Laboratory Photoionization Fronts in Nitrogen Gas: A Numerical Feasibility and Parameter Study

As discussed in Drake et al. (2016), photoionization events deposit considerable amounts of energy at the front. In addition, the photoionized medium downstream from the front has a much higher temperature than the material upstream. This allows the photoionization front to be described as a heat front. Astrophysical applications of heat fronts are considered in Kahn (1954), Goldsworthy (1958, 1961), Axford (1961), and Osterbrock & Ferland (2006). A similar analysis is useful in the laboratory system considered here.

We start by moving to a frame of reference that follows the front. A planar, steady front is assumed, and we label the conditions upstream of the front with the subscript 0 and those downstream with the subscript 1. The conservations of mass, momentum, and energy are then written as

$$\begin{eqnarray}&&{\rho }_{0}{u}_{0}={\rho }_{1}{u}_{1},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{P}_{0}+{\rho }_{0}{u}_{0}^{2}={P}_{1}+{\rho }_{1}{u}_{1}^{2},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{1}{2}{\rho }_{0}{u}_{0}^{2}+\displaystyle \frac{\gamma }{\gamma -1}{P}_{0}-{q}^{2}{\rho }_{0}\right){u}_{0}\\ &&\qquad =\,\left(\displaystyle \frac{1}{2}{\rho }_{1}{u}_{1}^{2}+\displaystyle \frac{\gamma }{\gamma -1}{P}_{1}\right){u}_{1},\end{eqnarray} \tag{ 3 }$$

where ρ is the mass density, u is the velocity, P is the pressure, γ is the adiabatic index of the gas, and q measures the internal energy deposited in the gas during photoionization and has units of velocity for convenience. Following Osterbrock & Ferland (2006), q, having units of velocity, is defined as

$$\begin{eqnarray}&&{\phi }_{i}\left(\displaystyle \frac{1}{2}{m}_{i}{q}^{2}\right)={\int }_{{\nu }_{0}}^{\infty }\displaystyle \frac{\pi {F}_{\nu }}{h\nu }(h\nu -h{\nu }_{0})d\nu ,\end{eqnarray} \tag{ 4 }$$

where m_i is the mean mass of the ionized gas, ϕ_i is the ionizing photon flux arriving at the front, and F_ν is the spectral energy flux of the source, a function of frequency ν. It is with this definition that we are allowed to write Equation (3). If the front were instead generated by a diffusive heat front, then the incoming energy flux would depend on the gradient of the temperature. This would lead to very different structure at the front compared to that generated by a photoionization front (Drake 2018).

Using Equations (1) and (2) to simplify Equation (3), one finds

$$\begin{eqnarray}&&{\rho }_{0}({u}_{0}^{2}+{a}_{0}^{2})={\rho }_{1}({u}_{1}^{2}+{a}_{1}^{2})\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}({u}_{1}^{2}-{u}_{0}^{2})+\displaystyle \frac{\gamma }{\gamma -1}({a}_{1}^{2}-{a}_{0}^{2})={q}^{2}.\end{eqnarray} \tag{ 6 }$$

In order to simplify this further and arrive at an equation for the density ratio, we solve Equation (5) for a₁,

$$\begin{eqnarray}&&{a}_{1}^{2}=\displaystyle \frac{1}{{\mu }^{2}}({u}_{0}^{2}(\mu -1)+\mu {a}_{0}^{2}),\end{eqnarray} \tag{ 7 }$$

where μ = ρ₁/ρ₀. Substituting this into Equation (6) and making use of Equation (1), we arrive at

$$\begin{eqnarray}&&{\mu }^{2}({u}_{0}^{2}+{{ka}}_{0}^{2}+2{q}^{2})-k({u}_{0}^{2}+{a}_{0}^{2})\mu +{u}_{0}^{2}(k-1)=0,\end{eqnarray} \tag{ 8 }$$

where k = 2γ/(γ − 1). Following Drake et al. (2016), the isothermal sound speed, a² = p/ρ, upstream of the front is independent of the front and is useful in parameterizing the flow. Dividing through by ${{\rm{a}}}_{0}^{2}$, Equation (8) becomes

$$\begin{eqnarray}&&{\mu }^{2}({M}^{2}+k+2{\bar{q}}^{2})-k({M}^{2}+1)\mu +{M}^{2}(k-1)=0,\end{eqnarray} \tag{ 9 }$$

where M is the upstream Mach number and $\bar{q}=q$/a₀. This can be solved for two real, positive roots,

$$\begin{eqnarray}&&\mu =\displaystyle \frac{{\rho }_{1}}{{\rho }_{0}}=\displaystyle \frac{{Ak}\pm \sqrt{{k}^{2}{A}^{2}-4{{BM}}^{2}(k-1)}}{2B},\end{eqnarray} \tag{ 10 }$$

where $A={M}^{2}+1$ and $B={M}^{2}+k+2{\bar{q}}^{2}$.

