The design of a photoionization front experiment using the Z-Machine as a driving source and estimated measurements

Radiation-flux-dominated systems are relevant to astrophysics, early universe cosmology, and a variety of high-energy-density-physics (HEDP) systems. In present-day astrophysics, radiation from the forward shock of supernovae (SN) heats the relatively stationary, circumstellar medium (CSM) through photoionization (PI) fronts.^1–6 Specifically, type IIn SN have narrow lines in their observed spectra that correspond to the recombination of the photoionized CSM,¹ but a variety of other SN show interactions with the CSM during their evolution, such as type IIP, superluminous SN, and type Ia-CSM SN.^5,7–11 The ionization states present in these narrow lines are higher than one would expect from ionization due to electron collisions, which indicates photoionization is the mechanism behind this heating. There are few places in the SN literature that discuss the physics occurring in photoionizing CSMs,^1,5 let alone discuss the properties of the resulting heat front. This is likely due to insufficient angular resolution in the current observatories to discern this behavior and the need to understand the wind structure for each SN discovered.⁷ 

In the setting of early universe cosmology, the first galaxies produced very large, zero-age main sequence masses greater than 100M_⊙, stars that ionized the surrounding nebula and are thought to have led to the reionization of the intergalactic medium (IGM) through photoionization fronts.^12–14 Knowledge of the fraction of ionizing radiation that escapes these galaxies is important for determining the cause of reionization at the end of the cosmic dark ages,¹² and current measurements can have high uncertainties.¹⁵ The typical method for identifying high redshift galaxies involves observing changes in emission across images using different filters in the infrared spectra called the Lyman or Balmer break technique, depending on the energy range of the filters.^12,16 This measurement exploits the change in escaping galactic emissions across the K or L edge in H for the Lyman and Balmer break galaxies, respectively, with Lyman break galaxies having more active star formation and Balmer break galaxies being less active.^16–19 However, the redshifted line and continuum emission from the heated nebula can affect the magnitude of this change in signal across the filtered images, which changes the model fits, significantly, and the stellar ages.^16–19 Specifically, the inferred ages of these galaxies depend on the inclusion of the nebular emission in the model fitting.^16–19 Measurements of high redshift galaxies are difficult due to the limited number of photons that arrive at earth from these distant galaxies. This also makes spectroscopic measurements very difficult and means the emission spectra used to fit the observational data are exclusively dependent on models.

Experiments in the field of high-energy-density physics (HEDP) can generate very strong ionizing radiation fields to create plasmas in extreme environments, such as indirectly driven inertial confinement fusion schemes where high-Z hohlraums convert laser energy into ionizing radiation, which implodes a fusion fuel capsule. Heat fronts are also of interest to HEDP and there have been numerous experiments exploring diffusive radiation transport in supersonic heat waves.^20–26 In addition, there are a variety of experiments that create radiative fluxes relevant to astrophysical phenomena such as radiative shocks, photoionized plasmas, and hydrodynamic instability experiments.^27–32 These experiments have regions in the flux-dominated regime that can have physics scaled to or at parameters similar to those described in the context of astrophysics and early universe cosmology. Laboratory experiments to measure photoionization (PI) fronts would be a useful contribution to this body of knowledge.

To conduct a PI front experiment, the basic setup is fairly simple. One requires an external radiation source and a medium to propagate through. Work by Drake and Gray^33–35 shows that it is viable to create the conditions necessary for a PI front with reasonable experimental parameters. Most of that work is either generalized to an arbitrary source or designed after a laser-irradiated foil source.^36,37 The source considered here is a nested wire array z-pinch with W wires and a CH₂ foam on axis at the Sandia National Laboratory Z-Machine (Z).^31,38,39 This produces over 2 MJ of soft x rays as the radiative shock launched in the CH₂ bounces after it stagnates on axis, and the trapped radiation inside the W heats the metal and the rebounding shock and heat front propagate through the pinch plasma, reducing the opacity, allowing the radiation to escape. This produces a peak emission after about 100 ns that lasts for approximately 3 ns at over 1 TW cm⁻².^31,38,39 Current experiments using this source are relevant to a variety of astrophysical issues,³¹ including Fe opacity measurements at solar conditions, photoionized Ne, white dwarf photospheres, and photoionized Si relevant to black hole accretion disks.^40–42 

