Ionization disequilibrium in K- and L-shell ions

Accurate models of atomic spectra enable the interpretation or design of systems where radiative energy transport is important, including astrophysical objects, high-Z ablation plasmas in laser-driven x-ray backlighters and inertial fusion hohlraums, and plasmas with highly transient excitation and ionization dynamics generated by intense x-ray free-electron laser pulses.¹ The accuracy of model calculations can only be verified by comparison to experimental spectra observed from a source with a characterized thermodynamic state.

The problem of model verification is particularly pronounced for non-local-thermodynamic-equilibrium (non-LTE) spectral models of multiple-electron ions. The complete set of atomic states increases exponentially with atomic number Z and exceeds current computational capabilities even for modest Z. Truncating schemes are, thus, employed to keep calculations tractable.² On the other hand, the included states must capture the dominant pathways of population flow, while maintaining sufficient detail of the quantum states to produce accurate spectral line energies and radiation transport. For the collisional-radiative rates themselves, either analytic approximations or empirical fits to data are used. Additional details have been previously reported.² Each of these approximations introduces uncertainty into the spectral model. Though great advances have been made in this field, recent international workshops have shown variation among different non-LTE codes when modeling spectra from open-shell configurations.^3,4

Though non-LTE spectra have been studied at high-energy-density (HED) facilities for decades, the creation and characterization of a plasma suitable for model verification has proven to be difficult. Few datasets exist in the literature as a consequence. More recently, experiments using a sub-micrometer sample layer centrally buried in a low-Z tamper have been used at high-power laser facilities to create a homogeneous, expanding plasma at non-LTE conditions.^5–9 Both sides of the disk-like target are irradiated by high-power lasers to rapidly heat and expand the sample. The centrally buried layer is confined in the radial direction by the surrounding low-Z material, mitigating radial gradients and resulting in uniaxial expansion along the target normal. Axial gradients (along the expansion axis) in temperature are reduced once the plasma expands to subcritical densities since laser energy deposition is nearly uniform in the underdense plasma.^10,11 Axial tamping helps homogenize the density within the sample layer and slows the rate of expansion. Creation of spatially uniform conditions simplifies comparison between observed spectra and spectral model calculations because only a single thermodynamic state needs to be considered.

However, there is a risk of ionization disequilibrium in the expanding plasma. When the thermodynamic timescale is faster than the ionization equilibration timescale, the ionization state will depart from its steady-state (“equilibrium”) distribution and must be treated with a time-dependent calculation. This time-dependent ionization manifests in the measured spectrum. Using time-dependent spectra to validate steady-state calculations is improper and will give inaccurate results. Previous buried-layer studies have explored time-dependent ionization kinetics, but the conclusions have been varied. The threshold for steady-state ionization is a function of plasma temperature $Te$⁠, ion density $ni$⁠, nuclear charge, and ionization state through the ionization and thermodynamic time scales. Due to this multivariate dependence, it is not necessarily contradictory that previous publications have reached different conclusions of time-dependence for the particular target geometry and elements considered. Some studies have argued for the sufficiency of a steady state,¹⁰ while others demonstrated the need for time-dependent ionization,¹² and still others indicated the presence of regions of steady-state and of time-dependent ionization within the same plasma.¹³ Whether or not the ionization is time-dependent is idiosyncratic to the given platform. The potential for time-dependence must be considered when using the ionization state as a thermodynamic diagnostic. To the best of our knowledge, no observation of K-shell disequilibrium in axially tamped buried-layer targets has been published.