Equation (10) is now dependent only on known quantities, the upstream Mach number M, the ratio of specific heats γ, and $\overline{q}$. As mentioned above, q is related to the excess energy deposited in the front by photons. Parameters q and, conversely, $\bar{q}$ can be estimated for astrophysical environments by assuming an idealized hydrogen nebula surrounding an O-type star. The upstream sound speed is roughly a₀ ∼ 10 km s⁻¹ for T = 10,000 K. The energy deposited by the photons implies approximately ${q}^{2}\sim 2{k}_{{\rm{B}}}{T}_{s}/\overline{M}$, where T_s is the temperature of the source, k_B is Boltzmann's constant, and $\overline{M}$ is the average atomic mass. For a 52,000 K O-type star, q ∼ 30 km s⁻¹ and $\overline{q}=3$. For laboratory experiments, Drake et al. (2016) estimated $\overline{q}\sim 10$.

Figure 1 shows the density ratios for two cases for Equation (10). The $\bar{q}=0$ case represents the classic jump conditions without radiation, while the $\bar{q}=10$ case represents the laboratory conditions discussed above. A γ = 4/3 is assumed for both cases.

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Figure 1 also shows the four classic classifications of shock fronts. The D-type, or "dense," fronts are fronts that move subsonically with respect to the upstream gas and are shown in the left panel. Strong D-type fronts are supersonic rarefactions where the density drops by some amount at the front. Weak D-type fronts are fronts with subsonic rarefactions. In the absence of radiation, the density is essentially unchanged across the front. With radiation, there is some modest heating, which leads to lower densities behind the front. The right panel shows the R-type, or "rarified," fronts. The strong R-type is a modified shock wave where the radiation lowers the compression in the shock. The weak R-type is the classic, simple ionization front where the photons are absorbed in a thin layer near the front but leave the density largely unchanged. In nature, strong R-type fronts are unstable and are not seen.

It should be noted that the heat-front type evolves as the front moves outward. As discussed by Osterbrock & Ferland (2006), if we consider a uniform neutral medium and instantaneously turn on a large source of photons, the heat front generated will start as a weak R-type, that is, a photoionization front. Eventually the flux of ionizing photons drops low enough, due to geometric reduction and recombinations, that an R-type front is unsustainable and a shock is formed, followed by a strong D-type front.

The extent of a photoionized region depends on the flux of photoionizing photons and the recombination rate. Drake et al. (2016) used a two-level model atom to study photoionization fronts and found that the structure of the ionization front is strongly determined by the ratio of the recombination rate to the photoionization rate and the ratio of the electron-impact ionization rate to the recombination rate. Here we expand this model to three levels in order to evaluate the effects of multilevel atoms and additional energy levels on photoionization front structures.

Consider a model atom with three distinct energy levels with increasing ionization energies. Between three adjacent ion stages, three atomic processes are considered: ionization by electron collision, electron recombination, and photoionization. Figure 2 shows a schematic view of this model. We consider a planar model and that photoionization is the only mechanism that decreases the photon flux, i.e., geometric effects are ignored. The level population for each ionization state is determined by, along with equations for charge conservation and mass conservation,

$$\begin{eqnarray}\displaystyle \frac{{{dn}}_{1}}{{dt}} & = & -{\sigma }_{12}{n}_{1}{n}_{e}-{{\rm{\Gamma }}}_{{\ell },12}{n}_{1}\\ & & +{K}_{21}{n}_{2}{n}_{e}+{T}_{21}{n}_{2}{n}_{e}{n}_{e}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{dn}}_{2}}{{dt}}=-\left(\displaystyle \frac{{{dn}}_{1}}{{dt}}+\displaystyle \frac{{{dn}}_{3}}{{dt}}\right)\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}\displaystyle \frac{{{dn}}_{3}}{{dt}} & = & {\sigma }_{23}{n}_{2}{n}_{e}+{{\rm{\Gamma }}}_{{\ell },23}{n}_{2}\\ & & -{K}_{32}{n}_{3}{n}_{e}-{T}_{32}{n}_{3}{n}_{e}{n}_{e}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{n}_{e}={Z}_{1}{n}_{1}+{Z}_{2}{n}_{2}+{Z}_{3}{n}_{3}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{n}_{T}={n}_{1}+{n}_{2}+{n}_{3},\end{eqnarray} \tag{ 15 }$$

