The Key to Understanding Supersonic Radiative Marshak Waves using simple models and advanced simulations

This article studies the propagation of supersonic radiative Marshak waves. These waves are radiation dominated, and play an important role in inertial confinement fusion and in astrophysical and laboratory systems. For that reason, this phenomenon has attracted considerable experimental attention in recent decades in several different facilities. The present study integrates the various experimental results published in the literature, demonstrating a common physical base. A new simple semi-analytic model, is derived and presented along with advanced radiative hydrodynamic implicit Monte Carlo direct numerical simulations, which explain the experimental results. This study identifies the main physical effects dominating the experiments, notwithstanding their different apparatuses and different physical regimes. ∗ avnerco@gmail.com † highzlers@walla.co.il 1 ar X iv :1 91 1. 07 09 3v 1 [ ph ys ic s. co m pph ] 1 6 N ov 2 01 9


I. INTRODUCTION
Radiative heat (Marshak) waves play an important role in many high energy density physics phenomena, such as inertial confinement fusion (ICF) and astrophysical and laboratory plasmas [1,2]. In recent decades, several experiments using supersonic Marshak waves propagating through low-density foams have been performed and reported. These experiments facilitating high energy lasers, typically using hohlraums as a drive energy generator [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Typically, the drive energy in these experiments is transferred in the form of a heat wave into a low-Z foam, coated with a high-Z envelope (e.g. Au). The radiative waves are radiation dominated and approximately supersonic (i.e. hydrodynamic motion is negligible), and can be described by the Boltzmann equation. Nevertheless, the high-Z walls are optically thick, and thus affect the system through their ablation into the foam.
Consequently, hydrodynamics should be taken into account, in order to model their effect correctly.
The common numerical schemes for radiation transport usually employed in order to solve these problems, the implicit Monte-Carlo (IMC), and methods of discrete ordinates (The S N method), have been compared and validated with simple exact benchmarks on several occasions. However, the principal goal of these experiments has been to validate the macroscopic models for radiative hydrodynamics against real experiments, as opposed to simple benchmarks [16,18]. Hench, the theoretical understanding of these systems is still incomplete, also because of uncertainties in the input microscopic databases for these models, as opacity and equation-of-states (EOS) [17,18].
Most of the experiments examined in the present study were analyzed separately (at different levels) with theoretical models and/or simulations. Still missing, nevertheless, is a unified theoretical modeling and understanding of the different class of all the experiments. Accordingly, the main goal of the present study is to gain a comprehensive understanding and modeling of these experiments, allowing the derivation of a common base ground. In this work we integrate all the different experiments (that possess sufficient data for modeling), in order to significantly increase understanding of the radiative phenomena at hand. We present a simple semi-analytic model which takes into account the main physical aspects of the problem. This model yields both qualitative and quantitative results. Nevertheless, we use exact 2D Implicit Monte-Carlo (IMC), coupled to hydrodynamics simulations, in order to attain a detailed reconstruction of the experiments. These two building blocks enable a comprehensive understanding of the physical mechanisms dominating this type of experiments. The simple model was in part previously published in [19], a paper that included discussion of some of the main physical aspects of the problem. In the present study, the model is fully derived, including all the main physical procedures. Both the model and the exact simulations are examined against all the experimental results. As will be discussed further below, we demonstrate that although the different experiments were carried out with diverse apparatuses, diagnostics, and target fabrication methods, they share several features, and the main physics governing the system is very similar.

