Atomic modeling of the plasma EUV sources

We present the development of population kinetics models for tin plasmas that can be employed to design an EUV source for microlithography. The atomic kinetic code is constrained for the requirement that the model must be able to calculate spectral emissivity and opacity that can be used in radiation hydrodynamic simulations. Methods to develop compact and reliable atomic model with an appropriate set of atomic states are discussed. Specifically, after investigation of model dependencies and comparison experiment, we improve the effect of configuration interaction and the treatment of satellite lines. Using the present atomic model we discuss the temperature and density dependencies of the emissivity, as well as conditions necessary to obtain high efficiency EUV power at λ = 13.5 nm.     

Detailed Term Accounting calculations of plasma opacity

Radiative opacity is a key parameter to understanding the transport of energy in high energy density plasmas. Many calculations of opacity apply approximations that do not fully resolve the true line structure of the frequency dependent absorption spectrum and the accuracy of such results is uncertain for simulations containing many complex ions. The DAVROS opacity code has been developed to perform Detailed Term Accounting (DTA) calculations with the aim of resolving the fine structure resulting from the myriad line transitions due to arbitrarily complex atomic configurations. By making use of High Performance Computing (HPC), such calculations, whilst not yet routine, are now more feasible. Some illustrative results from DAVROS are presented and comparisons made with experiment data.     

The effect of unresolved transition arrays on plasma opacity calculations

Unresolved transition arrays (UTAs) are a method of approximating complex atomic physics in plasma opacity calculations, and as such are very important in modern plasma dynamic simulations. In this paper we use full atomic physics calculations to test various UTA models, paying particular attention to the lineshape and its effect on the mean opacity. We find that a Gaussian lineshape is sufficient provided that the line width is correctly determined. This width can be calculated using existing formulae, or approximated by neglecting correlations between term line energies and strengths or the selection rules on term - term transitions. We have quantified transition array narrowing due to correlations for a set of iron transitions and shown that a simple model for these incurs fairly large random errors. The neglect of the selection rules is also seen to result in random errors of up to an order of magnitude. These results may prove very useful in the future development of opacity codes, in particular those intended to run in line with hydrodynamic simulations.     

Simulations of Fe at 156 eV with configuration interaction and mixed UTA

Simulations of the transmission spectra measured by Bailey et al. (Phys. Rev. Lett. 99 (2007) 265002–265004), obtained with the newest version of the HULLAC, are presented. With this new version one can easily define which group of configurations will be computed to synthesize the spectrum. Moreover, one can now compute mixed UTA (MUTA). These modifications provide an extension of the older HULLAC codes. The aim of this work is to compare spectra with different ranges of configuration interactions and with different thresholds for treating separated lines in the MUTA model.     

Modeling of K-shell Al and Mg radiation from compact single, double planar and cylindrical alloyed Al wire array plasmas produced on the 1 MA Zebra generator at UNR

Radiative emission from alloyed Al single, double and compact cylindrical wire arrays have been studied using the 1 MA Zebra UNR generator. Single planar wire arrays using ten wires and double planar wire arrays and compact cylindrical wire arrays (CCWA) that both had sixteen wires were utilized. The wire composition is Al-5056 (95% of Al and 5% of Mg). We have observed that implosion of these alloyed Al wire loads generated optically thick Al plasmas that can be diagnosed using K-shell Mg lines. In particular, among the considered loads, the K-shell lines of Al from implosions of the double planar wire arrays have the highest optical depth for He-like Al resonance transitions, which occurred near the stagnation phase. X-ray time-gated and time-integrated spectra and pinhole images as well as photoconductive detectors signals were analyzed to provide information on the plasma parameters; electron temperatures and densities, implosion dynamics features and power and yields of the X-ray radiation. Previously developed non-LTE models were applied to model axially-resolved time-integrated, as well as time-gated spatially-integrated, K-shell spectra from Al and Mg. The derived time-dependent electron temperature, density and axial opacity were studied and compared. In addition, the wire ablation dynamics model (WADM) was used to calculate the kinetic energy of the plasma, which with the aid of a Local Thermal Equilibrium (LTE) magneto-hydrodynamics (MHD) simulation, allowed to estimate the precursor and stagnated z-pinch plasma electron temperatures from implosions of wire array loads.     

