Collisional-radiative Equilibrium Research Articles

Study of the 6.x nm short wavelength radiation spectra of laser-produced erbium plasmas for BEUV lithography

Beyond extreme ultraviolet (BEUV) radiation with a wavelength of 6.x nm for lithography is responsible for reducing the source wavelength to enable continued miniaturization of semiconductor devices. In this work, the Required BEUV light at 6.x nm wavelength was generated in dense and hot Nd:YAG laser-produced Er plasmas. The spectral contributions from the 4p–4d and 4d–4f transitions of singly, doubly and triply excited states of Er XXIV–Er XXXII in the BEUV band were calculated using Cowan and the Flexible Atomic Code. It was also found that the radiative transitions between multiply excited states dominate the narrow wavelength window around 6.x nm. Under the assumption of collisional radiative equilibrium of the laser-produced Er plasmas, the relative ion abundance in the experiment was inferred. Using the Boltzmann quantum state energy level distribution and Gram–Charlier fitting function of unresolved transition arrays (UTAs), the synthetic spectrum around 6.x nm was finally obtained and compared with the experimental spectrum. The spatio-temporal distributions of electron density and electron temperature were calculated based on radiation hydrodynamic simulation in order to identify the contributions of various ionic states to the UTAs arising from the Er plasmas near 6.x nm.

Probing atomic physics at ultrahigh pressure using laser-driven implosions

Spectroscopic measurements of dense plasmas at billions of atmospheres provide tests to our fundamental understanding of how matter behaves at extreme conditions. Developing reliable atomic physics models at these conditions, benchmarked by experimental data, is crucial to an improved understanding of radiation transport in both stars and inertial fusion targets. However, detailed spectroscopic measurements at these conditions are rare, and traditional collisional-radiative equilibrium models, based on isolated-atom calculations and ad hoc continuum lowering models, have proved questionable at and beyond solid density. Here we report time-integrated and time-resolved x-ray spectroscopy measurements at several billion atmospheres using laser-driven implosions of Cu-doped targets. We use the imploding shell and its hot core at stagnation to probe the spectral changes of Cu-doped witness layer. These measurements indicate the necessity and viability of modeling dense plasmas with self-consistent methods like density-functional theory, which impact the accuracy of radiation transport simulations used to describe stellar evolution and the design of inertial fusion targets.

Моделирование сильно фотоионизованной плазмы кремния

The work demonstrates the process of simulation a strongly photoionized silicon plasma with parameters close to astrophysical. The calculations were carried out within the model of collisional-radiative equilibrium. The obtained results were verified by comparison with the results of other participants of the codes comparison workshop. A comparative analysis of the results with the experimentally measured spectrum of soft X-ray transmission through a layer of photoionized silicon plasma is carried out.

Local measurements of the spatial magnetic field distribution in a z-pinch plasma during and near stagnation using polarization spectroscopy

We present here the detailed measurements of radial distribution of the magnetic field in a gas-puff z-pinch plasma at the final stages of the implosion phase and at stagnation. While the measurements are chordal, the radial distribution of different charge states was utilized to measure the magnetic field locally for certain radii, so that unlike chordal measurements in general, the magnetic field radial distribution was obtained with no need for the Abel inversion of the data. The distribution was measured using the Zeeman effect via a novel spectroscopic technique, at several axial locations, and demonstrates striking features such as the peak field remaining at a radius much larger than the stagnation radius at all times. Furthermore, while the distribution observed is sometimes monotonic with respect to the radius, it is often not, a behavior that can be linked to 2D features in the plasma column resulting from the Rayleigh–Taylor instability. The current flowing through the stagnating plasma was found to be a small fraction of the total current, resulting in clearly insufficient magnetic pressure to balance the plasma pressure at stagnation. The magnetic field data, taken over several axial positions, are used to obtain the true inductance in the imploding plasma for the first time; it is found that the data cannot explain the current turnover at stagnation. A simulation with the MACH2-Tabular Collisional-Radiative Equilibrium magnetohydrodynamics code in the r–z plane shows that the peak of the magnetic field pinches to a much smaller radius than is observed in the spectroscopic data. Furthermore, the depth of the computed current turnover at stagnation is smaller than the measured one. The two observed features of a radially extended magnetic field at stagnation together with a deep current turnover are a challenge to match in simulations. Various calculations and estimates of the inductive and resistive load voltages are examined to ascertain if they are responsible for the observed current notch. The results demonstrate that the knowledge of the true inductance in the driven load requires such magnetic-field-distribution measurements and that imaging data or electrical measurements are insufficient.