This article will show that an experiment to explore PI fronts using the emission from Z is possible using a N gas cell. The organization of this document is as follows. Section II describes the setup and physics in the photoionization front problem. Section III describes the nominal model and geometry used in the radiation hydrodynamics code used to design the experiment. Section IV will show the results from the nominal model and provide justification for the physics model used in the simulations. Section V will include 3D effects to identify the success of the experiment with the resulting lower values of the radiation flux. Section VI will provide estimated, time-resolved optical spectroscopy calculations to demonstrate the features useful in identifying a PI front. Section VII will present discussions and conclusions.

To conduct an experiment observing PI fronts, one requires a bright ionizing radiation source and a medium for it to propagate through, with Fig. 1(a) showing what that may look like for an experiment at a z-pinch. The source must be bright enough that the photoionization at the heat front that separates the hot and cold material is the dominant process of energy transfer to the plasma. There is the additional requirement that the radiation streams through the heated downstream region before being rapidly absorbed over a few mean free paths at the heat front, making the transport non-diffusive for a significant portion of the propagating medium. The implication of these features in PI fronts is that any modeling will need to consider the radiation hydrodynamics, radiation transport, and atomic kinetics in order to best treat the problem with the tools available. Additionally, this places a requirement on the experiment to evaluate the system in a way that provides adequate information about the regimes of radiation transport and dominant atomic processes to ensure the simulations and experiments are observing approximately the same physics.

Many authors have considered the radiation hydrodynamics of this problem,^{33–35,43–47} and this article will not reproduce these results as they are well documented. To provide context, using the nomenclature and results of Drake et al. and Gray et al.,^33–35 the simulations shown here will explore a weak-R type front, which is one of four solutions to the steady-state jump conditions of the radiation hydrodynamics equations under the assumption that the radiation pressure and energy density are negligible, but the radiation flux is not. A weak-R type front is a supersonic heat front that has no density change across the interface and limited hydrodynamic motion associated with its passage across a region of material.³³ 

As mentioned above, an analysis of this problem cannot neglect the atomic kinetics because this problem has the requirement that photoionization dominates without significant contributions from electron collisional ionization and the hydrodynamic equations alone cannot determine this. Drake et al. and Gray et al.^33–35 address this with two and three level atomic models, respectively, in an analytic solution of the steady-state rate equations^33,34 and the simulations in Gray et al. use an atomic model that is averaged over many of the available states.³⁵ The relevant results from the analytic solutions are two dimensionless parameters comparing the ratios of atomic rates. The first is

where n_i is the ion number density of the ith ionization state in cm⁻³, n_e is the electron number density in cm⁻³, $Ri+1,i$ is the recombination rate coefficient from the i + 1th to the ith ionization state in cm³ s⁻¹, and $Γi,i+1=∫FlσPIEdE$ is the photoionization rate from the ith to i + 1th ionization state in units of s⁻¹ with F_l the local radiation flux, σ_PI the photoionization cross section, and E the energy. The recombination rate coefficient $Ri+1,i$ is a sum of the three-body recombination rate coefficient obtained using the Lotz formula⁴⁸ and detailed balance, dielectronic recombination rate coefficient obtained using the fits described in Salzmann,⁴⁹ and radiative recombination rate coefficient obtained using the Verner and Ferland fits.⁵⁰ The dimensionless number α is the ratio of recombination to photoionization and determines if the plasma is actively photoionizing. For a PI front, it is necessary to have $α≪1$⁠, which indicates that the photoionization is occurring more rapidly than recombination. The second dimensionless number is

where $⟨σv⟩i,i+1$ is the electron collisional ionization rate coefficient using the Lotz formula⁴⁸ in cm³ s⁻¹ and the remaining variables are as defined in Eq. (1). β is one plus the ratio of the electron collisional ionization to the recombination and should be about unity in the case of a PI front. This is equivalent to saying that the ionizations due to electron collisions occur less frequently than recombinations, which are already much less frequent than photoionizations under the condition $α≪1$⁠. These dimensionless numbers take the place of the photoionization parameter, $ξ=Lner2$⁠, where L is the luminosity, n_e is the electron density, and r is the radius from the source, which is common in the astrophysics literature of photoionized plasmas.^51,52

