IMH F GPL F GPL F 25 IMH F GPL F GPL F poly fit(imh F ) 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] 25 IMH F GPL F 25 IMH F GPL F GPL F poly fit(imh F ) poly fit(gpl F ) GPL F poly fit(gpl F ) 2 2 temp [ev] 5 temp [ev] time [ns] time [ns] 25 IMH F GPL F GPL F 2 temp [ev] time [ns] Figure 4.23: Fitting the late in time points of GPL with the polynomial fit of IMH

124 3 Blackbody radiation temperature 2 IMH F GPL2 GPL2(98-99) IMH F (93-98) IMH F GPL F GPL2 GPL2(98-99) IMH F (94-99) temp [ev] 4 temp [ev] time [ns] time [ns] 8 IMH F GPL F GPL2 GPL2(98-99) IMH F (94-99) (GPL2-GPL F (99)) GPL F (98) 8 IMH F GPL F GPL2 IMH F (94-99) (GPL2-GPL F (99)) GPL F (98) 6 6 temp [ev] 4 temp [ev] time [ns] time [ns] IMH F GPL F GPL2 IMH F (94-99) (IMH F -GPL F (98)) GPL F IMH F GPL F (IMH F -GPL F (98)) temp [ev] 4 temp [ev] time [ns] time [ns] Figure 4.24: The second radiation temperature profile of the further away position fit using trends of second GPL drive and IMH in the further position.

125 4 IMH F GPL F IMH F GPL F temp [ev] 4 temp [ev] time [ns] time [ns] IMH F GPL F IMH F GPL F GPL F poly fit(imh) GPL F poly fit(imh F ) poly fit(gpl F ) temp [ev] 4 temp [ev] time [ns] time [ns] IMH F GPL F GPL F poly fit(gpl F ) IMH F GPL F GPL F temp [ev] 4 temp [ev] time [ns] time [ns] Figure 4.25: Polynomial fit of IMH further position used to extrapolate the late in time points of GPL.

126 5 Blackbody radiation temperature 3 5 IMH F GPL2 GPL2(98-99) IMH F (93-98) 5 IMH F GPL F GPL2 GPL F IMH F (93-98) GPL F (98) temp [ev] 2 temp [ev] time [ns] time [ns] 5 IMH F GPL F GPL2 GPL F GPL F (98) 5 IMH F GPL F GPL F GPL F (98) temp [ev] 2 temp [ev] time [ns] time [ns] 5 IMH F GPL F GPL F 5 IMH F GPL F GPL F 4 4 temp [ev] 3 2 temp [ev] time [ns] time [ns] Figure 4.26: The trends of the third and coolest temperature profile follow similar trends of a slow linear increase followed by a quick ramping up until the peak is reach and almost symmetric decrease.

127 6 5 IMH F GPL F 5 IMH F GPL F GPL F poly fit(imh F ) GPL F poly fit(imh F ) poly fit(gpl F ) temp [ev] 2 temp [ev] time [ns] time [ns] 5 IMH F GPL F GPL F poly fit(gpl F ) 5 IMH F GPL F GPL F 4 4 temp [ev] 3 2 temp [ev] time [ns] time [ns] 5 IMH F GPL F GPL F 4 temp [ev] time [ns] Figure 4.27: Matching ending point trends of IMH further to GPL further.

128 7 Further position a coefficients.9.8 a a2 a a a2 a3.7.4 IMH F a coefficient GPL F a coefficient time [ns] time [ns].2.8 IMH F GPL F f(x) f(x).9.8 IMH F GPL F f(x) f(x) a a coefficient.2..8 a 2 coefficient time [ns] time [ns].5 a a 3 coefficient IMH F GPL F f(x) f(x) time [ns] Figure 4.28: IMH and GPL further position a coefficient values and fits to values.

129 8 Starting with a known radiation drive of similar experimental conditions (IMH closer and further drives), the slopes of the radiation drive characterizing the z-pinch heating physics are approximated and then projected onto the known points of the GPL closer and further drives. This is done in four specific regions. ) The region from the start of the drive and leading up to approximately 9 ns. This is a zone where the heating occurs at a relatively slow and linear pace and is characteristic of the beginning ablation and implosion of the z-pinch wire arrays. 2) The second region from approximately 9 ns to the peak is a very rapid heating phase, but can still be linearly approximated. This around the time when the pinch begins its main convergence towards stagnation and in the experiments involving a low density foam core the radiating shock waves are launched. 3) The peak of the drive at approximately ns is determined by looking at peak brightness temperature and total integrated power measurements. 4) The final region occurs after the peak and stagnation of the pinch and any stable heating elements are broken down and begin cooling. This region is best approximated by a polynomial fit, and then for the extrapolated drive, the last time step is determined by the earliest minimum of the polynomial fit between the fits of the three temperatures, as the minimum is the last realistic point of the polynomial fit since there are no reheating mechanisms. The step by step process seen above shows how when a time resolved radiation drive is given, it is possible to expand a similar radiation drive that only contains a few data points. It is important to note the mechanisms and device creating this radiation drive are approximately the same, making the comparison possible. Just as the time history of the single temperature was shown above the same process is used for the other two temperatures and also for the all three a coefficients. What once was only a

130 9 time history of time resolved intensities as a given wavelength is now a time resolved, six parameter, known function. Now any degree of resolution or point on the time axis (x) can be approximated by using a simple linear interpolation scheme between the known parameters (x,y ),(x,y ) where x falls between x and x. y(x) = ( α)y +αy α = x x x x After all this has been accomplished and a radiation drive of sufficient fidelity has been generated by the three Planckian fit technique we can use blackbody relations to find more information. But first lets give some background. The specific energy density u ν can be defined as the energy per unit volume per unit frequency. If this is taken over all solid angles we get, u ν = u ν (Ω)dΩ = c I ν dω (4.5) Defining the mean intensity as the average of the specific intensity over all solid angles, J ν = 4π I ν dω (4.6) Then the total radiation density can be obtained by integrating over all frequencies, u = u ν dν = 4π J ν dν (4.7) c Defining the isotropic Planck function as B ν then, u = 4π B ν (T)dν (4.8) c But, the emergent flux, F, from an isotropically emitting surface is F = I cos θdω (4.9)

131 So that now F = F ν dν = π Bνdν = πb(t) (4.) This relates to the Stefan-Boltzmann law [83, 84] which states F = σt 4 so finally we have a relation giving us the brightness temperature of the planckian distribution. B(T) = σ π T4 (4.) For a more in depth description of these derivations see Mihalas [38]. The Stefan- Boltzmann law gives the total radiated power from a surface, P = σt 4 (4.2) but what is actually measured at a detector is in a given solid angle dω and what is emitted from an effective surface area is actually cosθ area, where θ is the angle between the radiating surface normal and the detector, and is related to the radiation temperature by, P detected dωcosθ area = σ π T4 (4.3) Similarly what is measured in spectral content is the blackbody radiation but divided by π. P ν(detected) dν dωcosθ area = 2hν3 c 2 dν (4.4) e hν kt So finally, the measurement made is not equated to σt 4 but rather to σ π T4. Once the data is acquired it can be turned into a temperature by taking, T 4 = πp σdωcosθ area (4.5) If the data is a spectral cut of a given range we take, πp(ν) σdωcosθ area (4.6)

132 and fit the spectrum to the following equation in order to get the temperature. 2πhν 3 c 2 e hν kt (4.7) Other important things to remember when treating real world examples is that the solid angle dω integrated over the sphere is 4π so dω is simply, area detector /distance 2 detector (4.8) Also, if a version of these equations is without the solid angle, it may differ by π; since solid angle in steradians is dimensionless, it can often times be left out.

133 2 Since the radiation drives are extrapolated from only a few points, it is important a systematic and controlled method is followed to minimize errors. The drives are also generated in order to mimic the specific drives created by z-pinch physics, specifically those that use nested wires arrays on the Z-machine at Sandia National Laboratory. The drives modelled here are then used to calculate the atomic kinetics of a neon gas system close to the source of radiation. The actual experiment uses a stainless steel box with thin silicone nitride windows to allow the radiation to propagate through the gas and give a path to the detector. These windows have been estimated to have an approximately steady attenuation of 8 percent of the incident radiation [85]. The following combination of steps is the process used to generate all the drives created for use in the atomic kinetics calculations. Fit the data points with the 3-Planckian fitting routine Extrapolate radiation temperatures of the fits in time by matching trends to previous similar drives Extrapolate a-coefficients of the Planckian fits for a full time history of all fitting parameters Using the new fit parameters create new photon energy dependent radiation drives for each step in time

134 GPLe closer position radiation drive fits Now that we have created a new full time history photon energy dependent radiation drive at a known location, we can use this information to look at the data we have generated and draw conclusions about the temperatures and behaviors of the x-ray radiation drive. Taking a look at the full time history of our extrapolated radiation drives, wecanseehowthedistributionofthephotonenergychanges intime, asshown in Figures This is most important at the ramping up section around 9 ns where the change in temperature with respect to time increases dramatically. The original points of the radiation drive at 99-3 ns used experimental data comprising of time resolved measurements of the power generated by the pinch, radius of the pinch and intensity along the radiation of the pinch [86]. These constraining measurements give a good description of the pinch as a function of time. Note the scale changes in the intensities of the drives. Also, since we have two different radiation drives at the same known location, we can use these distributions to run atomic kinetics calculations to draw conclusions about the sensitivity and effect the difference have on low density plasmas.

135 4 6 4 ns BB BB2 BB3 GPLe C F ν 5e+6 4.5e+6 ns BB BB2 BB3 GPLe C F ν intensity [J/cm2/s/eV] intensity [J/cm2/s/eV] 4e+6 3.5e+6 3e+6 2.5e+6 2e+6.5e+6 e e+7 3 ns BB BB2 BB3 2.5e+7 4 ns BB BB2 BB3.2e+7 GPLe C F ν 2e+7 GPLe C F ν intensity [J/cm2/s/eV] e+7 8e+6 6e+6 4e+6 intensity [J/cm2/s/eV].5e+7 e+7 5e+6 2e e+7 2.5e+7 5 ns BB BB2 BB3 GPLe C F ν 4.5e+7 4e+7 3.5e+7 6 ns BB BB2 BB3 GPLe C F ν intensity [J/cm2/s/eV] 2e+7.5e+7 e+7 intensity [J/cm2/s/eV] 3e+7 2.5e+7 2e+7.5e+7 e+7 5e+6 5e Figure 4.29: GPLe closer extrapolated radiation drive points with their three Planckian contribution breakdown for time steps.

136 5.4e+8.2e+8 7 ns BB BB2 BB3 GPLe C F ν.6e+8.4e+8 8 ns BB BB2 BB3 GPLe C F ν intensity [J/cm2/s/eV] e+8 8e+7 6e+7 4e+7 intensity [J/cm2/s/eV].2e+8 e+8 8e+7 6e+7 4e+7 2e+7 2e e+8 9 ns BB BB2 BB3 2.5e+8 9 ns BB BB2 BB3 GPLe C F ν GPLe C F ν 2e+8 2e+8 intensity [J/cm2/s/eV].5e+8 e+8 intensity [J/cm2/s/eV].5e+8 e+8 5e+7 5e e+8 9 ns BB BB2 BB3 3e+8 92 ns BB BB2 BB3 GPLe C F ν GPLe C F ν 2.5e+8 2.5e+8 intensity [J/cm2/s/eV] 2e+8.5e+8 e+8 intensity [J/cm2/s/eV] 2e+8.5e+8 e+8 5e+7 5e Figure 4.3: GPLe closer extrapolated radiation drive points with their three Planckian contribution breakdown for time steps.

137 6 4e+8 3.5e+8 93 ns BB BB2 BB3 GPLe C F ν 4.5e+8 4e+8 94 ns BB BB2 BB3 GPLe C F ν 3e+8 3.5e+8 intensity [J/cm2/s/eV] 2.5e+8 2e+8.5e+8 intensity [J/cm2/s/eV] 3e+8 2.5e+8 2e+8.5e+8 e+8 e+8 5e+7 5e e+8 5e+8 95 ns BB BB2 BB3 GPLe C F ν 9e+8 8e+8 7e+8 96 ns BB BB2 BB3 GPLe C F ν intensity [J/cm2/s/eV] 4e+8 3e+8 2e+8 intensity [J/cm2/s/eV] 6e+8 5e+8 4e+8 3e+8 2e+8 e+8 e e+9 97 ns BB BB2 BB3 2.5e+9 98 ns BB BB2 BB3.2e+9 GPLe C F ν 2e+9 GPLe C F ν intensity [J/cm2/s/eV] e+9 8e+8 6e+8 4e+8 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 2e Figure 4.3: GPLe closer extrapolated radiation drive points with their three Planckian contribution breakdown for time steps.

138 7 2.5e+9 99 ns BB BB2 BB3 2.5e+9 ns BB BB2 BB3 GPLe C F ν GPLe C F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e e+9 ns BB BB2 BB3 2.5e+9 2 ns BB BB2 BB3 GPLe C F ν GPLe C F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e e+9 3 ns BB BB2 BB3 2.5e+9 4 ns BB BB2 BB3 GPLe C F ν GPLe C F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e Figure 4.32: GPLe closer extrapolated radiation drive points with their three Planckian contribution breakdown for time steps.

139 8 2.5e+9 5 ns BB BB2 BB3 2.5e+9 6 ns BB BB2 BB3 GPLe C F ν GPLe C F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e Figure 4.33: GPLe closer extrapolated radiation drive points with their three Planckian contribution breakdown for time steps. From these drives we will now look at brightness temperature profile which relates to equations 4.3 and 4.2. Remembering that, F ν dν = σtb 4 = a σtr 4 +a 2σTr2 4 +a 3σTr3 4 T B = (a T 4 r +a 2T 4 r2 +a 3T 4 r3 )/4 We can now see the plots of power versus time and the brightness temperature versus time of the radiation drive in Figures 4.34 and Because the radiation drive is the heating source incident on the neon plasma it is important to understand the different characteristics of this radiation, not only the intensities and photon energy distribution, but also the time scales under which these different features occur.

140 9.2e+2 e+2 total integrated power [J/cm2/s] 8e+ 6e+ 4e+ GPLe C Power GPLe C Power GPLe C Power 2 GPLe C Power 3 2e time [ns] Figure 4.34: The total integrated power per unit time and the breakdown planckian power contributions to the total power. 6 5 brightness temp [ev] GPLe C T B GPLe C T B GPLe C T B 2 GPLe C T B time [ns] Figure 4.35: The brightness temperature profile as a function of time with individual contributions.

141 GPL further position radiation drive extrapolation The radiation drives generated for the further are very similar to those of the closer position; indeed the source itself is exactly the same-the difference comes from the decrease in the intensity of the radiation due to the inverse squared relation, e.g. geometry dilution effect. The radiation drives as seen by the front window at a further away position is also different due to its field of view. Being further away from the source, it has a wider field of view of the hardware surrounding the pinch. This hardware is heated by the main hot source and re-radiates at a cooler temperature. From a farther vantage point more of this cooler re-radiating hardware is seen and contributes to the drive. Similar to the IMH drives, this further GPLe drive has the shift of the distribution toward lower photon energies, as shown in Figures Due to this consistency in positions and distribution behaviors of the drives, the differences these drives have on the atom kinetics can yield valuable well characterized information about the atomic kinetics of these low density laboratory astrophysical plasmas.

142 2 9 ns BB BB2 BB3 GPLe F F ν 9e+6 8e+6 3 ns BB BB2 BB3 GPLe F F ν 8 7e+6 intensity [J/cm2/s/eV] intensity [J/cm2/s/eV] 6e+6 5e+6 4e+6 3e+6 2 2e+6 e e+7.2e+7 4 ns BB BB2 BB3 GPLe F F ν.8e+7.6e+7 5 ns BB BB2 BB3 GPLe F F ν intensity [J/cm2/s/eV] e+7 8e+6 6e+6 4e+6 2e+6 intensity [J/cm2/s/eV].4e+7.2e+7 e+7 8e+6 6e+6 4e+6 2e e+7 6 ns BB BB2 BB3 GPLe F F ν 8e+7 7e+7 7 ns BB BB2 BB3 GPLe F F ν 2e+7 6e+7 intensity [J/cm2/s/eV].5e+7 e+7 intensity [J/cm2/s/eV] 5e+7 4e+7 3e+7 5e+6 2e+7 e Figure 4.36: GPLe further extrapolated radiation drive points with their three Planckian contribution breakdown for time steps to 7 ns.

143 22.2e+8 e+8 8 ns BB BB2 BB3 GPLe F F ν.6e+8.4e+8 9 ns BB BB2 BB3 GPLe F F ν.2e+8 intensity [J/cm2/s/eV] 8e+7 6e+7 4e+7 intensity [J/cm2/s/eV] e+8 8e+7 6e+7 4e+7 2e+7 2e e+8.4e+8 9 ns BB BB2 BB3 GPLe F F ν.8e+8.6e+8 9 ns BB BB2 BB3 GPLe F F ν.2e+8.4e+8 intensity [J/cm2/s/eV] e+8 8e+7 6e+7 intensity [J/cm2/s/eV].2e+8 e+8 8e+7 6e+7 4e+7 4e+7 2e+7 2e e+8.6e+8 92 ns BB BB2 BB3 GPLe F F ν 2e+8.8e+8 93 ns BB BB2 BB3 GPLe F F ν.4e+8.6e+8 intensity [J/cm2/s/eV].2e+8 e+8 8e+7 6e+7 intensity [J/cm2/s/eV].4e+8.2e+8 e+8 8e+7 6e+7 4e+7 4e+7 2e+7 2e Figure 4.37: GPLe further extrapolated radiation drive points with their three Planckian contribution breakdown for time steps 8 to 93 ns.

144 23 2e+8.8e+8 94 ns BB BB2 BB3 GPLe F F ν 2.5e+8 95 ns BB BB2 BB3 GPLe F F ν.6e+8 2e+8 intensity [J/cm2/s/eV].4e+8.2e+8 e+8 8e+7 6e+7 intensity [J/cm2/s/eV].5e+8 e+8 4e+7 5e+7 2e e+8 96 ns BB BB2 BB3 6e+8 97 ns BB BB2 BB3 3e+8 GPLe F F ν 5e+8 GPLe F F ν intensity [J/cm2/s/eV] 2.5e+8 2e+8.5e+8 e+8 intensity [J/cm2/s/eV] 4e+8 3e+8 2e+8 5e+7 e e+9 98 ns BB BB2 BB3.2e+9 99 ns BB BB2 BB3 GPLe F F ν GPLe F F ν e+9 e+9 intensity [J/cm2/s/eV] 8e+8 6e+8 4e+8 intensity [J/cm2/s/eV] 8e+8 6e+8 4e+8 2e+8 2e Figure 4.38: GPLe further extrapolated radiation drive points with their three Planckian contribution breakdown for time steps 94 to 99 ns.

145 24.2e+9 ns BB BB2 BB3.2e+9 ns BB BB2 BB3 GPLe F F ν GPLe F F ν e+9 e+9 intensity [J/cm2/s/eV] 8e+8 6e+8 4e+8 intensity [J/cm2/s/eV] 8e+8 6e+8 4e+8 2e+8 2e e+9 2 ns BB BB2 BB3.2e+9 3 ns BB BB2 BB3 GPLe F F ν GPLe F F ν e+9 e+9 intensity [J/cm2/s/eV] 8e+8 6e+8 4e+8 intensity [J/cm2/s/eV] 8e+8 6e+8 4e+8 2e+8 2e e+9 4 ns BB BB2 BB3.2e+9 5 ns BB BB2 BB3 GPLe F F ν GPLe F F ν e+9 e+9 intensity [J/cm2/s/eV] 8e+8 6e+8 4e+8 intensity [J/cm2/s/eV] 8e+8 6e+8 4e+8 2e+8 2e Figure 4.39: GPLe further extrapolated radiation drive points with their three Planckian contribution breakdown for time steps to 5 ns.