II. THE EXPERIMENTS
During the past decades, several experimental campaigns were performed and published in literature. The different experiments studied in this paper possess a common procedure, which is presented schematically in Fig. 1(a). High energy laser (1kJ-350kJ) is delivered into a small (∼1-3mm), high-Z cavity (usually made of gold), i.e. hohlraum, used as an X-ray source. This shot is represented by the blue beams in Fig. 1(a). The hohlraum walls absorb the laser energy, heated and re-emits soft X-ray, with radiation temperature of about 100-300eV (red arrows in Fig. 1(a)). A physical package, made of a dilute (low-density) foam cylinder is pinned to the hohlraum (through a hole in the hohlraum walls), so X-rays are delivered into the foam (i.e. Drive temperature), generating a heat wave (Marshak wave) that propagates down stream. The foam is usually coated with gold which is optically-thick for X-rays, minimizing possible radiation leakage (the yellow lines around the gray foam in Fig. 1(a)). Note, that since this basic design include interaction between heat waves and several materials inside the physical package, one must consider the possible hydrodynamic effects of radiation-material interaction. We discuss this further below in Sec. III Table I summarize the different experiments, have been published in literature and analyzed in this paper. For each experiment we specify the material of the foam, its density, and the maximal drive temperature. Table I demonstrates the large range of temperatures, materials (low and high-Z) and densities investigated in this study. Some of the experiments [9,10,14] use a small amount of higher-Z doping foams that are compared to "pure foams". This allows a study of the sensitivity of the Rosseland opacity, (almost) without foam lengths, The out-coming flux is measured via XRD or an X-ray steak camera. In the figure, images of the breakout radiation flux from the foam in different lengths by an X-ray streak camera.
(c) An alternative experimental technique tracks the heat wave, using a number of slits (holes) in the gold wall, allowing the measurements of the radiation wave emission as it propagates inside the tube. The figures are taken from [20], [5] and [23] respectively.
alteration the heat capacity.
In these experiments, different techniques are employed to examine the heat wave propagation in time. The most popular is to measure the flux breaking through the edge of the foam as a function of time using an X-ray steak camera [3][4][5][6][8][9][10][11] or an X-ray diode (XRD) array [15,16], using different foam lengths. An example of measuring the radiative flux using an X-ray streak camera is presented in Fig. 1(b) taken from [5].
Another diagnostic tool used, has been to measure the heat wave radiation perpendic- ular to the heat wave propagation, through a set of small slits (holes) in the gold tubes (see Fig. 1(c)), tracking the heat wave Eulerian position [15,16]. Another version of this diagnostics technique is to use one long window [13,14]. Alternatively the self emission of the foam can be tracked by a back-lighter, which was the technique employed for example in [12]. However, in this technique the foam should be bare, allowing the radiation of the back-lighter to pass through the material.
In the different experiments, the radiation drive temperature in the hohlraum is measured as a function of time, which allowing an estimation of the incoming temporal flux into the foam. Fig. 5(a) (green curve) shows a typical example of the radiation drive temperature (T D ), of the high-energy Back et al. experiment [5]. It should be noted that the radiative temperature in the hohlraum, is usually measured via the laser entrance hole (LEH) of the hohlraum (the LEH is shown in Fig. 1(a)). Interpolation to the exact drive temperature that enters into the foam is not trivial [16,21,22]. In this paper we assume that the temperature which was measured via the LEH is equal to the exact drive temperature (T D ).
The governing equation that describes the behavior of radiative heat waves is the radiative transport equation (RTE), also known as the Boltzmann equation (for photons) [24]: where I(Ω, r, t) is the specific intensity of the radiation at position r, propagating in thê Ω direction, at time t. B(T m ( r, t)) is the thermal material energy, while the material temperature is T m ( r, t), c is the speed of light and S(Ω, r, t) is an external radiation source.
σ a (T m ( r, t)) and σ s (T m ( r, t)) are the absorption (opacity) and scattering cross-sections, respectively. For the experiments discussed in this paper the Boltzmann equation should be coupled to the material energy equation: Where C V is the heat capacity, and a is the radiation constant. When the heat wave velocity is close enough to the speed of sound inside the material, hydrodynamics cannot be neglected (i.e. the flow becomes subsonic), and the radiation equations should be coupled to the hydrodynamic equations. In the examined experiments, the heat wave propagating in the foam is supersonic. However, when the gold tube is heated the heat wave within the gold walls become subsonic.
An exact solution for the transport equation is hard to obtain, especially in multidimensions. The most well-known exact approaches are the P N approximation, the S N method and Monte-Carlo techniques [24]. In the P N approximation, we solve a set of moments equations when I(Ω, r, t) is decomposed into its first N moments. The S N method solves the transport equation in N discrete ordinates. These two approaches yield an exact solution of Eq. 1 when N → ∞. Alternatively, a statistically implicit Monte Carlo (MC) approach can be used [25]. It is also exact when the number of histories goes to infinity.
In the present work, the radiative transfer in the different experiments is modeled via a full 2D IMC model, coupled to the hydrodynamics equations. We now turn to describing the numerical simulations.