The simulation study to design a PI front experiment on Z used the Helios-CR code by Prism Computational Sciences.⁵³ This is a 1D Lagrangian radiation-magnetohydrodynamics code that has the ability to use tabular equations-of-state, tabular or inline detailed atomic calculations for the opacities, and multi-group diffusion or discrete ordinates radiation transport. The inline, frequency-resolved, and non-equilibrium atomic physics calculations generate population distributions, which the code uses to generate opacities and emissivities. The code uses these material properties in the solution of the radiation transport equation using the method of short characteristics from the model of Olson and Kunasz⁵⁴ for angle-resolved radiation transport. The radiation flux used in each zone is the angle-averaged value of the specific intensities calculated using the short characteristics method with the selected number of rays. As Gray showed,³⁵ a more detailed physics model using inline atomic calculations for opacities and multi-angle radiation transport produces significant changes in the propagation of the front compared to tabular opacity values on a coarse frequency grid and diffusive radiation transport. Additionally, Helios-CR allows for user-input flux boundaries and previous work using the Z radiative properties platform^31,39 produced a spectrally resolved input file of the Z emission, in units of J cm⁻² s⁻¹eV^–1, at a radial location 45 mm from the source. The source has a color temperature of 63 eV and its spectral content is best fit by the sum of three geometrically diluted Planckians due to emission from the pinch plasma and the heated, surrounding load hardware. Figure 2 shows the calibrated input file at three times over the peak emission and Fig. 1 in Ref. 38 shows the contributions of the three Planckians to the total flux. The peak photon energy at the time of maximum emission is about 180 eV and the spectrum has significant emission up to about 1 keV, which allows for direct photoionization of all the ionization states of N. The H-like state of N has an ionization energy of 667 eV, which supports the claim of direct photoionization.

The nominal setup of the simulations uses five zones of mylar with a thickness of 1.44 μm that uses quotidian equations-of-state tables, using the model of More et al.⁵⁵ and local thermodynamic equilibrium, tabular opacities from the Prism Computational Sciences code Propaceos. Following the mylar, there are 100 zones of N with a thickness of 1.35 cm, which utilizes the non-equilibrium atomic physics, inline detailed configuration accounting (DCA) opacity calculations on a fine frequency grid that uses more resolution around line features. The simulation for radiation transport uses the angle-resolved, short characteristics method for the entire simulation domain. The nominal atomic model for N excludes all autoionizing states from the calculated energy levels in the ATBASE database. The simulation for radiation transport uses a S_n model with five angles. Figure 1(b) shows an overview of the physics model and the zoning configuration.

Helios-CR does not output the atomic rate coefficients needed to calculate the values of α and β, therefore it is necessary to post-process the simulation output. There is the possibility of using an atomic kinetics code, such as PrismSPECT, to determine the rates and rate coefficients for this analysis, but this is computationally expensive and using the calculated plasma parameters to evaluate these values is sufficient for the design of an experiment under the assumption of a one-dimensional system. To calculate the rate coefficients, this analysis uses fits to the radiative and dielectronic recombination rate coefficients^49,50 and the Lotz formula for the electron collisional ionization rate, which combined with detailed balance and the Saha equation provides the three-body recombination rate coefficient.⁴⁸ The photoionization rate is slightly more complicated. Gray et al. derived a scheme for approximating the photoionization rate from the discrete ordinates radiation transport in Helios-CR as