Study of the possibility for increasing the emission of soft x-rays from the plasma of a low-energy vacuum discharge triggered by a laser

The spectral parameters of x-ray radiation in the wavelength range from 25 to 450 Å emitted by plasma of a vacuum discharge with a storage energy ≤45 J, induced on the Fe cathode by radiation from a neodymium laser with a power density up to 5 · 1012 W cm−2, are studied. On the basis of the collisional-radiative equilibrium model, a method was developed to produce a sufficiently accurate approximation of the experimental spectrum by a linear combination of the calculated spectra of plasma components with a given electron temperature and density. It was shown that the main part (>80%) of the radiation energy in the experimental spectra is provided by relatively cold peripheral plasma with a temperature of no more than 10 eV and a density of 5 · 1017 cm−3. It was established experimentally that a decrease in the laser power density due to the defocusing of the laser beam leads to an increase in the intensity of plasma radiation in the entire spectral range and a change in the spectral structure in the shortwave range. According to simulations, these changes are caused by an increase in the temperature and density of the micropinch formed in the discharge plasma up to 160 eV and 3 · 1021 cm−3, respectively.

Degeneracy and relativistic microreversibility relations for collisional-radiative equilibrium models.

We present the relativistic expressions of standard nonrelativistic microreversibility relations that can be used in collisional-radiative equilibrium models to calculate the transition rates including the free electron degeneracy for collisional excitation and deexcitation, collisional ionization and three-body recombination, dielectronic capture and autoionization, photoexcitation and photodeexcitation, and radiative recombination and photoionization. Semiempirical expressions or more refined calculations can be used for the cross sections of interest as long as they are calculated by taking into account either nonrelativistic, relativistic, or ultrarelativistic effects for both the bound and free electrons. The bound and the free electrons should be treated on the same footing. This is crucial for the internal consistency of the approach valid at arbitrary degeneracy and relativistic degrees.

Simulations of Ar gas-puff Z-pinch radiation sources with double shells and central jets on the Z generator

Radiation-magnetohydrodynamic simulations using the non-local thermodynamic equilibrium Mach2-Tabular Collisional-Radiative Equilibrium code in (r, z) geometry are performed for two pairs of recent Ar gas-puff Z-pinch experiments on the refurbished Z generator with an 8 cm diameter nozzle. One pair of shots had an outer-to-inner shell mass ratio of 1:1.6 and a second pair had a ratio of 1:1. In each pair, one of the shots had a central jet. The experimental trends in the Ar K-shell yield and power are reproduced in the calculations. However, the K-shell yield and power are significantly lower than the other three shots for the case of a double-shell puff of 1:1 mass ratio and no central jet configuration. Further simulations of a hypothetical experiment with the same relative density profile of this configuration, but higher total mass, show that the coupled energy from the generator and the K-shell yield can be increased to levels achieved in the other three configurations, but not the K-shell power. Based on various measures of effective plasma radius, the compression in the 1:1 mass ratio and no central jet case is found to be less because the plasma inside the magnetic piston is hotter and of lower density. Because of the reduced density, and the reduced radiation cooling (which is proportional to the square of the density), the core plasma is hotter. Consequently, for the 1:1 outer-to-inner shell mass ratio, the load mass controls the yield and the center jet controls the power.

2-D RMHD Modeling Assessment of Current Flow, Plasma Conditions, and Doppler Effects in Recent Z Argon Experiments

By varying current-loss circuit parameters, the Mach2-tabular collisional radiative equilibrium 2-D radiation magnetohydrodynamic model was tuned to reproduce the radiative and electrical properties of three recent argon gas-puff experiments (same initial conditions) performed on the Z machine at Sandia National Laboratories. The model indicates that there were current losses occurring near or within the diode region of the Z machine during the stagnation phase of the implosion. The “good” simulation reproduces the experimental K-shell powers, K-shell yields, total powers, percentage of emission radiated in α lines, size of the K-shell emission region, and the average electron temperature near the time-of-peak K-shell power. The calculated atomic populations, ion temperatures, and radial velocities are used as input to a detailed multifrequency ray-trace radiation transport model that includes the Doppler effect. This model is employed to construct time-, space-, and energy-resolved synthetic spectra. The role the Doppler effect likely plays in the experiments is demonstrated by comparing synthetic spectra generated with and without this effect.