IV. 2D SIMULATIONS
This section describes the full 2D simulations use for the present study. Demonstration of a 2D radiative hydrodynamics simulation of a propagating heat wave for the high-energy Back et al. experiment [5] can be seen in Fig. 2 (Temperature maps in (a), and density maps in (b) different times). Fig. 3 shows similar maps of the SiO 2 experiment conducted by Moore et al. [16]. In both examples, the two-dimensional effects can be seen clearly at late times (especially in 2.5nsec in Fig. 2 or 3nsec in Fig. 3), as the heat front is bent along the r direction [26] due to energy loss to the gold walls. Far from the center the sample become denser, due to the hydrodynamic ablation of the opaque walls into the foam (the black line in Figs. 2 and 3 shows the boundary between the foam and the gold). Another effect that can be clearly seen, is the reduction in cross section area on the hohleraum side (right side in the images) due to wall lateral movement.
As discussed above, the most popular diagnostics used in the different experiments, was a measurement of the flux that breaks out from the foam as function of time, by an X-ray The different experiments that are studied in this work are compared to the simulations by following heat front positions, and in the relevant experiments (when the heat-wave is studied via several foam lengths), the out-coming flux is also examined.

V. SIMPLE (SEMI-ANALYTIC) MODEL
As already noted, one of the main objective of the present study is to introduce a simple approximate semi-analytic model, for the purpose of analyzing, the experimental results.
This model is based on the 1D heat-wave propagation analytic model of Hammer and Rosen (HR) [27], while a primary version, that includes only some of the physical phenomena was introduced in [19]. In the present work, we expand the model, and derive a full version that take into account all the main physical phenomena that affects the general propagation of a radiative heat wave inside a finite tube. We identify the following four separate physical mechanisms that affect the system, each of which is itemized more fully below: (1) the correct incoming energy flux into the foam; (2) the experimental diagnostics cut off; (3) the energy loss to wall heating; (4) the effect of wall ablation. It should be noted that the first two mechanisms are one dimensional in nature, while the last two must consider the two dimensional nature of the problem. Therefore, we separate the discussion to 1D aspects/corrections to the model from the 2D effects that have to be taken into account.
Using this model we study all relevant experimental results that were summarized in Table. I in Sec. II. We demonstrate the model using as an example the Back et al. high-energy SiO 2 experiment [5]. The full comparison with the experimental results is presented below in Sec. VI. In the next section we present a detailed description of the four physical mechanisms listed above.