where $TR,l$ is the local radiation temperature from Helios-CR, $TR,s$ is the radiation temperature of the source, $Bν$ is a Planckian at the source temperature, $σPI,ν$ is the photoionization cross section using the fits of Verner et al.,^56,57h is the Planck constant, and ν is the photon frequency.³⁵ This was reasonable due to the use of a temperature boundary in that simulation study, but, here, there is not one temperature to prescribe to the input flux file, as mentioned above. The analysis below modifies Eq. (3) using the brightness temperature, $TR,B$⁠, of the input flux file as follows:

where σ_SB is the Stefan–Boltzmann constant and $Fs,ν$ is the frequency-resolved flux from the input file. Using Eq. (4), the resulting photoionization rate is

This calculation neglects the thermal, self-emission from the plasma as Gray et al. showed that it is not a significant contribution to the radiation field.³⁵ Post-processing the radiation transport calculations to calculate the photoionization rate is necessary because Helios-CR only outputs the radiation flux at each material boundary. The code determines the spectral grid used in the radiation transport calculations in each zone during the time step from the inline opacity calculations and uses the hydrodynamic zones to determine the spatial boundaries. The user cannot retrieve the frequency grid in each zone at a particular time step from the code output.

The parameter space for the experiment design spans the thickness of the mylar window and the pressure of the N gas. It is important to note that an experiment would initially fill a gas cell with N₂ gas, which the emission from the rise of the z-pinch would break these molecules into N atoms. The simulations use a volume of N gas because there is no mechanism in Helios-CR to break down molecules into their constituent atoms. The initial pressure input into the simulations does account for the factor of two increase in particle density when the N₂ breaks into two atoms. There is also the freedom to change the gas species, but this study does not explore that aspect of the problem. The decision to use N gas allows the atomic physics to be reasonably simple while having the photoionization opacity significantly overlap with the peak of the source spectrum. This deviates from the astrophysical case where the tail of the photon distribution provides the energy to drive the PI front, but it is necessary to change this in the experiment to generate a large enough photoionization rate. The window thickness has a limited role in the PI front physics, other than altering the incident radiation flux on the N, but it is necessary to determine the effects of thicker material for constructing a gas cell in a real experiment. The ablation pressure on the front window does launch a shock into the N, indicated by the peak in temperature around 0.175 cm in Figs. 3(a)–3(f), but this should not significantly impact the measurements as it is several millimeters behind the PI front.

Using two images as an example, the interpretation of these figures is as follows: Fig. 3(b) shows the log of parameter α, from Eq. (1), for a selection of N ionization states calculated at each of the 100 spatial zones of N in a single time step. Figure 3(e) shows the calculation of β, from Eq. (2), under the same conditions as Fig. 3(b). These figures allow one to observe where the dimensionless parameters satisfy the conditions for a PI front. To relate this to Figs. 3(b) and 3(e), the former shows values of logα, for this time step, are at a minimum much less than 1 for the NII and NIII states around 1.1 cm with the higher ionization states having minima closer toward the source. The corresponding locations in Fig. 3(e), which shows the calculated values of β, show that this parameter is unity at the location of minimum α for each ionization state. The α and β figures in this article show the results from the NII to NVI states because the preheat from the rise time of the pinch emission begins to ionize the neutral N and the photon mean free path of the NVI state is quite large. In order to assist the reader in understanding the structure of the heat front, the figures overlay the electron temperature in red, dashed lines over the calculated dimensionless parameters. Figures like this allow for comparisons of the ability to generate PI fronts with different gas pressures and front window thicknesses.

The mylar window thicknesses in this study are 1.44, 2.16, and 2.88 μm with gas pressures of 0.5, 0.75, 1.5, 2.5, 5, and 10 atm. The design parameters of interest are the propagation length of the PI front, which should propagate about the length of the N in the 3 ns of peak emission, and the dimensionless numbers α and β.