where σ₁₂ and σ₂₃ are the electron collision ionization rate coefficients for levels 1 → 2 and levels 2 → 3, respectively, ${{\rm{\Gamma }}}_{{\ell },12}$ and ${{\rm{\Gamma }}}_{{\ell },23}$ are the local photoionization rates, K₂₁ and K₃₂ are the recombination rate coefficients, and T₂₁ and T₃₂ are the three-body recombination rate coefficients. Parameter n_e is the electron number density, while n₁, n₂, and n₃ are the number densities in each level. Z₁, Z₂, and Z₃ are the degree of ionization for each level.

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Equations (11) and (13) show the recombination terms in the traditional form, respecting that the leading behavior involves the product of two densities for two-body processes and three densities for three-body processes. In this case, the two-body coefficients are independent of density as one might hope, but the three-body coefficients retain some density dependence. Informed by the determination in Drake et al. (2016) that three-body recombination is not dominant in laboratory systems that produce a photoionization front, we choose to slightly overstate the total rate of recombination as follows. We rewrite the recombination terms between states i and j as ${{\rm{R}}}_{{ji}}\,=({K}_{{ji}}+{T}_{{ji}}{Z}_{3}{n}_{T}){n}_{j}{n}_{e}$, in which T_ji is evaluated at an electron density of ${Z}_{3}{n}_{T}$. This gives a conservative result, in the sense that conditions found to produce an ionization front of a certain size will in actuality produce one at least that large.

Equations (11) and (13)–(15) can be rewritten as, assuming steady state,

$$\begin{eqnarray}&&-{f}_{1}({\alpha }_{{\ell },1}({\beta }_{1}-1){f}_{e}+1)+{\alpha }_{{\ell },1}{f}_{2}{f}_{e}=0\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{f}_{2}({\alpha }_{{\ell },2}({\beta }_{2}-1){f}_{e}+1)-{\alpha }_{{\ell },2}{f}_{3}{f}_{e}=0\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{f}_{1}+{f}_{2}+{f}_{3}-1=0,\end{eqnarray} \tag{ 18 }$$

where

$$\begin{eqnarray}&&{\beta }_{1}={\sigma }_{12}/{R}_{21}+1\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{\beta }_{2}={\sigma }_{23}/{R}_{32}+1\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{\alpha }_{{\ell },1}={R}_{21}{n}_{T}/{{\rm{\Gamma }}}_{{\ell },12}\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&{\alpha }_{{\ell },2}={R}_{32}{n}_{T}/{{\rm{\Gamma }}}_{{\ell },23}\end{eqnarray} \tag{ 22 }$$

${f}_{e}={Z}_{1}{f}_{1}+{Z}_{2}{f}_{2}+{Z}_{3}{f}_{3}$, and f_i = n_i/n_T.

In order to complete our system of equations, we need to account for the decrease in photon flux due to photoionization events. The spatial evolution of the radiation flux is defined as

$$\begin{eqnarray}&&\displaystyle \frac{{{dF}}_{R}}{{dx}}=-{\sigma }_{1}{F}_{R}{n}_{1}-{\sigma }_{2}{F}_{R}{n}_{2},\end{eqnarray} \tag{ 23 }$$

where σ₁ and σ₂ are the spectrally averaged photoionization cross sections (see, e.g., Equation (30) below) for ionization state n₁ and n₂, respectively. This can be recast in terms of the initial optical depth $\tau ={n}_{T}\sqrt{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}$. The local radiation flux is related to the source radiation flux by F_Rℓ = F_Rg(τ), which recasts Equation (23) as

$$\begin{eqnarray}&&\displaystyle \frac{{dg}(\tau )}{d\tau }=-\displaystyle \frac{({f}_{1}{\sigma }_{1}+{f}_{2}{\sigma }_{1})g(\tau )}{\sqrt{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}}.\end{eqnarray} \tag{ 24 }$$

Finally, in order to relate the local photoionization rate to the rate at the source, we define α_ℓ,i as ${\alpha }_{{\ell },i}={\alpha }_{i}/g(\tau )$.