146 Flux comparisons Comparing the overall flux distribution of the closer position versus the further away position, in Figures , it is evident how a further away target sees a less intense distribution of energy. The distribution shape itself does not change; only intensity of the photons themselves drop due to the geometry dilution effect. This trend is seen in all the drives. The difference in the shape of the drives is seen between the IMH and GPLe drives. The IMH drives more heavily weigh high energy photons with those most important for this experiment being the ones higher than 95 ev in energy. This is important for our experiment because this is the ionization potential of helium-like neon ions, and there existence has been experimentally verified. With such a distinct difference between the two drives, one would expect completely different level populations of the neon depending on which drive is being used, which makes using an appropriate radiation drive critical when trying to match experimental level populations. The source of the difference in the shapes of these two drives could be a result of the dynamics of the source which created the radiation [87]. While both use a double nested array of tungsten wires, one of the implosion dynamics of the GPLe source used a low density foam to mitigate instabilities.

147 26 IMH closer vs. IMH further 8 ns IMH C F ν IMH F F ν 6e+6 3 ns IMH C F ν IMH F F ν 7 5e+6 6 intensity [J/cm2/s/eV] intensity [J/cm2/s/eV] 4e+6 3e+6 2e+6 2 e e+6 4 ns IMH C F ν IMH F F ν 2e+7 5 ns IMH C F ν IMH F F ν 8e+6.8e+7 7e+6.6e+7 intensity [J/cm2/s/eV] 6e+6 5e+6 4e+6 3e+6 intensity [J/cm2/s/eV].4e+7.2e+7 e+7 8e+6 6e+6 2e+6 4e+6 e+6 2e e+7 6 ns IMH C F ν IMH F F ν 6e+7 7 ns IMH C F ν IMH F F ν 2.5e+7 5e+7 intensity [J/cm2/s/eV] 2e+7.5e+7 e+7 intensity [J/cm2/s/eV] 4e+7 3e+7 2e+7 5e+6 e Figure 4.4: IMH C vs IMH F flux distributions: to 7 ns.

148 27 7e+7 8 ns IMH C F ν IMH F F ν.4e+8 9 ns IMH C F ν IMH F F ν 6e+7.2e+8 intensity [J/cm2/s/eV] 5e+7 4e+7 3e+7 2e+7 intensity [J/cm2/s/eV] e+8 8e+7 6e+7 4e+7 e+7 2e e+8 9 ns IMH C F ν IMH F F ν.8e+8 92 ns IMH C F ν IMH F F ν.4e+8.6e+8.2e+8.4e+8 intensity [J/cm2/s/eV] e+8 8e+7 6e+7 intensity [J/cm2/s/eV].2e+8 e+8 8e+7 6e+7 4e+7 4e+7 2e+7 2e e+8 93 ns IMH C F ν IMH F F ν 3.5e+8 94 ns IMH C F ν IMH F F ν 2.5e+8 3e+8 intensity [J/cm2/s/eV] 2e+8.5e+8 e+8 intensity [J/cm2/s/eV] 2.5e+8 2e+8.5e+8 e+8 5e+7 5e Figure 4.4: IMH C vs IMH F flux distributions: 8 to 94 ns.

149 28 5e+8 95 ns IMH C F ν IMH F F ν 6e+8 96 ns IMH C F ν IMH F F ν 4.5e+8 4e+8 5e+8 intensity [J/cm2/s/eV] 3.5e+8 3e+8 2.5e+8 2e+8.5e+8 intensity [J/cm2/s/eV] 4e+8 3e+8 2e+8 e+8 e+8 5e e+8 97 ns IMH C F ν IMH F F ν e+9 98 ns IMH C F ν IMH F F ν 7e+8 6e+8 8e+8 intensity [J/cm2/s/eV] 5e+8 4e+8 3e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e+8 e e+9 99 ns IMH C F ν IMH F F ν e+9 ns IMH C F ν IMH F F ν 8e+8 8e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e Figure 4.42: IMH C vs IMH F flux distributions: 95 to ns.

150 29 e+9 ns IMH C F ν IMH F F ν e+9 2 ns IMH C F ν IMH F F ν 8e+8 8e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e e+9 3 ns IMH C F ν IMH F F ν e+9 4 ns IMH C F ν IMH F F ν 8e+8 8e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e e+9 5 ns IMH C F ν IMH F F ν e+9 6 ns IMH C F ν IMH F F ν 8e+8 8e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e Figure 4.43: IMH C vs IMH F flux distributions: to 6 ns.

151 3 GPLe closer vs. GPLe further 6 ns GPLe C F ν GPLe F F ν.4e+7 3 ns GPLe C F ν GPLe F F ν 4.2e+7 intensity [J/cm2/s/eV] intensity [J/cm2/s/eV] e+7 8e+6 6e+6 4e+6 2 2e e+7 4 ns GPLe C F ν GPLe F F ν 3e+7 5 ns GPLe C F ν GPLe F F ν 2e+7 2.5e+7 intensity [J/cm2/s/eV].5e+7 e+7 intensity [J/cm2/s/eV] 2e+7.5e+7 e+7 5e+6 5e e+7 6 ns GPLe C F ν GPLe F F ν.4e+8 7 ns GPLe C F ν GPLe F F ν 4e+7.2e+8 intensity [J/cm2/s/eV] 3.5e+7 3e+7 2.5e+7 2e+7.5e+7 e+7 5e+6 intensity [J/cm2/s/eV] e+8 8e+7 6e+7 4e+7 2e Figure 4.44: GPLe C vs GPLe F flux distributions: to 7 ns.

152 3.6e+8 8 ns GPLe C F ν GPLe F F ν 2.5e+8 9 ns GPLe C F ν GPLe F F ν.4e+8.2e+8 2e+8 intensity [J/cm2/s/eV] e+8 8e+7 6e+7 intensity [J/cm2/s/eV].5e+8 e+8 4e+7 5e+7 2e e+8 9 ns GPLe C F ν GPLe F F ν 3e+8 92 ns GPLe C F ν GPLe F F ν 2.5e+8 2.5e+8 intensity [J/cm2/s/eV] 2e+8.5e+8 e+8 intensity [J/cm2/s/eV] 2e+8.5e+8 e+8 5e+7 5e e+8 93 ns GPLe C F ν GPLe F F ν 4.5e+8 94 ns GPLe C F ν GPLe F F ν 3.5e+8 4e+8 3e+8 3.5e+8 intensity [J/cm2/s/eV] 2.5e+8 2e+8.5e+8 intensity [J/cm2/s/eV] 3e+8 2.5e+8 2e+8.5e+8 e+8 e+8 5e+7 5e Figure 4.45: GPLe C vs GPLe F flux distributions: 8 to 94 ns.

153 32 6e+8 95 ns GPLe C F ν GPLe F F ν 9e+8 96 ns GPLe C F ν GPLe F F ν 8e+8 5e+8 7e+8 intensity [J/cm2/s/eV] 4e+8 3e+8 2e+8 intensity [J/cm2/s/eV] 6e+8 5e+8 4e+8 3e+8 2e+8 e+8 e e+9 97 ns GPLe C F ν GPLe F F ν 2.5e+9 98 ns GPLe C F ν GPLe F F ν.2e+9 2e+9 intensity [J/cm2/s/eV] e+9 8e+8 6e+8 4e+8 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 2e e+9 99 ns GPLe C F ν GPLe F F ν 2.5e+9 ns GPLe C F ν GPLe F F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e Figure 4.46: GPLe C vs GPLe F flux distributions: 95 to ns.

154 33 2.5e+9 ns GPLe C F ν GPLe F F ν 2.5e+9 2 ns GPLe C F ν GPLe F F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e e+9 3 ns GPLe C F ν GPLe F F ν 2.5e+9 4 ns GPLe C F ν GPLe F F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e e+9 5 ns GPLe C F ν GPLe F F ν 2.5e+9 6 ns GPLe C F ν GPLe F F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e Figure 4.47: GPLe C vs GPLe F flux distributions: to 6 ns.

155 34 IMH closer vs. GPLe closer 6 ns IMH C F ν.4e+7 3 ns IMH C F ν GPLe C F ν GPLe C F ν 4.2e+7 intensity [J/cm2/s/eV] intensity [J/cm2/s/eV] e+7 8e+6 6e+6 4e+6 2 2e e+7 4 ns IMH C F ν 3e+7 5 ns IMH C F ν GPLe C F ν GPLe C F ν 2e+7 2.5e+7 intensity [J/cm2/s/eV].5e+7 e+7 intensity [J/cm2/s/eV] 2e+7.5e+7 e+7 5e+6 5e e+7 6 ns IMH C F ν.4e+8 7 ns IMH C F ν GPLe C F ν GPLe C F ν 4e+7.2e+8 intensity [J/cm2/s/eV] 3.5e+7 3e+7 2.5e+7 2e+7.5e+7 e+7 5e+6 intensity [J/cm2/s/eV] e+8 8e+7 6e+7 4e+7 2e Figure 4.48: IMH C vs GPLe C flux distributions: to 7 ns.

156 35.6e+8 8 ns IMH C F ν 2.5e+8 9 ns IMH C F ν GPLe C F ν GPLe C F ν.4e+8.2e+8 2e+8 intensity [J/cm2/s/eV] e+8 8e+7 6e+7 intensity [J/cm2/s/eV].5e+8 e+8 4e+7 5e+7 2e e+8 9 ns IMH C F ν 3e+8 92 ns IMH C F ν GPLe C F ν GPLe C F ν 2.5e+8 2.5e+8 intensity [J/cm2/s/eV] 2e+8.5e+8 e+8 intensity [J/cm2/s/eV] 2e+8.5e+8 e+8 5e+7 5e e+8 93 ns IMH C F ν 4.5e+8 94 ns IMH C F ν GPLe C F ν GPLe C F ν 3.5e+8 4e+8 3e+8 3.5e+8 intensity [J/cm2/s/eV] 2.5e+8 2e+8.5e+8 intensity [J/cm2/s/eV] 3e+8 2.5e+8 2e+8.5e+8 e+8 e+8 5e+7 5e Figure 4.49: IMH C vs GPLe C flux distributions: 8 to 94 ns.

157 36 6e+8 95 ns IMH C F ν 9e+8 96 ns IMH C F ν GPLe C F ν GPLe C F ν 8e+8 5e+8 7e+8 intensity [J/cm2/s/eV] 4e+8 3e+8 2e+8 intensity [J/cm2/s/eV] 6e+8 5e+8 4e+8 3e+8 2e+8 e+8 e e+9 97 ns IMH C F ν 2.5e+9 98 ns IMH C F ν GPLe C F ν GPLe C F ν.2e+9 2e+9 intensity [J/cm2/s/eV] e+9 8e+8 6e+8 4e+8 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 2e e+9 99 ns IMH C F ν 2.5e+9 ns IMH C F ν GPLe C F ν GPLe C F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e Figure 4.5: IMH C vs GPLe C flux distributions: 95 to ns.

158 37 2.5e+9 ns IMH C F ν 2.5e+9 2 ns IMH C F ν GPLe C F ν GPLe C F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e e+9 3 ns IMH C F ν 2.5e+9 4 ns IMH C F ν GPLe C F ν GPLe C F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e e+9 5 ns IMH C F ν 2.5e+9 6 ns IMH C F ν GPLe C F ν GPLe C F ν 2e+9 2e+9 intensity [J/cm2/s/eV].5e+9 e+9 intensity [J/cm2/s/eV].5e+9 e+9 5e+8 5e Figure 4.5: IMH C vs GPLe C flux distributions: to 6 ns.

159 38 IMH further vs. GPLe further 35 ns IMH F F ν 4.5e+6 3 ns IMH F F ν GPLe F F ν GPLe F F ν 3 4e+6 3.5e+6 intensity [J/cm2/s/eV] intensity [J/cm2/s/eV] 3e+6 2.5e+6 2e+6.5e+6 e e+6 4 ns IMH F F ν 9e+6 5 ns IMH F F ν GPLe F F ν GPLe F F ν 6e+6 8e+6 7e+6 intensity [J/cm2/s/eV] 5e+6 4e+6 3e+6 2e+6 intensity [J/cm2/s/eV] 6e+6 5e+6 4e+6 3e+6 2e+6 e+6 e e+7 6 ns IMH F F ν 5e+7 7 ns IMH F F ν.2e+7 GPLe F F ν 4.5e+7 4e+7 GPLe F F ν intensity [J/cm2/s/eV] e+7 8e+6 6e+6 4e+6 intensity [J/cm2/s/eV] 3.5e+7 3e+7 2.5e+7 2e+7.5e+7 e+7 2e+6 5e Figure 4.52: IMH F vs GPLe F flux distributions: to 7 ns.

160 39 7e+7 8 ns IMH F F ν.2e+8 9 ns IMH F F ν GPLe F F ν GPLe F F ν 6e+7 e+8 intensity [J/cm2/s/eV] 5e+7 4e+7 3e+7 2e+7 intensity [J/cm2/s/eV] 8e+7 6e+7 4e+7 e+7 2e e+8 9 ns IMH F F ν.2e+8 92 ns IMH F F ν GPLe F F ν GPLe F F ν e+8 e+8 intensity [J/cm2/s/eV] 8e+7 6e+7 4e+7 intensity [J/cm2/s/eV] 8e+7 6e+7 4e+7 2e+7 2e e+8 93 ns IMH F F ν.6e+8 94 ns IMH F F ν GPLe F F ν GPLe F F ν e+8.4e+8.2e+8 intensity [J/cm2/s/eV] 8e+7 6e+7 4e+7 intensity [J/cm2/s/eV] e+8 8e+7 6e+7 4e+7 2e+7 2e Figure 4.53: IMH F vs GPLe F flux distributions: 8 to 94 ns.

161 4 2.5e+8 95 ns IMH F F ν 3e+8 96 ns IMH F F ν GPLe F F ν GPLe F F ν 2e+8 2.5e+8 intensity [J/cm2/s/eV].5e+8 e+8 intensity [J/cm2/s/eV] 2e+8.5e+8 e+8 5e+7 5e e+8 97 ns IMH F F ν e+9 98 ns IMH F F ν GPLe F F ν GPLe F F ν 3e+8 8e+8 intensity [J/cm2/s/eV] 2.5e+8 2e+8.5e+8 e+8 5e+7 intensity [J/cm2/s/eV] 6e+8 4e+8 2e e+9 99 ns IMH F F ν e+9 ns IMH F F ν GPLe F F ν GPLe F F ν 8e+8 8e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e Figure 4.54: IMH F vs GPLe F flux distributions: 95 to ns.

162 4 e+9 ns IMH F F ν e+9 2 ns IMH F F ν GPLe F F ν GPLe F F ν 8e+8 8e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e e+9 3 ns IMH F F ν e+9 4 ns IMH F F ν GPLe F F ν GPLe F F ν 8e+8 8e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e e+9 5 ns IMH F F ν e+9 6 ns IMH F F ν GPLe F F ν GPLe F F ν 8e+8 8e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 intensity [J/cm2/s/eV] 6e+8 4e+8 2e+8 2e Figure 4.55: IMH F vs GPLe F flux distributions: to 6 ns.

163 Power comparisons Remembering the Stefan-Boltzmann relation, relating power and temperature in equation 4.2, we can look at the individual contributions of each Planckian fit to the total power of the system, shown in Figures As can be seen from the figures the individual contributions to the total power of the IMH drives are markedly different. Where the main deviation in the GPLe drives are only seen in the coolest 3rd Planckian fit. Also, even with significant differences between the individual Planckian contributions the total power differences between the IMH and GPLe drives are small. This lends to a sensitivity in the photon energy distribution created by the radiation drives as seen by the neon gas and this effect is seen in past results from Hall [65, 66, 22] and the atomic kinetics results shown here in later chapters.

164 43 IMH closer vs. IMH further 4e+ e+2 3.5e+ 9e+ 8e+ total integrated power [J/cm2/s] 3e+ 2.5e+ 2e+.5e+ e+ IMH F Power IMH F Power 2 IMH F Power 3 IMH F Power total integrated power [J/cm2/s] 7e+ 6e+ 5e+ 4e+ 3e+ IMH C Power IMH C Power 2 IMH C Power 3 IMH C Power 2e+ 5e+ e time [ns] time [ns] 7e+ 3e+ 6e+ 2.5e+ total integrated power [J/cm2/s] 5e+ 4e+ 3e+ 2e+ IMH F Power IMH C Power total integrated power [J/cm2/s] 2e+.5e+ e+ IMH F Power 2 IMH C Power 2 e+ 5e time [ns] time [ns] 4e+ e+2 3.5e+ 9e+ 8e+ total integrated power [J/cm2/s] 3e+ 2.5e+ 2e+.5e+ e+ IMH F Power 3 IMH C Power 3 total integrated power [J/cm2/s] 7e+ 6e+ 5e+ 4e+ 3e+ IMH F Power T IMH C Power T 2e+ 5e+9 e time [ns] time [ns] Figure 4.56: IMH C vs IMH F Power as a function of time.

165 44 GPLe closer vs. GPLe further 5e+.2e+2 4.5e+ 4e+ e+2 total integrated power [J/cm2/s] 3.5e+ 3e+ 2.5e+ 2e+.5e+ GPLe F Power GPLe F Power 2 GPLe F Power 3 GPLe F Power total integrated power [J/cm2/s] 8e+ 6e+ 4e+ GPLe C Power GPLe C Power 2 GPLe C Power 3 GPLe C Power e+ 2e+ 5e time [ns] time [ns] 4.5e+ 4.5e+ 4e+ 4e+ total integrated power [J/cm2/s] 3.5e+ 3e+ 2.5e+ 2e+.5e+ e+ GPLe F Power GPLe C Power total integrated power [J/cm2/s] 3.5e+ 3e+ 2.5e+ 2e+.5e+ e+ GPLe F Power 2 GPLe C Power 2 5e+ 5e time [ns] time [ns] 2.5e+.2e+2 2e+ e+2 total integrated power [J/cm2/s].5e+ e+ GPLe F Power 3 GPLe C Power 3 total integrated power [J/cm2/s] 8e+ 6e+ 4e+ GPLe F Power T GPLe C Power T 5e+ 2e time [ns] time [ns] Figure 4.57: GPLe C vs GPLe F Power as a function of time.

166 45 IMH closer vs. GPLe closer e+2.2e+2 9e+ 8e+ e+2 total integrated power [J/cm2/s] 7e+ 6e+ 5e+ 4e+ 3e+ IMH C Power IMH C Power 2 IMH C Power 3 IMH C Power total integrated power [J/cm2/s] 8e+ 6e+ 4e+ GPLe C Power GPLe C Power 2 GPLe C Power 3 GPLe C Power 2e+ 2e+ e time [ns] time [ns] 7e+ 4.5e+ 6e+ 4e+ total integrated power [J/cm2/s] 5e+ 4e+ 3e+ 2e+ IMH C Power GPLe C Power total integrated power [J/cm2/s] 3.5e+ 3e+ 2.5e+ 2e+.5e+ e+ IMH C Power 2 GPLe C Power 2 e+ 5e time [ns] time [ns] 2.5e+.2e+2 2e+ e+2 total integrated power [J/cm2/s].5e+ e+ IMH C Power 3 GPLe C Power 3 total integrated power [J/cm2/s] 8e+ 6e+ 4e+ IMH C Power T GPLe C Power T 5e+ 2e time [ns] time [ns] Figure 4.58: IMH C vs GPLe C Power as a function of time..

167 46 IMH further vs. GPLe further 4e+ 5e+ 3.5e+ 4.5e+ 4e+ total integrated power [J/cm2/s] 3e+ 2.5e+ 2e+.5e+ e+ IMH F Power IMH F Power 2 IMH F Power 3 IMH F Power total integrated power [J/cm2/s] 3.5e+ 3e+ 2.5e+ 2e+.5e+ GPLe F Power GPLe F Power 2 GPLe F Power 3 GPLe F Power e+ 5e+ 5e time [ns] time [ns] 2.5e+.6e+.4e+ 2e+ total integrated power [J/cm2/s].5e+ e+ IMH F Power GPLe F Power total integrated power [J/cm2/s].2e+ e+ 8e+ 6e+ 4e+ IMH F Power 2 GPLe F Power 2 5e+ 2e time [ns] time [ns] 9e+ 5e+ 8e+ 4.5e+ total integrated power [J/cm2/s] 7e+ 6e+ 5e+ 4e+ 3e+ 2e+ IMH F Power 3 GPLe F Power 3 total integrated power [J/cm2/s] 4e+ 3.5e+ 3e+ 2.5e+ 2e+.5e+ e+ IMH F Power T GPLe F Power T e+ 5e time [ns] time [ns] Figure 4.59: IMH F vs GPLe F Power as a function of time.

