Modelling of spectral properties and population kinetics studies of inertial fusion and laboratory-astrophysical plasmas

Fundamental research and modelling in plasma atomic physics continue to be essential for providing both basic understanding and advancing on many different topics relevant to high-energy-density systems community [1]. In the most general scenario, the plasmas are not in local thermodynamic equilibrium (LTE), which means many atomic properties—such as cross-section of all relevant collisional and radiative atomic processes and atomic energy levels—need to be known to properly describe and diagnose the plasma conditions. The basis for most non-LTE (NLTE) simulations is the collisional–radiative (CR) model [2], which describes each ion charge state in the plasma in terms of a number of atomic levels. The distribution of atomic populations among the levels is determined by solving a set of coupled rate equations whose generation requires calculating all transition rates among the atomic levels. Once the population distributions have been obtained, material and radiative properties—such as opacities, emissivities, specific energies and other quantities of interest for the equation of state—can be calculated.

In this connection, the Atomic Physics Group at the Institute of Nuclear Fusion (DENIM) in Spain has accumulated experience over the years in developing a collection of computational models and tools for determining the atomic energy structure, ionization balance and radiative properties of, mainly, inertial confinement fusion (ICF) and laser-produced plasmas in a variety of conditions. In this work, we will focus on some of the latest advances and results of our research, with emphasis on ICF and, also, laboratory-astrophysical applications. In the following section we briefly describe the most recently developed models and codes, i.e. ABAKO/RAPCAL [3, 4] and ATMED [5]. ABAKO/RAPCAL was designed to perform detailed atomic kinetics and radiative properties calculations over a wide range of temperature and density, including the coronal, LTE and non-LTE regimes. Meanwhile, the new average-atom screened hydrogenic model, ATMED, was aimed for providing fast estimates and potential in-line hydrodynamical calculations of emissivities and opacities for plasmas under LTE conditions. In sections 3 and 4, we report on recent applications of ABAKO/RAPCAL to two completely different kind of laboratory-plasma experiments. In the first one, connected to laboratory astrophysics, radiative properties calculated from RAPCAL are used to characterize the radiative blast waves launched in xenon clusters. In the second one, relevant for the ICF community, we discuss the use of ABAKO as part of physics model employed for the spectroscopic analysis of shock-ignition implosions. Finally, conclusions are presented.

ABAKO is a CR model that can be applied to low-to-high Z ions for a wide range of laboratory-plasma conditions: coronal, LTE or NLTE, and optically thin or thick plasmas. It combines a set of analytical approximations to atomic rates, which yield substantial savings in computer running time, still comparing well with more elaborate codes and experimental data. Autoionizing states are explicitly included in the kinetics in a fast and efficient way. Radiation transport effects due to line trapping in the plasma are taken into account via geometry-dependent escape factors. Also, since the atomic kinetics problem often involves very large sparse matrices, an iterative method is used to perform the matrix inversion. It has been shown that ABAKO results compare well with customized models and simulations of ion population distributions. A detailed description of the model can be found in [3].

Meanwhile, the RAPCAL [4] code determines several relevant plasma radiative properties such as the frequency-dependent opacities and emissivities, mean and multigroup opacities, source functions, radiative power losses, specific intensities and plasma transmission. For line transitions, RAPCAL uses a Voigt profile including natural and Doppler broadenings, unresolved-transition-array (UTA) corrections, as well as an approximate formula to account for the electron-impact width.

In the following two sections, we discuss the use of ABAKO/RAPCAL for the study and investigation of two completely different kinds of laboratory-plasma experiments, one connected to laboratory astrophysics and the other one to ICF.