These equations are solved for values relevant in laboratory experiments, α ∼ 10⁻²–10⁻⁴, β₁ = β₂ = 1, σ₁ = σ₂, and Z_i = 1, and are shown in Figure 3. An iterative approach is taken in solving the above equations. First, using an initial estimate for g(τ), the ionization state fractions are updated using standard root-solving algorithms. These are then used to update g(τ) using a common ODE solver. The left panel of Figure 3 shows results for α_i = 10⁻⁴, while the right panel shows α₁ = 10⁻⁴ and α₂ = 10⁻³. For equal values of α the system quickly transitions from a fully ionized state to a fully neutral state, that is, there is essentially no region where the singly ionized state is present. This is in contrast to the unequal α case. Here there is an appreciable region where the singly ionized state is dominant. This suggests that for photoionization experiments with multilevel atoms, one could see many ionization fronts, one for each ionization state. In practice, this depends on the ionization energy of each state or the electron configuration of the atom.

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Each simulation is run in two dimensions with Cartesian coordinates, with the x coordinate ranging between 0 and 7 mm and the y coordinate ranging between 0 and 3.5 mm. Figure 4 schematically shows the target design. The outer tube, shown in gray, is made of an acrylic plastic. The gold region contains the gold foil and supporting structures, which, once heated, provides the radiation drive into the nitrogen gas. The impinging lasers are represented by the green triangle. The nitrogen gas is held within the target by a gas window consisting of a thin polyimide film supported by a stainless steel grid. The crosshatch pattern shows the simulated region. There is a small region below the outer tube and to the left of the gas window. This is a vacuum region in the experiment and is modeled here as a void region filled with xenon gas at very low density.

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For simplicity and to save on computation time, we make two simplifications in modeling this experiment. First, we ignore the gold foil. The resolution requirements are reduced since the gold foil is very thin compared to the size of the nitrogen gas cell. The gas window, however, is included and is modeled as a 5 μm piece of acrylic plastic. Second, the laser heating and subsequent emission from the gold foil are modeled as a radiation temperature boundary condition.

Davis et al. (2016) studied the emission from laser-heated gold foils. These results serve as the basis for our radiation temperature boundary condition. These authors measured the X-ray emission using the DANTE X-ray spectrometer and derived the radiation temperature as a function of time by fitting the DANTE spectra to a Planckian profile. The left panel of Figure 5 shows the results of this experiment for a 0.5 μm gold foil. Also shown is the parameterized radiation temperature boundary conditions used in the Omega-like simulations. The radiation temperature rises linearly with time, reaching the peak by 1 ns. The temperature is then held fixed for the next 5 ns. The temperature then falls linearly in time to zero over the last 2 ns. Finally, since the gold foil is located in front of the entrance to the gas cell, the radiation boundary condition is only implemented over the entrance to the gas cell. For all other hydrodynamic variables a reflecting boundary condition is used. Reflecting boundary conditions are used for the upper and lower y boundaries, while outflow conditions are used for the right x boundary.

We report results from a total of 20 Omega-like simulations. The incident radiation temperature, and consequently the incident laser intensity, is varied between 80 and 140 eV. This can be achieved by varying the intensity of the incident lasers. The pressure in the gas cell is varied between 5 and 40 atm. This corresponds to a number density of nitrogen atoms between 2.5 × 10²⁰ and 2.0 × 10²¹ cm⁻³. Each simulation is run for a total simulation time of 2 ns, which allows the radiation drive to peak and emit at its peak temperature for 1 ns. In the discussion presented below, the nominal run is a model with a nitrogen pressure of 10 atm, or equivalently a density of 5.0 × 10²⁰ cm⁻³, and a drive temperature of 100 eV, matching the ideal setup as described in Drake et al. (2016).

The right panel of Figure 5 shows the X-ray drive shape as a function of time derived using the X-ray power profile from the ZAPP platform, as shown in Figure 3 of Rochau et al. (2014). The drive is mirrored around the peak in order to obtain a symmetric profile. This profile is then fit with a Cauchy profile, providing the drive shape as a function of time used for the radiation boundary condition.