We present Sc K-shell and Ge L-shell spectra from a characterized buried-layer plasma. L-shell spectroscopy was recently proposed by Marley et al. as a thermodynamic diagnostic that is more sensitive to the plasma conditions than traditional K-shell measurements.¹⁴ Additionally, the two species have different ionization equilibration time scales due to their different nuclear charge and ionization state. To discriminate between the plasma temperature $Te$ and the observable ionization balance, we define $TZ$ to be the temperature required for a steady-state atomic kinetics calculation to reproduce the ionization balance evident in the observed spectrum. By fitting each spectrum with steady-state calculations from the atomic kinetics model SCRAM,¹⁵ the $TZ$ of each species can be inferred. By construction, $TZ=Te$ for each species whose ionization is in thermal equilibrium with the circumfluent plasma. Spectra from the two species will test the spectral model more stringently than a single spectrum, while also providing a means to determine if the ionization distributions have attained a steady state.

The paper is organized as follows. In Sec. II, we describe the experimental geometry and the suite of diagnostics used. Constraints on the plasma conditions are derived in Sec. III, where we outline the data reduction and analysis of each diagnostic. In Sec. IV A, we demonstrate that the Sc and Ge spectra obtained early during the expansion show good agreement with steady-state calculations constrained by the measured ion density, while spectra obtained once the plasma has expanded to $ni≈1019$ cm⁻³ are not reproduced by steady-state calculations. In Sec. IV B, we demonstrate that ionization time scales exceed the hydrodynamic timescale, indicating that the Sc ionization has departed from its steady state distribution. After this time, the Sc spectra must be represented by time-dependent calculations.

Targets consisted of a 250-μm-diam Ge:Sc sample centrally buried in a 1000-μm-radius, 10-μm-thick Be tamp (see Fig. 1). The buried layer consisted of sequentially deposited layers of $0.03 μ$m Sc, $0.06 μ$m Ge, and $0.03 μ$m Sc. The buried layer was kept thin to mitigate axial gradients and help maintain uniform conditions throughout the Ge:Sc sample. To determine the background contribution of the Be tamp and its impurities to the data, targets fabricated without a buried layer were also shot.

The OMEGA-60 laser facility¹⁶ was used to deliver a total of 7.5 kJ of laser energy to both sides of the targets over 3.5 ns using a constant-irradiance drive. Beams were propagated through focus to achieve an approximately 600-μm-diam spot size on the target, providing a drive irradiance of $1014 W/cm2$ on each face of the target.

The lasers ablate, burn through, and cause the entire target package to expand primarily along the target normal and heat to keV temperatures. A uniform and slowly decreasing temperature is established within the sample once deposition of laser energy and conduction from the surrounding hot Be reservoir are balanced by mechanical losses from expansion and radiative losses from emission.^10,11 Two-dimensional simulations of similar experiments indicate a high degree of radial and axial uniformity within the sample after the laser burns through the buried layer.¹⁷ 

An array of 16 pinholes was coupled to a four-strip x-ray framing camera¹⁸ with a 50-ps integration window to capture time-gated x-ray images of the plasma emission. Two such cameras were deployed looking nearly parallel and perpendicular to the target normal, yielding face-on and side-on views, respectively, of the expansion, as shown in Fig. 1. These orthogonal views of the x-ray emission of the Ge and Sc are used to determine the average ion density and its evolution during expansion.

Sc K-shell and Ge L-shell spectra are obtained simultaneously with the x-ray images. The spectra were collected by separate multipurpose spectrometers (MSPECs),^19–21 time-gated by x-ray framing cameras with a 200-ps integration window, and recorded on Kodak T-Max 400 film. For convenience, we will refer to the two spectrometers according to the species they were configured to observe, i.e., the Sc spectrometer and the Ge spectrometer. All imaging and spectral measurements were obtained during the laser drive.