Furthermore, the ATMED code was aimed for providing fast calculations of the photon energy dependent opacity as well as the Rosseland and Planck mean opacities for single element and mixture hot dense plasmas under LTE. Typical ATMED running times are of the order of seconds. The code was developed within the framework of an average atom model. The needed atomic data are computed using a relativistic screened hydrogenic model based on a new set of universal screening constants including j-splitting [5]. These screening constants were obtained from the fit to a wide database of high-quality atomic energies, ionization potentials and transition energies taken from [6, 7]. The opacity calculations take into account the bound–bound, bound-free, free–free contributions, as well as scattering processes. The line absorption cross-sections are computed using a new analytical expression for oscillator strengths based on relativistic screened-hydrogenic wave functions. Line shapes include natural, Doppler and electron collisional broadenings. To obtain a more realistic value of the Rosseland mean opacity, an additional broadening of the bound–bound transitions has been included by considering the fluctuations of the occupations numbers into the atomic shells. ATMED also provides the plasma equation of state and shock Hugoniot curves. The model has been successfully tested and, in spite of its simplicity, it is able to predict the correct order of magnitude of quantities of interest, so it can be used to model radiative transport phenomena in hydrodynamic codes and experiments in an approximate way. A detailed discussion of many applications of ATMED can be found in [8]. Here, as a case of illustration, we show in figure 1 the frequency-dependent opacity corresponding to a Be–Cu plasma mixture. A calculation like this could be of interest for simulations of indirect-drive fusion capsules, in which Cu-doped beryllium has been used as ablator material.

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The main goal of laboratory astrophysics is to design and perform reproducible and well-characterized experiments, and then use hydrodynamical simulations and scaling arguments to both explain and predict actual astrophysical phenomena. Among many others, one particular area of interest within laboratory astrophysics is that of radiative shocks [10, 11], which are observed around astronomical objects in a wide variety of forms, e.g. accretion shocks, pulsating stars, supernovae in their radiative cooling phase, etc.

In this work, we report on atomic kinetics and radiative properties calculations in connection to radiative blast waves launched in xenon clusters [12, 13]. Ordinarily gases absorb only a small fraction of the laser energy. However, if atomic clustered gases are used, the absorption increases dramatically and a hot (T_e ∼ keV), high-energy-density plasma is formed and, eventually, a cylindrical blast wave generated [14]. Using clusters as a target medium, radiative blast waves can be launched even with laser energies as low as 100 mJ, provided that the pulse duration is short enough for an efficient energy coupling into clusters before they explode (which requires sub-picosecond lasers).

In the experiments here discussed, xenon clusters were irradiated at average gas densities of 0.16 and 0.25 kg m⁻³ with femtosecond laser pulses of 440 mJ and 360 mJ, respectively. The evolution and morphology of the blast waves were monitored using imaging interferometric and Schlieren techniques. Time histories of electron density radial profiles were also determined from interferometric measurements. ABAKO/RAPCAL was used for a better understanding and characterization of these radiative blast waves [15, 16]. Typical mass densities and electron temperatures of these xenon plasmas are expected to be in the range 10⁻⁵–10⁻³ g cm⁻³ and 1–20 eV, respectively. A first step for the characterization of the xenon plasmas was to perform a population kinetics study and compute basic quantities such as the average ionization and ion population distribution for such plasma conditions. For the given ranges of temperature and density, and within the steady-state assumption, it is difficult to know a priori whether the population distributions of xenon plasmas will be those characteristic of a coronal, LTE or NLTE regime. This fact prevents the use of a simplified coronal model as well as the Saha–Boltzmann (SB) equations, at least until a first atomic kinetics study has been carried out based on collisional–radiative calculations. At this point, it must be noted that such xenon atomic kinetics calculations are challenging, since for the temperatures and densities of interest, xenon will be only a few times ionized and the calculations will involve complex ions with a large number of bound electrons. As the number of bound electrons increases, the number of energy levels and electronic configurations do dramatically and some averaging technique must be used to keep the problem manageable. Here, we took advantage of the proper balance existing in ABAKO between accuracy and computational cost to obtain reasonable estimates of average ionization and ion population distributions. Additionally, we made a careful selection of electronic configurations to be included following the criteria discussed in [16]. For each CR calculation, the model included about ten xenon ions, which yields a CR matrix with ∼70000 energy levels. As illustration, in figure 2(a) the charge state distribution (CSD) of xenon is plotted as a function of the electron temperature for a fixed mass density of 5 × 10⁻⁴ g cm⁻³, characteristic of blast waves in xenon clusters. As seen, population is mainly distributed among four different charge states for a given temperature value within the range of interest.