Figure 6 of Rochau et al. (2014) gives insight into the spectrum and intensity of the drive. A typical photoionization experiment on the Sandia Z machine is placed between 45 and 100 mm from the pinch, which generates a near-Planckian emission source (e.g., Foord et al. 2004). The drive spectrum is well fit by a single temperature of T_R ∼ 93 eV. For a circular source of radius R_s, the radiation flux is geometrically reduced with distance, D, from the source. The total flux as a function of distance from the sources, where we assume that the distance D is some small multiple of the source size, is given by

$$\begin{eqnarray}&&F(D)=\sigma {T}_{R}^{4}\left(\displaystyle \frac{{R}_{s}^{2}}{{R}_{s}^{2}+{D}^{2}}\right)\sim \sigma {T}_{R}^{4}{\left(\displaystyle \frac{{R}_{s}}{D}\right)}^{2},\end{eqnarray} \tag{ 25 }$$

where σ is the Stefan–Boltzmann constant and T_R is the radiation temperature. Thus, the radiation at D is reduced relative to the blackbody flux at the source by a factor of ${({R}_{s}/D)}^{2}$, defining a "geometrical reduction factor" at D. For a radiation temperature of T_R = 93 eV, the surface flux is 7.7 TW cm⁻². The measured flux at a distance of D = 45 mm from the ZAPP source is 1.7 TW cm⁻², which gives a reduction factor of 0.22. The source size, R_s, is defined by the left y boundary and is set to 3.5 mm. Therefore, to achieve the proper flux at the opening of the target, the left x boundary is set to a distance of 7.45 mm. In addition, the radiation temperature source extends over the entire boundary instead of over a small portion as in the Omega-like simulations described above. Schematically, this is shown in Figure 6.

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We report a total of six z-pinch simulations, which vary the nitrogen pressure (and equivalently the nitrogen density). This is in contrast to the Omega-like simulation, where the peak drive temperature is also varied. This is due to the ZAPP platform target placement and pinch design. Each simulation is run for 10 ns, much longer than the Omega-like simulation, which allows the radiation drive to reach its peak value. Finally, a wider range of nitrogen densities are studied with the ZAPP setup versus the Omega-like setup.

Because the simulations reported here are intended to be directly relevant to experiments, it is sensible to discuss anticipated observables and diagnostics. The primary diagnostic is anticipated to be streaked X-ray absorption spectroscopy. This allows for measurement of the velocity of the front in addition to the plasma parameters, i.e., the electron temperature, electron number density, and average charge.

X-ray absorption spectroscopy provides information on the density and temperature of the target based on the photons absorbed by the target. This requires a stable X-ray source having a relatively flat spectrum. Such a source is known as a backlighter. The backlighter provides a 300 ps, 2–4 keV source for the absorption spectroscopy using a 1% Argon dopant in the nitrogen gas (Keiter & Drake 2016).

A secondary measurement will use the streaked optical pyrometer (SOP) diagnostic (Miller et al. 2007). This device is typically used to measure the self-emission in laser-driven shocks. There are various options, under consideration, to couple this line of sight to an interferometer, which might provide further evidence regarding electron density and front speed. Additional details of the experiment results and diagnostics are given in H. LeFevre (2018, in preparation).

Figure 7 shows the two-dimensional density and radiation temperature plots for the nominal model (T_R = 100 eV, 10 atm of nitrogen, Omega-like setup). The top row shows the mass density, while the bottom row shows the radiation temperature. The first column shows the results at 1 ns, and the second column shows the results at 2 ns. Very little density evolution is seen between these two times. On the other hand, there is clear evolution of the radiation temperature. To highlight the slight density evolution, Figure 8 zooms in on the region near the entrance to the nitrogen gas cell. At 1 ns the density shows some compression at the interface between the nitrogen gas cell and the void region at 0.3 mm. This is mostly from the destruction of the plastic gas window. The nitrogen gas density is essentially unchanged. The radiation temperature, however, has penetrated much further along the gas cell.

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In order to illustrate this more clearly, Figure 9 shows line-out profiles through the nitrogen gas cell whose path is shown as the black line in Figure 7. The left panel of Figure 9 shows the density ratio, defined as the density divided by the ambient density, as a function of time. The x-axis is the optical depth through the system assuming that the gas is composed of only neutral nitrogen and does not take into account any evolution of the ionization state of the gas. Once we have accounted for the plastic gas window, there is essentially no evolution in the nitrogen gas density at all times. In fact, even at 2 ns, the maximum compression in the nitrogen gas is only a few percent at an optical depth of τ ∼ 50. The total, initial optical depth through the nitrogen gas is τ ∼ 700, which is much larger than optical depths studied here. This is to allow experimental diagnostics to study the front as a function of time as it moves through the gas cell.