Coupling of an average-atom model with a collisional-radiative equilibrium model

We present a method to combine a collisional-radiative equilibrium model and an average-atom model to calculate bound and free electron wavefunctions in hot dense plasmas by taking into account screening. This approach allows us to calculate electrical resistivity and thermal conductivity as well as pressure in non local thermodynamic equilibrium plasmas. Illustrations of the method are presented for dilute titanium plasma.

Effective versus ion thermal temperatures in the Weizmann Ne Z-pinch: Modeling and stagnation physics

The difference between the ion thermal and effective temperatures is investigated through simulations of the Ne gas puff z-pinch reported by Kroupp et al. [Phys. Rev. Lett. 107, 105001 (2011)]. Calculations are performed using a 2D, radiation-magnetohydrodynamic code with Tabular Collisional-Radiative Equilibrium, namely Mach2-TCRE [Thornhill et al., Phys. Plasmas 8, 3480 (2001)]. The extensive data set of imaging and K-shell spectroscopy from the experiments provides a challenging validation test for z-pinch simulations. Synthetic visible images of the implosion phase match the observed large scale structure if the breakdown occurs at the density corresponding to the Paschen minimum. At the beginning of stagnation (−4 ns), computed plasma conditions change rapidly showing a rising electron density and a peak in the ion thermal temperature of ∼1.8 keV. This is larger than the ion thermal temperature (<400 eV) inferred from the experiment. By the time of peak K-shell power (0 ns), the calculated electron density is similar to the data and the electron and ion thermal temperatures are equilibrated, as is observed. Effective ion temperatures are obtained from calculated emission line widths accounting for thermal broadening and Doppler velocity shifts. The observed, large effective ion temperatures (∼4 keV) early in the stagnation of this Ne pinch can be explained solely as a combination of compressional ion heating and steep radial velocity gradients near the axis. Approximations in the modeling are discussed in regard to the higher ion thermal temperature and lower electron density early in the stagnation compared to the experimental results.

Recent progress in source development for lithography at 6.x nm

In 2009 industry announced that sources would be needed at 6.x nm for future lithography. The brightest sources in this wavelength region are plasmas containing gadolinium and terbium. The strongest lines result from 4d–4f lines in the spectra of Ag-like Gd XVIII and Tb XIX through Rh-like Gd XX and Tb XXI. Assuming collisional radiative equilibrium, the optimum plasma temperature for producing these species is expected to be in the range 100–130 eV. Time integrated experimental spectra have been recorded with a variety of lasers and a maximum measured conversion efficiency (CE) for 150 ps pulses was 0.4%. The extreme ultraviolet (EUV) emission was observed to be anisotropic while the ion energy decreases with decreasing pulse length. The optimum laser intensity for efficient 6.7 nm EUV emission was determined to be close to 7 × 1013 W cm−2, which gives an electron temperature of ∼ 130 eV. The use of prepulses increases the CE which is limited by plasma opacity. To improve radiation transport low initial density targets and/or low electron density plasmas such as CO2 LPPs are required.

Does atomic polarizability play a role in hydrogen radio recombination spectra from Galactic H II regions?*This paper is dedicated to the memory of my esteemed senior colleague Professor Eduard Hintz (1929–2012), in appreciation of invaluable insights provided into atomic processes in the edge plasma of the tokamak.