Equation of state calculations for hot,dense matter at arbitrary densities and temperatures

Within the approximations of spherical lattice cell, central-field, and relativistic Fermi statistics, an algorithm with average atom model is presented to calculate the electronic energy levels and equation of state for hot and dense matter at arbitrary densities and temperatures. Choosing Zink's analytical potential as initial potential, we have solved the Dirac-Slater equation which satisfies the Weigner-Seitz boundary condition. The electronic energy bands are not taken into account. Taking energy level degeneracy as a continuous function of density, we have considered the pressure ionization effects for highly dense matter. Results for¹³Al atom are shown.     

A review of new wire arrays with open and closed magnetic configurations at the 1.6 MA Zebra generator for radiative properties and opacity effects

The studies emphasize investigation of plasma formation, implosion, and radiation features as a function of two load configurations: compact multi-planar and cylindrical wire arrays. Experiments with different Z-pinch loads were performed on 1.6 MA, 100 ns, Zebra generator at University of Nevada, Reno. The multi-planar wire arrays (PWAs) were studied in open and closed configurations with Al, Cu, brass, Mo and W wires. In the open magnetic configurations (single, double, triple PWAs) magnetic fields are present inside the arrays from the beginning of discharge, while in closed configurations (prism-like PWA) the global magnetic field is excluded inside before plasma flow occurs. The new prism-like PWA allows high flexibility in control of implosion dynamics and precursor formation. The spectral modeling, magneto-hydrodynamic (MHD) and wire ablation dynamic model (WADM) codes were used to describe the plasma evolution and plasma parameters. Experimentally observed electron temperature and density in multiple bright spots reached 1.4 keV and 5 × 10²¹ cm^?3, respectively. Two types of bright spots were observed. With peak currents up to 1.3 MA opacity effects became more pronounced and led to a limiting of the X-ray yields from compact cylindrical arrays. Despite different magnetic energy to plasma coupling mechanisms early in the implosion a comparison of compact double PWA and cylindrical WA results indicates that during the stagnation stage the same plasma heating mechanism may occur. The double PWA was found to be the best radiator tested at University scale 1 MA generator. It is characterized by a combination of larger yield and power, mm-scale size, and provides the possibility of radiation pulse shaping. Further, the newer configuration, the double PWA with skewed wires, was tested and showed the possibility of a more effective X-ray generation.     

Ignition and growth of a detonation by a high energy plasma

Many experimental and numerical studies have been achieved to describe the transition process of deflagration to detonation when a projectile impacts an explosive. Also a large work has been done for the determination of various parameters — such as the impact pressure, the efficiency factors, etc. of the laser — material interaction. When a laser beam impacts an explosive, the P² criterion, characteristic of shock initiated detonations, is no longer valid due to the generated hot plasma whose effect is to decrease the DDT (Deflagration to Detonation Transition) duration. The present paper deals with a modelling of the plasma-explosive medium allowing the determination of distances and times of the DDT process. The two phase modelling of the granular explosive takes into account the creation of hot spots. The pressure of the plasma is computed using a semi empirical model, while the temperature is obtained from Maxwell Boltzmann statistics. The authors focused their attention on the equation of state for the detonation products and the numerical process.     

Spectroscopic observations of Fermi-degenerate aluminum compressed and heated to four times solid density and 20 eV

Shock waves generated by temporally shaped laser ablation compressed and heated Al to ρ = 11 ± 5 g/cm³ and 20 ± 2 eV. The inferred density and temperature demonstrate that highly compressed, Fermi-degenerate plasma can be created by tuning the temporal pulse shape of the laser drive intensity. The density and temperature of these plastic-tamped Al plasmas in the warm dense matter regime were diagnosed using the Stark-broadened, Al 1s–2p absorption spectral line shapes. These observations represent the forefront of opacity measurements for warm dense matter and are important for high energy density physics and inertial confinement fusion.     

