Study of the radiation temperature on ablator by using the shock wave technique

Accurate knowledge of the x-ray flux on ablator is crucial to analyze the mass ablation rate. The radiation source is usually measured by F-XRD and M-XRD through LEH. Previous study shows that the peak radiation temperature measured by F-XRD may deviate with the real radiation source inside the hohlraum [19]. The shock wave technique deduces the radiation temperature by the experimental result of Al sample which is standard material. Its opacity and equation of state (EOS) have been extensively studied. And the radiation temperature 'felt' by the Al sample and the ablator is almost the same. The peak radiation temperature obtained from shock wave technique is more accurate. Thus the shock wave technique is used to deduce the peak radiation temperature in this experiment. The experimental results show that the two peak radiation temperatures obtained by F-XRD and the shock wave technique are different in the two-LEHs spherical hohlraum.

In the experiment, nearly constant radiation temperature was obtained in the spherical hohlraum. The time behavior of radiation temperature (${T_{\text{R}}}$) and M-band fraction (${f_{\text{m}}}$) were measured by F-XRD and M-XRD, respectively. In figure 3, the measured temporal behaviors of ${T_{\text{R}}}$ and ${f_{\text{m}}}$ are shown. The peak radiation temperature is about 136 eV, and the error bar of ${T_{\text{R}}}$ is taken as 3% [26]. The M-band fraction is about 2.5%. The radiation temperature is nearly constant from 0.8 ns to 2.8 ns. The radiation temperature merely raises ∼10 eV.

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The step thicknesses of Al sample were measured before the experiment. With the step thicknesses information and the breakout time intervals deduced by the SOP image, the average shock velocities can then be obtained. With the measured shock velocities, the radiation temperature 'felt' by the Al sample can be obtained according to the relation between the radiation temperature and the shock velocity. The relation is found by numerical simulation. The detailed analysis is similar with that in [10]. The temporal behavior of the radiation temperature and the M-band fraction in simulation is taken from the experiment, as shown in figure 3. Numerous peak ${T_{\text{R}}}$ and ${f_{\text{m}}}$ have been used in simulation. The shock velocities vary with peak ${T_{\text{R}}}$ and ${f_{\text{m}}}$. When the calculated velocities of different steps with some particular ${T_{\text{R}}}$ and ${f_{\text{m}}}$ agree with the measured velocities, then these ${T_{\text{R}}}$ and ${f_{\text{m}}}$ are considered as the radiation source which are 'felt' by the Al sample.

The observed shock velocities ${V_{\text{s}}}$ in Al witness plates are shown in table 1. The laser energy in table 1 is without the backscattered laser energy. The backscattered laser energy fraction was ∼10%. As the last step was not so smooth, the average shock velocity from the fourth step to the fifth step has not been list. Shock velocity ${V_{{\text{s}}(i - (i + 1))}}$ represents the average shock velocity from step ($i$) to step ($i$+1), and $i$ = 1, 2, and 3. The average shock velocities were about 28–31 km s⁻¹ in the experiment. The uncertainty of the shock velocities is lower than 3%. The thickness and the time uncertainties have been taken into consideration to analyze the velocity uncertainties.

Table 1. The measured shock wave velocities in the Al witness plates.

Laser energy (kJ)	37.5	37.6	38.8
${V_{{\text{s}}(1 - 2)}}$	29.24 ± 0.82 km s−1	28.99 ± 0.91 km s−1	30.61 ± 0.84 km s−1
${V_{{\text{s}}(2 - 3)}}$	29.27 ± 0.84 km s−1	28.63 ± 0.78 km s−1	29.12 ± 0.78 km s−1
${V_{{\text{s}}(3 - 4)}}$	29.57 ± 0.82 km s−1	28.31 ± 0.78 km s−1	29.64 ± 0.81 km s−1

The average shock velocities of Al under numerous groups of (${T_{\text{R}}}$, ${f_{\text{m}}}$) have been calculated by using the multi-group radiation hydrodynamic code MULTI-1D [27], which solves the one-dimensional planar hydrodynamic equations coupled with the radiation transfer equation. It is widely used in the ICF study, such as laser target coupling [28], hohlraum physics and x-ray matter interaction [29]. The EOS of Al is from the sesame library (material CODE = 41) and the multi-group opacity of Al is from Thermos.

In figure 4, it is contour lines of the Al shock velocities in the ${T_{\text{R}}}/{f_{\text{m}}}$ plane. The black lines are contour lines of ${V_{{\text{s}}(1 - 2)}}$. The red lines are contour lines of ${V_{{\text{s}}(2 - 3)}}$. The blue lines are contour lines of ${V_{{\text{s}}(3 - 4)}}$. The solid lines are the observed shock wave velocities, and the dashed lines are obtained after considering the uncertainties of the shock wave velocity. For a given shock wave velocity, the radiation temperature rises slowly as ${f_{\text{m}}}$ increases from 0 to 0.05. The M-band fraction is taken as 0.25 ± 0.25. It is of small influence on the shock wave velocity. The overlapping region of the dashed lines agrees with the observed ${V_{{\text{s}}(1 - 2)}}$, ${V_{{\text{s}}(2 - 3)}}$ and ${V_{{\text{s}}(3 - 4)}}$ in the Al witness plates. And ${T_{\text{R}}}$ in that region gives the peak radiation temperature 'felt' by the Al samples. In figure 4, the peak radiation temperature is 157.5 ± 2.3 eV. On the other hand, radiation temperature between Al and Au sample is of a little difference. It also varies along the left-right direction (x-axis), as shown in figure 2. And these two factors may lead to uncertainty of ∼1 eV. Thus the peak radiation temperature on the ablator sample is about 157.5 ± 3.3 eV, and its uncertainty is less than 2%.