An example of the orthogonal x-ray image sequence is shown in Fig. 2. After subtracting emission from the images due to the Be tamp, the height of the expansion visible in the side-on images is inferred as the full-width at half-maximum (FWHM) of a Gaussian profile. An azimuthally averaged diameter is inferred from the face-on images as the FWHM of a fourth-order super-Gaussian. The Ge and Sc are assumed to homogeneously occupy a cylindrical volume given by the measured height and diameter. The spatially averaged total ion density $ni=nSc+nGe$ is calculated from this volume and the characterized initial mass of the buried layer assuming conservation of particles. Due to the high signal-to-noise ratio in the images, the statistical uncertainty in $ni$ from fitting the emission volume produces a fractional random uncertainty of $≤2.5%$⁠. Metrology of the initial buried-layer mass gives a fractional systematic uncertainty in $ni$ of 11%. Uncertainties in the absolute experimental timing are taken as the integration time of the measurement, of order 50 ps. Uncertainties in the time delay between measurements is constrained by the framing camera electronics and assumed negligible compared to the absolute timing and integration time.

Extracting ion density from self-emission images using this method has been previously validated against spectroscopic line ratios of Fe:V buried-layer targets.^22,23 In that work, radiative-transfer calculations were used to predict the enhancement of the He-like resonance line $(1P1−1S0)$ over the intercombination line $(3P1−1S0)$ as a function of target viewing angle and aspect ratio of the expanding buried layer. The relative enhancement between spectra obtained from face-on and side-on views of the layer required an ion density consistent with that measured by self-emission images.

The steady-state mean ionization state $Z¯$ of Sc and Ge at relevant conditions is shown in Fig. 3. Above 1 keV, the Sc $Z¯$ is relatively insensitive to density, while the Ge $Z¯$ retains sensitivity to both density and temperature. Though the Sc $Z¯$ changes little with temperature, the Sc K-shell emission spectrum changes dramatically (Fig. 4), most notably in the intensity of the Ly_α due to an increasing population of the H-like state. Combined with a relative insensitivity to density, the Sc K-shell spectrum is a compelling temperature diagnostic in this regime.

Sc K-shell spectra are processed by accounting for the spectral sensitivity of the spectrometer and framing camera.^21,24,25 Raw and processed data are shown in Fig. 5. Notable lines in the Sc spectra include the H-like series $np−1s$⁠, the He-like series $1s np−1s2$⁠, and the Li-like satellites $1s2p nℓ−1s2nℓ$ on the low-energy side of the He_α complex $1s2p−1s2$⁠. We infer the ionization temperature $TZ$ of the Sc by fitting optically thin regions of the observed Sc K-shell spectra with spectra calculated by SCRAM, assuming steady-state populations. The Li-like satellites and intercombination line of the He_α complex, and the Ly_β complexes were estimated by SCRAM to be optically thin. The Sc Ly_α line has an optical depth of 0.9 along the spectrometer line of sight during the earliest measurement, but quickly becomes optically thin due to expansion. The resonant line (w) of the He_α complex has an optical depth always exceeding 2. The optically thick resonant line and the entire He_β complex, as well as the intervening continuum regions, are ignored in the fits.

In the hybrid level structure of SCRAM, level energies and transition rates calculated in an isolated-atom framework by the fully relativistic flexible atomic code²⁶ are separated into detailed fine structure levels and averaged non-relativistic configurations and superconfigurations.²⁷ The collisional-radiative rate matrices are then constructed with the level and rate data and solved to determine populations needed to generate spectra. SCRAM generally exhibits good agreement with more detailed codes,²⁷ though emission features from transitions between averaged levels are less accurate.²⁸ The neglect of dense plasma effects implicit in using isolated-atom data is justified at the low electron densities of the experiment.²⁹ 

The inferred abundance distributions that best fit each spectrum assign all of the abundance into a single temperature or two neighboring temperatures. Such unimodal and narrow abundance distributions indicate that each spectrum is well-described by a single-temperature plasma. Figure 5 reveals the ability of a single-temperature spectrum to reproduce the data. For the remainder of the manuscript, the plasma will be treated as single-temperature. $χ2$ uncertainty analysis assuming a uniform plasma suggests a 10% uncertainty in the inferred temperature. Synthetic spectra at 1700 and 2200 eV (⁠ $≈±10%$ of the best-fit temperature) are overlayed in Fig. 5 to demonstrate the sensitivity of this diagnostic to temperature. This analysis is repeated for Sc spectra obtained at multiple times throughout the experiment to determine $TZSc(t)$⁠.