A. 1D corrected HR Model
When the problem contains several mean free paths (mfp), the specific intensity becomes close to isotropic, and the exact Boltzmann solution (Eq. 1) tends to a diffusion approximation [24]. In specific, the system tends to a local thermodynamic equilibrium (LTE), so both Eqs. 1 and 2 can be described by one equation for the temperature of the matter (which is equal to the radiation temperature, under the local thermodynamic equilibrium (LTE) assumption [28,29]). In this case, the heat wave is characterized by a sharp front, due to the nonlinear behavior of the opacity and the heat capacity.
As a first step before engaging the problem, one must first cover the basic microscopic physical qualities of the material in hand, i.e. opacity and EOS. Numerous studies cover selfsimilar solutions of both supersonic and subsonic radiative (Marshak) heat-waves [27,28,[30][31][32][33][34][35]. In these solutions, one assumes that the Rosseland mean opacity κ (which is connected to the absorption cross-section σ a (T m ( r, t)) = (κρ), when ρ is the material's density) and the internal energy e(T, ρ) can be approximated in a power-law form (using [27] notations): In the current work, we are interested in foam parameters (rather then solid gold as in [27]), so g, α and λ were extracted by fitting Eq. 3a to the opacity spectrum calculated using CRSTA [36,37]. f , β and µ were extracted by fitting Eq. 3b to the EOS from SESAME tables (when appears) [38], or QEOS [39] tables in the relevant regime of the experiment (by mean of temperatures and densities). The different parameters for the different material are presented in the Appendix. The parameters for Au were taken from [27].
Hammer and Rosen (HR) calculated an exact analytic solution for 1D LTE supersonic diffusion equation using a perturbation expansion theory, for a general surface boundary condition T S (t) [27]. The heat front position, x F (t), as a function of time is solved analytically and can be expressed as: where: Using the HR solution demands the surface temperature T S (t) as an input. A naive assumption, that the surface temperature is equal to the radiation drive (hohlraum) temperature here. This deviation caused earlier studies presented in literature to use an ad hoc factor to decrease the effective T S (t) to yield an agreement between the theory and the experiments [3,16]. Below we consider the physical phenomena which dominates the process of obtaining the correct boundary condition without ad hoc coefficients or free parameters.   The 2D model which includes energy losses to the gold wall is in the orange dashed curve. The 2D model which also includes the gold ablation that blocks part of the energy that enters the foam is in the red dashed curve. The full 2D simple model that includes all the gold ablation effects is in the dashed black curve.

The different radiation temperatures
Analyzing the problem, one must distinguish three different radiation temperatures: The drive (hohlraum) temperature T D (t), the surface temperature T S (t) and the brightness temperature of the re-emitted flux, T obs (t), that a detector will measure [21,22]. The latter (T obs (t)) is the temperature in ≈ 1mfp optical depth (2/3 mfp, assuming LTE diffusion behavior) [22].
Assuming LTE diffusion, the Marshak boundary condition at the surface of the material yields [21,22,24,28,40]: Yielding the correct T S (t) via the given T D (t) is due to knowing F (0, t), which is the time dependent energy flux on the boundary. HR yields also the total stored energy inside the material [27], recalling that F (0, t) ≡Ė(t): One can solve Eqs. 4, 6 and 7 as a closed set of equations, yielding a correct solution for T S (t). Following [19], we use a simple algorithm (detailed in the appendix there), in order to compute T S (t).
An example of the corrected T S (t), for the high-energy Back et al. experiment is shown in is yields a considerable improvement of the HR model (compared to the naive assumption in green curve). However, the model yields ≈ 1.3 faster x F (t) than the experiment.

Heat-wave position correction due to experimental cut-off
Another significant 1D feature that needs to be taken into account when comparing the theoretical heat front position, x F (t), to the experiments, is the exact definition of the "experimental front". This is important especially when measuring the heat flux from the side of the foam (via a hole or bare foam as in [12][13][14] ). The experimental heat front is usually set as a finite value of the maximal emitted flux. For example, in the Keiter et al.
experiment it is determined where the radiative energy is 40% of the highest flux [14], and in Ji-yan et al. experiment it is in 50% of the highest flux [12].
In order to calculate the appropriate theoretical estimation, we assume that the temperature profile of the heat wave inside the foam is a Henyey-like profile [21,27], with the corrected x F (t), yields from Eq. 4: In [19] we show that Eq. 8 provides good temperature profile estimation for any given boundary condition with a corrected x F (t). Assuming that the experimental heat position is determined where the radiative energy is f of the highest flux, the effective energy density The effective heat front position can be obtained using Eq. 8: This modification is used in sections VI F and VI G to analyze the experiments conducted by Keiter et al. [14], and Ji-yan et al. [12], where the tracking on the heat front was from the side of the foam (see Fig. 1(c)).