Computation of normal impinging jets in cross-flow and comparison with experiment

The PHOENICS code has been used to model the flow field surrounding subsonic and underexpanded jets impinging on a ground plane in the presence of a cross-flow, for cases with both a fixed ground plane and a ‘rolling road’. The standard k-ε turbulence model is used, without correction factors. It is confirmed that this overpredicts the free jet entrainment rate; the wall jet spreading rate is slightly underpredicted but the initial thickness is too high. Agreement with experiment is, nevertheless, much better than for previous calculations, showing the importance of the extent of the grid used. The ground vortex formed in cross-flow is shown to move with varying effective velocity ratio and with rolling road operation in the same manner as experimentally observed. Ground vortex self-similarity is also accurately predicted with the numerical modelling.     

Transport properties in mixtures involving carbon dioxide at low and moderate density: test of several intermolecular potential energies and comparison with experiment

It is the purpose of this paper to extract unlike intermolecular potential energies of five carbon dioxide-based binary gas mixtures including CO₂–He, CO₂–Ne, CO₂–Ar, CO₂–Kr, and CO₂–Xe from viscosity data and compare the calculated potentials with other models potential energy reported in literature. Then, dilute transport properties consisting of viscosity, diffusion coefficient, thermal diffusion factor, and thermal conductivity of aforementioned mixtures are calculated from the calculated potential energies and compared with literature data. Rather accurate correlations for the viscosity coefficient of afore-cited mixtures embracing the temperature range 200 K < T < 3273.15 K is reproduced from the present unlike intermolecular potentials energy. Our estimated accuracies for the viscosity are to within ±2%. In addition, the calculated potential energies are used to present smooth correlations for other transport properties. The accuracies of the binary diffusion coefficients are of the order of ±3%. Finally, the unlike interaction energy and the calculated low density viscosity have been employed to calculate high density viscosities using Vesovic–Wakeham method.     

Rheological aspects of dense lignite-water suspensions; time dependence, preshear and solids loading effects

An investigation of the rheological properties of dense lignite-water suspensions is reported here in order to evaluate the parameters that affect the rheology of these suspensions. Different types of particle size distributions were achieved via wet grinding with particle sizes down to 1 μm. Polyelectrolytes and surfactants were used to improve the stability and rheology of suspensions. The achieved solid volume fraction, φ, with acceptable rheological behavior was 0.45, while the ratio of φ/φ _m was close to 0.85; φ _m is the maximum solid volume fraction. Various types of flow tests were performed on lignite-water suspensions with various preshear times and stress levels. Typical shear rates varied in the range between 10^–2 to 10² s^–1. The low shear behavior was found to be quite different from the high shear behavior for the most concentrated suspensions, with a plateau value at low shear rates indicating the appearance of a yield stress. These plateau values, however, depend on shear history, which is responsible for the development of different structures in the sample. The viscosity curves corresponding to the ascending and descending parts of the flow curve were found to be different; these flow curve parts can be described either by well established models or by modified ones. The non-Newtonian time dependent behavior of the lignite-water suspensions is attributed to the high value of the ratio φ/φ _m , the polydispersity of the particle size distribution, and the non-spherical shape of the lignite particles.     