The simulations at 2.5, 5, and 10 atm suggest that these densities are too high for the radiation to penetrate a significant distance into the gas and this article does not include them in further discussions. The 0.5 atm simulations show the heat front propagating through the gas cell before the peak flux arrives, which is also not a useful design as this will not take advantage of the largest possible photoionization rate. Comparing the electron temperature profiles in addition to the α and β values for the remaining pressures in Figs. 3(a)–3(f) shows that while all seem to produce PI fronts, 0.75 atm does so in the most convincing fashion. It also seems that the PI front reaches the end of the gas cell during peak emission, while having sufficient spatial extent between the different ionization states to differentiate them in measurement.

Changing the window thickness reduces the temperature and penetration of the PI front, which makes sense as more of the source energy goes into heating and expanding the window when there is more material. It is important to note that the reduction in temperature is due to the lag of the front as the window thickness increases not a decrease in the final temperature. This is evident from comparing the electron temperatures in Figs. 4(a)–4(c) where for varying window thickness at 0.5 atm of initial gas pressure there is no significant change in the magnitude of temperature, but the spatial location changes significantly. This indicates that the temperature is affected by the spectral content of the source more than the total energy of the source and that the window thickness only acts to delay the development of the front. The fact that the inflection point in the electron temperature in Figs. 4(d) and 4(e), which have an initial gas pressure of 0.75 atm, is again not significantly different from the values at 0.5 atm further confirms this interpretation. Figures 4 and 5 show the results for the α and β calculations, respectively, over the range of window thicknesses and a small sample of gas pressures. These have the same interpretation as the values in Fig. 3. Increasing the window thickness acts to reduce the radiation flux that is available to support the PI front. This results in lower PI front velocities and therefore smaller propagation distances into the gas cell. Viewing the rows of Figs. 4(a)–4(c), 4(d)–4(f), and 4(g)–4(i), one sees a similar trend of the minima of logα shifting to positions closer to the source as the window thickness increases at this simulation time step. Additionally, these values of logα are sufficiently less than one, indicating a photoionizing plasma. The corresponding values of β in Fig. 5, similarly arranged in rows 5(a)–5(c), 5(d)–5(f), and 5(g)–5(i), show that at the locations of minimum α, the values of β are unity, which shows that all of these window thicknesses allow for sufficient radiation fluxes to support PI fronts without generating significant electron heating.

Gray et al. justifies the physics included in the simulations by comparing different models, and this study goes further by probing the effects of changing the resolution on the results. A resolution study, atomic model study, and angular grid for radiation transport study show that the nominal model reasonably captures the physics of interest. Sections IV A–IV C describe these investigations in more detail. The studies compare values from two different simulations using the following metric:

where δ is the percent difference between the deviated model and the nominal model, ε_d is the value from the deviated model, and ε_n is the value from the nominal model.

The nominal spatial resolution, in the N, is 135 μm. Since the inline atomic calculations produce a fine frequency grid, there is a significant amount of line structure captured by the model that would have large opacity values, which could then have photon mean free paths that are less than the grid spacing of the simulation. To try and capture the possible effects of this, a systematic increase in the number of zones from 100 zones to 200, 400, 800, and 1000 zones decreased the minimum spatial element to 13.5 μm. Since Helios-CR is a Lagrangian code, the zone boundaries are able to change with each time step, which would change the spatial resolution. However, there are no significant changes in the zone boundaries as a function of time in the N. This makes sense because the PI front is supersonic and results in little to no change in mass density in the propagating medium. Examining the difference plots in Figs. 6(a)–6(d) shows that there is less than 10% difference between the deviated model and the nominal model for any possible number of zones. This suggests that to spatial scales of 13.5 μm, the nominal model captures all of the relevant physics reasonably well. To improve mass matching, the simulations with 800 and 1000 zones of N used 20 and 40 zones of mylar, respectively, instead of the 5 mylar zones used in the nominal model. These simulations also used a simplified atomic physics model with a maximum principal quantum number of n = 4 to reduce the computation time, which the next section justifies.