168 Brightness temperature comparisons The brightness temperature is another parameter of the Stefan-Boltzmann relation in equation 4.2. This temperature is what is used to characterize many astrophysical and high energy density systems. Though, there are many different temperatures characterizing a plasma, electron, ion, color, and radiation, to name a few, using a single metric can be a power tool when simplifying the description of a plasma. The rationalization for using this parameter is described previously in this chapter and now the results of this are shown. Using the convolution of three Planckian distributions to describe the plasma allows us to take the single brightness temperature extracted previously from the arbitrary radiation drive and break it into its three components. As can be seen in Figures the brightness temperatures as seen at the further position are lower than those at the closer position. This is consistent with the geometry dilution effect, discussed previously. An interesting results is seen when though the individual contributions to the total brightness temperature can be different the end results, total brightness temperature, is very similar. This can lead to information breaking down the specific components of the source radiation that may not have been available, previously.

169 48 IMH closer vs. IMH further brightness temp [ev] IMH C T B IMH F T B brightness temp [ev] IMH C T B 2 IMH F T B time [ns] time [ns] brightness temp [ev] 5 IMH C T B 3 IMH F T B 3 brightness temp [ev] IMH C T B IMH F T B time [ns] time [ns] brightness temp [ev] IMH C T B IMH C T B IMH C T B 2 IMH C T B 3 brightness temp [ev] IMH F T B IMH F T B IMH F T B 2 IMH F T B time [ns] time [ns] Figure 4.6: IMH C vs IMH F Brightness temperature as a function of time.

170 49 GPLe closer vs. GPLe further brightness temp [ev] GPLe C T B GPLe F T B brightness temp [ev] GPLe C T B 2 GPLe F T B time [ns] time [ns] brightness temp [ev] GPLe C T B 3 GPLe F T B 3 brightness temp [ev] GPLe C T B GPLe F T B time [ns] time [ns] brightness temp [ev] GPLe C T B GPLe C T B GPLe C T B 2 GPLe C T B 3 brightness temp [ev] GPLe F T B GPLe F T B GPLe F T B 2 GPLe F T B time [ns] time [ns] Figure 4.6: GPLe C vs GPLe F Brightness temperature as a function of time.

171 5 IMH closer vs. GPLe closer brightness temp [ev] GPLe C T B IMH C T B brightness temp [ev] GPLe C T B 2 IMH C T B time [ns] time [ns] brightness temp [ev] GPLe C T B 3 IMH C T B 3 brightness temp [ev] GPLe C T B IMH C T B time [ns] time [ns] brightness temp [ev] GPLe C T B GPLe C T B GPLe C T B 2 GPLe C T B 3 brightness temp [ev] IMH C T B IMH C T B IMH C T B 2 IMH C T B time [ns] time [ns] Figure 4.62: IMH C vs GPLe C Brightness temperature as a function of time.

172 5 IMH further vs. GPLe further brightness temp [ev] GPLe F T B IMH F T B brightness temp [ev] GPLe F T B 2 IMH F T B time [ns] time [ns] brightness temp [ev] 2 5 GPLe F T B 3 IMH F T B 3 brightness temp [ev] GPLe F T B IMH F T B time [ns] time [ns] brightness temp [ev] GPLe F T B GPLe F T B GPLe F T B 2 GPLe F T B 3 brightness temp [ev] IMH F T B IMH F T B IMH F T B 2 IMH F T B time [ns] time [ns] Figure 4.63: IMH F vs GPLe F Brightness temperature as a function of time.

173 Summary In summary, we have found a well characterizable method of matching an arbitrary radiation drive produced by the Z-machine by the combination of three Planckian radiation distributions of a given temperature and dilution factor. This is a powerful method due to the importance of understanding the photon energy distribution of a radiation drive as well as its overall intensity. For example, a radiation drive can have relatively similar overall brightness temperatures, relating to σt 4, while the color temperatures can be different to the point of completely changing the photon energy distribution of the problem, i.e. a larger contribution to the exponential decaying high energy tail of the distribution may seem minimal, but it makes a large difference to ions with a greater ionizing potential. These high energy photons are needed to break the high ionization threshold of the He-like Neon ions and without this high energy we would not see them. The experimental data, though, shows a significant contribution of He-like neon so we know there is a level of ionization radiation available from the radiation distribution needed to created these ions. Correctly establishing what this level is, is important and with the collection of radiation drives now available of differing distributions the study can now be done to show how sensitive the He-like ions are the different radiation drives through a series of atomic kinetics calculations.

174 53 Chapter 5 Electron kinetics simulations 5. Introduction When the neon gas is irradiated by the energetic flux of x-rays, as previously mentioned the neon gas starts to go through a series of burn- through stages. These burn-through stages simply refer to the isoelectronic sequence of ionization stages. So beginning as neutral neon gas or neon-like neon, then as the first electron is burnt off or given sufficient energy that it breaks the potential holding it in place the neon ion then becomes fluorine-like neon, or neon +. From there oxygen-like neon, or neon +2, and so on and so forth. These electrons that are free from the neon atoms comprise the free electron pool. This group of free electrons all have an initial energy beyond their binding energy and then begin to thermalize. For example, if we consider helium-like neon, neon +8, it has an ionization potential of 95 electron volts. So if this ion collides with an incoming photon of 295 ev then the ionized electron will be freed with an energy of ev. As has been predicted by simulations the average charge state distribution that the neon gas achieves is Z 8. This means that for a 3 Torr fill pressure with an atom number density of 8 atoms per cm 3 we now have 8 8 free electrons per cm 3. Now, these electrons once freed are

175 54 still undergoing all the atomic process interactions with ions and the other free electrons [88, 89]. This is a significant group of particles whose interactions, behaviors, and characteristics need to be accounted for if the neon photoionized plasma is to be properly described and characterized. 5.2 Overview of Boltzmann electron kinetics model In order to understand the behavior of these electrons we turned to formulae that has been developed to model the electron kinetics, including free-free, bound-free, and bound-bound processes [9, 9, 92, 93]. Traditionally equilibrium equations and atomic kinetics codes when treating the distribution of electrons do so by assuming a Maxwell-Boltzmann distribution, f(v) = 4πv ( 2 m 2πkT )3 e mv 2 2kT (5.) Making this assumption allows the free electrons in the system to be characterized by a single temperature. This works well for plasmas that are very collisional but for non-collisional this assumption does not hold. As just discussed the atomic kinetic rate equations determine the atomic state populations. Some of these rate equations though are directly linked to the pool of free electrons. For example, upward processes like photoionization and collisional ionization add to the pool, while downward processes such as 3-body recombination and radiative recombination remove electrons from the pool. These processes not only remove the electron from the pool of free electrons but also the energy contribution that electron made to the total pool. This makes it evident that the pool of free electrons and the processes governing the bound electrons are connected very closely. This is why the code solves simultaneously the

176 55 atomic kinetic rate equation and the Boltzmann electron kinetics. The method for solving the Boltzmann equation was developed by Bretagne [94, 95] and allows the distribution function of electrons to propagate in time, ( ) ( ) ( ) f f f t f(u,t) = + + t t t ee en in +A(t)S(E p,u) (5.2) This distribution function f(u,t) is made up of elastic processes such as electronelectron collisions (ee) and electron-neutral collisions (en), and inelastic collisions (in). The last term in the equation relates to the primary particle flux (A(t)) and ionization cross sections (S(E p,u)), where E p and u are energies of the primary and secondary particles, respectively. It was shown by Abdallah and Colgan [93] that the Boltzmann equation and solution for the time-dependent atomic kinetic rate equations could be simplified to a system of coupled first-order differential equations, df i dt =S i(f,n,g(hν)) dn j =R j (f,n,g(hν)) (5.3) dt Here f i = f(e i,t), with f being the electron energy distribution function(eedf) and E i is the electron energy. S is the functional term comprising of all the elements of the electron energy distribution function as shown in equation 5.2. The terms f and N represent all the components of f i and N i, and are the population densities of all the atomic states used in the collisional-radiative model. Finally, G(hν) is the radiation field incident upon the plasma comprising of photons with energy hν. 5.3 Electron energy distribution function When the radiation drives described in Chapter 4 are used, specifically GPLe, in the Boltzmann equation solver we get an estimation of both the behavior of the free

177 56 electrons and the behavior of the states of the ions in the neon plasma. Firstly, the distribution of the free electrons is calculated. This distribution of the free electrons with their associate energy is called the electron energy distribution function (EEDF) When the equations in 5.3 are solved we obtained a series of EEDF s as a function of time. The time evolution of these functions are shown in figure 5.. The peak of the EEDF and the peak of the radiation drive have their colors changed from the collection of red traces to green and blue, respectively. The shift of the distribution from lower energy electron to higher energy electrons from the time when the EEDF peaks (at 8 ns) to when the intensity of the radiation drive peaks (ns) is clearly visible. This shift in energy to higher energy electrons along with figure 5.2 shows that not only are we developing a pool of higher energy electrons but we are also gaining more of them. The average charge state shown in figure 5.2 plateau s at the closed shell of Helium-like neon for a charge of +8. For the higher filling pressures at each moment in time there are less freed electrons. Since high filling pressure contain more particles the collisionality of these higher pressures drive recombination, thus reducing the free electron pool.

178 57 3.5e+7 3T C 3T C 8ns 3T C ns 3e+7 2.5e+7 EEDF [cm -3 ev - ] 2e+7.5e+7 e+7 5e Electron energy [ev] Figure 5.: The series of EEDF s at different time steps for a filling pressure of 3 Torr in the closer position, the EEDF at 8 ns is highlighted in green while the blue is the EEDF at the peak of the radiation drive, ns p5 C 7p5 C 5 C 3 C 7 6 Zbar Figure 5.2: The average charge state of the neon gas as a function of time for different filling pressures in the closer position.

179 58 From these EEDF s a temperature can be ascertained to describe the plasma conditions. Since no assumptions were made as to the form of the free electron distribution there are no temperature parameters governing its behavior. So in order to quantify the distribution by a single temperature an assumption must be made. The standard and most common method is to apply a Maxwell-Boltzmann distribution, though other distribution have been used such as Druyvesteyn [96]. Viewed on a log scale and fit with a Maxwell-Boltzmann distribution of a single temperature the EEDF and the Maxwell-Boltzmann fit look like figure 5.3. e+8 e+6 log(f) [cm -3 ev - ] e+4 e+2 e+ e+8 e Electron energy [ev] Figure 5.3: The EEDF distribution calculated in the blue, along with the red line of the Maxwell-Boltzmann fit. As can be seen in the figure the Maxwell-Boltzmann distribution fits the low energy side of the EEDF well. When a temperature is assigned to electrons this is generally the method used. As seen in the figure, though, there is a portion of the free electrons that do not fit well to the Maxwell-Boltzmann distribution. Specific-

180 59 ally shown here the deviation begins just before 25 ev and increases thereon. This deviation, in this case, is representative of a second group of electrons of much higher energies. Figures 5.4 and 5.5 show the characterization of the neon plasma by the thermal Maxwell-Boltzmann distributions fit to the EEDF as functions of time. Figure 5.4 shows the trends of the closer position, while Figure 5.5 shows the trends of the further position. Both closer and further position values from inception to 9 ns, when the radiation drive starts the ramp up phase before peaking, are very similar and approximately linear. From that point, the higher the filling pressure the higher the achieved electron temperature. Looking at the individual filling pressures compared against one another for closer and further positions in Figure 5.6 we can see the specific differences that occur to the neon plasma by moving it a greater distance away from the ionization radiation source. For lower filling pressure the temperature of the further position is consistently below the closer position as one would expect from a larger incident flux on the closer position, but as the filling pressure increases this difference decreases until 3 Torr when early in time the closer and further position temperature are intertwined indicating a stronger dependence on the number of electrons than on the source of heating radiation. Also, for lower filling pressures, late in time, plateau at similar final electron temperature where again at higher filling pressure this convergence decreases and begins to diverge as seen in plots (c) and (d) of Figure 5.6, where at the final time step the difference between T e for closer and further is only a few ev for 5 Torr but for 3 Torr this had increased to almost ev.

181 p5 C 7p5 C 5 C 3 C 4 Electron temperature [ev] Figure 5.4: The thermal electron temperature as a function of time for the closer position radiation drive p5 F 7p5 F 5 F 3 F 4 Electron temperature [ev] Figure 5.5: The thermal electron temperature as a function of time for the further position radiation drive.

182 6 5 3p5 C 3p5 F 5 7p5 C 7p5 F Electron temperature [ev] Electron temperature [ev] (a) (b) 5 5 C 5 F 5 3 C 3 F Electron temperature [ev] Electron temperature [ev] (c) (d) Figure 5.6: Comparing the thermal electron temperatures as functions of time for the closer and further positions each of a different fill pressure: (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr Two temperature model approximation It is from this behavior that we now apply a second Maxwell-Boltzmann distribution. Fitting a secondary Maxwell-Boltzmann distribution implies a plasma of two distinguishing pools of electrons, one of hot high energy electrons and a second of cold electrons that have had time to relax into a thermal distribution. Fitting these two Maxwell-Boltzmann distribution to the EEDF then takes the form, ( c)f TC (hν)+cf TH (hν) (5.4)

183 62 Here, f is the Maxwell-Boltzmann distribution of a cold (T C ) group of electrons and a hot (T H ) group of electrons and c is the scaling factor representative of the portion of hot vs cold electrons. This process was carried and out and the results are shown in Figures 5.7, 5.8, 5.9, and 5.. Each figure is a collection of fits to the four different filling pressures at a certain position at a certain time, specifically 95 and ns. e+8 3.5T C 95ns MB T e [ev]=4.3 MB T e [ev]=28 e+8 7.5T C 95ns MB T e [ev]=3.7 MB T e [ev]=3 e+6 e+6 e+4 e+4 log(f[cm -3 ev - ]) e+2 e+ log(f[cm -3 ev - ]) e+2 e+ e+8 e+8 e Electron energy [ev] e Electron energy [ev] (a) (b) e+8 5T C 95ns MB T e [ev]=6. MB T e [ev]=3 e+8 3T C 95ns MB T e [ev]=22.5 MB T e [ev]=4 e+6 e+6 e+4 e+4 log(f[cm -3 ev - ]) e+2 e+ log(f[cm -3 ev - ]) e+2 e+ e+8 e+8 e Electron energy [ev] e Electron energy [ev] (c) (d) Figure 5.7: Fitting the EEDF generated by the closer radiation drive at 95 ns with a two temperature Maxwell-Boltzmann model where the colder Maxwell-Boltzmann distribution is the red trace and the hotter distribution is the green trace. Where the individual plots correspond to (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr. Table 5. summarizes the hot and cold temperatures, associated with each EEDF s filling pressure at each time step, each characterized by a Maxwell-Boltzmann distribution. As can be seen from the figures a two temperature model fits these calculated

184 63 EEDF s very well. It follows from this that there is a population of electrons that have thermalized to a Maxwell-Boltzmann distribution and a population of energetic electrons that have yet to relax into this thermalized distribution. These hot electrons are continually generated by the high energy tail of the radiation distribution and are done so at a rate that, though small, is continually populated. e+8 3.5T F 95ns MB T e [ev]=2.6 MB T e [ev]=35 e+8 7.5T F 95ns MB T e [ev]=.6 MB T e [ev]=35 e+6 e+6 e+4 e+4 log(f[cm -3 ev - ]) e+2 e+ log(f[cm -3 ev - ]) e+2 e+ e+8 e+8 e Electron energy [ev] e Electron energy [ev] (a) (b) e+8 5T F 95ns MB T e [ev]=2.5 MB T e [ev]=45 e+8 3T F 95ns MB T e [ev]=4.7 MB T e [ev]=55 e+6 e+6 e+4 e+4 log(f[cm -3 ev - ]) e+2 e+ log(f[cm -3 ev - ]) e+2 e+ e+8 e+8 e Electron energy [ev] e Electron energy [ev] (c) (d) Figure 5.8: Fitting the EEDF generated by the further radiation drive at 95 ns with a two temperature Maxwell-Boltzmann model where the colder Maxwell-Boltzmann distribution is the red trace and the hotter distribution is the green trace. Where the individual plots correspond to (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr.

185 64 Looking at the electron energies at which the EEDF begins to deviate from a Maxwell-Boltzmann distribution for 95 ns this area is very similar for 3.5 Torr and 7.5 Torr but then for 5 Torr and 3 Torr and obvious shift towards higher energies is seen. This indicates that as the neon plasma is continually heated by the ever increasing radiation drive higher energy electrons are continually generated. e+8 3.5T C ns MB T e [ev]=6.5 MB T e [ev]=5 e+8 7.5T C ns MB T e [ev]=7.7 MB T e [ev]=6 e+6 e+6 e+4 e+4 log(f[cm -3 ev - ]) e+2 e+ log(f[cm -3 ev - ]) e+2 e+ e+8 e+8 e Electron energy [ev] e Electron energy [ev] (a) (b) e+8 5T C ns MB T e [ev]=23.7 MB T e [ev]=22 e+8 3T C ns MB T e [ev]=35.9 e+6 e+6 e+4 e+4 log(f[cm -3 ev - ]) e+2 e+ log(f[cm -3 ev - ]) e+2 e+ e+8 e+8 e Electron energy [ev] e Electron energy [ev] (c) (d) Figure 5.9: Fitting the EEDF generated by the closer radiation drive at ns with a two temperature Maxwell-Boltzmann model where the colder Maxwell-Boltzmann distribution is the red trace and the hotter distribution is the green trace. Where the individual plots correspond to (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr. This is also the case for the EEDF s at ns. The deviation between the hot and cold electron distribution also becomes smaller as the radiation drive reaches peak temperatures. The shift towards higher electrons energy distributions is still

186 65 evident at the peak of the and even more obvious so as the scale for which the EEDF energy ends at 5 ev. Which, for 3 Torr in both the closer and further positions at ns the hot electron Maxwell-Boltzmann approximation to this outside this range. e+8 3.5T F ns MB T e [ev]=6.4 MB T e [ev]=24 e+8 7.5T F ns MB T e [ev]=7.6 MB T e [ev]=38 e+6 e+6 e+4 e+4 log(f[cm -3 ev - ]) e+2 e+ log(f[cm -3 ev - ]) e+2 e+ e+8 e+8 e Electron energy [ev] e Electron energy [ev] (a) (b) e+8 5T F ns MB T e [ev]=22.8 MB T e [ev]=46 e+8 3T F ns MB T e [ev]=3.6 e+6 e+6 e+4 e+4 log(f[cm -3 ev - ]) e+2 e+ log(f[cm -3 ev - ]) e+2 e+ e+8 e+8 e Electron energy [ev] e Electron energy [ev] (c) (d) Figure 5.: Fitting the EEDF generated by the further radiation drive at ns with a two temperature Maxwell-Boltzmann model where the colder Maxwell-Boltzmann distribution is the red trace and the hotter distribution is the green trace. Where the individual plots correspond to (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr.