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The peak radiation temperatures obtained by F-XRD and the shock wave technique are different. For comparison, the peak radiation temperatures in three shots are presented in figure 5. The peak radiation temperature uncertainty measured by F-XRD is about 3%. It is larger than that obtained by the shock wave technique, which is less than 2%. The peak radiation temperatures obtained by the shock wave technique are of ∼20 eV's higher than those measured by F-XRD. This result is in consistent with [18]. The difference is due to the different field of views between F-XRD and the sample. F-XRD views the hohlraum from LEH at 42°. Field of view of F-XRD is also shown in figure 5. In the two-LEH spherical hohlraum, it cannot view the laser spot. And it is also influenced by the cold plasmas outside the hohlraum [11]. Thus the measured F-XRD radiation temperature is lower. However the sample was mounted over the diagnostic hole on the waist. The laser spots were also located at the waist. And large amounts of the radiation flux from the laser spots could irradiate the interior surface of the sample. Thus the radiation temperature felt by the sample is higher than that measured by F-XRD.

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In the experiment, one successful shot was performed and obtained a high quality burn-through image of a two-step-shaped Au sample. In the shot, the peak radiation temperature obtained by F-XRD and the shock wave technique were ∼136.5 ± 4.1 eV and ∼157.5 ± 3.3 eV, respectively. The burn-through image is shown in figure 6(a). The time fiducial was used to correlate the radiation temperature history with the burn-through image. Time interval was 0.5 ns between two neighboring time fiducial points. The normalized intensity-time line-outs taken from the burn-through image is shown in figure 6(b). Marshak wave in the ablation region is self-similar. And the ablation rates in the two steps were nearly the same. The difference of mass thickness is equal to the corresponding mass ablation rate times the difference of burn-through time between the two steps. Since the ablator density and the step thickness were measured before the experiment, the mass ablation rate (${m_{\text{a}}}$) can then be inferred. It is written as:

$$\begin{align}{\mathop m\limits^ \bullet}_a = \frac{{\rho \Delta d}}{{\Delta t}}.\end{align} \tag{ 1 }$$

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Particularly, $\Delta d$ = 0.2 μm is the delta thickness of the two steps, and $\Delta t$ is the burn-through time interval. The ablation front is measured at 0.2 peak. The two steps were burned through at ∼1.715 ns and ∼2.153 ns, respectively. And $\Delta t$ is about 0.438 ns. Thus the mass ablation rate of Au is (18.93 g cm⁻³)(0.2 μm)/(0.438 ns) ≈ 0.86 mg cm⁻² ns⁻¹. For the two radiation sources, the temperature is different during the burn-through time interval. As the peak radiation temperature being ∼136.5 ± 4.1 eV and ∼157.5 ± 3.3 eV, the radiation temperature at that period is 131.5 ± 5.5 eV and 152.8 ± 4.7 eV, respectively.

The uncertainty of the mass ablation rate is due to the step sample density and thickness uncertainties and the burn-through time uncertainty. The uncertainty in density is ∼1.5%. The step sample thickness uncertainty is ∼0.7%. And the uncertainty of $\Delta d$ is ∼1%. The burn-through time uncertainty comes from the streak camera sweep rate and the rough ablation front. The streak camera sweep rate is about 2.72 ± 0.07 ps/pixel. The delta pixel of the ablation front between the two steps is 161 ± 3.08 pixel. Thus the uncertainty of $\Delta t$ is ∼3.2%. Then the mass ablation rate uncertainty is ∼3.8% by equation (1).

The mass ablation rate of Au is calculated by using the radiation hydrodynamic code MULTI-1D. The EOS of Au and CH is from the sesame library and the multi-group opacity of Au and CH is from the Thermos and SNOP code [27], respectively. The x-rays ranging from 0 to 5 keV are divided into 75 groups. With the peak radiation temperature being 136.5 ± 4.1 eV and 157.5 ± 3.3 eV, the mass ablation rate at energy band 195–227 eV has been calculated. To simplify the analysis, the M-band fraction is the same with that in figure 3. In figure 7, it is the measured and simulated mass ablation rate. In simulation, the peak radiation temperature uncertainty is taken into consideration to analyze the ${\mathop m\limits^ \bullet }_a$ uncertainty. The mass ablation rate calculated by using the peak radiation temperature obtained from the shock wave technique agrees well with the experimental result. However the lower peak radiation temperature measured by F-XRD results in about 20%'s decreases of mass ablation rate in simulation. And the shock wave technique is crucial to provide the 'real' radiation source data. In this way, we can obtain more accurate peak radiation temperature on ablator.

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