Ge L-shell spectra were collected and processed in the same manner as the Sc spectra. The processed spectra are shown as black traces in Fig. 6, and identified lines are listed in Table I. A few lines in the Ge spectrum are optically thick. During the earliest measurement, the most optically thick line is a $2p23d−2p3$ line of N-like Ge, with a transition energy of 1563.4 eV and an optical depth of 1.7. The subsequent ionization rapidly populates the Li-like state, making a strong $3d−2p$ Li-like line at 1730.6 eV the most optically thick line, with an optical depth of 3.4 that decreases in time due to expansion. Additionally, a line emission feature at 1570 eV has been removed from the Ge spectra, identified as He_α emission from Ar (⁠ $hν=3140 eV$⁠) in the Be tamper that is reflected from the spectrometer crystal in the second order.

We now consider the consistency of the atomic model by calculating steady-state Ge spectra, given the $ni$ measured from the x-ray images and assuming $Te=TZSc$⁠. The steady-state calculations are compared to the processed spectra in Fig. 6. Note that complexity of the structure and kinetics calculations for the L-shell spectra is considerably more challenging than for K-shell spectra, and reproduction of line positions and intensities of the L-shell spectra will not be as high of a quality as for the K-shell. At the earlier times (1.8–2.2 ns), the predicted envelope agrees with the observed spectra, and line positions are accurately reproduced, most notably the distinct Be- and Li-like lines above $hν=1700 eV$ that emerge at 2.2 ns. The kinetics model assigns consistent ionization temperatures to the Ge and Sc during the early expansion phase of the plasma. It is likely that both Sc and Ge have equilibrated to the local plasma conditions, and $TZSc=TZGe=Te$⁠.

After 2.2 ns, the Ge spectra calculated at $TZSc$ deviate substantially from the observed spectra. Once $TZSc$ and the observed Ge ionization plateau, the SCRAM-predicted Ge ionization distribution recombines in response to the decreasing density of the expanding plasma. Consequently, the strong $3d1→2p1$ Li-like lines at 1730.6 and 1766.0 eV in the synthetic spectra decrease in intensity as the Li-like population decreases, and the envelope of the emission shifts to lower photon energies as less-ionized charge states are populated. In the observed spectra, however, the Li-like lines persist, and the emission envelope does not shift.

The late-time Ge and Sc spectra cannot be modeled with steady-state calculations at a single temperature. In the preceding analysis, we have generated synthetic Ge L-shell spectra at $Te=TZSc$⁠, and we find disagreement with the observed Ge spectra. We can instead identify temperatures $TZGe$ that describe the Ge spectra (see Fig. 6). Determination of $TZGe$ from the Ge spectra is negligibly influenced by the few optically thick lines because the majority of the spectrum remains optically thin. Uncertainty in $TZGe$ is estimated to be 200 eV. Even accounting for uncertainties, $TZGe>TZSc$ at late times. We note that Sc spectra generated at $TZGe$ substantially overpredict the observed intensity of the Ly_α feature.

Combining the ion density and ionization temperature analyses, we can reconstruct the evolution of the buried layer through $TZ−ni$ space (Fig. 7). Acquisition of $(TZ,ni$⁠) begins at 1.5 ns, after the laser has ablated through the Be tamper and into the buried layer. Arrival of the ablation front at the Ge:Sc layer is signaled both by the bright emission from the Ge:Sc layer visible in the x-ray images and by the increasing ionization evident in the spectra. Direct illumination by the laser causes the buried layer to rapidly heat and expand, as is evident in the $ni$ and $TZ$ measurements plotted vs time in Fig. 7. An accelerated decrease in density at 2.5 ns is caused by radial expansion as the pressure equilibrates between the expanding Ge:Sc region and the surrounding Be.¹⁷ This expansion follows a radially inward-propagating density perturbation launched by an increased pressure on the sides of the Ge:Sc layer once the laser burns through the Be tamp but has yet to fully burn through the Ge:Sc. This density perturbation leads to a slight enhancement in x-ray emission, visible as the faint ring of emission inside the Ge:Sc layer at 2.2 ns. Because emissivity scales with $ne2$⁠, the density perturbation is estimated from the ratio between the signal in the ring to the signal interior to the ring to be less than 5%. The radial expansion is visible in the face-on images of Fig. 2.