187 66 The estimates to the hot electron temperatures, Table 5., are generalized to the nearest ten s of degrees, but still reasonable as the differences between the hot and cold temperatures are still over an order of magnitude in difference. There does appear to be a trend allowing higher electron temperature with higher filling pressure. The columns labelled c are representative of the fraction of the hot electrons in total electron population, as seen in equation 5.4. Looking at the pressures 3.5 and 7.5 Torr at ns is when we see the largest fraction of hot electrons. Even at their largest they are still only a very small fraction of the total population while 99.98% are the thermal Maxwellian electrons. This is very strong evidence for treating the free electrons with Maxwell-Boltzmann distributions in other calculations. This evidence from the solving the Boltzmann solver corroborates the trends seen in the hydrodynamics simulations from Helios-CR in Chapter 6. Table 5.: Maxwell-Botlzmann EEDF fits summary Fill[Torr] T C [ev] T H [ev] c T C [ev] T H [ev] c Closer Further 95 ns e e e e e e e e-5 ns e e e e e e ns e e e e e e

188 Electron-electron equilibration times When considering the behavior s of the hot and cold electron pools. The first notion to consider is how long it takes the hot electrons to thermalize and relax to the temperature of the cold electrons. This is known as the electron-electron equilibration time and has been well documented [97]. We consider two groups of particles each haveatemperaturecharacterizing amaxwell-boltzmann distribution, namely, T C and T H. The determination for the rate of change between the two temperatures takes the general form, This then leads to, dt dt = T C T H t eq (5.5) ( ) α (T C +T H ) 3/2 (5.6) N e lnλ Where, α is a constant relating to the fundamental values of the particles involved, like mass and charge, N e is the electron number density, lnλ is the coulomb logarithm [98]. After hashing out all these details we cancreate table 5.2 that contains the times it takes for the hot electrons to thermalize to the temperature of the cold electrons listed in table 5.. These equilibration times show a clear trend indicating faster thermalization at higher filling pressures. This thermalization is driven by collisions so having the process occur at a quicker rate in environments where this processes has a higher frequency of occur is consistent. It is also consistent that the closer gas cell position, with its higher radiation flux, enable a faster equilibration time. This is because the higher radiation flux generates more energy electrons than a lower radiation flux (further position). More energy is indicative of higher speeds causing collisions to occur more frequently. These collisions are the heart of the thermalization process and lend to a faster thermalization for the closer position.

189 68 We have now shown that the vast majority of the free electron distribution, in our experiment, is well characterized by a Maxwellian distribution. Hydrodynamic and atomic kinetic simulations generally take on the assumption that the free electrons follow this Maxwellian distribution. Now that we have verified this point we can use these hydrodynamic and atomic kinetic simulations with more confidence. Table 5.2: Equilibration times between hot and cold temperatures of Maxwell-Boltzmann distributions. Fill[Torr] T eq [ps] Closer Further 95 ns ns ns Population distributions As described earlier the Boltzmann model self-consistently solves the atomic kinetic rate equations along with the electron kinetics. This allows us to visualize the progression throughthe different charge states of the neon gas as it is heated by the radiation drive. The key difference between this atomic kinetic solver and other is in the fact that the free electron distribution is simultaneously solved for. This is important be-

190 69 cause both the free and bound electrons are connected through various processes and rely on one another. Figures 5. and 5.2 are consistent with the previous figures in each containing four plots with four filling pressures. These plots are only briefly mentioned here and will be used in comparison later with atomic kinetics calculations computed by separate, independent codes. These plots, as a function of time, show how the different charge states are populated and in turn de-populated. As can be seen from the figures the different filling pressures have widely varying differences in how each individual ion stage behaves. The similarities are found when considering the Helium-like and Hydrogen-like neon shells. First looking at the Helium-like neon shell it is shown that as the filling pressure increases the rate at which it populates increases dramatically beyond 8 ns. If just this time step is investigated from 3.5 Torr to 3 Torr the here drops from just under 4 % to almost 5 %. While they all reach above 95 % just before the peak of the radiation drive at ns. This behavior is also observed in the population distribution of the further position drive, though not as pronounced as in the closer position. The Hydrogen-like neon level as expected are lower in the further position as the ionization potential of this electron is 362 electron volts. The band of time, in which, this charge states populates is the same though due to the selective band of photon able to ionize this electron out of its shell.

191 Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (a) (b) Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (c) (d) Figure 5.: Isoelectronic sequence distributions as functions of time for different filling pressures at the closer position: (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr.

192 Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (a) (b) Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (c) (d) Figure 5.2: Isoelectronic sequence distributions as functions of time for different filling pressures at the further position: (a) 3.5 Torr, (b) 7.5 Torr, (c) 5 Torr, (d) 3 Torr.

193 72 Chapter 6 Heating simulations 6. Introduction This section discusses simulations of the hydrodynamics of a plasma created from a distribution of photons that are generated by a z-pinch implosion experiment using the radiation-hydrodynamics code HELIOS-CR [29, 99]. We begin with a description of the simulation method used by the code and the physics model that is included in the code. Next, the results from simulations performed using a radiation drive and target parameters which correspond to experimental conditions are presented. 6.2 HELIOS-CR radiation-hydrodynamic model HELIOS-CR is a -D Lagrangian radiation-magnetohydrodynamics code that is suitable for modelling the complex time-dependent systems of laser-produced plasmas and z-pinch plasmas. This code has been used to simulate a variety of plasmas across different physics communities [,, 2]. Though, focus in recent years has been on studying high energy density physics and inertial confinement fusion it is also applicable to industrial modeling applications and astrophysics [3, 4, 2, ].

194 73 A target in HELIOS-CR consists of one or more layers of material in either planar, cylindrical, or spherical geometries. Each layer or region of material is divided into a number of cells. The thickness, number of cells in the simulations, and material used for each layer are all user specified parameters. This allows freedom to generate complex targets of varying materials and geometries. The construction is done in this manner because the mathematical hydrodynamic description of a target be expressed in different forms. The two standard forms are Eulerian and Lagrangian. In Eulerian formalism a reference frame is fixed around a position vector r and any changes in fluid properties are observed at that position. In the Lagrangian approach the fluid is broken into cells (method used in HELIOS-CR) and the evolution of the cell changes with a new independent variable given by dm = ρ(r)r δ dr (6.) Here, dm is the Lagrangian mass and within each zone the mass remains constant. The zone boundaries radii, however, are functions of time. To show the conservation of momentum we begin with the differential form of Euler s equation, ρ d v dt = P (6.2) P is the total pressure of the system, which is the sum of the contributions of pressure from the ions, electrons and the artificial viscosity which is added to alleviate discontinuities. Multiplying both sides by the differential of geometric volume of the system and then integrating over the entire volume of the system we obtain, ρ d v dt dṽ = Pd Ṽ (6.3) (6.4)

195 74 here Ṽ is the surface surrounding the volume V. Remembering that the pressure at the boundary is zero makes the surface integral zero on the boundary and using Reynolds transport theorem the derivative can be removed from the integral leaving, d dt ρ vdṽ = d (6.5) dt where is the total momentum of the fluid, which in the absence of external pressure in the region of the contour conserves the total momentum. To understand the energy ofthesystem, ortheheatingandcoolingoftheelectronsandions, weneed toevaluate the first law of thermodynamics, which states that the change in the internal energy of a system (de) is equal to the heat added (dq) to the system minus the work done (PdV) by the system. de = dq PdV (6.6) This means that we assume our system evolves through a series of local thermodynamic equilibrium stages. Considering a stage of thermodynamic equilibrium being spaced by a time interval dt we can rewrite the first law as, de dt +PdV dt = d Q (6.7) Here δ Q is the specific heat absorbed per unit time and includes any of the physical processes that are of interest to the problem being considered. In the presence of external radiation sources the intensity of the external field needs to be accounted for. The specific intensity of radiation I(r,n,ν,t) at a position r in a direction n, with frequency ν defined as the amount of energy δǫ that is carried by radiation in frequency interval (ν,dν), across the surface element ds into a solid angle dω in a time interval dt: δǫ = I(r,n,ν,t)n dsdωdνdt (6.8)

196 75 It follows that the mean intensity J(r,ν,t) is the average of the specific intensity over all solid angles: J(r,ν,t) = 4π I(r, n, ν, t)dω (6.9) Two other ideas are used extensively are optical depth and the source function. Optical depth, τ ν, at a position r and frequency ν is defined as a dimensionless quantity given by the absorption coefficient α ν of the material integrated along the line of sight from an outside surface, τ ν (r) = r r α ν (r )dr (6.) Conceptually, the optical depth is the number of mean free paths a photon of frequency ν can travel along the line of sight from r to r max (end of material). Traditionally, an optical depth greater than one is said to be optically thick and less than one optically thin. The second concept is that of the source function S ν which is the ratio of the emission coefficient,j ν and the absorption coefficient,α ν, S ν = j ν α ν (6.) The emission coefficient is related to the spectral emissivity of an object and is defined as the radiant exitance of a real body (M) to that of an ideal body (M BB ), ǫ(ν) = M(ν) M BB (ν) (6.2) Comparing expressions for energies from an isotropically emitting medium ǫ ν and j ν hold the relation, j ν = ǫ νρ 4π (6.3) Here ρ is the mass density, and a similar formalism can be carried out for the absorption of a material through Kirchhoff s law. With this cursory treatment of radiation

197 76 from a emitting medium the transport equation can then be written in terms of optical depth along side the source function, where µ = cosθ for a planar geometry assumption, µ di(ν) dτ(ν) = I(ν) S(ν) (6.4) The inclusion of radiative processes in collisional-radiative model calculations is computationally expensive because it involves iterative techniques between the rate equations and the radiation transport equation. This leads to the need for different models that can be used to handle the radiation transport in hydrodynamic codes. The transport of radiation can be handled in HELIOS-CR by two different models: one model, a flux-limited, multi-group diffusion model and a second multi-group and multi-angle radiation transport model. The multi-group model is used as a simplification to the radiative transfer and energy equations. The radiative transport equation is solved individually for energy density, radiation conductivity, radiation emission, Planck absorption, Planck emission, and Rosseland opacity each for a specified photon energy group. The frequency space is defined by the problem at hand. Once these terms are solved for they are then explicitly included in the electron energy equation and are followed by a flux-limiting term when solving for the energy densities of adjacent groups. The radiation transport can be approximated when the mean free path of the photons is much smaller than the path length of the material they are traveling through. The multi-angle radiation transport model, schematically show in Figure 6., calculates specific intensities in the + r and - r directions at a frequency ν, a position given by optical depth τ ν, and along a ray defined by the cosine of the angle with respect to the surface normal. These different intensities, with respect to angle, are then averaged and used to calculate a flux at a given position. In the case of external radiation fields non-zero boundary conditions are applied at τ ν = and τ ν = T ν

198 77 - r +r L r } Figure 6.: Schematic of geometry used for the multi-angle diffusion approximation. The different atomic models used are the standard local thermodynamic equilibrium (LTE) and an inline collisional-radiative model (CRM). The LTE model holds the assumptions that under steady-state conditions the level population densities are filled according to the Boltzmann relation and the distribution of ions are filled by the Saha equation. Other assumptions included are that the electron collisional rate is greater than the ionic collisional rate. It follows from this that the energy distribution of the plasma is dominated by the electrons and follows a Maxwellian relation. The crux of the problem comes when we are concerned with the accuracy of using the Saha-Boltzmann relation for level populations when not all of the parameters of the system are in a state of LTE with a common temperature. The necessary condition when studying the dynamics of photoionized plasmas is to have the collisional rates for a given transition be dominated by the corresponding radiative rate. The CRM approach strongly contrasts with the LTE model in that the population distribution at a point does not depend only upon the plasmas parameters at that point. The population distribution is determined by balancing collisional processes and radiative processes. This is done by self-consistently solving multi-level, atomic

199 78 rate equations while including a radiation field, dn i dt = n i N L j i N L W ij + n j W ji (6.5) Here i and j correspond to the lower and upper atomic levels, respectively, and N L is the number of levels included in the calculation. HELIOS-CR solves this coupled set of atomic rate equations at each time step in the simulation accounting for the rates of populating (W ij ) and depopulating (W ji ) processes. Included in the W ij and W ji terms are the following atomic process. n e independent processes j i Table 6.: Atomic processes n e dependent processes spontaneous emission collisional excitation stimulated absorption/emission collisional deexcitation photoionization collisional ionization stimulated recommbination collisional recombination autoionization electron capture radiative recombination Due to the inclusion of the external radiation field the population distributions can have a spatial gradient even when there is no gradient in the temperature and density of the plasma. The radiation field is included by the use of frequency averaged mean intensity J ν as seen earlier in this chapter and in Mihalas [38]. Since the CRM method uses atomic data like energy level structure, cross sections, and oscillator strengths one of the limiting factors of the CRM is the quality of the atomic data used.

200 Motivation for the simulations Previous computational and experimental work has shown qualitative agreement between the data and simulated results [65, 66, 22, 67]. Simulations in planar geometry have shown electron temperatures that are in excess of those inferred from absorption line feature measurements. Running hydrodynamic simulations allows the testing of sensitivities between the atomic kinetics of the photoionized neon gas and the radiatively driven hydrodynamics of the neon gas. 6.4 Simulation parameter description 6.4. Radiation drive characterization The radiation environment used was modeled and simulated using the 3-D view factor code VISRAD [8]. A geometric model of the experimental setup was implemented and further constrained by the total bolometric power as emitted by the dynamic z-pinch hohlraum system measured off-axis and by a time-resolved evolution of the source size. Figure 6.2 shows an example of the time-resolved history of brightness temperature extracted from two different radiation drives. The temperatures are extracted using the Stefan-Boltzmann relation, relating the total integrated flux to the brightness temperature of the emitting medium. F hν = σtb 4 (6.6) The two drives are reflected in the traces of brightness temperature labeled IMH C (red) and IMH F (green). These two drives are representative of a radiation drive in

201 8 6 IMH C IMH F 5 4 T B [ev] 3 2 2e-8 4e-8 6e-8 8e-8 e-7.2e-7 time [ns] Figure 6.2: The brightness temperature extracted from two different radiation drives, closer position (red), further position (green). the closer position, IMH C and a radiation drive in the further away position IMH F. The further away position drive is always less bright than the closer position drive, clearly showing the geometry dilution effect. With tighter characterizations of the radiation incident on the neon gas the dynamics of photoionization can be more accurately assessed. Figure 6.3 shows a photon energy dependent plot of a radiation drive (VISRAD) and that same drive after it has passed through a.4 micron slab of mylar (mylar attenuated VISRAD), because the VISRAD model was created using time-dependent experimental data it generates a time-dependent radiation drive. The radiation drive shown in Figure 6.3 was taken from the ns timestep. This is the timestep signifying the peak of emitted x-ray power. Integrating the total flux from Figure 6.3 and using the Stefan-Boltzmann relation gives the brightness temperature estimation at ns. It is important to note that the red trace is not a strict specific intensity but an angle averaged mean intensity that takes into account the self

202 8.2e+9 VISRAD Mylar attenuated VISRAD e+9 flux[j/cm 2 /s/ev] 8e+8 6e+8 4e+8 2e hv[ev] Figure 6.3: Attenuation of radiation through a planar slab of mylar emission of the heated mylar Electron temperature and average charge state Figure6.4 shows a plot of the average charge state distribution, Z, of theneon plasma in the simulation, as the plasma is heated by the radiation drive. For a homogeneous plasma charge neutrality demands that, n e = Z zn z (6.7) z= where z is the charge of the individual ions. This leads to the simple relation for the average charge state of a partially ionized single-element plasma to be, Z = n e n a (6.8) Where n e is the electron number density and n a is the total atom number density of the species. The LTE trace of Figure 6.4, shown in red, uses tabulated values that are

203 82 generated using the assumptions of thermodynamic equilibrium. Namely, that the atomic rates are dominated by collisions and that the ratio of population densities of two states of an ion, i and j, where i < j, with energies E i and E j is determined by the Boltzmann distribution, n i = g [ i exp E ] i E j n j g j k B T (6.9) and the population distribution of adjacent ions is given by the Saha equation, n z+ n e = 2 g [ z+ λ 3 B n z g exp E ] z z k B T (6.2) Helios-CR also has the capability of using a non-lte atomic level model where the atomic level populations are solved by collisional-radiative, time-dependent atomic rate equations This was described previously and is represented by equation 6.5. The green trace labelled NLTE shows the Z for simulations completed using this model. As can been seen from the traces the qualitative behavior between the LTE and NLTE simulations are very similar throughout the time evolution of the simulation. The Z of the NLTE case is consistently lower than the LTE case at every time step, but approaching it at the peak of the radiation drive. A possible explanation of this could be the explicit treatment of the inline time-dependent atomic rates used in the NLTE case. As shown in equation 6.9 the level populations are determined by the degeneracy, energy level structure, and temperature. Where in the NLTE model each atomic process has either a corresponding populating or depopulating rate. In the two stage process of dielectronic recombination a free electron is captured by an ion into an excited state and in doing so also transfers some of its energy to a second bound electron causing it to become excited, leaving the ion in a doubly excited state.

204 LTE NLTE 7 6 Z bar Time[ns] Figure 6.4: Average charge state distribution vs time for an LTE vs NLTE neon plasma This doubly excited ion can then either radiatively stabilize by decaying to a lower level or it may autoionize returning to its original charge state. In the NLTE model dielectronic recombination can be treated using an explicit autoionizing level and an electron capture rate coefficient instead of a dielectronic rate coefficient. The next two Figures 6.5 and 6.6 are both plots of the electron temperature as a function of time. There are two systems employed by these figures. The first, shown in both Figure 6.5 and 6.6, is a system comprising of a layer neon gas that is trapped between two layers of mylar (mylar-neon-mylar). The second contains only the layer of neon, shown in Figure 6.6. The temperature that is shown is always that of unshocked neon. When radiation is incident on a material is has the potential to

205 Electron temperature [ev] Figure 6.5: Average electron temperature in unshocked neon as a function of time with LTE (red) and NLTE (green) treatement for a mylar-neon-mylar plasma launch a shock that travels through the system. In the cases where hydrodynamic motion is not taken into account this shock wave does not occur. For hydrodynamic systems, though, this is a possibility. So to extract our temperatures we look at only the portion of the neon that has not undergone this density perturbation. As can be seen in the mylar-neon-mylar system the temperature is very similar up until the main heating by the radiation drive at 95ns. From here, the LTE temperature ramps up significantly while the NLTE case maintains its steady increase. Looking next at Figure 6.6 and comparing the two systems it can be seen, in both the LTE and NLTE case, the temperatures of the neon only system are substantially higher before the peak of the radiation drive and stays hotter through the peak. This can be explained by the attenuation the radiation achieves when passing through the mylar

206 85 slab. For the neon only case the radiation is unattenuated by the windows and higher temperatures are expected, though no shock or compression heating is occurring from the mylar windows. As was seen with a lower average charge state fromfigure 6.4 the simulation using the NLTE atomic kinetics model generates a lower average electron temperature in the neon. A more detailed and complete set of calculations are shown at the end of this chapter. 5 4 Electron temperature [ev] Figure 6.6: Average electron temperature of unshocked neon gas in mylar-neon-mylar (solid) and neon only system (dashed) as a function of time with LTE (red) and NLTE (green) treatment Figure 6.6 displays the electron temperature in a neon only case. This was done as to most closely resemble the simulation using the Boltzmann model.

207 Simulation parameter explanation Geometrical setup of experimental parameters Planar geometry was chosen due to the slab design of the gas cell. The gas cell can be broken into three slab-like regions. The first region is a thin planar sheet of material which acts as a window transmissive enough for x-rays to penetrate but structurally sound so that it seals the neon gas into the cell. The second region is a block of an ideal gas, specifically, neon. The third region is similar to the first region, another thin planar sheet of material used to trap the gas. Spatial grid and material properties The spatial grid sets up the specific details of the geometry and the way in which the Lagrangian zones will be initialized. The first parameter to specify is the thickness of the slabs and the associated number of zones for the specified thickness. The regions are determined by their distance fromthe origin set as R (min). The first regionis the transmissive window closest to R (min). For the front window.4 micron thick mylar (C H 8 O 4 ) was used. The second region was one centimeter of neon gas followed by another.4 micron layer of mylar enclosing the neon. These values are summarized in table 6.2. Properties of the materials that make up the regions also need to be specified, for each region, the mean atomic weight [amu], a constant mass density [g/cm 3 ], initial temperature [ev], and initial velocity [cm/s].