Helios-CR allows the user to vary the number of energy levels in the atomic model used when solving the collisional radiative (CR) equations for the inline opacity calculations. There is always a trade-off between adding more detail to models used for experiment design and being able to do enough runs to explore a broad enough parameter space. The goal is not to have a perfect computational model in advance of the experiment, but to produce an experiment that obtains data useful for further, more-detailed analysis. That being said, the nominal atomic model, used in all of the simulations described above, includes term and fine structure splitting with 1168 energy levels over all of the ionization states of N. This model only uses electron configurations with principal quantum numbers up to 10 and energies that are below the ionization potential of the ions and therefore excludes only the available autoionizing states. Autoionizing states are important for accurately determining the ionization state distribution and the opacity of a plasma.^58–60 This section intends to justify the use of a simplified atomic model that excludes autoionizing states for the purposes of designing an experiment. To test if the nominal model is sufficient for this design study, a series of simulations systematically increased and decreased the number of levels in the atomic model to see how this changes the plasma parameters from the nominal model. It is important to note that these atomic models do not account for core electron photoionization of nitrogen ions with four or more electrons. This absorption mechanism should not have a strong effect on the velocity of the PI front, which is dependent on source flux and gas density.³³ However, the increased opacity from these transitions will lead to a shorter photon mean free path at the PI front. This will likely have the effect of reducing the spatial extent of the PI fronts between the different ionization states. This is acceptable for a design study but any future analysis of experimental results will need to adjust the atomic model to account for inner shell photoionization of the Be-, B-, C-, and N-like states, which may require a different approach for truncating the atomic model in order to keep the computation time reasonable.

This analysis uses different processes for increasing and decreasing the complexity of the atomic model for comparison with the nominal atomic model. To decrease the model complexity, this article reduces the maximum principal quantum number to remove states with excited electrons at high n. To increase the model complexity, a process added electron configurations, by energy, until electrons are further excited beyond a configuration selected for comparison. The following paragraphs describe both processes in more detail.

To systematically decrease the complexity from the nominal model, the analysis reduces the maximum principal quantum number in steps of one until the differences in plasma parameters exceed 20%. For example, if a model had n = 5 as the highest principal quantum number, the next simplified model would have n = 4 as the highest principal quantum number for the available states. This produces a model without electron configurations that have spectator electrons in the n = 5 state. The largest principal quantum number used here was n = 10, as mentioned above. Figures 7(a)–7(g) show that these changes in the atomic model produce parameter variations that stay under 10% in the nitrogen zones during the peak emission time of the Z radiation flux for a maximum principal quantum number of 4 or greater and generally under 20% over longer periods. It is interesting to note that for Figures 7(a)–7(f), there is a reduction of the plasma parameters before about 0.5 cm and an increase after, but that changes in Fig. 7(i) where the maximum principal quantum number is 2 and the opposite behavior occurs. It is not clear why this is so, but it may be due to the calculated opacity values when the atomic model is so drastically simplified.

To systematically increase the complexity of the atomic model, the process for adding levels compares the lowest energy level excluded from the previous model, called the comparison level, to the remaining excluded electron configurations, for each ionization state. Energy levels are added from the energy-ordered ATBASE database until the next electron configuration has an additional electron outside of the ground state orbital. For example, in the nominal model N-like atom, the comparison level is $1s22s22p26h1$⁠. The last electron configuration added in the first promoted electron model for this ionization state is $1s22s22p28d1$ because the next electron configuration, in energy order, is $1s22s12p33s1$⁠, so there is an additional electron excited out of the ground state orbital. The second iteration of this process for the N-like atom uses $1s22s12p33s1$ as the comparison level and stops when the next electron configuration, ordered by energy, is $1s22s22p13s2$⁠. This process gets repeated for each ionization state except for the H-like ion, which uses all available electron configurations in the nominal model. Table I shows the maximum electron configuration energies in the ATBASE database for each ionization state in the nominal, first promoted electron, and second promoted electron models. This analysis used two iterations of this process before the computation time was exceptionally long. Figures 7(h) and 7(i) show there are no variations beyond about 10% from the nominal model, suggesting that this level of atomic complexity is sufficient for this design study. These atomic models add autoionizing states to the nominal model. Figure 7(h) shows minimal changes to the plasma conditions at peak emission compared to the nominal model after increasing the number of energy levels to 1556 over all of the ionization states. In Fig. 7(i), the next increase in atomic model complexity used 4103 energy levels across all ionization states and found differences from the nominal model of less than 10%. These differences are mostly in the post-front region, where the electron temperature is larger than in the nominal model. However, the mean charge is minimally perturbed in the post-front region, but the pre-front region has a reduction in all parameters—except the mass density, which stays unchanged. This reduction suggests that the gas now attenuates higher energy photons in the post-front region that were previously passing through to the upstream material. This explains the increased electron temperature in the post-front region and further indicates that the added opacity from the autoionizing states is mostly affecting the higher ionization states far behind the front, at the less than 10% level.