Prior to 2.4 ns, $TZSc=TZGe$⁠. From the definition of $TZ$⁠, we deduce that both Sc and Ge have equilibrated to the local plasma conditions, and $TZSc=TZGe=Te$⁠. However, at 2.4 ns $TZSc$ plateaus at 2000 eV, while $TZGe$ continues to increase to 2700 eV at 2.8 ns. We attribute this to the onset of a time-dependent, nonequilibrium ionization state in the Sc (see Sec. IV B). We have investigated the effects of the assumptions made in this analysis to confirm that the discrepancy in $TZ$ is not artificially introduced by the assumptions. We discuss these below.

The absolute density $ni$ does not affect our conclusions. We have inferred density from the volume of the emitting plasma. Images clearly indicate that the plasma is expanding, so the density must be decreasing throughout the measurements. Steady-state SCRAM calculations predict a lower ionization with decreasing density in the range of 10¹⁷–10²² ions/cm³. As such, a different $ni(t)$ (with the requirement $dnidt<0$⁠) would still yield recombination in the synthetic spectra. On the contrary, the observed spectra do not change appreciably after 2.2 ns, indicating that the true ionization of Ge does not change in response to the decreasing density. We conclude that an error in the absolute density would not cause the observed inconsistency.

Additionally, the observed inconsistency is not caused by axial gradients. Targets in which the Ge and Sc layer order was inverted also exhibited $TZGe>TZSc$ during the late-phase expansion.

Comparison between τ and the smaller of τ_hydro and τ_th indicates when a system in ionization equilibrium will depart from equilibrium. If plasma conditions change rapidly such that $τ>min(τhydro,τth)$⁠, the ionization will take some time (of order τ) to equilibrate to the new state. Thus, the ionization will “lag” the changing conditions. Prior to equilibrating, diagnosis of plasma conditions from the ionization will yield a result intermediate to the old and new conditions. In the extreme case $τ→∞$ at some time t₀, the ionization will not respond to an evolving thermodynamic state, instead appearing to “freeze in” at the state at t₀. For example, if the plasma is then heated, the ionization will indicate too low of a temperature. This is liable to happen in an expanding plasma because the collisional ionization rate $neS$ and the radiative recombination rate $neα$ decrease with lower density, lengthening the ionization timescale.

We can evaluate τ for the potential limiting ionization and recombination processes in our system. For Sc, we consider if the Sc ionization lags the steady-state value during the heating process, leading to $TZSc<Te$⁠. Since the observed Sc spectrum is predominantly He-like and the steady-state ionization is anticipated to be further ionized, we will evaluate τ between He-like and H-like Sc. For Ge, likewise, we consider if the ionization lags the steady-state value as the plasma expands. A predominant Li-like charge state of Ge and a steady-state ionization that recombines with decreasing density indicates that the recombination of Li-like to Be-like Ge could limit this system. Tabulated values of S and α of the proposed limiting ionization and recombination processes are taken from the OPEN-ADAS database.^31,32 Rate coefficients for Ge were linearly extrapolated from coefficients for Fe through Zn. Collisional-radiative rates within SCRAM were not used in order to make this metric accessible to a broader set of researchers. The only dependence on SCRAM is through determination of the measured $TZ$⁠. To calculate $ne$ in Eq. (6), a constant $Z¯=24.5$ is assumed, being the average ionization of He-like Sc and Li-like Ge.