208 87 Table 6.2: Region spacing Region name R(min)[cm] R(max)[cm]. mylar neon mylar.4.28 Table 6.3: Region parameters Region name mean at.wt.[amu] mass den[g/cm 3 ] init T[eV] init v[cm/s]. mylar neon e mylar The mean atomic weight used for the mylar was calculated as, mylar = C H 8 O 4 C = 2.7[amu] H =.79 O = at.wt. = (2.7) + 8(.79) + 4(5.9994) = = mean at.wt. = 92.68/22 =

209 88 It is important to note that this value is not the one used in table 6.3. The reason for the extra division by two is that the equation of state was calculated for C 5 H 4 O 2. This is appropriate because the proportions are still the same. The mean atomic weights for neon is, neon = Ne = 2.797[amu] The next consideration is for the mass density of the neon in the central region. The table value of is specific to the mass density of neon gas at 3 Torr of pressure. For mass densities at different pressures see table 6.4. Table 6.4: Neon densities Torr Mass Density( MP RT )[g/cm3 ] Number Density[#/cm 3 ].56e-4 3.3e e-5.65e e e e e e e e-6.5e7 These densities were calculated using the ideal gas law. Given a gas at room temperature the state variables of pressure P, volume V, and temperature T are characterized by the relationship, see appendix C: PV = nrt (6.2)

210 89 Here n is the number of moles in the gas which can be found by the ratio of the (Mass)/(Molar Mass) and R is the universal ideal gas constant. Given the specific example of 3 Torr neon gas at room temperature as seen in the table 6.4 the mass density is calculated as, PV = nrt = m M RT P = m RT V M ρ = MP RT = (2.797[g/mol])(3999[Pa]) (8.34[ m3 Pa Kmol ])([29K]) = 3.347e-5[g/cm 3 ] Hydrodynamic fluid zones Two zoning models were tested: automatic zoning and simple zoning. Automatic zoning uses algorithms that maximize the mass matching between zones. This significantly increased computational time without any major deviations in the results. The standard zoning setup can be seen in table 6.5. Table 6.5: Region zoning Zones Thickness[cm] Density[g/cm 3 ] Mass[g/cm 2 ] < mass/zone > [g/cm 2 ]..4e-4.397e+.9558e e-6 2..e e e e e-4.397e+.9958e e-6

211 9 Equation of state and opacity The equation of state (EOS) and opacity information are calculated using the Prism code PROPACEOS. Newer versions of PROPACEOS using a Quotidian EOS were found to be inconsistent with physics results. This QEOS is a methodology designed for hot dense matter at low temperatures [5] and is not suitable for our case. Atomic kinetics model Local thermodynamic equilibrium (LTE) models are used to calculate LTE opacities. Here, the level populations are determined by the Boltzmann and Saha equations. When an NLTE model is chosen a detailed configuration approach is used and the atomic kinetic rate equations relating each specific atomic process is solved for. The variationintreatment oflteandnltephysics wasonlyappliedtoneon. TheMylar was always treated with LTE atomic kinetics, in order to save on computational time and not be the main target of our simulations. Radiation transport model Radiation transport model options are either none, diffusion, or multiangle. The radiation transport can be approximated by a standard diffusion model. This is when the mean free path of the photons is smaller than the path length of the material they are traveling through. This is not the case for many transitions in the specific neon cases we are looking at so the multi-group, multi-angle radiation transport model was be used, with five angles. This model calculates the specific intensities in the + r and - r directions at a frequency ν, a position given by optical depth τ ν, and along a ray defined by the cosine of the angle with respect to the surface normal.

212 9 These intensities are then angle averaged and used to calculate the flux at a given position. In the case of external radiation fields non-zero boundary conditions are applied at τ ν = and τ ν = T ν. There is also the option to change how the frequencies are binned. For our calculation we specified the frequency groups to ensure proper treatment and resolution was given to the photon energy of interest to our specific problem. Table 6.6 is organized such that the number of groups and the upper bound of each group is specified, with a minimum photon energy, hν [ev] of.. Table 6.6: Frequency groups # of groups max hν[ev] e e e+3 Driving radiation source The radiative heating of the plasma due to an external radiation field is simulated using a time and frequency dependent non-planckian radiation field. These drives and the method used to create them are discussed in chapter 4. Imposed time constraints The maximum simulation time can be varied but for this case was set to ns with a maximum of million cycles. The original radiation drive modelled by in reference [65] had experimental data up to ns. This made ns a natural stopping time

213 92 for the simulation. Times beyond this were not relevant to experimental parameters, and are thus not representative of the experiment. The drive extrapolated using the radiation drives from [65] data and using updated experimental variables from more recent Z facility shots as discussed in 4 stop at 6 ns. To keep symmetry with the original drives the simulations were allowed to extend to ns. Table 6.7: Output times beg.time[sec] output every[sec]..e-9 e e-8 5e-9 3..e-7 e-9 4..e-7 5e-9 Simulations output The output mode selected was time-based, with a binary plot file (.bpf) to be read by Hydroplot, a PCS Inc. program for extracting information from these binary files. The time-based output intervals were selected as shown in table 6.7. The table shows, beginning at one nanosecond, data that is output every ten nanoseconds.

214 6.6 Impact of hydrodynamics and atomic models on plasma heating 93 This first important point to note is the sensitivity of the heating due to the different radiation drives. A significant amount of work was made to produce the GPLe drive, as discussed in chapter 4, to most accurately model the radiation seen by the neon gas. Figure 6.8 shows the results of the heating calculations using the two different radiation drives, IMH and GPLe. The IMH drive, using both LTE and NLTE atomic kinetics, is seen to significantly overestimate the temperature calculated using the GPLe drive and the temperature extracted from the experiment of 9 ± 2 ev [6] shown in Figure 6.7 and Table 6.8. As was seen in chapter 4 the radiation drive of IMH has a much larger contribution from the high energy tail of the distribution. Table 6.8: Average electron temperatures for each fill pressure Pressure[Torr] < T e >[ev]. 3 8 ± ± ± 4 The parameters varied to create the following figures from the hydrodynamic simulations are the pressure used to fill the gas cell and the radiation drive seen by the gas cell. Four different variations were carried out on two different material setups. The first arrangement of materials is as close to the experimental arrangement as possible. This is a layer of mylar followed by a body of neon gas followed

215 94 Figure 6.7: Electron temperature as a function of filling pressure. The shot number associated with each data point is also shown. by a final layer of mylar. These two mylar layers are the windows used in the experiment to keep the neon gas enclosed within the cell. The second arrangement of materials takes away the mylar leaving only the section of neon gas. With these two material configurations four variations of the simulations combine how the atomic level populations are treated and if hydrodynamic motion is included. First, the LTE approximation, this as discussed previously, assumed the level and ionic populations are determined by the Saha-Boltzmann equation. Second, following the LTE simulations the level populations are then calculated using the NLTE approximation for the neon gas which employs a collisional radiative model to solve the populations due to

216 95 8 GPLe LTE IMH LTE 8 GPLe NLTE IMH NLTE Electron temperature [ev] Exp: 9±2eV Electron temperature [ev] Exp: 9±2eV Peak drive Peak drive (a) (b) Figure 6.8: Heating from the GPLe (red) and IMH (green) drives using LTE (a) and NLTE (b) atomic kinetics. the rates of individual processes. Within each of these two kinetics assumptions the option to turn, either on or off, the hydrodynamic motion of the system comprising the third and fourth variations. These options are summarized in table 6.9 Table 6.9: Helios-CR run parameters. Closer Further 2. My-Ne-My Ne-only 3. LTE NLTE 4. Hydro No-Hydro Figure 6.9 contains four plots containing the simulations using the LTE assumption for level populations. Within these different models of running the simulations four different filling pressures were used: 3, 5, 7.5, and 3.5 Torr. Each of these four filling pressures are those represented in the plots where the red trace = 3 Torr, green trace = 5 Torr, blue trace = 7.5, and the pink trace = 3.5 Torr. The second parameter available to change is the intensity of the flux on the mylar and neon, this

217 96 is manifest through the two different radiation drives used as discussed in chapter 4, the radiation drive in the closer position (stronger flux) and the radiation drive in the further position (weaker flux) T LTE 5T LTE 7.5T LTE 3.5T LTE 8 7 3T LTE no-hydro 5T LTE no-hydro 7.5T LTE no-hydro 3.5T LTE no-hydro 6 6 Electron temperature [ev] Electron temperature [ev] (a) (b) 8 7 3T LTE Ne 5T LTE Ne 7.5T LTE Ne 3.5T LTE Ne 8 7 3T LTE Ne no-hydro 5T LTE Ne no-hydro 7.5T LTE Ne no-hydro 3.5T LTE Ne no-hydro 6 6 Electron temperature [ev] Electron temperature [ev] (c) (d) Figure 6.9: Plots each containing the same four filling pressures with the four variations of simulations using the closer radiation drive (a) LTE of mylar-neon-mylar, (b) LTE of mylar-neon-mylar without hydro motion, (c) LTE with only a neon slab, (d) LTE with only a neon slab and no hydro motion. The electron temperature labelled by the y-axis in all of the following plots corresponds to the electron temperature of the unperturbed section of neon. This means neon that has not undergone an asymptotic perturbation of its density. This can happen when the intense distribution of photons hits the mylar or the neon. Especially, in this case, when the radiation is incident upon the mylar, these mylar

218 97 layers heat up and send a shock inward through the neon. This shock, however, does not converge till much after the peak of the radiation drive or after the duration of the experimental measurement [65, 22]. This is an important characterization because when for the neon to be classified as photoionized means the density needs to remain low enough that the collisional processes do not overwhelm the radiative processes. When this asymptotic shock front travels through the neon it creates regions of very high density. Again, looking at the Figure 6.9 plots (a) and (b) for the full system (mylar-neon-mylar) and treating neon in LTE with and without hydro motion, respectively, it can be seen that the hydro motion plays a role in the temperature of the electrons. If next, plots (c) and (d) are considered which simulate the neon without the mylar windows it can be seen that the mylar windows are the main cause for the differences seen in the full system treatment. For the LTE cases at the peak of the radiation drive the electron temperatures range around the following values, Table 6.: Closer LTE peak drive electron temperatures Filling pressure[torr] My-Ne-My T e [ev] Ne T e [ev] LTE LTE no hydro LTE LTE no hydro The next series of simulations treated the atomic kinetics by solving the NLTE

219 98 atomic kinetic rate equations. Qualitatively the behavior of the plots are similar to the LTE cases with some scaling considerations. Though, on closer inspection it is seen that the trends associated with the filling pressures are inverted. For the LTE cases, as the filling pressure is increased the electron temperature (around the peak of the radiation drive) drops. Where, in the NLTE cases the electron temperatures increase as a function of filling pressure (around the peak of the radiation drive). This behavior appears to be different than that found in [22] where the electron temperature was found to be invariant of the filling pressure T NLTE 5T NLTE 7.5T NLTE 3.5T NLTE 8 7 3T NLTE no-hydro 5T NLTE no-hydro 7.5T NLTE no-hydro 3.5T NLTE no-hydro 6 6 Electron temperature [ev] Electron temperature [ev] (a) (b) 8 7 3T NLTE Ne 5T NLTE Ne 7.5T NLTE Ne 3.5T NLTE Ne 8 7 3T NLTE Ne no-hydro 5T NLTE Ne no-hydro 7.5T NLTE Ne no-hydro 3.5T NLTE Ne no-hydro 6 6 Electron temperature [ev] Electron temperature [ev] (c) (d) Figure 6.: Plots each containing the same four filling pressures with the four variations of simulations using the closer radiation drive (a) NLTE of mylar-neon-mylar, (b) NLTE of mylar-neon-mylar without hydro motion, (c) NLTE with only a neon slab, (d) NLTE with only a neon slab and no hydro motion.

220 99 Table 6.: Closer NLTE peak drive electron temperatures Filling pressure[torr] My-Ne-My T e [ev] Ne T e [ev] NLTE NLTE no hydro NLTE NLTE no hydro Those simulations were done using a different radiation drive which could explain differences as the neon plasma is sensitive to the radiation drive both to the total intensity and to the photon energy distribution of the drive. Figure 6. shows the calculations using NLTE atomic kinetics. This correlates to Figure 6.9 that used LTE atomic kinetic. The most obvious differentiating difference between the two figures is the overall lower temperature ranges of the NLTE calculations. Since LTE calculation assume Saha-Boltzmann statistics which have the built in assumption of being collisional in nature this is evidence for our plasmas being dominated by radiative processes. Even for the highest density case of 3 Torr the electron temperature is forced down by over ev for the NLTE case. Again, the heavy influence of the windows in the hydrodynamics of the simulations can be seen in plots (a) and (b) of Figure 6.. Where, in the neon only NLTE simulations the hydrodynamics appear to have very little influence upon the electron temperature of the system. The windows inthe system to appear to forcethe electron temperatures of different filling pressures into a tighter group. Where, in the case

221 2 without the windows the electron temperatures of the different filling pressures are more spread out and sectioned off trending higher filling pressures (more electrons) to have higher temperatures T LTE 5T LTE 7.5T LTE 3.5T LTE 8 7 3T LTE no-hydro 5T LTE no-hydro 7.5T LTE no-hydro 3.5T LTE no-hydro 6 6 Electron temperature [ev] Electron temperature [ev] (a) (b) 8 7 3T LTE Ne 5T LTE Ne 7.5T LTE Ne 3.5T LTE Ne 8 7 3T LTE Ne no-hydro 5T LTE Ne no-hydro 7.5T LTE Ne no-hydro 3.5T LTE Ne no-hydro 6 6 Electron temperature [ev] Electron temperature [ev] (c) (d) Figure 6.: Plots each containing the same four filling pressures with the four variations of simulations using the further radiation drive (a) LTE of mylar-neon-mylar, (b) LTE of mylar-neon-mylar without hydro motion, (c) LTE with only a neon slab, (d) LTE with only a neon slab and no hydro motion. Figure 6. shows the plots of LTE calculations using the further radiation drive. All the same variations used in the previous two figures driven by the closer radiation drive wereusedforthethisandfigure6.2. Aswaspreviously shownwiththefurther radiation drive there is not only a drop in total intensity of radiation but by moving the gas cell back the view factor of the gas cell is changed. This in turn changes what

222 2 Table 6.2: Further LTE peak drive electron temperatures Filling pressure[torr] My-Ne-My T e [ev] Ne T e [ev] LTE LTE no hydro LTE LTE no hydro radiating hardware the gas cell sees and there is a different distribution of energies among the photons. This said the qualitative trends between the full mylar-neonmylar simulations and between the neon only system are still evident. In plots (a) and (b) of Figure 6. shows the influence of the mylar windows on this further gas cell position. In both cases the electron temperatures of all filling pressures grouped vary close together. If the temperatures at the peak of the drive are looked at, as in Table 6.2 slight trends in relation to filling pressures may be inferred from the upward and downward temperatures. In the neon only simulations the influence from hydrodynamic motion is very minimal. The final series of plots in the Figure 6.2 use NLTE atomic kinetics for the further gas cell. In the first two plots (a) and (b) we note an inversion of the trends with filling pressure between 95 and ns. Before this period higher filling pressures denote lower electron temperatures and after this period the trend of higher filling pressures is with higher temperatures. This is true for cases of mylar-neon-mylar with and without hydro motion. When considering only neon cases it is evident that as

223 22 Table 6.3: Further NLTE peak drive electron temperatures Filling pressure[torr] My-Ne-My T e [ev] Ne T e [ev] NLTE NLTE no hydro NLTE NLTE no hydro the radiation drive reaches the largest influx of photons the higher the filling pressure the higher the electron temperature. Especially when inspecting the values at the peak of the radiation drive as in Table 6.3, aside from NLTE with no hydro motion, the trends of increasing temperature with increasing filling pressure are evident.

224 T NLTE 5T NLTE 7.5T NLTE 3.5T NLTE 8 7 3T NLTE no-hydro 5T NLTE no-hydro 7.5T NLTE no-hydro 3.5T NLTE no-hydro 6 6 Electron temperature [ev] Electron temperature [ev] (a) (b) 8 7 3T NLTE Ne 5T NLTE Ne 7.5T NLTE Ne 3.5T NLTE Ne 8 7 3T NLTE Ne no-hydro 5T NLTE Ne no-hydro 7.5T NLTE Ne no-hydro 3.5T NLTE Ne no-hydro 6 6 Electron temperature [ev] Electron temperature [ev] (c) (d) Figure 6.2: Plots each containing the same four filling pressures with the four variations of simulations using the further radiation drive (a) NLTE of mylar-neon-mylar, (b) NLTE of mylar-neon-mylar without hydro motion, (c) NLTE with only a neon slab, (d) NLTE with only a neon slab and no hydro motion.

225 24 Chapter 7 Atomic kinetics simulations 7. Introduction In this chapter, attention is given to a series of atomic kinetics calculations made using a Boltzmann electron/atomic kinetics model that (discussed in Chapter 5), the astrophysics code Cloudy [7, 32], and the non-equilibrium, collisional-radiative atomic kinetics code PrismSPECT [3, 8]. The motivation to use these codes to model the atomic kinetics relates to data taken from experimental measurements and to determine whether or not the atomic kinetics are in steady-state. From the data absorption spectra relating to specific line transitions is taken, specifically hydrogen-, helium-, lithium- and beryllium-like neon. These absorption features give information about the atomic ionic level populations, which is what the codes solving the atomic kinetics give. Comparing these correlations can be made of the experimental data against the codes that are used to model the physics that are believed to occur in the experiments. The atomic level structure model used was a reduced detailed configuration accounting model and is discussed in [9] and []. Cloudy is described on its website ( as..a spectral synthesis code designed to simulate conditions in interstellar matter under a broad range of conditions. While

226 25 the Boltzmann model and PrismSPECT were designed with specific interest to the high energy density laboratory community, Cloudy was designed for application in photoionized astrophysical plasmas. This large difference in the development considerations for the codes make them interesting to compare with one another. Being designed for plasmas of different natures make this an interesting study. Cloudy treats hydrogen and helium ions with the greatest precision. It treats the atomic level structure of the hydrogen-like neon ions with nl with n <=. It treats helium-like neon ions in this way also except that it is resolved into singlet and triplet energy terms. It also treats the free electron distribution as Maxwellian differing from the Boltzmann model. The PrismSPECT code written and developed by Prism Computational Sciences, Inc. is used to calculate the atomic kinetic of a test plasma independently of hydrodynamic data. Like Cloudy, PrismSPECT assumes a Maxwellian distribution for the electrons. As mentioned this study was designed to simulate a plasma under conditions similar to the experimental neon gascell plasma created on the Sandia Z-machine. This will give insight into the experimental plasma conditions. The PrismSPECT model gave us the most control of the atomic level structure used and the ability to test sensitivities to this atomic level structure. After much study about the appropriate atomic level structure needed while still accurately representing the plasma and its conditions the following structure was attained [].