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The right panel of Figure 9 shows the line-out profile for both the electron temperature (dashed lines) and the radiation temperature (solid lines). The difference in evolution between the two temperatures is striking. At early times, t = 0.5 ns, the rise in radiation temperature can be seen with almost no change in electron temperature. However, by t = 1.5 ns, both the radiation and electron temperatures are ∼90 eV near the entrance to the nitrogen gas. The difference between these two temperatures may be explained by looking at the photoionization timescales. A simple estimate for the photon flux near the source is

$$\begin{eqnarray}&&{F}_{\gamma }=\displaystyle \frac{{\sigma }_{\mathrm{SB}}{T}_{s}^{4}}{2.7{k}_{{\rm{B}}}{T}_{s}}=2.36\times {10}^{23}{T}_{s}^{3},\end{eqnarray} \tag{ 27 }$$

where σ_SB is the Stefan–Boltzmann constant, k_B is Boltzmann's constant, and T_s is the source temperature in units of eV. The photoionization timescale is then

$$\begin{eqnarray}&&{\tau }_{\mathrm{PI}}=\displaystyle \frac{{10}^{-12}}{\bar{\sigma }{F}_{\gamma }},\end{eqnarray} \tag{ 28 }$$

where τ_PI is the photoionization timescale in units of ps and $\bar{\sigma }$ is the photoionization cross section and is on the order of 10⁻¹⁸ cm². At t = 0.5 ns, as the source is ramping up, it has a value of T_s = 50 eV and gives a photoionization timescale of 30 ps. At t = 1.5 ns, after the source has reached its peak, the timescale has decreased to 6 ps.

The temperature profiles at late times have T_R well above T_e. The ionizing photons from the source that penetrate the shocked plastic have begun to ionize the gas in front of the shock. This creates a slight knee in the radiation temperature profile at t = 1.5–2 ns.

The results shown in Figure 9 can also be used to estimate the photoionization front velocity. The front velocity is estimated by

$$\begin{eqnarray}&&{v}_{f}=\displaystyle \frac{{F}_{\gamma }}{n}=\displaystyle \frac{D{\rm{\nabla }}{E}_{R}}{2.7{T}_{R}n},\end{eqnarray} \tag{ 29 }$$

where F_γ is the photon flux, n is the particle number density, and T_R is the local radiation temperature in units of eV. At t = 1.5 ns, the front is located at ∼840 μm, while at t = 2.0 ns the front is at ∼1100 μm. This gives a front velocity of ∼500 μm ns^–1. The radiation temperature near the front is ∼40 eV, which, by using the flux given by Equation (27), gives a front velocity of ∼300 μm ns^–1, consistent with the simulation result. The photoionization front therefore travels ∼3 μm in a single photoionization timescale. Thus, near the leading edge of the front one would expect the nitrogen to become multiply ionized over a short distance.

It is also important to note the separation between electron and radiation temperatures in Figure 9. As mentioned in Drake et al. (2016), electron heat conduction can significantly contribute to the overall energy flux when T_e ∼ T_R and would produce nearly identical T_R and T_e profiles. Electron heat conduction is only important when the electron temperature is greater than the radiation temperature and when the particle density is greater than 10²¹ cm⁻³. The results shown here for our nominal model show that electron heat conduction does not play an important role here.

In this section, we turn our attention to determining whether a photoionization front is established in any of our simulations. There are two dimensionless quantities that are important in characterizing the photoionization front: the ratio of the recombination rate to the photoionization rate, denoted above as α_i (Equation (21)), and the ratio of the coefficient of electron-impact (also known as collisional) ionization to that of recombination, denoted by β_i (Equation (19)).