B. 2D corrections to the 1D estimation
Observiation of the 2D advance simulations (see Sec. IV, Fig. 2, 3) demonstrates why the heat front propagation slows at late times. This is due to two phenomena. One is the energy loss to the walls (usually gold in most of the experiments) that coat the foam. The other is the ablation of heated walls, blocking part of the energy that enters into the foam from the hohleraum. The ablation of the walls (when it appears) increases the density of the foam, as shown in Figs. 2(b) and 3(b). We note, that one dimensional simulations, in which no energy is lost to the walls and where the wall ablation is not present, show no such slowing. We also note, that similar phenomena were observed in previous hydrodynamic experiments [41,42].
A schematic diagram for this 2D physics is shown in Fig. 6. The cold low-density foam and walls are shown in blue and yellow, respectively. The orange area is the heated area inside the foam, when the yellow-orange pattern is the heated area inside the gold walls, that also ablates into the foam. The ablated wall blocks part of the energy that enters the foam due to smaller hohlraum-foam interface area on one hand, and compress the foam on the other hand, via an ablation shock, laterally propagating inward, towards the tube axis.
Thus, the heat wave propagation slows due to the density-dependency in Eq. 5b.
FIG. 6. Schematic diagram of 2D slice of the heat wave. The cold foam and the gold walls are in blue and yellow, respectively (the wall area is larger than in reality). The orange area is the heated foam, while the yellow-orange pattern is the heated part of the gold. The heat wave loses energy to the gold walls, and therefore its velocity becomes slower. The walls also ablates into the foam, blocks part of incoming energy and make the foam denser, and thus, slows the heat wave propagation.

Energy wall losses treatment
For evaluating the energy losses to the walls, one can use the self-similar solutions of the 1D slab-geometry subsonic Marshak waves [27,[31][32][33][34]. Each spatial element is exposed to the heat front in time t 0 (x) for a period t − t 0 (x), so the total energy that leaks to the wall is the sum of all the energy spatial stored elements (see also [19]): We need an expression for E mat (t), which is the energy in one such element. In most experiments, the walls are made from gold, so we can use the expressions from [34]. Here we use the expression for constant boundary temperature (The dependency of the results to specific BC is small): Note that although the general form of Eq. 11 is correct, the numerical coefficients should be calibrated for other materials than gold. For example, in one experiment the foam was coated with Beryllium (low-Z material) sleeve, exploring the sensitivity to the heat wave propagation to the wall looses (comparing to the high-Z gold walls) [6,26,43]. The parallel expression for Be is: We note that this expression is for supersonic heat-wave (without hydrodynamic motion), since the Be is optically thin, and in ≈ 200eV it is mostly supersonic. As in the foams, the parameters for the opacity was fitted to CRSTA tables [36,37], and for the EOS, SESAME table [38]. The different parameters for the different materials (Gold, Be) are presented in the Appendix.
In some of the experiments, such as the Ji-yan et al. experiment [12], the tube had no walls, so the leakage can be approximate as an emission to a vacuum [28,40], instead of Eq. 10: We note that considering the 2D effects (as in Eq. 10), we assume that the heat wave has a flat-top shape inside the foam, i.e. a constant temperature until x F (t) for simplicity.
In [19] this assumption was checked against a more exact Henyey-like temperature profile, but the effect was small.
In Fig. 5(a) we can see (red curve) the T S (t) that is yielded by taking into account the energy losses to the gold walls. We can see the non-negligible difference between the 1D-T S (t) and the 2D-T S (t) (≈ 10eV). In Fig. 5(b) (dashed orange curve) we can see the major improvement achieved by taking into account the 2D effect of energy losses to the walls, that covers about 2/3 of the difference between the 1D-model prediction and the experimental results.