227 26 PrismSPECT atomic model []: neutral fully stripped neon ions FS() - H-like(I.P. (ev) 362.6) - 25 levels nl n <= 5,l = spdfg He-like(I.P. (ev) 95.8) - 29 levels snl n <=,l = spdfgh 2lnl n <= 2,l = sp,l = sp Li-like(I.P. (ev) 239.9) - 48 levels s 2 nl n <=,l = spdfgh snln l n <= 2,l = spd,n <= 7,l = spdf Be-like(I.P. (ev) 27.27) - 76 levels s 2 nl n <= 3,l = spd snln l n <= 2,l = sp,n <= 3,l = spd B-like(I.P. (ev) 57.93) - 5 levels s 2 nl n <= 2,l = sp C-like(I.P. (ev) 26.2) - 2 levels s 2 nln l n <= 2,l = sp,n <= 2,l = sp N-like(I.P. (ev) 97.) - 8 levels s 2 nln l n l n <= 2,l = sp,n <= 2,l = sp,n <= 3,l = s O-like(I.P. (ev) 63.45) - levels s 2 nln l n <= 2,l = s,n <= 2,l = sp F-like(I.P. (ev) 4.96) - 3 levels s 2 nln l n <= 2,l = s,n <= 2,l = sp Ne-like(I.P. (ev) 2.56) - levels s 2 2s 2 sp 6

228 27 AllofthecodesuseaNLTEtreatmenttofilltheatomiclevelpopulations. Cloudy and PrismSPECT give SS calculation results, meaning that the time derivative of the solution for each time step is set to zero, so that the results of each time step are independent of one another. The Boltzmann model PrismSPECT give TD calculation results. This allows us to have two SS codes comparisons and two TD code comparisons, where PrismSPECT being able to have the atomic kinetic rate equations solved in both fashions also a more complete comparison to be made between the models. As mentioned in the previous chapter the Boltzmann model calculates, in line, the atomic kinetics and the electron kinetics. Using this method to solve for the electron distribution and use it to a find a temperature gives an estimation of the temperature that uses experimental parameters, making it as close an approximation to the electron temperature since it was solved for directly without a Maxwellian assumption. The atomic kinetic rate equation assumptions require an input electron temperature. In order to model the calculations as closely as possible to the experiment this electron temperature evolution found from the experimental parameter was then used in the Cloudy and PrismSPECT calculations. This now allows all of the parameters required by the atomic kinetics calculations to be representative of data extracted from the experiment: radiation drive conditions, neon sample electron temperature evolution, neon sample size, and neon sample particle densities. Before showing the results from each code it is important to note the results from the work done in Chapter 4. This work to find the most accurate representation of the radiation drive have effects shown here. As mentioned before the IMH radiation distribution has a larger tail contribution to high energy photons. The hydrogen-like neon ionization potential (95 ev) falls on this high energy tail. With a larger distribution of flux above this threshold one would expect, as seen in Figure 7., to see a larger fraction

229 IMH C TD GPLe C TD IMH C SS GPLe C SS time [ns] Figure 7.: The hydrogen-like neon s from the two different drives IMH (red) and GPLe (green) using both the steady-state (dashed) and time-dependent (solid) treatment. of hydrogen-like neon. With the more accurate representation of the radiation drive we can more accurately estimate the level populations of the neon plasma. 7.2 Steady-state results 7.2. Cloudy results The first item to notice in the Cloudy treatment are the initial conditions. Cloudy initializes the populations based on an initial calculation of the incident flux. This is seen by looking at the neutral and fluorine-like populations. The populations in these levels still hold a majority of the initial populations but it is not a purely cold gas,

230 29 which explains the percentage of population in the fluorine-like neon, seen in Figures 7.2 and 7.3. Other simulations like the Boltzmann model, shown later, initialize all the population in the neutral neon atom Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (a) 3.5 Torr (b) 7.5 Torr Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (c) 5 Torr (d) 3 Torr Figure 7.2: Charge state populations as a function of time as calculated by Cloudy using the radiation drive in the closer position. The next major difference is seen in the helium-like and hydrogen-like fractions. Cloudy is a code designed to model astrophysical space plasmas. In the TD Boltzmann model the hydrogen-like fraction is barely noticeable in the plots, where in the high density case of the Cloudy simulations it reaches almost 3 %. Due to this populating of the hydrogen fraction a dip is seen, in the 3.5 Torr closer position case, Figure 7.2 (a),(b) and 7.3 (a) this is seen in the helium-like population. In the

231 2 other cases this is very evident in the lithium-like populations Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (a) 3.5 Torr (b) 7.5 Torr Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (c) 5 Torr (d) 3 Torr Figure 7.3: Charge state populations as a function of time as calculated by Cloudy using the radiation drive in the further position. This information sheds light on the variations of the measurement in the experiment. In the experiment, the measured time-integrated spectra are taken at a time just around the peak of the drive, at ns. As seen from these Cloudy values this is a very dramatically changing time with the lithium-like, helium-like, and hydrogenlike features changing very rapidly. Seeing how it is also the lithium-like, helium-like, and hydrogen-like ions that are measured in the experiment the level of difficult is only increased. Luckily, though, all three ion stages are measure simultaneously to be compared with one another.

232 PrismSPECT SS results The PrismSPECT SS simulations show similarities to the Cloudy results. Like Cloudy, PrismSPECT initializes the populations based on the calculation from the initial radiation distribution Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (a) 3.5 Torr (b) 7.5 Torr Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (c) 5 Torr (d) 3 Torr Figure 7.4: Charge state populations as a function of time as calculated by PrismSPECT using the radiation drive in the closer position. As seen in Figures 7.4 and 7.5, the competition between populating the top three charge states is also very apparent in these calculations. Although, in the PrismSPECT calculations the helium-like fractions maintain a high level of population unlike that of Cloudy where the helium-like fraction is filled quickly and then just as quickly emptied. In the PrismSPECT calculations the helium-like fraction gives its

233 22 population to hydrogen-like and fully stripped neon but then as the plasma cools and the radiation flux lessens the electrons quickly recombine and refill the helium-like fraction. The other interesting case here is the generation of fully stripped ions which were not seen in either of the other calculations Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (a) 3.5 Torr (b) 7.5 Torr Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (c) 5 Torr (d) 3 Torr Figure 7.5: Charge state populations as a function of time as calculated by PrismSPECT using the radiation drive in the further position.

234 Steady-state Li- through H-like ions Now, inspecting the Li-, He-, and H-like population fractions closer and further plotted against one another we can see the specific variations between the calculations a little better. The colors and line styles of the plots are labelled in the figure captions, but each figure includes plots of each of these three charge states from Cloudy and PrismSPECT SS calculations. These ions species are shown because they are view in experimental measurements. As seen in Figures 7.6 and 7.7 all the codes have similar quantitative behavior. For the closer position we see a more pronounced depopulation of helium-like neon intothehydrogen-like ions. Theislesssointhefurtherpositionaswouldbeexpected. With a lower amount of ionizing flux, less hydrogen-like ion would be generated. As a function of time the He-like populations around ns are increasing for all codes as a function of fill pressure with the largest change occurring from PrismSPECT in the closer position at 3.5 Torr with an increase from under 4 % to over 8 % at 3 Torr. The Li-like evolution has the most similarities between the codes. Cloudy predicts the slowest rate of population increase as a function of time but at ns an increase is seen. For PrismSPECT the Li-like populations are below 5 %, and Cloudy at 3 Torr just breaks 5 % of the population in Li-like neon. In the further position, Figure 7.7, the radiation drive is geometrically diluted. The less intense radiation lends to a prediction of less ionizing power. The H- and He-like populations fractions show similar trends to populations seen in the closer positions, only yielding less due to the dilution effect. For He-like populations this is apparent, for example, looking at Cloudy, the populations continually drop over % when the fill pressure increases from 3.5 Torr to 3 Torr. More electrons means the higher probability for collisions

235 24 and recombination effects. These recombination processes suppress the excitation and ionization of electrons into higher levels and charge states. This is even more prominent when there is a smaller amount of radiating flux to drive upward processes. This geometrical dilution effect is also seen in the Li-like fractions. Again, looking at Cloudy for 3 Torr in the closer position at ns this level is just above %, where in the further position it is at just under 3 %. In this further position if there was more ionizing flux exciting these electrons out of the Li-like neon shell the populations would be lower like that of the closer position. It is also made obvious by looking at the H-like fractions that are continually decreasing for both codes at each fill pressure. Being the maximum allowed charge state this populations represents the maximum level of ionization achieved by each calculation.

236 25.9 C Li C He C H P Li P He P H.9 C Li C He C H P Li P He P H (a) 3.5 Torr (b) 7.5 Torr.9 C Li C He C H P Li P He P H.9 C Li C He C H P Li P He P H (c) 5 Torr (d) 3 Torr Figure 7.6: Lithium-, Helium-, and Hydrogen-like neon fractions as a function of time for each code calculation in the closer position: Cloudy(dash), PrismSPECT(dot); Li-like(red), He-like(green), H-like(blue).

237 26.9 C Li C He C H P Li P He P H.9 C Li C He C H P Li P He P H (a) 3.5 Torr (b) 7.5 Torr.9 C Li C He C H P Li P He P H.9 C Li C He C H P Li P He P H (c) 5 Torr (d) 3 Torr Figure 7.7: Lithium-, Helium-, and Hydrogen-like neon fractions as a function of time for each code calculation in the further position: Cloudy(dash), PrismSPECT(dot); Lilike(red), He-like(green), H-like(blue).

238 Time-dependent results The feature of the Botlzmann model and PrismSPECT is their ability to solve the atomic rate equations in a time-dependent fashion. dn k dt = i R ik N i N k R ki +N k F k (7.) i where, in the previous calculations the time derivative of the population density, N k of level k with rates, R, between levels i and k were forced to zero, dn k dt = (7.2) For more information on these equations and the rates associated with them refer to Chapter Boltzmann results The Boltzmann model results in Figures 7.8 and 7.9 show the charge state distributions of each isoelectronic sequence in time for each filling pressure. When first looking at the closer position results for 3.5 Torr and 7.5 Torr are very similar especially for the neutral to nitrogen-like ions. From there, for carbon-like to lithium-like the most noticeable difference is in the maximum populations and the width of the carbon feature. This could be indicative that there are more particles driving recombination in these ions. The effect of the high ionization potential on neon s helium-like shell can also begin to be seen in the lithium-like populations just before ns. The heliumand hydrogen-like features are very similar in all four cases. The change seen in the rate at which the helium-like shell is filled is influenced by the increasing number of electrons.

239 Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (a) 3.5 Torr (b) 7.5 Torr.9.8 Ne F O N C B Be Li.9.8 Ne F O N C B Be Li He H FS He H FS (c) 5 Torr (d) 3 Torr Figure 7.8: Charge state populations as a function of time as calculated by the Boltzmann model and using the radiation drive in the closer position. Now looking at the results of the populations in the gas cell further position, in Figure 7.9 we see a more distinct and marked difference between each of the fill pressures. Again, in the 3.5 and 7.5 Torr case the results are qualitatively very similar, though in the further case the populating and depopulating of each ion stage is more distinctly individual and sequential in the early ion stages. The bunching up against the closed shell helium-like neon ion is also very more apparent as the filling pressure increases, especially in the lithium-like fraction at 3 showing the population being quickly depopulated. Investigating these top ion stages more closely, like was done with the SS res-

240 Ne F O N C B Be Li He H FS Ne F O N C B Be Li He H FS (a) 3.5 Torr (b) 7.5 Torr.9.8 Ne F O N C B Be Li.9.8 Ne F O N C B Be Li He H FS He H FS (c) 5 Torr (d) 3 Torr Figure 7.9: Charge state populations as a function of time as calculated by the Boltzmann model and using the radiation drive in the further position. ults, we look at the Li-, He-, and H-like neon populations fractions. As can be seen by looking at Figures 7. and 7. the TD results are very tell by the minimal change between the codes in both the helium- and hydrogen-like neon fractions. The level that the lithium-like neon fraction is able to achieve is very similar between the codes. The difference in the ion lies in how quickly it ionizes. For lower filling pressures the Boltzmann model ionizes much faster than the PrismSPECT model. The PrismSPECT model using the more detailed treatment of the atomic kinetics accounts for the difference. Where the Boltzmann model uses a simple detailed configuration averaged representation of the energy levels. The quantitative behavior

241 22.9 P Li B Li P He B He P H B H.9 P Li B Li P He B He P H B H (a) 3.5 Torr (b) 7.5 Torr.9 P Li B Li P He B He P H B H.9 P Li B Li P He B He P H B H (c) 5 Torr (d) 3 Torr Figure 7.: Lithium-, Helium-, and Hydrogen-like neon fractions as a function of time for each code calculation in the closer position: PrismSPECT(dash), Boltzmann(dot); Lilike(red), He-like(green), H-like(blue). of the results in the further position are very similar to those in the closer position with the main difference being the overall level of the H- and He-like population fraction weighted lower and the Li-like fractions filled more. This is consistent with the geometrical dilution effect of the radiation drive.

242 22.9 P Li B Li P He B He P H B H.9 P Li B Li P He B He P H B H (a) 3.5 Torr (b) 7.5 Torr.9 P Li B Li P He B He P H B H.9 P Li B Li P He B He P H B H (c) 5 Torr (d) 3 Torr Figure 7.: Lithium-, Helium-, and Hydrogen-like neon fractions as a function of time for each code calculation in the further position: PrismSPECT(dash), Boltzmann(dot); Li-like(red), He-like(green), H-like(blue).

243 222 Turning to the results of these time-dependent (TD) solutions we look at the next four Figures, , that show the four fill pressure results of each charge state as a function of time. We investigate PrismSPECT more closely due to the high level of resolution in the atomic level structure used compared to the other codes. Figure 7.2 shows the average charge state, Zbar, and the first five charge states neutral to carbon-like neon for the closer position. The simulations all used the same parameters as the SS calculation including electron time history and atomic level structure, labelled atm2 in the figure plot legends. Looking first at Figure 7.2(a) for each step in time the level of ionization is lower as an inverse function of fill pressure until just before ns when the neon reaches the hard to break closed shell of neon and all fill pressures plateau. This inverse relation of the average charge state is explainable by the number of particles in the neon plasma. While there is a source of flux exciting and ionizing these particles these particles are also constantly colliding and recombining with one another impeding the upward processes. This makes it so that the higher density of a material the more work it will require to become ionized, given the same radiation source. This idea is also seen in the other charge states, for both the closer and further positions. Looking at a particle charge state, 7.2 as an example, shows 3.5 Torr peaks at 7 ns, 7.5 Torr at 23 ns, 5 Torr at 32 ns, and 3 Torr at 4 ns. It is also evident that the level of each of these successive filling pressures peak is continually dropping. With the level peaking at early time illustrates that it is easy to ionize and reach the peak level. Figures 7.2 and 7.3 show the atomic level populations as a function time for each fill pressure in the closer position while Figures 7.4 and 7.5 are for the further position. It is seen in the Z- bar plots of the average charge state distribution that at each step in time, till the

244 223 plateau at ns, the lower filling pressure correlates to a higher average charge state. For a more specific example illustrating this behavior we can look at the O-like neon fraction seen in plots (d) of Figures 7.2 and 7.4. The lowest filling pressure in both these cases peaks before every other filling pressure. The smaller amount of particles account for a lack of recombination effects that would suppress the ionization of this charge state.

245 Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Zbar (a) Zbar (b) Neutral Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD (c) F-like (d) O-like Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD (e) N-like (f) C-like Figure 7.2: PrismSPECT time-depedent calculations using the closer position radiation drive and filling pressures in Torr: 3.5(red), 7.5(green), 5(blue), 3(pink). Zbar, Neutral through Carbon-like neon.

246 Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD (a) B-like (b) Be-like Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD (c) Li-like (d) He-like Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD (e) H-like (f) Fully stripped Figure 7.3: PrismSPECT time-dependent calculations using the closer position radiation drive and filling pressures in Torr: 3.5(red), 7.5(green), 5(blue), 3(pink). Boron-like through fully stripped neon.

247 Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD 8.8 Zbar (a) Zbar (b) Neutral Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD (c) F-like (d) O-like Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD (e) N-like (f) C-like Figure 7.4: PrismSPECT time-dependent calculations using the further position radiation drive and filling pressures in Torr: 3.5(red), 7.5(green), 5(blue), 3(pink). Zbar, Neutral through Carbon-like neon.

248 Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD (a) B-like (b) Be-like Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD (c) Li-like (d) He-like Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD Torr atm2 TD 7.5 Torr atm2 TD 5 Torr atm2 TD 3 Torr atm2 TD (e) H-like (f) Fully stripped Figure 7.5: PrismSPECT time-dependent calculations using the further position radiation drive and filling pressures in Torr: 3.5(red), 7.5(green), 5(blue), 3(pink). Boron-like through fully stripped neon.

249 7.4 Steady-state & time-dependent comparisons 228 All of the TD simulations were initialized using the SS populations as calculated by the SS simulations at each of their initial times. The populations generated at the closer position are shown in Figures 7.7 through The further position results are shown in Figures 7.29 through Each of the figures contain four plots, where each plot is indicative of a different filling pressure. The first Figures 7.7 and 7.29 show timeevolution of Z. The first difference toobserve isthat theydiffer throughthetime history of the drive. This difference shows that there are significant time-dependent effects that are effecting the level populations in this experiment. These transient effects are seen because of the time scales at which different events are occurring. There is the evolution of the radiation drive incident on the neon plasma and there is the hydrodynamic and environment time scales of the plasma. Then within the plasma there are the time scale of each of the individual processes of the atomic kinetics. Closely related to this are the relaxation time scales of the free electrons which are ionized out of the neon ions. All of these events, with their associated time scales interactinsuchaway tocreateaverydynamic andcomplicated system. Thisis the reason for the need of both experimental results and verification and confirmation by model calculations made across many different aspects of the physics involved in the experiment. A convenient result comes around the peak of the radiation drive. As the fill pressure is increased the differences between the TD and SS results decrease. The effects in this experiment can be simplified to expressions of radiation effects and collisional effects. With the purpose of creating a photoionized plasma the goal is to maximize the radiation effects. Each of the different filling pressures, plots (a)-(d), use the same radiation drive. That means, in the simplified view, that for each plot

250 229 the effects from radiation are the same and all that is changing are the effects from collisions. Early in time then when the radiation field is weak collisions dominate the level populationsandastheradiationfieldincreases itsplaysmoreofaroleinthelevel populations. Late in time as the filling pressure increases the differences between TD and SS level populations are minimal suggesting a collisional effects being minimized by the radiation effects lending to the definition of a photoioinzed plasma. The second noticeable effect is the time scales that each ion stage burns through, meaning all the electrons comprising a particular ion stage get ionized out of their potential. The SS cases always begin to ionize sooner. This is seen both in the Z-bar plots and all ion stages, though the effect becomes minimized at higher fill pressures as seen in the He-like neon plots of Figures 7.25 and The time difference between the SS and TD ionization fronts, leading increasing slope of ionization, decreases both with increase ionization stage and with fill pressure. The decrease in time for ionization stages can be accounted for by the increasing level of radiation from the pinch, while the decrease in time with fill pressure can be accounted for by remembering that the more collisional a plasma is the more quickly it is able to relax into a steady state solution. The most applicable result to the experiment is seen in the values of the H-like population fraction. Seen in Figures 7.26 and 7.38 the SS solutions vary from around 4 % at 3.5 Torr to 5% at 3 Torr. The TD solutions, however, at the peak of the drive maintain a steady half percent. As will be seen in Chapter 8 the H-like fraction is more consistent with the TD results than the SS results.

251 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Zbar Zbar Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Zbar Zbar Figure 7.6: Steady-state and time-dependent calculations using closer position radiation drive, Zbar. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.7: Steady-state and time-dependent calculations using closer position radiation drive, neutral neon.

252 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.8: Steady-state and time-dependent calculations using closer position radiation drive, F-like. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.9: Steady-state and time-dependent calculations using closer position radiation drive, O-like.

253 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.2: Steady-state and time-dependent calculations using closer position radiation drive, N-like. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.2: Steady-state and time-dependent calculations using closer position radiation drive, C-like.

254 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.22: Steady-state and time-dependent calculations using closer position radiation drive, B-like. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.23: Steady-state and time-dependent calculations using closer position radiation drive, Be-like.

255 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.24: Steady-state and time-dependent calculations using closer position radiation drive, Li-like. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.25: Steady-state and time-dependent calculations using closer position radiation drive, He-like.

256 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.26: Steady-state and time-dependent calculations using closer position radiation drive, H-like. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.27: Steady-state and time-dependent calculations using closer position radiation drive, fully stripped neon.

257 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Zbar Zbar Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Zbar Zbar Figure 7.28: Steady-state and time-dependent calculations using further position radiation drive, Zbar. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.29: Steady-state and time-dependent calculations using further position radiation drive, neutral neon.

258 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.3: Steady-state and time-dependent calculations using further position radiation drive, F-like. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.3: Steady-state and time-dependent calculations using further position radiation drive, O-like.

259 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.32: Steady-state and time-dependent calculations using further position radiation drive, N-like. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.33: Steady-state and time-dependent calculations using further position radiation drive, C-like.

260 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.34: Steady-state and time-dependent calculations using further position radiation drive, B-like. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.35: Steady-state and time-dependent calculations using further position radiation drive, Be-like.

261 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.36: Steady-state and time-dependent calculations using further position radiation drive, Li-like. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.37: Steady-state and time-dependent calculations using further position radiation drive, He-like.

262 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.38: Steady-state and time-dependent calculations using further position radiation drive, H-like. 3.5 Torr atm2 SS 3.5 Torr atm2 TD 7.5 Torr atm2 SS 7.5 Torr atm2 TD Torr atm2 SS 5 Torr atm2 TD 3 Torr atm2 SS 3 Torr atm2 TD Figure 7.39: Steady-state and time-dependent calculations using further position radiation drive, fully stripped neon.

