The hydrodynamic and radiative properties of low-density foams heated by x-rays

Low-density polymer foams are widely used in experiments related to inertial confinement fusion (ICF) driven by lasers [1] and high power pulsed generators [2]. Application of foam structures for smoothing the laser energy deposition onto the ICF-target shell was proposed by Dunne [3] for laser driven experiments at the national ignition facility (NIF). In order to ensure high symmetry of the integral radiation flux at the D–T capsule ablation surface, foams, surrounding the capsule, have been used in the hohlraum target design at the 20 MA Z-machine in Sandia [4]. Laser imprint reduction in fusion scale plasma of mm size was reported in experiments on the LIL facility at the laser energy level of 12 kJ [5]. In experiments, where under-critical density foams were used, plasma-induced smoothing was attributed to laser light multiple scattering on laser induced density perturbations due to the stimulated forward Brillouin scattering [6]. Increased stimulated Raman scattering (SRS) was measured in experiments where low-density polymer foam layers, ionized by a nanosecond laser pulse, have been used as plasma targets to measure SRS backscattering of the picosecond laser pulse [7]. These observations are of prime interest to the ICF community, as they contribute to explaining the large level of stimulated Raman backscattering measured in laser experiments at a MJ level. X-ray heated low-density foam layers are promising candidates for the creation of plasmas with near critical electron density (NCD), which can be used in relativistic laser experiments. Recently, many novel phenomena have been discovered when ultra-intense laser pulses interact with NCD plasmas of electron density n_e in the range of 0.1 n_c  <  n_e  <  n_c, where n_c is the critical plasma electron density associated with the laser frequency. Effects such as relativistic self-focusing [8], pulse steepening [9] and resonant [10] or stochastic [11] electron acceleration can greatly improve the laser to particle energy conversion efficiency.

Another important application of x-ray heated low-Z foam materials was found in ICF-relevant experiments on the heavy ion energy loss in ionized matter. These experiments, which use a unique combination of a high power laser and a heavy ion linear accelerator, have been carried out at the Gesellschaft für Schwerionenforschung, GSI Darmstadt [12, 13]. Increase of the ion energy loss up to a factor of 2 was demonstrated in a plasma target, created due to laser–matter interaction. Since the ion stopping in plasma depends on the plasma ionization degree, which on the other hand is a function of plasma electron density and temperature, hydrodynamic stability during the ion beam–plasma interaction time of 3–5 ns and a homogeneous plasma layer is required for the correct interpretation of measurements. Such homogeneous, ns-long living plasma layers have been obtained by irradiation of foam targets with soft x-rays, generated due to interaction of the laser pulse with a gold hohlraum [14]. The combined target design for ion stopping experiments was proposed by Vasina and Vatulin [15].

In high energy density experiments using laser or pulsed power generators, a large variety of foam substances with different mean densities and types of microstructure are used. The mean densities of polymer foams range from 0.1 g cm⁻³ down to 2 mg cm⁻³, up to 10³ times less than a solid. Their microstructure can be described by randomly distributed thin fibers or walls, like in agar-agar (C₈H₈O₃), the quasi regular 3D sponge-like structure in polystyrene (CH)_n and cellulose triacetate (TAC) with C₁₂H₁₆O₈ chemical composition. The wall or fiber thickness and the pore size strongly depend on the foam type and fabrication conditions. The TAC aerogels (further referred to as foams or TAC-foams) present the finest structure with an average thickness of solid fiber of 100 nm and a pore size of around 1 μm [16, 17]. These proved to be suitable for contemporary laser plasma application [18–20] and experiments on heavy ion stopping.

In this paper, we report on experimental results obtained at the PALS laser facility and results of numerical simulations aimed at creation of hydrodynamic stable homogeneous plasma layers. In these experiments, soft x-ray radiation was generated due to interaction of the laser pulse with a gold hohlraum. X-rays heat the foam layer, which was attached to the open side of the cylindrical hohlraum, and transfer it from solid into the plasma state.

The experimental set-up used to characterize the hohlraum-based x-ray source and the radiative and hydrodynamic properties of the created polymer plasmas are described in section 2. In section 3, measurements of the hohlraum radiation spectral distribution, duration of the x-ray pulse, and the size of the x-ray source are discussed. All these parameters are important for characterization of the energy of the hohlraum based x-ray sources.