In principle, the ionization time scales τ_Sc and τ_Ge, hydrodynamic timescale τ_hydro, and thermodynamic timescale τ_th can be calculated directly from the measured $Te$ and $ni$ of Fig. 7. However, two issues arise when analytically propagating errors from the measurements. First, systematic uncertainties and temporal correlations artificially inflate the uncertainty of the time derivatives in τ_hydro and τ_th. Second, use of the tabulated α and S prevent use of analytic approximations of error propagation. Instead, all time scales and their uncertainties were calculated from Monte Carlo error propagation using synthetic $Te$ and $ni$ datasets inferred from the experimental measurements.³³ The synthetic datasets were constructed to incorporate temporal correlations between the $Te$ and $ni$ measurements of Fig. 7, critical to correctly determine uncertainties of the time derivatives in τ_hydro and τ_th.

The ionization of both Ge and Sc begin in steady-state due to the higher density and shorter ionization time scales early during the expansion. Because measurements were performed once the temperature approached a steady-state, τ_th is the longest timescale for all times. Relative to the ionization rate, the temperature is slowly varying. The rate of change of the temperature does not prevent the ionization from reaching a steady state.

As the plasma expands, $τhydro<τSc$ for t > 2.4 ns, suggesting that the Sc ionization will not equilibrate with the evolving plasma conditions. If the plasma temperature $Te$ continues to increase while $τGe<τhydro<τSc$⁠, a situation will arise where $TZSc<TZGe$⁠, consistent with the inferred $TZ$ for these times. Ge, having a faster τ by virtue of the smaller ionization potential of its dominant charge state, will exhibit an ionization that follows $Te$ more closely than that of Sc. Because no independent measurement of $Te$ was made, we cannot conclusively determine the relationship between $Te$ and $TZGe$ or the time-dependence of the Ge ionization.

We note that the comparison between τ_hydro and τ will indicate when a system initially in ionization equilibrium departs from equilibrium, but it will not indicate how long the system will take to return to equilibrium. To estimate a lower bound for the time to re-establish equilibrium, one can take the value of τ when it falls back below both τ_hydro and τ_th. For instance, Fig. 8 shows that τ_hydro exceeds τ_Sc around 3.0 ns; since $τSc≈1.5$ ns at this time, we can expect it would take at least 1.5 ns for the Sc ionization to equilibrate with the plasma conditions. Likewise, $τhydro>τGe≈0.5$ ns at a time of 2.8 ns, and the Ge ionization will take at least 0.5 ns to equilibrate.

As an additional check of the time-dependence, we have performed time-dependent calculations of the atomic state populations and resulting spectra using SCRAM. Time-dependent calculations of the Ge spectra using $TZGe(t)$ closely resemble the corresponding steady-state calculations. According to these calculations, it is unlikely that the Ge ionization is time-dependent. Time-dependent calculations of the Sc spectra were performed for $TZGe(t)$⁠. The calculations indicate a departure from steady state but fail to reproduce the observed Sc spectra. The calculated time-dependent ionization is too slow, and the Ly_α intensity is lower than observed for all but the latest time. However, Fig. 8 indicates that the Ge ionization may also become time-dependent after 2.6 ns, in which case $TZGe$ becomes an inaccurate metric of $Te$⁠. To determine a $Te−ni$ trajectory consistent with all measurements, the time-dependent Sc and Ge spectra would need to be generated along an ensemble of plausible $Te−ni$ trajectories and analyzed for goodness of fit to the observed Ge and Sc spectra. Such multi-dimensional, multi-objective optimization is beyond the scope of this paper.