Ablation of the wall -velocity effects
By taking into account the inward lateral ablative motion of the wall one should estimate the rate of ablation. For this purpose, one can use, the subsonic self-similar solution, for the ablation velocity. For gold, the ablation velocity is [34]: u(ρ gold ) is a factor betweenũ = 1 (at the surface) andũ = 0 (at the heat front inside the gold), which is determined by the self-similar velocity profile. The exact self-similar function, as a function of the density is given in Fig. 7. However, the self-similar solution assumes a free surface (when the density goes to zero), while here the ablative wall in restrained by the finite density foam. Therefore, we should take the value ofũ(ρ) for the density of the same order of the foam's density. The approximate value is calibrated from the full 2D simulations and found to be ρ ≈ 4ρ f oam , for the different experiments. The coefficients in Eq. 14 should be replaced in those cases the foam is coated with material other than gold.
Specifically, in the Be sleeve experiment [6,43], we set u = 0 since the heat wave inside the sleeve is almost supersonic.
Knowing the surface velocity as a function of time, the radius of the foam cylinder for each space interval x at time t is: As a result, the inlet cross section of the tube decrease in time, decreasing the energy flux from the hohlraum into the foam. The modified entered energy is thus: In addition to this effect, the foam becomes denser due to the wall ablation: The time-dependent effective mean density of the foam is: Since the heat front propagation velocity depends upon the density in the foam (Eq 5b), this increase in foam density results with a decrease in the heat front velocity. Eq 5b can be rewritten as: In Fig. 5(b) we can see the effects of the ablation on the heat wave front. In the red-dashed curve we can see the effect of the energy-blocking on the foam (which is quite small), while the black dashed curve is the full 2D model, that takes into account the compression effect of the foam. This yields results that are very close to those in the experimental data (The effect of the ablation of the T S (t) itself is small, see black vs. blue curves in Fig. 5(a). In the next section we examine this model and the 2D simulations in all experimental outlines.

VI. ANALYZING EXPERIMENTS
In this section we will review the different experimental measurements, and analyze them by the simple model along with 2D radiative hydrodynamics simulations.  It can be seen that the 1D simple model (solid curves in Fig. 9) yields good agreement with the experimental results for the low drive temperatures, while in the high-drive (T D = 150eV) temperature (blue squares in Fig. 9), the experimental data has a large spread, preventing a clear comparison. Even so, the basic trends are captured and reproduced by the model.
We note that using the original naive HR estimations yields much faster velocities than our corrected model. Deviations of the naive estimations from the experimental data are about 20%. For that reason, Massen et al. reported an ad hoc correction (that was calibrated from a full simulation) for the heat-wave propagation velocity by a factor of ∼ 0.8, taking into account the re-emitted flux from the foam, back to the hohlraum [3]. In this work we have presented the physical explanation for this correction, quantitatively.
Since in this experiment the drive temperature was given as constant, and the foam diameter was not reported, we limit the analysis in this experiment to the 1D simple model.
We also note that the difference between the simulations and the model are much smaller than the scatter of the experimental data.