Effective x-ray heating of any material requires proper matching of the x-ray source spectral distribution and the photon energy dependent absorption properties of heated matter. Measurements of the fraction of x-ray energy absorbed by foam layers are presented in section 3.1.

Depending on the x-ray photon mean free path in matter, foam heating can occur via propagation of the radiation driven heat waves or volumetrically. In section 3.2, measurements of the x-ray heat wave propagation velocity are presented. The hydrodynamic response of the foam layer on x-ray heating has been investigated and is compared with those of solid foil of approximately the same areal density. The results are presented in section 3.3.

Over a few years, extended numerical simulations for different parameters of the hohlraum-based x-ray source and polymer foam layers have been performed [21–25] and were used for optimization of the conversion efficiency of the foam indirect heating process. In section 4 we present results of the simulations, which are relevant for experiments carried out at the PALS laser facility and can be helpful for better understanding of the indirect matter heating process.

Experiments on the investigation of the indirect laser heating of low-density polymer foams have been carried out at the PALS laser facility in Prague (www.pals.cas.cz). In this scheme, a laser pulse interacts with a gold wall of a cylindrical converter (hohlraum) where the laser energy is effectively converted into soft x-rays with a spectral distribution close to the Planckian one. Hohlraum generated x-ray radiation heats volumetrically a low-density foam layer creating a hydrodynamic stable plasma layer, which can be used, for example, in experiments on heavy ion stopping in plasma and relativistic laser based electron acceleration.

Investigations of the foam heating by x-rays include three important steps. The first is the characterization of the hohlraum based x-ray source and the conversion efficiency of the laser energy into energy of the soft x-ray radiation. This includes measurements of spectral and spatial distributions of the convertor radiation, duration of the x-ray radiation used for the foam heating and evaluation of the hohlraum radiation temperature. The next important step is investigation of the mechanism of the foam heating and characterization of TAC-plasma absorption properties. This gives information about a fraction of the x-ray energy absorbed by a low-density foam layer and converted into the plasma internal energy. The last step includes characterization of the plasma parameters reached.

The iodine laser with a fundamental wavelength of 1.315 μm and 0.35 ns pulse duration has been operated in a 3ω-option depositing into the hohlraum up to 300 J energy focused into an 800 μm spot. The laser intensity was kept moderate at the level of (2–4)  ×  10¹⁴ W cm⁻² in order to suppress hot electron generation.

Two types of combined converter-foam targets used in experiments are shown in figures 1(a) and (b). Combined targets of the type 1 have been used for measurements of the photon energy dependent TAC-plasma x-ray absorption properties and plasma expansion dynamics. The type 2 has been designed for the investigation of the radiation driven heat wave propagation in the TAC-foam and later for experiments on heavy ion stopping in plasmas [12, 13]. In both target types a cylindrical geometry of the hohlraum was used. A hole of 1.3–1.4 mm in diameter, drilled in the 1.9 mm thick Al-plate, was coated with a 12.5 μm thick Au-foil. In some experiments, the open end of the hohlraum, opposite to the foam, was covered by a 100 nm thin gold layer in order to increase the efficiency of x-ray generation. Depending on the experimental goals, a Cu-holder with a 2 mg cm⁻³ TAC-foam layer was attached in two different ways as shown in figures 1(a) and (b). The diameter of the foam layer was 2.5–2.8 mm, the thickness was varied from 0.5 to 1.5 mm.

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The experimental set-up includes a variety of plasma diagnostics such as x-ray diodes, a transmission grating spectrometer and x-ray pinhole cameras for measurements of the hohlraum x-ray emission with spectral, spatial and temporal resolutions. For characterization of a hydrodynamic behavior of x-ray heated foam layers an optical streak-camera was used. Propagation of supersonic radiation heat waves from the hohlraum exit into the foam was monitored by means of a microchannel plate (MCP)-based four-frame gated pinhole camera with a nanosecond time resolution. Positions of x-ray detectors relative to the combined targets are indicated in figures 1(a) and (b).