The development of accurate non-LTE atomic spectral models relies on experimental spectra from a characterized source. Few datasets exist in the literature due to the difficulty of creating and diagnosing non-LTE plasmas. We have presented simultaneous Sc K-shell and Ge L-shell spectra from an expanding buried-layer plasma with measured ion density. The two spectra not only test predictions of the ionization balance of two materials with appreciably different Z, but also provide an in situ determination of ionization equilibrium. Sc K-shell and Ge L-shell spectra obtained during the early-phase expansion are successfully reproduced by a steady-state calculation by the atomic kinetics model SCRAM at the measured density and a single temperature. This demonstrates the accuracy of the atomic kinetics model within SCRAM. As the plasma expands, steady-state calculations fail to match both spectra at the same plasma conditions, and we find that $TZGe>TZSc$⁠. Calculations of the thermodynamic and ionization time scales throughout the experiment suggest that $TZGe$ and $TZSc$ diverge once the Sc ionization timescale exceeds the hydrodynamic timescale. The difference in $TZ$ is, thus, most likely due to a time-dependent ionization state of Sc. The Sc spectra lags behind the rising plasma temperature, resulting in the observed $TZGe>TZSc$⁠. Time-dependent SCRAM calculations were unable to reproduce both spectra.

Rather than signaling the end of a useful measurement, the onset of time-dependence of the Sc ionization poses an important observation for atomic kinetics models to reproduce. The thermodynamic time scales can be experimentally varied to force disequilibrium to occur earlier or later during the experiment to further test the accuracy of kinetics models. Additionally, time-dependent ionization increases the complexity of K-shell temperature diagnostics in expanding plasmas. The possibility of such must be considered carefully when using ionization-based plasma diagnostics at low densities.

The results of this study indicate the value of L-shell emission as a temperature diagnostic, with certain advantages over traditional K-shell diagnostics. The equivalence between $TZSc$ and $TZGe$ while the ionizations are in steady state validates the use of Ge L-shell spectra as a plasma thermometer. The proposed metric of ionization equilibrium indicates that an L-shell species will remain in equilibrium to lower densities than a K-shell species. Finally, L-shell emission spectra can be more sensitive to changes in temperature; L-shell ions exhibit a greater change in $Z¯$ and in the spectrum for a given change in temperature due to the similarity in ionization potential for L-shell ionization states of a given species. When density measurements are available, L-shell spectroscopy should be viewed as an appealing temperature diagnostic.

This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0003856, the University of Rochester, and the New York State Energy Research and Development Authority, and under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DEAC52–07NA27344. D.A.C. acknowledges the DOE NNSA SSGF support, which is provided under Cooperative Agreement No. DE-NA0003960. This work has also been supported by the Research Grant No. PID2019-108764RB-I00 from the Spanish Ministry of Science and Innovation.

This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof. LLNL-JRNL-848188.

The authors have no conflicts to disclose.

David Thomas Bishel: Data curation (equal); Formal analysis (lead); Methodology (supporting); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Gabriel Perez-Callejo: Writing – review & editing (supporting). Philip M. Nilson: Funding acquisition (equal); Supervision (equal); Writing – original draft (supporting); Writing – review & editing (supporting). David Alexander Chin: Formal analysis (supporting); Writing – review & editing (supporting). James Ryan Rygg: Supervision (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Gilbert Collins: Supervision (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Edward von Marley: Conceptualization (lead); Data curation (equal); Formal analysis (supporting); Funding acquisition (supporting); Investigation (lead); Methodology (lead); Project administration (lead); Supervision (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Marilyn Schneider: Conceptualization (supporting); Funding acquisition (equal); Resources (lead); Writing – review & editing (supporting). D. A. Liedahl: Conceptualization (supporting); Methodology (supporting); Writing – review & editing (supporting). Robert F. Heeter: Conceptualization (supporting); Methodology (supporting). Mark E. Foord: Conceptualization (supporting); Methodology (supporting). Gregory Elijah Kemp: Conceptualization (supporting); Methodology (supporting). Yechiel Frank: Methodology (supporting). James Emig: Investigation (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.