B. Xu et al. experiments
Several studies reported about a decade ago showed a set of heat wave measurements conducted at the SG-II facility using C 6 H 12 foams [8][9][10][11]. The radiation drive temperature T D that was reported and has been used for analysis in the current study is shown later in Fig. 11(b) (green curve). The experiments were carried out in 50mg C 6 H 12 with different lengths, 300µm and 400µm [8]. The main diagnostic that was used to track the heat wave propagation was also an XRSC in several different energy lines. The major line was around 210eV. An additional version of this experiment used a 300µm copper doped C 6 H 12 foam (C 6 H 12 Cu 0.394 ) [9,10].
A comparison of the intensity of the flux that leaks from the edge of the foam as a function of time, between the experimental data and the 1D IMC numerical simulation is presented in Fig. 10 (black curves). The simulations show good agreement with the data, with a difference of less than 50psec in breakout times, defined as the time the intensity is in half of the maximum intensity. Moreover, the variance between the 300µm and the 400µm, is very similar in the experimental results and in the simulations. We note that by the time the heat wave break out of the foam, the gold tube around the foam (600µm diameter) do not heat significantly. Therefore, simulations for this experimental setup demonstrate low sensitivity to 2D or hydrodynamic effects.
The experimental breakout times are given in Fig. 11(a)  In Fig. 11(b) the resulting 1D (solid blue) and 2D (dashed black) surface temperatures T S (t), are presented. It can be seen that as expected, the temperatures used for the model are significantly lower than the drive temperature T D (solid green), as was shown above in the Back et al. experiment (see Fig. 5(a) and the relevant discussion). We note again, that the T D is estimated rigorously, and not as a scaling factor, as was the case in previous works.
This results show good agreement, concerning the heat front breakout time ( Fig. 11(a)), whilst both the 1D and the 2D models yield close agreement to the experimental data (the experimental data lies between the 1D and the 2D models). Specifically, the models predict (as do the full simulations) the difference in break out time between the lengths that were These experiments were carried at the OMEGA-60 facility in 2000, and were then the most detailed supersonic heat wave experiments to date [5,6]. The radiative drive temperature was shown previously in Fig. 5 and reached the maximal temperature of ≈ 190eV. The experiment was carried out using two different foams, 50mg/cc SiO 2 and 40mg/cc Ta 2 O 5 in several lengths (0.25-1.25mm). The samples were fabricated inside a cylinder, 1.6mm diameter of different lengths, and were coated with 25µm thick gold. The Ta 2 O 5 foam has a higher Z than the SiO 2 foam, and thus is more opaque due to its smaller Rosseland mean free path. However, the SiO 2 , has a larger heat capacity. Hench, the heat wave propagation is similar in both foams.
A comparison between the experimental results, the simple model and the simulations, presented and discussed in Sec. V and Sec. IV VI, for the SiO 2 version of this experiment, demonstrating the importance of the 2D effects (energy wall loss and wall ablation) in this experimental setup.
The flux radiate that leaks from the edge of the foam was measured as a function of time for different lengths as in Fig. 4, showing good agreement between IMC simulations and experimental results, especially for the heat front breakout times (when flux reached half of its maximal intensity). In Fig. 12(a)   For the Ta 2 O 5 foam, an additional experiment was carried out, replacing the optically-thick gold sleeve with an optically thin Beryllium sleeve [6,26,43]. The full experimental data was not published, yet, it was reported that the heat front breakout times (with 0.5mm and 1mm foam lengths) are 10% later, compared to the gold data (orange circles in Fig. 12(a)) [6]. This is due to the increased leakage of energy through the optically thin medium. The heat wave in the Be is almost supersonic (see Sec. V) and energy leakage can be computed (see Eq. 18(b) in [34]). We observe that the simple analytic model (orange curve) predict the difference between the Be tube and the gold tube, in the 0.5mm foam it yields 10% delay, while in the larger foam it is somewhat overestimates the time. The laser pulse length was fairly long, ∼ 10nsec. As for the high energy experiment, the samples were fabricated as cylinders of 1.6mm in diameter, in different lengths (0.5, 1 and 1.5mm), and were coated with 25µm of gold.
Since this experiment was conducted in a different thermodynamic regime (low temperature, low density), a specific new set of parameters was established for the semi-analytic model (see table II in the appendix). A full comparison between the experimental flux that leaks from the edge of the foam and full 2D IMC simulations as a function of time is shown in Fig. 13(a). Good agreement is evident again between the data and the 2D simulations for the breakout times. The agreement between the data and the simulations for the shape of the signal is only fair, however within the typical agreement achieved in previous studies [4,7]. In Fig. 13 [15][16][17][18]. These experiments were conducted in the high-power NIF facility and the drive temperature had reached to ≈ 300eV and shown in [16]. The experiment was carried out with two different foams, Two main diagnostics were used in the experiments. First is the Dante (an array of X-ray diodes), for measuring the radiative flux on the foam back side. A detailed analysis of the results from this diagnostic was presented in [17,18], showing a non-negligible gap between the the experimental results and the theoretical predictions. This is because this experiment was designed in such a way that calculating the arrival time of the heat front to the end of the foam (2.8mm), is very sensitive to correct opacity and EOS [20]. However, the full 2D simple model (black curves) exhibits good agreement for both foams.
The full IMC radiative hydrodynamics simulations agree well with the experimental measurements. Two different IMC curves are presented in Fig. 14  In these experiment the heat wave front position was measured using small window in the gold coating the foam, tracking the self emission of the hot foam. The heat wave front was defined as the place where the radiative flux reaches 40% of its maximal value (matches to T (x) ≈ 0.75T max ). In this case, we use Eq. 9 with a Henyey-like profile (discussed in detail in Sec. V A 2). We note that in this experiment, 2D effects are extremely important and have to be take into account by using a 2D model.
The difference between pure versus doped foams can be clearly seen in Fig. 15 [14]).
The 2D model prediction is given in the red curve for the pure foam, and in the black curve for the doped foam. Good agreement is evident between experimental data and the model prediction. We note that the 1D model (without 2D corrections) significantly deviates from the experimental data. The 2D IMC simulations (blue and green curves for the pure and doped foams, respectively), agree very well with the experimental results.