3.1. Characterization of the hohlraum based x-ray source and polymer foam x-ray absorption properties

Effective x-ray heating of any material requires proper matching of the x-ray source spectral distribution and photon energy dependent absorption properties of heated matter.

Therefore, the characterization of the converter radiation field, its spectral and spatial distribution was one of the first important experimental goals. Measurements of the hohlraum generated x-ray pulse duration and the soft x-ray source size together with converter radiation temperature allowed estimation of the total x-ray energy irradiated by the source and the laser to x-ray energy conversion efficiency.

Temporal evolution of the soft x-ray flux generated in the Au-converter and the flux transmitted through a foam layer was measured by means of two vacuum x-ray diodes with carbon cathodes covered by 7 μm Mylar filters. Photons pass through the filter and the anode grid and interact with a photocathode. An induced photocurrent is registered in dependence on time. Voltage applied between the photocathode and the anode grid was 1 kV. In these experiments, type 1 targets were used.

X-ray flux from the open end of the cylindrical hohlraum was registered by the diode D1, as shown in figure 2(a), the x-ray diode D2 was placed from the opposite hohlraum side, which was covered by a 1 mm thick 2 mg cm⁻³ TAC-foam layer, and measured the x-ray flux transmitted through the TAC-plasma.

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Figure 2(b) shows the duration and the amplitude of the x-ray signals measured by D1 and D2. The fact that the duration of the signal measured by the diode D1 of 10 –15 ns is much longer than the laser pulse (0.35 ns) can be explained by the processes which take place in the hohlraum interior after laser irradiation. The laser energy deposited during 0.35 ns into the focal spot is redistributed inside the Au hohlraum via absorption and reemission processes and stays confined for a longer time. This is very important for maintaining a long-lasting process of foam heating by the hohlraum x-rays.

Comparison of the D1 and D2 photo-current signal amplitudes demonstrates up to tenfold attenuation of the x-ray signal due to absorption in the 1 mm thick layer of 2 mg cm⁻³ TAC-plasma. High x-ray absorption efficiency is of importance for effective conversion of the hohlraum x-ray energy into a plasma temperature.

In order to characterize the size of the hohlraum based soft x-ray source, a pinhole camera with a grazing incidence glass mirror has been used [14] with an angle of 4.6° with respect to the mirror surface. The reflectivity of the glass mirror for x-rays drops considerably with higher photon energies. This allows, in combination with a micrometer thin Mylar filter, cutting out of a narrow spectral window between 200 and 280 eV. Images were recorded by an absolutely calibrated UF-4 x-ray film. In the case of homogeneous hohlraum radiation field distribution, the x-ray source size is equal to the diameter of the hohlraum cylinder.

Characterization of the hohlraum radiation field and photon energy dependent TAC-plasma absorption properties has been carried out using a transmission grating spectrometer (TGS) with a 700 lines per mm gold-grating. For spectra registration in the photon wavelength range from 18 to 200 Å, the UF-4 x-ray film was used. The film was absolutely calibrated in a spectral range 0.28–2.3 keV, for lower photon energies the UF-4 sensitivity was extrapolated down to 0.06 keV.

In figure 3(a), raw spectra of the Au-hohlraum radiation (1), hohlraum radiation transmitted through 0.5 mm (2), 1 mm (3) and 1.5 mm (4) thick 2 mg cm⁻³ TAC-plasma versus the wavelength in angstroms are presented. Experiments were made with a combined target of type 1. Zero order of diffraction (at λ = 0) shows a de-magnified image of the x-ray source integrated over all photon energies. The spectral resolution was dominated by the large source size and reached Δλ = 21 Å. The procedure of spectrum reconstruction includes five diffraction orders of the grating.

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Absolute emissivity of the hohlraum plasmas in J/keV/sr/cm², reconstructed from the recorded spectra, is presented in figure 3(b (1)). Spectra can be well approximated by the Planckian distribution with the maximum in the photon energy range between 120 and 140 eV and a corresponding color temperature of 40–45 eV.