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Also, the conditions to define an optically thin, radiative regime for laboratory astrophysics experiments relevance to radiative supernova remnants were formulated in [17]: (a) for the radiative flux to escape from the blast wave, the mean free path of the photons, λ_ph, must be larger than a characteristic size scale of the system, h; and (b) the radiative cooling time, τ_rad, must be shorter than the convective transport time, τ_conv = h/s, where s is the plasma sound speed. In this regard, measurements indicate that blast waves in clustered xenon reached the needed conditions to enter the radiative flux regime—defined as the situation in which the radiative energy flux is greater than the material energy flux, thus playing an important role in the evolution of the blast wave. Nevertheless, a theoretical confirmation of this fact is desirable and here we show as ABAKO/RAPCAL becomes useful to carry out this study. The frequency-dependent free path of photons was computed for a mass density of ρ = 5 × 10⁻⁴ cm⁻³ and an electron temperature of T_e = 10 eV (see figure 2(b)), which are approximated values for the upstream medium according to measurements and hydrodynamical simulations [12, 13]. Characteristic size of the xenon blast waves discussed here is h ≈ 0.01 cm, so, as observed in figure 2(b), the upstream medium can be assumed as optically thin for almost the entire range of photon energies, although some absorption effects are expected in the interval from 60 to 100 eV. We also computed the radiative cooling time τ_rad [17] for several temperature and density values within the range of interest and for all cases τ_rad was found to be less than τ_conv ≈ 1 µs, which is the typical convective transport time of the xenon blast waves. Therefore, with the only exception of a photon energy interval of width ∼60 eV, the theoretical analysis performed using ABAKO/RAPCAL allows us to classify the xenon blast waves within the optically thin, radiative regime. This conclusion agrees with what had been previously suggested based on experimental data.

The results discussed so far were obtained assuming steady state. Therefore, for them to be meaningful, the validity of this assumption for the case of study must be checked and possible time-dependent effects discarded. A key quantity for this analysis is the time scale of the most frequent atomic process in the plasma, τ_a [18]. Time-independent models can be acceptably accurate if the characteristic time t_plasma for the change in the local plasma conditions is significantly larger than τ_a, i.e. t_plasma ≫ τ_a. When this inequality holds true, it can be considered that the populations distributions adjust almost instantaneously to the changing conditions of the plasma. For the case of blast waves experiments in xenon, it can be inferred from data that t_plasma ∼ 1 ns. Moreover, calculations made with ABAKO for densities from 10⁻⁴ to 10⁻³ g cm⁻³ and temperatures from 1 to 20 eV give τ_a values which are slightly less than t_plasma. This means that the results shown above must be taken with caution and considered only for the purpose of qualitative analyses. An accurate study would require full time-dependent calculations.

ABAKO has been used as part of the physics model employed for the first spectroscopic analysis of shock-ignition implosions. The results have been discussed and published in [19, 20]. Here, we briefly emphasize the main results. Shock ignition [21] is an approach to ICF that relies on the low-adiabat implosion of a thick-wall shell target that creates a high areal-density fuel assembly, which is subsequently driven to ignition by a spherically convergent shock wave. Initial shock-ignition experiments at the OMEGA laser facility were successfully performed [22]. Furthermore, shock-ignition has aroused significant interest in the ICF community and several recent studies have been performed for possible shock-ignition experiments at the National Ignition Facility [23] in the United States as well as in the future HiPER [24] and Laser Megajoule [25] facilities in Europe.