Since highly excited atoms, which contribute to the radio recombination spectra from Galactic H II regions, possess large polarizabilities, their lifetimes are influenced by ion (proton)–induced dipole collisions. It is shown that, while these ion–radiator collisional processes, if acting alone, would effectively limit the upper principal quantum number attainable for given plasma parameters, their influence is small relative to that of electron impacts within the framework of line broadening theory. The present work suggests that ion–permanent dipole interactions (Hey et al 2004 J. Phys. B: At. Mol. Opt. Phys. 37 2543) would also be of minor importance in limiting the occupation of highly excited states. On the other hand, the ion–induced dipole collisions are essential for ensuring equipartition of energy between atomic and electron kinetic distributions (Hey et al 1999 J. Phys. B: At. Mol. Opt. Phys. 32 3555; 2005 J. Phys. B: At. Mol. Opt. Phys. 38 3517), without which Voigt profile analysis to extract impact broadening widths would not be possible. Electron densities deduced from electron impact broadening of individual lines (Griem 1967 Astrophys. J. 148 547; Watson 2006 J. Phys. B: At. Mol. Opt. Phys. 39 1889) may be used to check the significance of the constraints arising from the present analysis. The spectra of Bell et al (2000 Publ. Astron. Soc. Pac. 112 1236; 2011 Astrophys. Space Sci. 333 377; 2011 Astrophys. Space Sci. 335 451) for Orion A and W51 in the vicinity of 6.0 and 17.6 GHz are examined in this context, and also in terms of a possible role of the background ion microfield in reducing the near-elastic contributions to the electron impact broadening below the predictions of theory (Hey 2012 J. Phys. B: At. Mol. Opt. Phys. 45 065701). These spectra are analysed, subject to the constraint that calculated relative intensities of lines, arising from upper states in collisional–radiative equilibrium, should be consistent with those obtained from Voigt profile analysis. It is shown that the experimental technique yields an excellent temperature diagnostic for the H II regions. On the other hand, strong evidence is not obtained, from those spectra which satisfy the above constraint on intensity, to indicate that the electron impact broadening theory requires substantial correction. The main grounds for attempting a revision of theory to allow for the influence of the ion microfield during the scattering processes on the upper and lower states of each line thus still appear to have a stronger theoretical (Hey 2007 J. Phys. B: At. Mol. Opt. Phys. 40 4077) than experimental basis.

Spatially-resolved spectra from a new uniform dispersion crystal spectrometer for characterization of Z-pinch plasmas

Based on self-focusing and uniform dispersion principle, a novel mica uniform dispersion crystal spectrograph (UDCS) was developed, whose linear dispersion was designed to be constant. An aluminum wire-array imploding experiment was performed at “Yang” accelerator and K-shell emission spectra were measured to investigate Z-pinch aluminum plasma characteristics by using the UDCS. The linear dispersion deduced from obtained spectra with linear fitting was in good agreement with the theoretical value. By using the collisional–radiative equilibrium (CRE) model, the intensity ratios of Ly-α/He-α line and Ly-β/He-ε were given as a function of electron temperature. The measured intensity ratios were compared to the computed ones and the electron temperatures were estimated to be 512eV and 489eV. The slope of the free-bound X-ray continuum provides a model-independent diagnostic of the core electron temperature which was around 1215eV for this implosion. Analysis of Z-pinch radial temperature gradient shows that the temperature decreases sharply near the core and slowly to the shell regions. Benefiting from the UDCS providing high spectral resolution (λ/Δλ∼2500), the existence of first-order Langmuir dips were observed and reliably identified in Al Ly-α and Ly-γ resonance line profiles. The relationship between the position from the line center of red Langmuir dips and electron density was studied. The electron densities deduced from these fine spectral features were 5.52×1021cm−3 and 4.57×1021cm−3 respectively, which compared well with those derived from measurements of the line shift. The ion temperature was determined from the Doppler broadening of Al He-α resonance, He-like intercombination and Li-like s, t (n=2) satellite lines.

K-Shell Spectra from CH Foam Tamped Ti Layers Irradiated with Nanosecond Laser Pulses

Hot dense titanium plasma was produced by irradiating a CH foam coated titanium layer with nanosecond laser pulses. The time-integrated emission spectra of He-like titanium were measured by using an absolutely calibrated flat crystal spectrometer. The synthetic spectra obtained by the steady collisional-radiative (CR) equilibrium model associated with the hydrodynamic simulations were compared with the experimental spectra. The results show that the electron density increases up to 5 × 1021 cm−3 with CH foam of density 40 mg/cc. This work indicates that an overlay of CH foam on a solid target is beneficial to create an optimum condition for K-shell emissions from a high density plasma region.

Two-dimensional radiation MHD modeling assessment of designs for argon gas puff distributions for future experiments on the refurbished Z machine