A calorimetric temperature of the Au-hohlraum plasma characterizes the integral of irradiated energy, which depends on the measured spectral emissivity, x-ray pulse duration and the source size. The evaluation of the calorimetric radiation temperature from the measured time integrated hohlraum spectrum supposes the Planckian spectral distribution of the converter radiation. This is not the case during the action of the laser pulse of 0.35 ns, but at later times (>1 ns) the equilibrium radiation field will be established and the Planckian approximation can be applied for description of the hohlraum spectra [26]. The evaluation procedure results in a calorimetric radiation temperature of ~35–40 eV for the 300 J laser shots and the x-ray source energy of 35–40 J. The diagnostics show stable results from shot-to-shot.

Comparison of the measured hohlraum spectra in the photon energy range from 0.06 to 1 keV with those transmitted through the foam layers of different thickness allows investigation of the TAC-plasma absorption properties in dependence on the photon energy. Features of the transmitted spectra, presented in figure 3(b, 2–4), demonstrate from two to tenfold attenuation of the hohlraum radiation by TAC-plasmas depending on the layer thickness and the photon spectral range. This strong attenuation is caused by the effective photo-absorption on L- and K- shell electrons of carbon and oxygen. This is in good qualitative agreement with calculated TAC-plasma opacities (see figure 6, section 4). For correct and detailed interpretation of measured data on the polymer plasma absorption coefficients, hydrodynamic simulations of the heating process accounting for radiation transport are required. This work is in progress.

3.2. Propagation of supersonic x-ray heat waves in TAC-plasmas

High absorption efficiency of the hohlraum radiation by a plasma layer ensures effective conversion of the x-ray energy into the plasma internal energy. Depending on the foam density and the x-ray source spectrum, foam heating can proceed in two different regimes, which are discussed in section 4.

In the described experiments, the process of foam heating was observed by means of a MCP-based 4—frame gated pinhole camera providing 2D x-ray images caused by plasma self-radiation [27]. The microchannel chip is divided into four sectors which can be gated independently from each other by applying high voltage pulses (see figure 4(a)). The exposition time and delay between frames can be varied. The size of four pinholes used in our experiment was 70 –100 μm; pinholes were not covered by any filters. The MCP-camera has a maximum quantum efficiency between 0.07 and 1.2 keV. The combined targets of type 2 were used for these measurements.

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X-ray hohlraum radiation enters the TAC-foam through the open end of the converter and is absorbed by a layer of cold matter with a thickness of the order of the mean free photon path. Ionization and heating processes result in the creation of plasma, which starts to irradiate in forward and back directions. Figures 4(a) and (b) show the self-radiation of the TAC-plasma heated by this mechanism at different times. Exposition time and delay between successive frames were 3 ns.

Averaged over the exposition time of 3 ns radiation heat wave velocities at different times after the laser pulse have been evaluated: $V\left(1\,\text{ns}\right)=1.2\times {{10}^{7}}\text{cm}\text{}{{\text{s}}^{-1}};~V\left(4\,\text{ns}\right)=0.8\times {{10}^{7}}\text{cm}\text{}{{\text{s}}^{-1}};$ $V\left(7.1\,\text{ns}\right)=0.4\times {{10}^{7}}\text{cm}\text{}{{\text{s}}^{-1}};$ $V\left(10.1\,\text{ns}\right)=0.1\times {{10}^{7}}\text{cm}\text{}{{\text{s}}^{-1}}$. An important characteristic feature of foam materials is that propagation of radiative heat waves has a supersonic character. This means that radiation energy transfer occurs faster than the plasma thermal expansion or expansion caused by shock waves. In this case, plasma density keeps its initial value during the heating process and the velocity of the heat wave front propagation can be used to deduce the TAC-plasma temperature (see further discussion in section 4).

3.3. Hydrodynamic stability of x-ray heated foam layers

In experiments with the combined targets type 1, the expansion dynamics of foams heated by x-rays has been investigated and compared with those of solid foils with nearly the same areal density. The optical radiation of expanded plasma was registered by means of an optical streak-camera in a photon range of 500–1500 nm. A wideband pass color filter OS-11 was used in order to suppress a laser radiation and gain maximum intensity to streak. A time fiducial at the streak picture has been produced by means of the glass fiber which transferred a small part of the laser beam to the slit of the streak camera. The length of the glass fiber was chosen so that the fiducial corresponds to the moment of the laser pulse interaction with the hohlraum. The delay between the laser impact and the moment when expansion of the most part of the target is launched depends on the target thickness. In experiments, the hydrodynamic response of the 1 mm thick 2 mg cm⁻³ TAC-foam on x-ray heating has been compared with those of the 1.4 g cm⁻³ 0.9 μm thin Mylar foil.