In order to estimate these ratios in our simulations, we have taken the electron-impact ionization rates from Voronov (1997), and the recombination rate coefficients are computed using radiative recombination rate coefficients from Badnell (2006) and the approximate three-body recombination rate coefficients from Lotz (1967) and Drake et al. (2016). Nikolić et al. (2013) showed that for high-density plasmas, n_e >10¹⁰ cm⁻³, dielectronic recombination rate coefficients are highly suppressed. Since the densities studied here are much higher than this threshold, we assume that the contribution from dielectronic recombination is zero. We note here that these rate coefficients are not necessarily the same rates computed by Propaceos and used in computing the opacity and equation-of-state tables used in the crash models. However, these rates are sufficient in estimating α and β. The photoionization rate is approximated as ${{\rm{\Gamma }}}_{i}={\bar{\sigma }}_{i}\phi$, where ${\bar{\sigma }}_{i}$ is the average photoionization cross section for ion i and ϕ is the radiation flux. The average photoionization cross section for ion i is computed as

$$\begin{eqnarray}&&\bar{{\sigma }_{i}}=\displaystyle \frac{{\int }_{{E}_{\mathrm{th}}}^{\infty }{\sigma }_{i}(\epsilon ){F}_{\epsilon }(T)d\epsilon }{{\int }_{0}^{\infty }{F}_{\epsilon }(T)d\epsilon },\end{eqnarray} \tag{ 30 }$$

where σ_i is the photoionization cross section for ion i, E_th is the threshold energy for photoionization, and F is the spectral photon flux. The photoionization cross sections are taken from Verner & Yakovlev (1995) and Verner et al. (1996). The flux is computed as $\phi =-(c\lambda /\kappa ){\rm{\nabla }}{E}_{R}$, where c is the speed of light, E_R is the radiation energy density, κ is the Rosseland mean opacity, and λ is the flux limiter. Opacities are computed using a gray approximation for nitrogen, provided by Propaceos.

The values of α and β are computed for each ionization state of nitrogen along the line shown in Figure 7. For a given ionization state, we determine the location where X_i/X_i,max =0.5, which defines the photoionization front, using the (reasonably accurate) approximation that only two states are populated significantly at any one location. X_i(=ρ_i/ρ) is the mass fraction for state i and is defined as the density of charge state i divided by the total ion density, and ${X}_{i,\max }$ is the maximum value for ionization state i. X_i is computed from the average ionization state as reported from the EOS tables, which ranges between zero for completely neutral nitrogen and seven for fully ionized nitrogen. If the average ionization state is reported as Z = 2.5, then the mass fractions for both N iii and N iv are set to 0.5. The mass fraction for each nitrogen ion is shown in Figure 10 for the nominal model at t = 1.5 ns. Note that the location where X_i/X_i,max = 0.5 is multivalued and slightly complicates the determination of the photoionization front. Therefore, only the point closest to the radiation source is considered.

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Figure 11 shows the α and β values for the nominal run at four times. At very early times (top left panel) only the first two ionization states of nitrogen are populated, but with very high α values, indicating a heat front and not a photoionization front. This is due to the relatively weak ionizing spectrum with a source temperature of T_R = 50 eV. However, after the source temperature has reached its peak value (t = 1.0 ns), almost every ionization state is populated, and the parameters are suggestive of a photoionization front. For example, at t =1.5 ns only the VI → V transition has a β value appreciably above 1. Error bars are also given in Figure 11. In most cases the α and β values for the 25% and 75% ionization values do not differ from those at 50%.

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Another important feature of Figure 11 is the location of each photoionization front as a function of time and optical depth. For example, at t = 1.5 ns, there are effectively two or three groups of points. Each group roughly corresponds to an electron orbital subshell of nitrogen, with the 2p (outermost) subshell found at larger optical depths, then the 2s subshell, and the 1s (innermost) subshell appearing last. This confirms the results found in Section 2.2, with multiple fronts being generated.

Three representative transitions are chosen to compare the results for the rest of the parameter study: II → III, IV → V, and VI → VII, as they represent transitions for each electron shell of nitrogen. As shown in Figure 11, IV → V represents a worst-case scenario in terms of β for all the states considered, that is, the values for β are consistently larger than other transitions, especially at late times. Four representative times are chosen to compute the α and β values for these transitions. The first time is chosen during the rise in source temperature (t = 0.5 ns), another once the source temperature reaches its peak (t = 1 ns), and two while the source temperature is held constant at the peak (t = 1.5 and 2 ns).

Figure 12 shows the results of Figure 11 in a slightly different manner, where each "pixel" gives the corresponding value for α and β. The interpretation is the same as in Figure 11. After the source has reached its peak value, the photoionization rate is higher than the recombination rate coefficients and gives values for α ≪ 1. Similarly, the electron collision ionization rate is smaller than the recombination rate and gives values for β ∼ 1.