The results of varying the atomic model in this way indicate that the nominal model is a reasonable approach to this atomic physics within the capabilities of the code used here. This is especially true as the analysis to calculate α and β uses post-processing of the Helios-CR output.

The multi-angle model of radiation transport in Helios-CR solves the transport equation along a ray through a slab of material on a frequency-resolved optical depth grid. In planar geometry, this model allows for 1-angle, 2-angle, or 5-angle calculations for solving the transport equation. Comparing the results of simulations using the different angles and an atomic model that has n = 4 as the maximum principal quantum number, which has less than 10% deviation from the nominal atomic model, with a window 1.44 μm thick and 0.75 atm N shows minimal difference between the 5-angle and 2-angle calculations. Using the same model configuration in a simulation except with 1 angle for radiation transport shows a drastic difference with the 5-angle calculation. Figures 8(a) and 8(b) compare the differences between the 1-angle and 2-angle calculations and the 5-angle model with the percent difference parameter described above. This study shows that the 1-angle calculation results in a much different distribution of the radiation energy over the spatial grid and that at least 2-angle calculations are necessary.

The results of the parameter space scan in Sec. III B are useful for initial design, but to gain an understanding of the real effects present in an experiment, multi-dimensional simulation tools are necessary. Unfortunately, at the time of this writing, there are not any tools that provide the detailed atomic model and non-diffusive radiation transport desired for a PI front with multi-dimensional spatial grids easily accessible to university users. However, there are ways of emulating 3D geometric effects to the source flux, which is of the most importance here, within the constraints of a 1D code. Since the radiation streams up to the heat front interface, the location of the front from the gas cell window is the relevant distance for geometric dilution. Using a geometric decay model of the on axis dilution of a finite extent source, the resulting attenuation factor is

where R is the radius of the source and D is the distance from the source. To apply this to the simulation, consider D to be a function of time equal to the position of the PI front. Then, using an iterative method, one can calculate the front locations to get D(t), use D(t) to calculate f(t), and apply f(t) to the input flux file at the appropriate times to get a geometrically attenuated flux. Figure 9 shows the attenuation of the source for two iterations at the peak emission time using this method. Applying this technique over two iterations in a simulation using 0.75 atm of N and a 1.44 μm window shows that the second iteration produces a PI front using a source radius of 2.82 mm, which is the radius of a circle with the same area as a 5 × 5 mm² as this is the smallest window an experiment would use.

This provides a lower limit on the flux in an experiment because the attenuation factor assumes the source is at the front of the mylar window when in reality it is 45 mm away, which makes the real source much more collimated than in this attenuation method. Figures 10(a) and 10(b) show the calculated values of α and β for the iterations of the artificially attenuated input flux. These simulations suggest that even under very conservative estimates of the flux, there is still a PI front with the value of β still unity and α about an order of magnitude larger than with the full source flux, which still makes it less than one. There is a reduction in velocity as one sees with increasing gas pressure and window thickness, but here there is also a significant increase in α due to the reduction in photon flux.