Figures 5(a) and (b) show the optical streak images of the expanded plasma, created by the x-ray heating of a thin Mylar foil and a TAC-foam in similar experimental conditions. Both targets have been observed in transverse direction (along the y-axis in figure 1(a)), so that at the beginning the observation of the foam was hidden by the Cu-holder. Plasma self-radiation in the optical range can be observed only after plasma expands out of the holder size. The average expansion velocities resulting from the optical streak images are ~9   ×   10⁶ cm s⁻¹ and ~5   ×   10⁶ cm s⁻¹ for Mylar and the TAC plasmas respectively. In contrast to Mylar foil, TAC-plasma expands with the 20 ns delay between the laser pulse and launching of the plasma expansion demonstrating slow hydrodynamic response to the energy deposition. These results show that the x-ray heated foam layer cannot be replaced by a solid foil of the same areal density if experiments require a hydrodynamic stability of the plasma target [12, 13]. Moreover, simulations, presented in the next section, show that in contrast to the x-ray heated foam layer, which demonstrates homogeneous distribution of plasma parameters, irradiation of a solid foil with the same areal density results in strong gradients of plasma temperature and density, which vary on the sub nanosecond time scale.

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Heating of a porous substance by hohlraum generated x-rays was simulated using 1D two-temperature hydrodynamic code RADIAN [21, 22]. In case the plasma energy balance is affected not only by external radiation, but also by plasma self-radiation, the process of plasma formation has to be described by a set of hydrodynamic equations supplemented by the equation of radiative transfer, where the interaction of the radiation with matter is nonlocal and nonlinear. In simulations, the phase of the target material homogenization caused by micro expansion of heated structured material was neglected and the equation of state for an ideal gas was used. This approach can be applied in the case of extremely fine porous matter whose homogenization occurs on a time scale shorter than the external energy deposition time.

Radiative transfer in plasma was considered in a quasi-diffusion approximation [28]. The method takes into account both the transport of the external radiation and plasma thermal self-radiation. The transport equation includes the spectral absorption coefficients which can strongly affect the results of numerical simulation. In this sense, correct optical constants, which take into account a chemical composition of heated matter, are of crucial importance. The optical constants for the TAC chemical composition have been calculated in a context with the 'ion model' described in [22–24] for plasma temperatures and densities typical for the discussed experiments. Using these constants provides the best agreement between experimental results and simulations [21, 22]. Figure 6 shows an example of the TAC spectral absorption coefficients, calculated for plasma of 15 eV temperature and 2 mg cm⁻³ density. The absorption peaks around 100, 300 and 600 eV photon energies correspond to the resonance photo-absorption into K- and L-shells of ionized carbon and oxygen ions. The spectral radiative transfer was considered in the multigroup approximation in the photon spectral range of 0.1–1000 eV. The minimum amount of spectral groups used in simulations was 130. The group absorption coefficients were determined by averaging with the Rosseland weight function [21, 22].

Using hydrodynamic simulations, spatial distribution of the plasma density and the temperature inside the foam layer at different times of the heating process were predicted. Results have been obtained for a variety of hohlraum and foam parameters which were important for optimization of the combined target design and interpretation of experimental data [12–14]. Density and temperature evolution inside the 1 mm thick 2 mg cm⁻³ TAC-foam layer, heated by converter generated x-rays is shown in figure 7. Experimentally measured radiation temperature and duration of the soft x-ray pulse have been used for the calculations. X-ray heating occurs from the right side.

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The analysis of the TAC-plasma temperature evolution presented in figure 7 shows that during the first 3 ns the energy transport occurs via propagation of the radiation heat wave from the right side (hohlraum exit) into the foam layer. The heat transport occurs via absorption of the external hohlraum radiation by TAC-foam, conversion of cold matter into the plasma state and subsequent plasma reemission in the forward and back directions. By application of extremely low densities of matter, like in the case of 2 mg cm⁻³ foams, one can approach a supersonic heat regime. In this case, the heat wave velocity, which is inversely proportional to plasma density, is higher than the shock velocity. This situation secures hydrodynamic stability of the plasma layer. The layer reaches homogeneous temperature distribution already at 3–4 ns after laser irradiation. At the same time practically 100% of the plasma keeps the initial average TAC-foam density of 3   ×   10²⁰ atoms cm⁻³. Later, at 10–25 ns, the foam undergoes expansion keeping rather flat density and temperature distributions.