G. Ji-Yan et al. experiment
This experiment was performed on the SG-II laser, with a maximal radiation drive temperature of ∼ 175eV. The sample was a bare (without coating) plastic cylinder (C 8 H 8 ), with a density of 160mg/cc, 0.2mm in diameter, and 0.3mm long. The heat wave propagation was diagnosed by an XRSC that measured the self emission from the plastic perpendicular to the heat wave propagating. Since this foam is bare, the 2D model uses the radiating flux from the plastic to the vacuum surrounding, using Eq. 13. In addition, The heat wave front was defined as the place where the radiative flux reaches 50% of the highest flux. Hench, we modify the heat front due to Eq. 9 with f = 0.5. The experimental data is shown by the black circles in Fig. 16, while the different simple models are shown in the solid curves. We can see that the full 2D simple model again yields results that are very close to the experimental data, while 1D is too fast. This is due to the significant leakage of energy to the vacuum.

VII. SUMMARY
In the present work, a simple semi-analytic model for the radiative heat wave propagation in low density foams was presented and validated against a variety of experimental data, from different experiments carried out over the past three decades. The experimental results were also used to validate 2D numerical simulations using IMC for the radiative processes.
Although the experimental setups that were examined employed different foams, densities and compounds, doping, dimensions, etc., the simple semi-analytic model reproduces the experimental data very well, with differences of less than 10%. The model is based on the Using the exact opacity, CRSTA tables [36,37] and SESAME tables [38], or QEOS [39] EOS tables, we have fitted for every material its numerical parameters in the relevant experimental regime (by mean of temperatures and densities), for using by the different simple semi-analytic models. The parameters for the foams are above the double-line, and for the coats are beneath the double-line. The parameters for Au were taken from [27].

ACKNOWLEDGMENTS
We acknowledge the support of the PAZY Foundation under Grant N  All the materials parameters that were used in this paper in the simple semi-analytic models. The parameters are fitted to exact opacity, CRSTA tables [36,37] and SESAME tables (when appears) [38], or QEOS [39] tables in the experimental relevant regime. The parameters for foams are above the double line, and for the coats are beneath it.
thanks to Mordecai (Mordy) Rosen from LLNL for the valuable conversations regarding the three different radiation temperatures and the wall-albedo.