263 242 Chapter 8 Calculation of ionization parameter Once the ionizing flux from a source is known, whether by finding the color and brightness temperatures from an inferred charge state distribution and ion density or from an estimation of the photon energy dependent distribution, a simple estimation to determine the relative importance between collisional and photoionization processes of plasma is given by ionization parameter [2, 26]. ξ = 4πF n e (8.) Where ξ is in units of erg cm s, F is the radiation energy flux in erg cm 2 s, and n e is the electron number density in cm 3. For more discussion about rates and the information that follows see Chapter 2. This balance between collisional process and radiative processes includes the photoionization rate, β i, the collisional ionization rate, C i, and the recombination rates α i+ (for both dielectronic and radiative) [3, 4, 5]. n e C i n i +β i n i = n e α i+ n i+ (8.2) Where here, n e and n i,n i+ are the electron number density and ion number density of the i th or i th + ion charge state, respectively. In more conceptual terms in local thermodynamic equilibrium collisional processes are the dominant atomic processes. The electron density must behigh enough that anionin anexcited statehas a greater

264 243 chance of returning to the ground state through collisional processes rather than through non-collisional processes. For coronal equilibrium the upward transitions are assumed to be caused by collisions between electrons and ions but the downward transitions are caused by spontaneous emission and not by collisional de-excitation. The two models derived from equation 8.2 are collisional ionization equilibrium (CIE) and photoionization equilibrium (PIE). For CIE it is assumed that the ionization caused by radiation is negligible and so the radiation term, β i, in equation 8.2 goes to zero. In contrast, for PIE radiation dominates the ionization events so that n e C i β i. Difficulties arise when comparing laboratory astrophysical plasmas with space plasmas due to the extremes of their different density regimes. For example, threebody recombination is also an inverse process of electron collisional ionization, but is generally only important at high densities. One of the purposes of all the previous chapters is to characterize the conditions of this particular laboratory astrophysical plasma so that appropriate comparisons can be made. 8. X-ray drive intensity profiles The x-ray drive intensity profiles integrated over a photon energy range seen by the gas cell in the closer and further positions are shown in Figure 8.. As would be expected the flux from a gas cell in a closer position is always higher. Moving the gas cell back generates a smaller view factor solid angel for the gas cell windows to see, thus having less photons pass onto it. This is classified as the geometry dilution effect.

265 244 e+2 Closer Further e+9 e+8 Flux [erg*cm 2 /s] e+7 e+6 e+5 e+4 e Figure 8.: Closer and further position photon-energy integrated flux profiles: closer (red), and further (green) 8.2 Electron density profiles The electron density profiles as functions of time are shown in Figure 8.2. This particular set of traces were calculated using the Boltzmann model as it gives the electrons the most detailed treatment. The neon atom having bound electrons each atom number density with its associated filling pressure has the potential to free possible electron from each atom for a total of times this atom number density total possible free electrons. This means the higher the filling pressure the more potential for a higher total electron number density. This is what is seen in the figure where the electron number density increases with filling pressure and the rate of increase of each filling pressure also increases.

266 245 9e+8 8e+8 3 Torr F 5 Torr F 7.5 Torr F 3.5 Torr F Electron number density [cm -3 ] 7e+8 6e+8 5e+8 4e+8 3e+8 2e+8 e Figure 8.2: Electron density profiles for each filling pressure in Torr: 3.5 (pink), 7.5 (blue), 5 (green), 3 (red) If we use the values of intensity and electron number density from these two plots we can use these to visualize the ionization parameter to gauge the relative importance of photoionization processes and collisional processes. 8.3 Time evolution of ionization parameter The ionization parameter calculated from the results of the Boltzmann model, Cloudy simulations, and PrismSPECT simulations are shown in Figures 8.3, 8.4, and 8.5, respectively. It is clear that the further position ionization parameters always fall below that of the closer position. This is due to the fact that the intensity of the ionizing flux is always lower in the further position. The case could also be made

267 246 for the electron density, but even in this case the amount of free electrons increasing the total electron number density as a function of time increases with with large radiation as seen from the results from the Boltzmann model in Chapter 5. As the filling pressure increases the ionization parameter decreases. With the radiation field held constant in the case of the closer and further positions, respectively, the only change is in the denominator of the ionization parameter. These are easily understood behaviors based on the details of the ionization parameter equation. Other laboratory astrophysical experiments have reported ionization parameters ranging from 6-25 erg cm s [6, 7, 8, 9, 2, 2, 22, 23]. Ionization parameter [erg*cm/s] Torr C 5 Torr C 7.5 Torr C 3.5 Torr C 3 Torr F 5 Torr F 7.5 Torr F 3.5 Torr F Time[ns] Figure 8.3: Boltzmann model ionization parameter results for each filling pressure in the closer (solid) and further (dashed) positions. Where 3.5 Torr (pink), 7.5 Torr (blue), 5 Torr (green), and 3 (Red) The individual charge state distributions as a function of time for each code vary widely, but the average charge state distributions are very similar. Each code also uses the same radiation drive distribution so as one would expect the variations in

268 247 Ionization parameter [erg*cm/s] Torr C 5 Torr C 7.5 Torr C 3.5 Torr C 3 Torr F 5 Torr F 7.5 Torr F 3.5 Torr F Time[ns] Figure 8.4: Cloudy ionization parameter results for each filling pressure in the closer(solid) and further(dashed) positions. Where 3.5 Torr (pink), 7.5 Torr (blue), 5 Torr (green), and 3 (Red) the ionization parameter are not very different, both qualitatively and quantitatively. Each code gives values from 7 to 4 at the peak of the radiation drive. Looking at each individual filling pressure in Figure 8.6 the relative differences between each code can be seen. There is little difference between ionization parameter values, and even less so as the fill pressure is increased. As the fill pressure decreases in the closer positions the PrismSPECT results deviate from the other two, with the biggest differences between the codes occurring at the lowest filling pressure. The differences in the further position values are not as obviously related to the filling as the differences between the codes are apparent and similar at every fill pressure.

269 248 Ionization parameter [erg*cm/s] Torr C 5 Torr C 7.5 Torr C 3.5 Torr C 3 Torr F 5 Torr F 7.5 Torr F 3.5 Torr F Time[ns] Figure 8.5: PrismSPECT ionization parameter results for each filling pressure in the closer(solid) and further(dashed) positions. Where 3.5 Torr (pink), 7.5 Torr (blue), 5 Torr (green), and 3 (Red)

270 B 3.5 Torr Cl C 3.5 Torr Cl P 3.5 Torr Cl B 3.5 Torr F C 3.5 Torr F P 3.5 Torr F 7 6 B 7.5 Torr Cl C 7.5 Torr Cl P 7.5 Torr Cl B 7.5 Torr F C 7.5 Torr F P 7.5 Torr F Ionization parameter [erg*cm/s] Ionization parameter [erg*cm/s] (a) 3.5 Torr (b) 7.5 Torr 35 3 B 5 Torr Cl C 5 Torr Cl P 5 Torr Cl B 5 Torr F C 5 Torr F P 5 Torr F 8 6 B 3 Torr Cl C 3 Torr Cl P 3 Torr Cl B 3 Torr F C 3 Torr F P 3 Torr F Ionization parameter [erg*cm/s] Ionization parameter [erg*cm/s] (c) 5 Torr (d) 3 Torr Figure 8.6: The ionization parameter as a function of time for each filling pressure from each of the atomic kinetics calculations: Boltzmann (B), Cloudy (C), PrismSPECT (P), in the closer (Cl) and further (F) positions

271 H-like neon and the ionization parameter The hydrogen-like neon population fraction is a value that we experimentally measure [3]. This makes it ideal to test against our atomic kinetics simulations. With the ionization parameter a calculation of the intensity and the electron number density we can also verify these values experimentally and use them in simulations. With these results we have a full experimental and computational suite of values to test the ionization parameter..3 PrismSPECT.25 H-like Neon fraction Ionization parameter [erg*cm/s] Figure 8.7: Steady-state calculation data (black) using PrismSPECT along with a polynomial fit to the calculation data (red) Both PrismSPECT and Cloudy output the hydrogen like level population. These populations are then taken and plotted against the experimental values of intensity and electron number density to calculate the ionization parameter. Figure 8.7 shows

272 25 the hydrogen-like values (black points) plotted against these ionization parameter values. These values are then fit with a polynomial fit to get a better understanding if any trends exist in the calculation data..3 Cloudy.25 H-like Neon fraction Ionization parameter [erg*cm/s] Figure 8.8: Steady-state calculation data (black) using Cloudy along with a polynomial fit to the calculation data (green) Figure 8.8 shows the results as calculated by Cloudy. The green trace in this figure is representative of the polynomial fit to the Cloudy calculation data. Both the PrismSPECT and Cloudy calculations were ran using the electron temperatures as calculated by the Boltzmann model and the side power adjusted intensities and fill pressures of the respective shots. Figures 8.7, 8.8, and 8.9 all use steady-state solutions to the atomic kinetic rate equations. As can be seen from Figure 8.9 the steady-state results are all above the values extracted from the experimental data (fit

273 252 to experimental points is the black-dashed trace)..3 Experimental data PrismSPECT Cloudy.25 H-like Neon fraction Ionization parameter [erg*cm/s] Figure 8.9: The fits to the calculations, PrismSPECT (red) and Cloudy (green), along with the experimental data points and the fit to those experiment data points(black-dashed) [3] PrismSPECT, also has the capability of solving the time-dependent solutions to the atomic kinetics. Figure 8. shows the steady-state results (red) and the timedependent results (green). It is clear from the figure that solving the time-dependent solutions more closely approaches the values of the experiment. This is more evidence towards a strong transient behavior in our experiment. As was seen previously the electron kinetics, very closely go through, a series of steady-state distributions. This however does not negate the transient effects in the atomic kinetics.

274 253.3 Experimental data PrismSPECT TD PrismSPECT SS.25 H-like Neon fraction Ionization parameter [erg*cm/s] Figure 8.: Time-dependent (green) and steady-state (red) PrismSPECT calculations along with the experimental data fit (black-dashed) [3]

275 254 Chapter 9 Conclusions In this dissertation we have discussed a series of modeling studies to understand the heating and ionization of a laboratory photoionized plasma experiment performed at the Z facility of Sandia National Laboratories. To this end, we have used several modeling tools: () a view-factor computer code constrained with x- ray power and gated monochromatic image measurements of the z-pinch to model the time-history of the photon-energy resolved x-ray flux driving the photoionized plasma, (2) a Boltzmann self-consistent electron and atomic kinetics model to simulate the electron distribution function and configuration-averaged atomic kinetics, (3) a radiation-hydrodynamics code with inline non-equilibrium collisional-radiative and radiation transport atomic kinetics to perform a comprehensive numerical simulation of the experiment and plasma heating, and (4) steady-state and time-dependent non-equilibrium, collisional-radiative atomic kinetics with non-equilibrium radiation transport calculations including fine-structure energy level description to assess transient effects in the ionization and charge state distribution of the plasma. Very little is known about the basic properties of photoionized plasmas since the majority of the work done on laboratory plasmas has been for the case of plasmas driven by a distribution of particles. Hence, the modeling and simulation results presented

276 255 here provide key information and guidance to both test theory assumptions and approximations as well as to develop confidence on data interpretation and analysis. Also, the comparison with preliminary experimental results suggests the importance of non-equilibrium collisional-radiative atomic kinetics in the plasma heating and of atomic-kinetics transient effects on plasma ionization. The first step is to characterize the photon-energy dependent x-ray drive impinging on the front window of the gas cell. This part of the work is discussed in Chapter 4. Since, in the experiments, the gas cell is located outside the return current cannister at 4.3 cm and 5.9 cm from the pinch axis, modeling the x-ray drive requires accounting for details of the x-ray emission of the z-pinch and the re-radiation effect of the pieces of hardware surrounding the pinch. This calculation is done with a view-factor code that considers the geometry of the actual setup employed in the Z experiments further constrained by x-ray side power and monochromatic image measurements. The result is a time-history of photon-energy resolved x-ray drives that is not characteristic of a sequence of single Planckian or geometry-diluted Planckian distributions. Instead, each time-dependent photon distribution is very well approximated by a linear combination of three geometry-diluted Planckian distributions that follow the time-evolution of the x-ray flux during a time interval of ns, including the run in and collapse phases of the pinch. The latter produces the brightest burst of x-rays that lasts for approximately 6 ns, during which, the plasma achieves the highest ionization. In this representation, the two Planckian distributions with the lowest radiation temperatures (color or spectral) can be interpreted as representative of the re-radiation effect of the hardware close to the pinch while the Planckian with the largest radiation temperature represents the radiation contribution from the pinch itself. These results also show the non-negligible contribution of hardware re-

277 256 radiation effects to the x-ray flux driving the plasma. In the experiments considered in this work, a careful estimation of the photon-energy dependence of the x-ray drive has turned out to be critical. Indeed modeling of plasma heating and ionization performed with an alternative x-ray flux distribution characteristic of almost the same brightness temperature (i.e. energy content) but different photon-energy distribution produces significant changes in both plasma heating and ionization. The properties of the energy distribution function of electrons in a photoionized plasma is a problem that has not yet been addressed in published literature. In a photoionized plasma, electrons are produced by direct photoionization and resonant photoexcitation followed up by spontaneous autoionization. These atomic processes are driven by the intense, external x-ray flux that creates and sustains the plasma. The non-thermal photoelectrons are then expected to thermalize by electron-electron elastic scattering while at the same time undergoing a number of inelastic processes that affect the atomic kinetics and charge state distribution of the plasma, including electron collisional excitation and de-excitation, electron collisional ionization and recombination, free-free Bremsstrahlung emission and free-free inverse Bremsstrahlung absorption, photoionization due to plasma self-emission and electron radiative recombination, autoionization and resonant electron capture, and photoexcitation driven by plasma self-emission and spontaneous radiative decay. This intertwined collection of processes changes both the energy state population and total number of free electrons in the plasma. We attack this problem in Chapter 5 by simultaneously and self-consistently solving the Boltzmann equation for the electron kinetics and a system of kinetic rate equations for the atomic kinetics. We refer to this model as the Boltzmann model. Under the assumptions of a uniform and optically thin (to its selfemission radiation) plasma, the model calculation tracks in time the evolution of the

278 257 electron distribution as it is being shaped by all these processes. The results indicate that the electron distribution function evolves through a series of equilibrium states characterized by a Maxwellian distribution with a high-energy tail that accounts for no more than.2 % of the population of free electrons. This is an important result since it justifies the concept of a single electron temperature to represent the vast majority of electrons in the photoionized plasma. It also supports the application of a radiation-hydrodynamic model to simulate photoionized plasmas where the electrons are characterized by a temperature and assumed to evolve in time through a sequence of equilibrium states. An analysis of the timescales associated with electron-electron thermalization, x-ray drive variation, and photoionization rates further supports this significant finding. These calculations also point to the existence of transient effects in the atomic kinetics, a fact that is confirmed by the detailed atomic kinetic modeling discussed in Chapter 7, and that indicates that the x-ray drive timescale is too short for the atomic kinetics to reach steady state. Furthermore, the electron temperatures produced by the Boltzmann model calculations suggest a relatively cold and thus over-ionized plasma. This is expected for a photoionized plasma where a distribution of photons is supposed to produce and sustain plasma ionization. The actual electron temperature values fall in a range that is consistent with experimental estimates. In Chapter 6 we perform radiation-hydrodynamics simulations of the Z experiment with the Helios-CR code, including inline collisional-radiative atomic kinetics. These calculations represent the most comprehensive effort to model the gas cell experiment at Z since they take into account self-consistently and simultaneously the hydrodynamic expansion, radiation absorption, transmission, and radiation reemission of the front and rear windows, as well as the radiation absorption, heating, and transient collisional-radiative atomic kinetics of the neon gas fill. These simula-

279 258 tions apply to the case of gas cells sealed with mylar windows. In this connection, one important consideration is the possible hydrodynamic perturbation of the neon gas due to the shocks launched into the gas by the expansion of front and rear windows. The simulations confirm that more than 8 % of the gas volume remains hydrodynamically unperturbed (i.e. motionless) at the time of peak x-ray drive, which is also the time when the photoionized plasma is probed by absorption spectroscopy. In an effort to test sensitivities, a series of systematic simulations were done where local-thermodynamic-equilibrium (LTE) vs. transient non-equilibrium, collisionalradiative atomic kinetics for the neon gas, hydrodynamics vs. no-hydrodynamics motion, and only-neon system vs. window/neon/window complete system were checked. The results of the simulations indicate that: () the use of an LTE approximation to model the emissivity and opacity of the neon plasma produces more heating than that obtained using the transient collisional-radiative model (as can be inferred from electron temperature values), (2) the radiation from the small layer of shocked neon close to the window surface contributes to the heating of the unperturbed neon plasma, and (3) the most realistic and complete simulation, including transient collisionalradiative atomic kinetics for neon, the complete system comprised of front window plus neon gas plus rear window, and allowing for hydrodynamic motion produce the best approximation to the electron temperature estimate obtained from data analysis of 9 ev ± 2 ev quite independent of filling pressure, and thus to the modeling of the heating of the photoionized plasma. The Boltzmann model calculations of Chapter 5 point out the presence of timedependent effects in the atomic kinetics. To investigate this issue further, we performed a series of collisional-radiative atomic kinetics simulations with the PrismSPECT code using a spectroscopic quality atomic model (i.e. using fine structure or J-

280 259 level description of the atomic energy level structure), as well as non-equilibrium radiation transport to take into account opacity effects on atomic kinetics. In this way we were also able to remove the optically thin approximation employed in the Boltzmann model. Modeling calculations were performed under the steady-state and time-dependent approximations using the same input time-histories and gas filling conditions. Thus, the comparison of the results from these two calculations represents a clear way to identify the importance of transient effects in the atomic kinetics. Indeed, as can be seen in Chapter 7, this comparison showed significant differences in the evolution of the average ionization state and the ion s. Furthermore, the importance of transient effects in the atomic kinetics is consistent with the comparison between the time-scales associated with the x-ray drive and the photoionization rates. The ionization state shows the characteristic lag, relative to the steady-state case, associated with a fast transient ionization. It is interesting to note that this effect is noticed in the run-in phase as well as at the peak of the x-ray drive, and it is larger for the lowest density gas cell fill. In particular, a transient ionization fraction of H-like neon is observed at the peak of the x-ray drive that is clsoer to the experimental observation while the result from the steady-state calculation significantly overestimates the H-fraction at peak x-ray drive. Finally, in Chapter 8 we discuss the idea of the ionization parameter that is of common use in astrophysics to characterize photoionized plasmas. The range of parameters covered in the Z experiments permits a systematic study of the photoionized plasma over an order of magnitude range in the ionization parameter. In turn, this makes it possible to look at trends in plasma ionization as a function of the ionization parameter. We illustrate this point by looking at the H-like neon ion fraction as a

281 26 function of the ionization parameter. The H-like neon fraction is the ceiling of the charge-state distribution in the plasma and thus a good measure of the ionization of the plasma. While the experimental values are considered preliminary and currently undergoing a refined analysis [24], the comparison with theory and modeling results computed with PrismSPECT as well as the astrophysical code Cloudy show similar qualitative trends but significant quantitative differences. Motivated by gas cell photoionized plasma experiments performed at the Z facility, this dissertation has presented a comprehensive modeling study of the heating and ionization of the plasma produced in those experiments. This is a challenging problem since photoionized plasmas are far-from-equilibrium systems and laboratory research on them is just beginning. Significant progress has been made though, and the results obtained in this dissertation have cut new ground and have been critical to sort out physics effects and guide the interpretation of experimental observations.