It is important to note that increasing the foam mean density by keeping the areal one, results very quickly in an under-sonic heat regime. According to simulations this already happens for the 10 mg cm⁻³ 200 μm thick foam layer [21] and strong plasma gradients begin to be characteristic for this regime.

Figure 8 represents, as a limit case, the expansion of a 0.9 μm thin TAC-foil of 2 g cm⁻³ density under the x-ray heating. One observes very different temporal evolution of the plasma density and temperature compared to the low-density foam layer (figure 7). Simuations have been made for a foil heated by the Planckian radiation of 30 eV temperature and 5 ns duration. At higher temperatures and longer x-ray pulses one would expect even more foil expansion dynamics. During the action of the x-ray radiation onto the μm-thin TAC-foil, plasma with strong gradients of the temperature and density is created. Already in a sub-nanosecond time window (0.1–0.9 ns in figure 8), the plasma's density drops more than 10 times on the plasma scale length of 5–10 μm. On this scale length, the plasma temperature varies from 20–25 eV down to almost 0 eV. Treatment of data, obtained in experiments using plasma targets with strong and fast varying gradients, are extremely complicated.

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The Mylar foil expansion velocity of 9   ×   10⁶ cm s⁻¹, resulting from the optical streak measurements (figure 5(a)) is in good agreement with simulation results. In simulations (figure 8), at 10 ns, the plasma expands up to 0.09–0.1 cm, factor 1000 of its initial thickness, and reaches a rather flat density profile with a density of about 1 mg cm⁻³. This results in the avarage expansion velocity up to 10⁷ cm s⁻¹.

As mentioned in the discussion of the optical streak measurements (figure 5(b)), the TAC-plasma created by the x-ray heating of the foam layer expands two times slower than the solid foil and with 20 ns delay between the laser pulse and launching of the plasma expansion. Simulations confirm this slow hydrodynamic response on the heating process. Unfortunately, a quantitative comparison between the experimentally measured TAC-plasma expansion velocity and results presented in figure 7 in this case is not possible, since simulations are limited by the time of 25 ns.

As noted in section 3, two different regimes of the foam layer heating, which depend on the target average density and the hohlraum temperature, can be realized. These two regimes are presented in figures 9(a) and (b). In the optically thin case (a), when the x-ray photon mean free path is larger than the layer thickness, the target will be partially transparent for the external x-ray source. This leads to a low level of radiation absorption and therefore to a moderate conversion efficiency of the external x-ray source energy into the TAC-plasma temperature. A big advantage of this regime is the volumetric character of the heating process, which provides homogeneous temperature distribution over the layer thickness on the very early stage of irradiation. In the optically thick case (b), when the photon mean free path in plasma is shorter than the plasma size, target heating occurs step by step via propagation of the radiation driven heat wave. This regime, where up to 90% of the external source energy can be used for the foam layer heating, can be achieved by increasing the target density or decreasing the hohlraum temperature. In some cases, at the beginning of the x-ray heating process, the plasma layer is optically thick for external radiation. But at later times, when the plasma temperature increases, it starts to be partially transparent since the plasma absorption coefficients drop with the temperature [23–25].

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The heating process, observed in experiments by means of a 4-frame MCP camera (figure 5), can be attributed to the optically thick case. The radiation heat wave propagates with a supersonic velocity and heats the foam target layer by layer. Marshak [29] was the first who considered phenomena associated with radiation heating a 1D semi-infinite medium. The extensive analytic analysis of the radiation driven heat process, which has a supersonic and diffusive nature, is presented in the classical textbook by Zeldovich and Reizer [30]. In close to LTE conditions, the position of the radiation heat wave front $x$ in time can be scaled by the following approximation [22, 30]

$$\begin{eqnarray}&&x\sim \frac{{{T}^{3.2}}}{{{\rho}^{1.1}}}\cdot \text{tim}{{\text{e}}^{7/8}}.\end{eqnarray} \tag{ 1 }$$

Here $T$ and $\rho$ are plasma temperature and density, respectively.