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Figure 13 shows the results for all the Omega-like runs where the nitrogen density varies from 2.5 × 10¹⁹ to 2 × 10²¹ cm⁻³, corresponding to an initial pressure of 5–40 atm, respectively, and the peak radiation source temperature varies between 80 and 140 eV. A 4 × 3 panel similar to that of Figure 12 represents each model. The source temperature T_R increases along the x-axis, while the nitrogen pressure (alternatively density) increases along the y-axis.

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As Figure 13 shows, photoionization fronts are generally expected for radiation source temperatures above 90 eV and nitrogen pressures between 5 and 10 atm. Both of these constraints are readily achievable for current experimental setups. For laser-driven experiments Davis et al. (2016) showed that radiation temperatures of 100 eV for 0.5 μm gold foils are possible; see Figure 5. Meanwhile, the ZAPP platform is well characterized in Rochau et al. (2014) and consistently produces 90 eV sources. In addition, the current experimental design makes use of a gas window that allows for gas pressures well within the ranges considered here. Therefore, it is likely that the photoionization fronts can be produced with the conditions considered here.

The right panel of Figure 13 suggests two important experimental considerations. First, at early times, β is very large, suggesting that these ionization states are populated through electron collisions instead of photoionizations. However, once the radiation source temperature has reached its peak, β values are close to 1 over a wide range of source radiation temperatures and nitrogen pressures. One concludes that data collected at early times, before 1 ns, are unlikely to see photoionization-dominated behavior. Second, as discussed above, the IV → V transition is a worst-case scenario among all possible transitions but remains close to 1, with peak values of β ∼ 3, suggesting that the photoionization front should be apparent for nearly every transition. Finally, this confirms the estimates made in Drake et al. (2016) and suggests that photoionization fronts should be produced with radiation temperatures of T_R > 90 eV and for nitrogen pressures between 5 and 20 atm.

Figure 14 shows similar α (top panel) and β (bottom panel) plots for the "z-pinch" models. The same atomic transitions are considered as in the Omega-like models. As shown in Figure 5, the X-ray drive profile takes 10 ns to reach peak. Therefore, α and β values are computed at 2.5, 5.0, 7.5, and 10 ns. Note that, due to the design of the ZAPP platform, the radiation source temperature is fixed at T_R ∼ 90 eV and only the initial nitrogen pressure is varied. The nitrogen pressure increases along the x-axis.

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Lower nitrogen densities are favored in "z-pinch" experiments as compared to Omega-like experiments. At very high nitrogen pressure (5–10 atm), both α and β are very large, which disfavors a photoionization front. In fact, a photoionization front is only favored at nitrogen pressures between 1.25 and 5 atm. The understanding that optimum gas densities for the case of Z are roughly three times smaller than those for Omega suggests that an actual experiment for Z should be physically three times larger than the example shown in Figure 6, if one desires for it to have the same scale in optical depths. Since the radiation drive takes time to develop, the photoionization front also takes some time to develop. In a real experiment, one would expect to observe photoionization fronts between 2.5 and 5.0 ns after the flux reaches the target. Similar to the Omega experiments, there is no preferred ionization state.

In this section, results from a parameter study looking for the formation of a photoionization front have been presented. Two dimensionless quantities are used to characterize the front α, which relates electron recombinations to the photoionizing flux, and β, which relates the electron-impact ionization to recombinations. A photoionization front is characterized by an α ≪ 1 and β ∼ 1. As shown in Figures 13 and 14, both the laser-driven and "z-pinch" experiments are capable of generating true photoionization fronts. In fact, photoionization fronts are generated over a wide range of radiation source temperatures and nitrogen pressures.

We now compare the theoretical results and numerical results. With regard to the hydrodynamical model developed in Section 2.1, we find very good agreement between the theory and simulations. For a weak "R"-type front (the classic, simple photoionization front), Figure 1 shows that the density is essentially unchanged. Figure 9 shows that the density is nearly unchanged, matching the theoretical predictions.

The three-level model of the atomic physics we developed in Section 2.2 proved useful in interpreting the simulations. We introduced α, the ratio of the electron recombination rate to the photoionization rate, and examined its relationship to the equilibrium ionization state fraction. As shown in Figure 3, different values of α result in each ionization charge state dominating in distinct regions. This result is confirmed in Figure 11, which shows that each ionization state of nitrogen is found at distinct optical depths.

We note that although adding additional levels to the approximation presented in Section 2.2 is straightforward, we find that three levels are sufficient for our present purposes.