282 26 Appendix A Atomic Kinetics A. Distribution equations A.. The Boltzmann Equation Gases can be studied by considering the microscopic action of individual particles or by considering the macroscopic action of the gas as a whole. Macroscopic actions of a gas can be directly measured, but to study the microscopic action of the particles a theoretical model must beused. This model is called the kinetic theory of gases and it assumes that particles are very small in relation to the distance between the particles, and that the particles are in constant, random motion and frequently collide with each other and with the walls of the container. The Boltzmann transport equation describes particle motion in kinetic theory [42, 4, 4]. Derivation The Boltzmann distribution enables us to determine how particles distribute themselves throughout a set of allowed energy levels. This arrangement is based on two factors: the capacity of the energy level and the number of ways the particles can legally arrange themselves in each energy level. Factor : The capacity of the energy level (g), degeneracy, or statistical weight raised

283 262 to the power of the number of particles in the energy level. The statistical weight is the relative capacity of the energy level, or the number of states that have a given energy g n i i (A.) n = number of particles i = energy level Factor 2: The number of ways a particle can distribute itself within an energy level (also known as the number of macrostates). n t! n i! n i+! n i+2!... (A.2) n t = total number of particles n i = number of particles in each energy level As a refresher in statistical macrostates lets look at an example. Four indistinguishable particles distribute themselves in a box with two compartments, one a quarter the size of the other. phase space internal states net probability 34 4! ! 3 3 4! ! 3! ! ! 2! 3 3 4! !! 4 4!.4 4 4! The net probability (W) of each arrangement are the two states (phase space and the internal state) multiplied together which shows the probability that n particles will the particular arrangement. [ ] W = g n n i t! i n i! n i+! n i+2!... (A.3)

284 263 The arrangement we are most likely to see is the arrangement with the highest probability so that is what we want to find. To do this the probability distribution W needs to be maximized using lnw and Stirling s approximation of lnx! = xlnx x. lnw = n i lng i +n i+ lng i lnn t! n i lnn i n i lnn i+...+n i +n i ( = n i +ln g ( i )+n i+ +ln g ) i lnn t! n i n i+ (A.4) (A.5) To solve for a maximum lnw a constant number of particles and a constant total energy of the system needs to be assumed, otherwise over all bound states (infinite) the partition function diverges (The partition function is the weighted sum of the number of ways an atom can arrange its electrons). These two constants are used in order to apply Lagrange s method of undetermined multipliers. () Constant number of particles n i = n t i (2) Constant total energy of the system, where ǫ i is the excitation energy and ǫ is the energy of the system in its ground state. Where E Thermal is the energy in excess that the system would have for all molecules in the lowest energy level. This is shown because at a given temperature the total energy of the system is also some fixed quantity. n ()+n (ǫ ǫ )+n (ǫ 2 ǫ )+... = i (ǫ i ǫ )n i = E Thermal These two condition are imposed for the use of Lagrange s method of undetermined multipliers. We can now determine the maximum probability with the new terms

285 264 multiplier terms added. lnw α n i β n i (ǫ i ǫ ) i i [ = lnw α n i β ] n i (ǫ i ǫ ) n i i i = lnw α n i β n i (ǫ i ǫ ) n i i i lnw = ( [n i +ln g )] i n i n i n i = (n i +n i lng i n i lnn i ) n i = +lng i lnn i n i n i = ln g i n i (A.6) (A.7) (A.8) For each energy level i: = ln g i n i α β(ǫ i ǫ ) n i = g i e α e β(ǫ i ǫ ) (A.9) (A.) To eliminate α consider a sample of given size, we will take Avogadro s number(n A ) for example and also define ǫ i ǫ as χ i, the excitation energy required to raise an electron from one energy level to another relative to the ground state. N A = i n i = e α i g i e β(χ i) e α = Substituting this into equation (A.) yields N A i g ie β(χ i) (A.) n i = N A i g ie β(χ i) g ie β(χ i ) (A.2)

286 265 The summation term shown above is not identified with any one state and can be rewritten to form the ratio of the populations of any two states i and j. n i /g i n j /g j = e β(χ i χ j ), n i n j = g ie β(χi) g j e β(χ i) (A.3) To solve for β the thermal energy due to the motion of the material can be used. Where E is the total energy of the system and E is the energy of the system with all particles in the lowest energy level. We also define the one-dimensional kinetic energy of translation to be kt (For a monatomic ideal gas, each of the three directions (x,y, 2 and z) contribute 2 kt per particle, for a total of 3 2 kt). E E = i n i (χ i ) (A.4) n i = N A i g ie β(χ i) g ie β(χ i ) Combining these two equations for one dimensional translational motion(no rotation) E E = N A i g ie β(χ i) g i (χ i )e β(χ i) i (A.5) The allowed energies can be found by solving Schrodinger equation in a one-dimensional square potential well which yields: ǫ i = i2 h 2 8ma 2 i =,2,3... (A.6) where h is Planck s constant, m is the mass of the particle, and a is the onedimensional track of fixed length (β = kt ). E E = N A i (i2 h 2 /8ma 2 )e β i2 h 2 8ma 2 i i2 h 2 e β 8ma 2 = 2 N A β = 2 RT = 2 N A(kT) = n i n j = g i g j e (χ i χ j )/kt = N A (i2 h 2 /8ma 2 )e β i i2h2 β e 8ma 2 dx 2 h 2 8ma 2 dx (A.7) (A.8)

287 266 This can also be re-written with the partition function U(T), the weighted sum of the number of ways an atom can arrange its electrons. Where N jk is the total number of atoms in a particular ionization state N jk = n jk = (n,j,k /g,j,k ) i g i,j,k e χ i,j,k/kt = (n,j,k /g,j,k )U j,k (T) (A.9) where U j,k (T) = i g i,j,k e χ i,j,k/kt (A.2) A..2 The Saha Equation Now that we have determined the arrangement of particles within set of energy levels with the Boltzmann equation (A.8) we need to find out the arrangement of these particles within different ionization states (relationship of free particles and the bound). We do this by considering two atoms of the same element. The Saha equation [25, 26] calculating the level of ionization is just an extension of the Boltzmann equation (A.8) so we can use this to determine the number ratio of ionization. Derivation Given the Boltzman equation from above and using different energy states A and B with their respective energies ǫ A and ǫ B we have n A n B = g A g B e (ǫ A ǫ B )/kt (A.2) In terms of the two ions mentioned above this transforms to the following equation where we use the notation used by Mihalas: i = excited state, j = ionization, k = species. n i,j+,k n i,j,k = g i,j+,k g i,j,k e (ǫ i,j+,k ǫ i,j,k )/kt Where i and i+ are two atoms of the same element, n i and n i+ are the number densities ofthe two types of atoms, g i andg i+ arethestatistical weights (degeneracy)

288 267 of the ground states of the two ions. Yet this equation is incomplete neglecting the free electrons from ionization. Consider that one of the atoms becomes fully ionized resulting in an ion in the ground state with a free electron in the continuum. This requires and energy of χ I + 2 mv2 = (χ I + p2 2m ), where χ I is the ionization potentional, the energy above the ground level at which the continuum begins and m and v are the mass and speed of the free electron. The statistical weight is fine the way it is but the degeneracy of the final state needs to be adjusted to account for the free electron g(v) = g i,j+,k g electron. Equation (A.2) can now be adjusted. n i,j+,k = g(v) e (χ I,j,k+ 2 mv2 )/kt n i,j,k g i,j,k (A.22) We identify g electron with the number of phase space elements available to the free electron, which, according to quantum statistics is g electron = 2(dxdydz dp xdp y dp z ) h 3 (A.23) Where the factor of 2 accounts for the two possible orientations of the electron spin. We can also choose a volume element to contain only one electron so that the electron density is dxdydz = n e. The momentum volume element can also be rewritten in terms of the electron s speed. Boltzmann s constant(k) is.38x 6 (ergs/k), Planck sconstant(h)is6.62x 27 (ergs s),theelectronrestmassis9.x 28 (grams) dp x dp y dp z = 4πp 2 dp = 4πm 3 v 2 dv (A.24) Equation (A.22) then becomes n i,j+,k = g i,j+,kg electron n i,j,k g i,j,k n i,j+,k = g i,j+,k8πm 3 n i,j,k g i,j,k h 3 e (χ mv I,j,k+ 2 2 )/kt d 3 x e (χ I,j,k+ mv2 2 )/kt v 2 dv

289 268 If using p2 2m for the energy of the free electron use the following equations for a substitution: y 2 = p2 2mkT 2ydy = p dppdp = 2mkTydy mkt If using mv2 2 for the energy of the free electron use the following equations for a substitution: x 2 = mv2 mv 2xdx = 2kT kt dvvdv = 2kT m xdx Now, summing over all final states by integrating over the electron velocity distribution we obtain: n i,j+,k n i,j,k = g i,j+,k g i,j,k 8πm 3 h 3 n e ( 2kT m )3/2 e χ I,j,k e x2 x 2 dx (A.25) Next, evaluating the integral e x2 x 2 dx = π 4 the equation becomes n i,j+,k = g 3/2 i,j+,k 2 2πmkT e χ I,j,k/kT n i,j,k g i,j,k n e h 2 (A.26) Taking the Saha equation one step further you can apply the Boltzmann equation (A.8) to get an expression for the occupation number of any state of ion in terms of the temperature, electron density, and ground state population of ion. Where the value of C I is 4.825x 5 n i,j+,k n i,j,k = g i,j+,k n e g i,j,k C I T 3/2 e χ I,j,k/kT (A.27) A..3 Planck Function A general definition of the Planck function is that it describes the spectral radiance (amount of light that passes through/emitted from a particular area) of electromagnetic radiation at all wavelengths from a black body at a certain temperature. First we need to answer the question what is a black body. First think of an object sitting

290 269 in space Planck s law [43, 27, 28] helps answer the question how do fixed bodies radiate? Looking at this very simply when an object is subject to radiation it heats up and then begins to re-radiate that heat. If this object were a black body it would absorb all the radiation that falls onto it meaning no radiation would pass through it and none would be reflected (a shiny metal is very reflective), thus the term black body. Kirchhoff [29, 3, 3] proved that a body emits radiation at a given temperature and frequency exactly as well as it absorbs the same radiation. This means that if a body could absorb better than it emitted, then in a room full of objects all at the same temperature the body would absorb radiation from the other bodies better than it radiates energy back to them. This means it will get hotter, and the rest of the room will grow colder, contradicting the second law of thermodynamics (law of increasing entropy (a measure of the amount of energy which is unavailable to do work). Derivation: Consider a photon of frequency ν propagating in a direction n inside a box with each side of length L (L x,l y,l z ) and wave vector k = ( ) ( 2π λ n = 2πν ) c n. Also assume that the length of the box is much greater than the wavelength of the photonso that it canbe represented by a standing wave. We also assume that they do not interact and therefore can be separated into three orthogonal Cartesian directions so that the allowable wavelengths are λ i = 2L n i (A.28) The number of nodes inthe wave in each direction x,y,z is forexample, n x = k x L x /2π, since there is one node for each integral number of wavelengths in given orthogonal

291 27 directions. Now, the wave can be said to have changed states in a distinguishable manner when the number of nodes in a given direction changes by one or more. If n i, we can thus write the number of node changes in a wave number interval as n x = L x k x 2π (A.29) Thus the number of states in the three-dimensional wave vector element k x k y k z d 3 k is N = n x n y n z = L xl y L z d 3 k (2π) 3 (A.3) Using the fact that L x L y L z = V, the volume of the container) and that photons have two independent polarizations (two states per wave vector k) we see that the number of states per unit volume per unit of three-dimensional wave number is 2/(2π) 3 which yields d 3 k = k 2 dkdω = (2π)3 ν 2 dνdω c 3 (A.3) we find the density of states (the number of sates per solid angle, per volume, per frequency) to be ρ s = 2ν2 c 3 (A.32) Next we ask what is the average energy of each state? We know from quantum theory that each photon of frequency ν has energy hν, so we focus on a single frequency ν and ask what is the average energy of the states having frequency ν. Each state may contain n photons of energy hν, where n =,,2,... Thus the energy may be E n = nhν. According to statistical mechanics, the probability of a state of energy E n is proportional to e βen where β = /(kt) and k is Boltzmann s constant. Therefore

292 27 the average energy is n= Ē = E ne βen U (β) U (β) = e βen n= = lnu (β) β (A.33) Where U(β) is the partition function, by the formula for the sum of a geometric series U (β) = e βen = e nhνβ = ( e βhν) n= n= (A.34) Thus we have the result Ē = hνe βhν e βhν = hν e hν/kt (A.35) Since hν is the energy of one photon of frequency ν, eq (A.35) says that the average number of photon of frequency ν, n ν, the occupation number, is n ν = [ e hν kt ] (A.36) Equation(A.35) is the standard expression for Bose-Einstein statistics with a limitless number of particles. The energy per solid angle, per volume, per frequency is the product of Ē and the density of states. However this can also be written in terms of u ν (ω) with u ν (ω) = Iν c ( ) sν 2 hν u ν (Ω)dVdνdΩ = c 3 e hν/kt dvdνdωu ν (Ω) = 2hν3 /c 3 e hν/kt (A.37) replacing I ν with B ν we have the relation of Planck s law B ν (T) = 2hν3 /c 2 e hν/kt (A.38) in terms of unit wavelength instead of frequency B ν (T) = 2hc2 /λ 2 e hc/λkt (A.39)

293 272 Appendix B Spectroscopy notation primer When speaking about spectroscopy in this Appendix it will be referring to the spectroscopy terms and definitions as applied to the realm of plasmas or ionized gases. Though a majority of the terms and definitions carry over into other subsets of physics such as atomic and molecular physics this section refers directly to the field of high and low density plasma physics, both through astrophysics and laboratory astrophysics. Spectroscopy here refers to a general technique for analyzing the ionic structure populated by radiation that is either absorbed or emitted from a plasma. The collection of ion edge and level signatures from a plasma is called a spectrum (plural:spectra). Analyzing the spectrum can produce certain parameters of the plasma, such as temperature or density. Historically a nomenclature was introduced by M. Siegbahn [32, 33] in the 92 s used for X-ray spectra. There has recently been an initiative to define a new nomenclature based on the systems used by the International Union of Pure and Applied Chemistry or IUPAC. This appendix will focus on the traditional Siegbahn nomenclature.

294 273 B. Basic definitions A spectrum can be defined as electromagnetic radiation spread across a range of energies or wavelengths and frequencies. The spectrum corresponds to photons or waves where ν is the frequency, λ is the wavelength and the energy is found using Planck s constant h. λν = c E = hν (B.) (B.2) Here, c is the speed of light in vacuum. Another usual definition is to relate temperature and energy which is simply done using Boltzmann s constant k. T = E k (B.3) The elucidation of quantities used to quantify and detail radiation will be saved for a well established and detailed text such as that by Cowan [34]. Generally speaking when radiation interacts with matter it can be classified in two ways: continuous or characteristic. Continuous radiation is generated, for example, by the retardation (Bremsstrahlung) or the accelerations of charged particles in electric and magnetic fields (Synchrotron). Characteristic radiation occurs from the transition of electronic states within an ion. It is these electronic transitions which we wish to plainly establish for use in this Appendix. The hierarchy is as follows with the corresponding splitting terming and quantum number additions below, respectively: Configuration Term Level State Configuration-averaged L-S term split Fine-structure Hyperfine n L and S J M or m j

295 274 As will be stated these hierarchical definitions are characterized by the use and arrangement of principle quantum numbers and their possible values. Table B.: Quantum numbers n l L s S J m M principal (radial) quantum number azimuthal quantum number total orbital angular momentum spin quantum number absolute value of the total electron spin total angular momentum magnetic orbital quantum number magnetic quantum number Here, n, defines the distance of the electron from the nucleus in the Bohr model and the azimuthal angular momentum l describes the shape of the orbits of the electrons in Sommerfeld s generalization of the Bohr model [35]. It is also important to note that the lower case terms refer to individual electrons. L for H-like ions L=l where cases with more than one electron L takes on all possible values of l i. s describes the spin of the electron and is either spin up (+/2) or spin down (-/2). J is the vector sum L and S. m, in Sommerfeld s formulation describes the orientation of the orbits described by l.

296 275 Table B.2: Possible values n l L,2,3,...,n,,2,...,n-2,n- li s -/2,+/2 S s i J m M L-S, L-S+,..., L+S -l,-(l-),...,-,,,...,(l-),l J,J-,...,,...,-J-,-J B.. Configuration The electronic configuration is defined as the distribution of the electrons in the orbitals or shells of an ion. This electronic configuration is defined and labeled by the principle quantum number n and the azimuthal quantum number l, where l can be l =,...,n- and l =,,2,3,4 values correspond to s,p,d,f,g. The two other important quantum numbers for defining the configuration are the magnetic orbital quantum number m or m l, which can take the values -l, -(l-),...,-,,,..., (l-),l, and s the spin quantum number which can either be +/2 or -/2. For example, the ground state configuration of an Oxygen atom is: s 2 2s 2 2p 4 Meaning for s 2 : two electrons fill the n= (l=) orbital, for 2s 2 : two electrons fill the n=2 (l=) orbital, and for 2p 4 : four electrons have started filling the n=2 (l=)

297 276 Table B.3: Orbital configurations breakdown n l m Number of orbitals Orbital name s 2 2 2s 2 -,,+ 3 2p 6 3 3s 2 -,,+ 3 3p 6 2-2,-,,+,+2 5 3d 4 4s 2 -,,+ 3 4p 6 2-2,-,,+,+2 5 4d 3-3,-2,-,,+,+2,+3 7 4f 4 Max number of electrons orbital. Here the terms s,2s,2p are the orbital designations, where the superscript defines the number of electrons within an orbital. B..2 Term Like the configuration the term is defined by a set of quantum numbers, here L, the total orbital angular momentum and S, the absolute value of the total electron spin. Here, like l, L=,,2,3,4 values refer to S,P,D,F,G terms. B..3 Level The level is the set of 2J+ states with specific values of L,S, and J, where J represents the total angular momentum of the ion and is the vector sum of L and S. Example The full description of an atomic level can now be described by: nl X(2S+) L J

298 277 Here X is the number of electrons in that orbital, for example, or 2 for the s orbital, -6 for the p orbital, etc. Above the simple example of ground state Oxygen was shown, but for simplicity we will now look at the Carbon atom which contains two less electrons than Oxygen: Carbon atom = s 2 2s 2 2p 2 Here, both s orbitals (s,2s) are filled leaving two electrons in the valence orbital. These two electrons yield the values: l = l 2 = L = l +l 2 with possible values L = from - to + L =,,2 and S has values S = s +s 2 S =, For the deviation from the configuration-averaged model the L and S terms were added so we look at only the L and S parts of the full description of an atomic level: (2S+) L So for the Carbon atom with L-S splitting we have the following possibilities: We have now gone from configuration averaging to L-S term splitting. The next step is to take into account fine-structure splitting, which from the flow chart above

299 278 Table B.4: LS term splitting L 2 2 S S 3 S P 3 P D 3 D means with need to add the J quantum number into the mix. This takes us to the level declaration terminology. Table B.5: Fine structure splitting L 2 2 S J S 3 S P 3 P 2 D 2 3 D 3 3 S P 3 P 2 D 3 D 2 3 P D 3 D 3 P 3 D From the L-S term splitting terms we have no shown how those terms separate in levels due to fine-structure splitting by looking at quantum number J. After that the levels split into states due to the hyperfine structure of the atom. This comes from the coupling of the magnetic moment of the electron and the nuclear magnetic s 2 2s 2 2p 2 S D 3 P

300 279 s 2 2s 2 2p 2 S D 3 P S D 2 3 P 2 3 P 3 P moment, hence the quantum number m j is now taken into account. B..4 Selection rules In ions there can be a myriad of possible transitions, but not all of these possible transitions are observed. That is because there are rules formulated and observed showing that some transitions are more likely than others. Selection rules, more specifically, electric or magnetic dipole selection rules, were created empirically to describe changes in quantum numbers (transitions) that were observed and to explain why some where not observed (forbidden transition). As stated, selection rules apply to transition and state the probability that a transition will occur. Table B.6: Selection rules L=(not allowed for one electron), +/- l= J=, +/-, except J= is not allowed S= M=, +/-, except M= gives J= which is not allowed

301 28 B..5 KLM notation The next common usage for describing emission transitions is labelled with the K,L,M lettering. As can be seen in the figure below K,L,M,... lettering has a direct connection with the usage of the principle quantum number n. Here K corresponds to n=, L corresponds to n=2, and M corresponds to n=3, etc. Any transition to a level is designated by the principle quantum number n of the level transitioned to. Where the structure is similar, for example, the n=2, L shell is broken up into three sub sections labelled with Roman numerals I,II,III. This follow from the breakdown explicitly shown previously that the n=2, l= p orbital is broken up into three sub orbitals Figure B.: Atomic structure model showing a conceptual model of the KLM notation [4]