While the simulations suggest that it is possible to generate a PI front with the Z flux, an experiment has to use the measurement tools available to observe the front. Since α and β are not directly observable values, it is necessary to measure the plasma parameters that these quantities depend on, which in this case are electron temperature, density, and ionization, in addition to knowledge of the source flux. Additionally, the propagation velocity is a useful parameter to measure for characterizing the PI front behavior. To make this measurement, the most relevant diagnostic technique available on Z is temporally resolved visible light spectroscopy using a streak camera coupled to a grating spectrometer. This provides spectral information from 1.56 eV to 3.5 eV over timescales of 30–100 ns depending on the streak camera settings. A fiber optic cable couples the emission from the heated N plasma to the grating, which provides spectral information about one location in space as a function of time. The line emission will provide information about the ionization states present in the N at the location of the fiber and fitting this line emission provides information about the electron temperature and density. If thermal emission dominates the plasma, it may still be possible to collect a temperature measurement by fitting the spectrum to a blackbody. To measure the velocity, it is necessary to have spectral information from at least two different locations because of an about 3 ns jitter in the peak current on Z.

To estimate the measured self-emission from the plasma, the SPECT3D code⁶¹ takes the output from Helios-CR and places it on a 3D spatial grid to calculate the emission and radiation transport in the desired geometry. SPECT3D calculations using the output from the nominal model described above using the unattenuated and second iteration attenuated input flux file coupled to a streaked spectrometer one meter away from the axis of the simulations show the anticipated spectra over the peak emission times. Figures 11(a) and 11(b) show the two spectra with line emission early in time that decays to thermal emission after the front passes the location of the fiber. As stated above, this allows for measurements based on line emission early in time and thermal emission later in time.

Figures 11(c) and 11(d) show the line structure early in time before the transition to thermal emission as the PI front heats the location observed with the spectrometer in the nominal and attenuated input flux, respectively. There are limited differences in the line structure between the brightness of the sources, which is a good indication that photoionization is driving this behavior as it seems more dependent on the spectral content of the source than the brightness, with the caveat that the source must be bright enough to sustain the state of the plasma. The strongest lines in Figs. 11(c) and 11(d) come from the NII and NIII states, with four transitions identified that inform the evolution of the system. The lines identified here satisfy the dipole selection rules and the labels that show n = 2 orbitals have a second excited electron. The selected transitions all have a $ΔJ=0,±1$ and opposite parity to satisfy the dipole selection rules. The atomic physics model in SPECT3D averages over the different magnetic and spin quantum numbers, which means transitions between even and odd orbital angular momentum states satisfy the dipole selection rules. The earlier time spectra show bright NII transitions from the 9f to 4d orbital at 3.42 eV and the 3p to the 3s with an additional electron in the 2p orbital at 2.23 eV that decrease in relative intensity over the nanosecond time interval shown. At the same time, the two, identified NIII lines, one at 1.67 eV showing the 7p to 6d transition and the other at 2.74 eV showing the 3p to 3S transition, increase in relative intensity over the one nanosecond time interval. This indicates that the plasma is evolving to a condition where the NIII ionization state is more prevalent than the NII state. It is important to note that there are other lines and ionization states than those identified here, but these show the decrease in the NII and increase in the NIII populations.

An experiment to observe PI fronts is relevant to the understanding of astrophysics, cosmology, and HED physics. The simulation study shown here suggests that the emission from Z is sufficient to produce the correct conditions for a PI front, even with consideration of 3D source effects. The front window and distance of the gas cell from the pinch plasma produce a reasonably collimated source to drive the PI front. This allows for a system that more closely resembles the 1D simulations than a smaller emitter that has to be closer to the gas cell. Estimates of streaked visible spectroscopy measurements show that there is a line structure that one can use to find transitions between the NII and NIII states. Then, a temperature measurement could be possible after thermal emission dominates the line emission.

The conversations with Igor Golovkin and Prism Computational Sciences were very useful for learning to use HELIOS-CR. We would also like to thank our collaborators on the Z astrophysical properties platform for their helpful comments and advice.

This work is funded by the U.S. Department of Energy NNSA Center of Excellence under Cooperative Agreement No. DE-NA0003869.

The authors have no conflicts to disclose.

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