Wear Mechanisms of Electrical Steel Sheets,Hard Metal and High Speed Steel Pins Coated with TiAlN and TiCN

The wear behaviour of electrical steel sheets, hard metal plasma nitrided and high speed HSS steel M2 coated with TiAlN and TiCN are investigated and compared, using the pin-on-disk standard test under different loads. The coating process has been performed in an industrial equipment unit for maximised conditions and the coated layers were measured and characterised. The load, sliding distance and velocity were kept constant during the sliding test in order to control these variables. The counterface disks used were electric steel sheets. The lost volume and temperature at the pin end have been measured during the wear test and comparisons for the coated HSS steel and hard metal were done, using a plot of lost volume versus sliding distance for both materials. A brief review of the main abrasion mechanisms of metals is presented. The pin worn surfaces were observed using a Scanning Electron Microscope. Hard metal plasma nitrided has shown superior wear resistance than WC with no surface treatment. The HSS steel coated with TiAlN had a wear resistance similar to the WC.     

Variational Average-Atom in Quantum Plasmas (VAAQP) – Recent progress, virial theorem and applications to the equation-of-state of warm dense Be

The Variational Average-Atom in Quantum Plasmas (VAAQP) code is based on a fully variational theory of dense plasmas in equilibrium in which the neutrality of the Wigner-Seitz ion sphere is not required, contrary to the Inferno model. We report on some recent progress in the VAAQP model and numerical code. Three important points of the virial theorem derivation are emphasized and explained. The virial theorem is also used as an important tool allowing us to check the formulas and numerical methods used in the code. Applications of the VAAQP code are shown using as an example the equation-of-state of beryllium in the warm dense matter regime. Comparisons with the Inferno model, and with available experimental data on the principal Hugoniot are also presented.     

Dynamic compressive strength properties of aluminium foams. Part II—‘shock’ theory and comparison with experimental data and numerical models

One-dimensional ‘steady-shock’ models based on a rate-independent, rigid, perfectly-plastic, locking (r-p-p-l) idealisation of the quasi-static stress-strain curves for aluminium foams are proposed for two different impact scenarios to provide a first-order understanding of the dynamic compaction process. A thermo-mechanical approach is used in the formulation of their governing equations. Predictions by the models are compared with experimental data presented in the companion paper (Part I) and with the results of finite-element simulations of two-dimensional Voronoi honeycombs.A kinematic existence condition for continuing ‘shock’ propagation in aluminium foams is established using thermodynamics arguments and its predictions compare well with the experimental data. The thermodynamics highlight the incorrect application of the global energy balance approach to describe ‘shock’ propagation in cellular solids which appears in some current literature.     

Simulation of elastic wave propagation using cellular automata and peridynamics,and comparison with experiments

Peridynamics is a non-local continuum mechanics formulation that can handle spatial discontinuities as the governing equations are integro-differential equations which do not involve gradients such as strains and deformation rates. This paper employs bond-based peridynamics. Cellular Automata is a local computational method which, in its rectangular variant on interior domains, is mathematically equivalent to the central difference finite difference method. However, cellular automata does not require the derivation of the governing partial differential equations and provides for common boundary conditions based on physical reasoning. Both methodologies are used to solve a half-space subjected to a normal load, known as Lamb’s Problem. The results are compared with theoretical solution from classical elasticity and experimental results. This paper is used to validate our implementation of these methods.     

The back and forth nudging algorithm applied to a shallow water model,comparison and hybridization with the 4D‐VAR

We study in this paper a new data assimilation algorithm, called the back and forth nudging (BFN). This scheme has been very recently introduced for simplicity reasons, as it does not require any linearization, or adjoint equation, or minimization process in comparison with variational schemes, but nevertheless it provides a new estimation of the initial condition at each iteration. We study its convergence properties as well as efficiency on a 2D shallow water model. All along the numerical experiments, comparisons with the standard variational algorithm (called 4D‐VAR) are performed. Finally, a hybrid method is introduced, by considering a few iterations of the BFN algorithm as a preprocessing tool for the 4D‐VAR algorithm. We show that the BFN algorithm is extremely powerful in the very first iterations and also that the hybrid method can both improve notably the quality of the identified initial condition by the 4D‐VAR scheme and reduce the number of iterations needed to achieve convergence. Copyright © 2008 John Wiley & Sons, Ltd.