The shock-ignition implosions discussed here and the associated spectroscopic analysis were performed at the OMEGA laser facility and led by Mancini from the University of Nevada, Reno. They employed plastic shell targets that had an internal radius of 387 µm, a wall thickness of 40 µm and an outer aluminum coating of 0.1 µm for sealing purposes. They were filled with 20 atm of D₂ and 0.072 atm of Ar, which was used for spectroscopic diagnosis. The observed argon radiation emission includes K-shell x-ray line transitions that span the photon energy range from 3000 to 4400 eV.

The model used to analyse the data is schematically illustrated in figure 3(a). It consists of a uniform sphere to characterize the state of the implosion core surrounded by a uniform concentric shell to account for the compressed shell confining the core. The parameters of the model are the core radius R, electron temperature T_c, electron density N_c, the compressed shell thickness ΔR, electron temperature T_s and electron density N_s. On the one hand, the value of R can be computed from N_c assuming mass conservation; this assumption relies on the approximation that all of the core mass contributes to the emission of radiation. On the other hand, the value ΔR is calculated from the values of R and N_s, as well as the fraction η of shell mass that is imploded, which is estimated from hydrodynamic simulations of the implosions [19]. The uniform model approximation for the spherical shell requires that the same imploded shell mass that contributes to the attenuation of the core radiation also contributes to the self-emission of the shell. To calculate the emergent intensity distribution for this model, we integrate the radiation transport equation for each photon energy along chords parallel to the direction of observation, and then further integrate all individual chord contributions to obtain the space-integrated spectrum for comparison with the experimental spectrum.

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The frequency-dependent emissivity and opacity of the core and shell were calculated with population number densities from ABAKO. All bound–bound, bound-free and free–free contributions from the argon–deuterium plasma in the core and the carbon–hydrogen plasma in the compressed shell within the spectral range of the measurements were included. For a given set of plasma temperature and density conditions, ABAKO takes into account all non-autoionizing and autoionizing states consistent with the continuum-lowering. Radiation transport effects on level population kinetics were considered via escape factors [26]. Detailed Stark-broadened line shapes including the ion-dynamics effect [27–29] were used for parent and satellite transitions of the argon spectrum. For completeness, natural and Doppler line broadening were included as well, but they are small compared with Stark broadening. Additionally, a plasma broadening effect on the bound-free emissivity and opacity was included according to the approximation discussed in [30].

The model geometry shown was integrated numerically and employed to perform the data analysis. However, to gain insight into the model characteristics and the formation of the emission spectrum, we used a simplified model geometry, which corresponds to the case of a spherical plasma source (core) seen through a plasma slab (shell)—see figure 3(b). The advantage of this simplified model geometry is that it has the same underlying physics of the model in figure 3(a) but the radiation transport equation can be integrated analytically, namely,

$$\begin{equation} F^\nu=F^\nu_{\rm c} \rme^{-\kappa_{\rm s}^\nu \Delta R}+F^\nu_{\rm s}, \end{equation} \tag{ 1 }$$

with

$$\begin{equation} F^\nu_{\rm c}= \pi R^2 \frac{\varepsilon_{\rm c}^\nu}{\kappa_{\rm c}^\nu}\left[1+\frac{\rme^{-2\kappa_{\rm c}^\nu R}}{\kappa_{\rm c}^\nu R}-\frac{\left(1-\rme^{-2\kappa_{\rm c}^\nu R}\right)}{2\left(\kappa_{\rm c}^\nu R\right)^2}\right], \end{equation} \tag{ 2 }$$

$$\begin{equation} F^\nu_{\rm s}= \pi R^2\frac{\varepsilon_{\rm s}^\nu}{\kappa_{\rm s}^\nu}\left(1-\rme^{-\kappa_{\rm s}^\nu\Delta R}\right), \end{equation} \tag{ 3 }$$

where ν is the photon frequency, F^ν is the final intensity flux and $\varepsilon_{\rm c}^\nu$, $\kappa_{\rm c}^\nu$, $\varepsilon_{\rm s}^\nu$ and $\kappa_{\rm c}^\nu$ stand for the temperature, density, and photon energy dependent emissivity and opacity of the core and the shell, respectively.