Argon Z-pinch experiments are to be performed on the refurbished Z machine (which we will refer to as ZR here in order to distinguish between pre-refurbishment Z) at Sandia National Laboratories with a new 8 cm diameter double-annulus gas puff nozzle constructed by Alameda Applied Sciences Corporation (AASC). The gas exits the nozzle from an outer and inner annulus and a central jet. The amount of gas present in each region can be varied. Here a two-dimensional radiation MHD (2DRMHD) model, MACH2-TCRE, with tabular collisional radiative equilibrium atomic kinetics is used to theoretically investigate stability and K-shell emission properties of several measured (interferometry) initial gas distributions emanating from this new nozzle. Of particular interest is to facilitate that the distributions employed in future experiments have stability and K-shell emission properties that are at least as good as the Titan nozzle generated distribution that was successfully fielded in earlier experiments on the Z machine before it underwent refurbishment. The model incorporates a self-consistent calculation for non-local thermodynamic equilibrium kinetics and ray-trace based radiation transport. This level of detail is necessary in order to model opacity effects, non-local radiation effects, and the high temperature state of K-shell emitting Z-pinch loads. Comparisons of radiation properties and stability of measured AASC gas profiles are made with that of the distribution used in the pre-refurbished Z experiments. Based on these comparisons, an optimal K-shell emission producing initial gas distribution is determined from among the AASC nozzle measured distributions and predictions are made for K-shell yields attainable from future ZR experiments.

Gd plasma source modeling at 6.7 nm for future lithography

Plasmas containing gadolinium have been proposed as sources for next generation lithography at 6.x nm. To determine the optimum plasma conditions, atomic structure calculations have been performed for Gd11+ to Gd27+ ions which showed that n = 4 − n = 4 resonance transitions overlap in the 6.5–7.0 nm region. Plasma modeling calculations, assuming collisional-radiative equilibrium, predict that the optimum temperature for an optically thin plasma is close to 110 eV and that maximum intensity occurs at 6.76 nm under these conditions. The close agreement between simulated and experimental spectra from laser and discharge produced plasmas indicates the validity of our approach.

Space-Resolved Diagnosis for Electron Temperature of Laser-Produced Aluminum Plasma

The K-shell emission spectra of laser-produced aluminum plasma are measured by a space-resolved spectrometer consisting of a flat crystal spectrometer with a 20 μm wide space-resolved slit. By using the approximation of the steady collisional-radiative equilibrium model, the interstage line intensity ratios of Ly-α resonance line to He-α resonance line are given as a function of electron temperature. The spectra profiles are decomposed to resonance line and the overlapped high-order satellites manifold. Thus reliable electron temperature is deduced from the interstage line intensity ratios of the decomposed resonance lines. The results of spaced-resolved temperature are compared with the hydrodynamic simulations. The diagnostics of electron temperature for laser-produced plasma is developed.

Simulations of the spectrum from a photoionized Si plasma

In order to interpret the spectrum of a photoionized Si plasma, we calculate the level populations and emissions using a collisional-radiative equilibrium model with detailed atomic data. In our calculations, levels with principal quantum number up to 9 are included and the electron impact processes are also taken into account. The peak around 1855 eV, which is not reproduced in the previous simulation, is obvious in the present results. Spectra analyzing shows that this peak mainly comes from the intercombination line of He-like Si. Our calculations indicate that the electron impact processes enhance this intercombination line about once, although the electron temperature is pretty low.

Density-induced continuum resonances and quasi-bound states in the collisional-radiative equilibrium of dense plasmas

The aim of this work is to improve the treatment of density effects in non-local thermodynamic equilibrium plasmas. The density effect on atomic structure (wavefunctions and energy levels) is modelled by an ion-sphere potential. The modification of the atomic potential, continuum lowering and appearance of resonances are presented. In particular, we show that the continuum resonances are linked to the electrons in the subshells passed into the continuum. Their presence determines the existence of partially bound configurations, which must be taken into account in the collisional-radiative model. We introduce in the set of rate equations a supplementary ionization process due to the plasma environment. This process (and its inverse) enters into the balance of all the other processes. It is equivalent to tunnelling ionization where an outer electron located above the ionization threshold (and trapped by the potential barrier) crosses the barrier. As an application, we studied the influence of temperature and density on the average ionization and the ionic populations of a carbon plasma. We compared these calculations with the traditional method based on the chemical picture with continuum lowering.

Theoretical calculations of X-ray spectra of Au plasma

By using a detailed configuration accounting method in the collisional radiative equilibrium model, and Cowans relativistic atomic structure code with respect to ionic averaged configurations and Hartree-Fock-Slater potential, we solve the rate equations for populations of all ionic and level populations of all excited states, in which main consider main atomic dynamic processes in a non-local thermodynamic equilibrium plasma can be considered, thereby obtaining the ionic populations and energy level populations and the ionic charge stage distribution and the average charge for Au. We also simulate and calculate the X-ray emission spectra of Au 5f—3d transition arrays. The calculated results are in good agreement with the experimental data for a well-characterized highly ionized Au plasma.