Due to the high hydrodynamic stability of foam layers in the time interval of at least 10 ns after irradiation, the plasma density can be considered as a constant value and equal to 2 mg cm⁻³ (3   ×   10²⁰ atoms cm⁻³) the initial foam density. Therefore, one obtains strong dependence of the radiation heat wave velocity on plasma temperature: $v\sim {{T}^{3.2\,}}$.

Results of the 4-frame MCP-camera have been used to evaluate radiation heat wave velocities at different times. Since the x-ray flux of the external source (hohlraum) varies in time (see XRD results in figure 2(b)), the heat wave velocity is not constant and drops in time with hohlraum temperature. 1D numerical simulations have been carried out for 1–3 mg cm⁻³ TAC-layers [22]. The 1D approach can be applied if the x-ray source diameter is larger than the size of the x-ray heated plasma region in the direction of the heat wave propagation. In our case, the diameter of the hohlraum cylinder was 1.3–1.4 mm and the corresponding plasma size up to 0.8–1 mm. Moreover, the pinhole images presented in figure 4(a) show a rather flat form of the plasma front. Results published in [22] and the temperature scaling (1) have been used for the TAC-plasma temperature evaluation behind the front of the radiation driven heat wave. Depending on time it varies from 25 to 17 eV. A mean charge of x-ray heated TAC-plasma in this temperature range and plasma particle density of 3   ×   10²⁰ cm⁻³ have been estimated by means of the generalized population kinetic code FLYCHK [31]. This results in an average ion charge Z = 2–3 and a plasma electron density of n_e ~ 10²¹ cm⁻³. Low-Z polymer plasmas at these parameters are in thermodynamical equilibrium, their charge state distribution is defined by the Saha equation and the radiation temperature is equal to those of electron and ion components. Such plasmas have an intermediate coupling parameter Γ = 0.2–0.5.

An advanced type of hydrodynamic stable plasma target with homogeneous distribution of plasma parameters has been proposed and characterized for applications in experiments on the heavy ion energy loss in ionized matter and experiments on relativistic laser interaction with near critical density plasmas.

Plasma was created via x-ray heating of polymer aerogels with a mean density 10³ times lower than those of solid matter. X-rays have been generated due to laser interaction with a gold hohlraum. Hohlraum radiation close to the Planckian spectral distribution with a temperature of 30–35 eV was effectively absorbed by the foam layer. Measurements of the foam layer absorption efficiency using x-ray diodes and a transmission grating spectrometer demonstrate up to tenfold attenuation of the hohlraum radiation by CHO-plasmas.

The hydrodynamic stability of plasma produced by the x-ray heating of a 1 mm thick foam layer was demonstrated using an optical streak camera and was compared with the hydrodynamic response of a micrometer thin solid foil of near the same areal density. A 20 ns delay between the laser pulse and launching of the plasma expansion in the case of the foam target is observed. This is 10 times longer than in the case of a solid foil. The hydrodynamic stability of the x-ray heated foam layer ensures high conversion efficiency of the absorbed x-ray energy into plasma temperature since plasma cooling, caused by expansion, starts with a measurable delay as compared to the solid foil.

Velocities of the supersonic radiation driven heat waves propagating in polymer foams have been measured by means of a 4-frame MCP-based pinhole camera. Strong dependence of the heat wave velocity on plasma temperature and the assumption of a constant plasma density result in 17–25 eV TAC-plasma temperature.

Heating of polymer foam layers by an external x-ray source has been simulated using the 1D hydrodynamic code RADIAN, which includes transport of radiation of the external source and self radiation of the created plasma. Results of the simulations [21–25] have been used for optimization of the combined target design and for the interpretation of experimental results obtained at the PALS laser system.

The research program was supported by DFG 584677, ISTC 2264, LG13029, and LD14089, and in part by the Access to Research Infrastructure activity in the 7th framework Program of the EU contract No. 284464, Laserlab Europe 111, and by the Ministry of Education, Youth and Sports of the Czech Republic under PALS RI project LM2010014.