The first term in equation (1), $F^\nu_{\rm c} \rme^{-\kappa_{\rm s}^\nu \Delta R}$, represents the radiation emitted by and transported through the core further attenuated by the transmission through the shell. The various contributions to $F^\nu_{\rm c}$ within the square brackets emerge after considering the intensity which results from the integral of the radiation transport equation computed along a chord inside a spherical, uniform plasma source and further integrating the intensity flux along the given line of sight over the surface of the sphere—details can be found in [31]. The intensity flux $F^\nu_{\rm c}$ has the small and large optical depth limits given by $F^\nu_{\rm c}\rightarrow\frac{4\pi}{3}R_{\rm c}^3\varepsilon^\nu_{\rm c}$ and $F^\nu_{\rm c}\rightarrow\pi R^2\frac{\varepsilon^\nu_{\rm c}}{\kappa^\nu_{\rm c}}$ for $2\kappa^\nu_{\rm c}R_{\rm c}\ll1$ and $2\kappa^\nu_{\rm c}R_{\rm c}\gg1$, respectively. As stated in [31], the physical interpretation of both limits is clear: in the optically thin case the final result is the volume integral of the emissivity, and in the high optical depth limit the result corresponds to a surface radiator of effective area given by the cross-section of the sphere.

The second term in equation (1), $F^\nu_{\rm s}$, gives the self-emission of the shell. Hence, the emergent intensity includes both the argon and the plastic emissions. Because the shell self-emission decreases with hν mainly as exp(−hν/T_s), its effect is mostly noticeable in the low energy side of the observed photon energy range. Figure 4 graphically illustrates the details of the formation of the spectrum according to equation (1) for a given set of model parameters.

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Fitting the observed spectrum by means of a weighted least-squares minimization procedure yields both core and shell temperature and density conditions. Furthermore, from N_c, R, N_s and R the areal density ρR of the imploded target can also be obtained. Details of this procedure and comparisons with hydro-simulations and experimental data can be found in [20]. From the analysis of the space- and time-integrated spectrum recorded in the referred shock-ignition experiments, the core and compressed shell temperature and density conditions values were extracted: T_c = 930 ± 5% eV, N_c = 1.0 × 10²⁴ ± 20% cm⁻³, T_s = 285 ± 6% and N_s = 3.8 × 10²⁵ ± 25% cm⁻³. Uncertainties in the parameters were determined from a confidence interval statistical study that takes into account the correlations between model parameters [32]. Furthermore, the spectroscopic analysis also provides an estimate of ρR = 0.17 ± 20% g cm⁻² for the target's areal density at the collapse of the implosion, which is consistent consistent with the results of early experiments based on particle diagnostics [22].

Fundamental research and modelling in plasma atomic physics play an important role to understand different phenomena in a wide variety of laboratory and astrophysical plasmas. In this regard, the codes and models developed by the Atomic Physics Group at the Institute of Nuclear Fusion (DENIM) combine accuracy and celerity in a proper balance and can be a valid tool to design and/or interpret experiments. As illustration, we discuss the use of the ABAKO/RAPCAL computational package, first, for the characterization of radiative blast waves launched in xenon clusters and, second, as part of the physical model for the spectroscopic diagnosis of shock-ignition implosions.

This work has been supported by the Spanish Ministry of Science and Innovation Grant No ENE2009-11208 and the Keep-in-Touch Project of the EU. The shock-ignition spectroscopic analysis was supported by Grant DE-FG52-09NA29042 of the NLUF program of the US Department of Energy.