3.1. Plasma Generation and Measurement of Radiative Properties

Since the heating process of the foam sample is determined by the properties of the soft X-ray flux from the hohlraum converter, the characterization of the hohlraum X-ray radiation is a necessary first step, the result is shown in Figure 2(a). The duration and amplitude of the soft X-ray signal indicate that the effective heating time of the X-rays is about 6 ns for the 2 ns/150 J laser pulse. The pinhole camera shown in the experimental setup of Figure 1 was fitted with a calibrated Fuji image plate and served to take an X-ray image of both the ionized foam plasma and the hohlraum radiation. Since the pinholes are not covered by any filters and all photons within the response range of the image plate are collected. The pinhole images are presented in Figure 2(b).

Figure 2: (a) The duration of the converter radiation pulse recorded by the X-ray diode is 5–7 ns; (b) the image of the pinhole camera with the outer features of the target geometry from the side at an angle of about 50° with respect to the X-axis.

The X-ray spectrum (see Figure 3, red line) emitted from the hohlraum was measured with a transmission grating spectrometer (TGS) coupled with a single-order diffraction grating to effectively exclude the influence of high-order diffraction. The blue line in Figure 3 is a black body radiation fit to the X-ray spectrum corresponding to a radiation temperature of 20 eV. The deviation from the black body radiation spectrum is mainly due to contributions from the radiation from the gold hohlraum.

Figure 3: Measured spectrum of the gold hohlraum. (a) Raw spectra of the hohlraum radiation recorded by TGS and (b) the reconstructed spectral distribution of the 20-eV black body radiation curve.

3.2. The Plasma Diagnostic

Figure 4 shows the high-resolution emission spectrum emitted from the foam target turned into plasma. In general, the energy of free electrons is described by a distribution. In the ideal case of thermal equilibrium, it is a Maxwell–Boltzmann distribution. Therefore, a continuous spectrum is emitted, originating from “free-bound” and “free-free” transitions, constituting the Bremsstrahlung background. Electronic transitions between ionic or atomic energy levels, so-called “bound-bound” transitions, are the origin of isolated ionic or atomic emission lines. The measured spectrum, therefore, is a superposition of isolated emission lines and a continuous background and contains a wealth of physical information due to the complex atomic processes and structures of plasma [Reference Zhao, Zhang and Cheng31, Reference Ciricosta, Vinko and Chung32].

Figure 4: Plasma emission spectra are dominated by oxygen lines. Here the continuous radiation background is already subtracted. The intensity was normalized at the maximum value, and the main lines were denoted by wavelength and transition.

The resonant lines of helium-like carbon ions (4.025 nm, 1s-2p) dominate the short-wavelength part of the spectrum, while at the long-wavelength position, Beryllium-like (62.973 nm, 2s-2p) and Boron-like (55.450 nm, 2s-2p) oxygen ions emerge. Some well-resolved and high-intensity characteristic lines in the middle-wavelength range were selected as wavelength references, such as 15.032 nm (OVI 2s-3p), 17.320 nm (O VI 2p-3d), and 19.253 nm (O V 2p-3d), which were also selected in previous projects for spectral lines identification [Reference Del Zanna, Dere, Young and Landi33–Reference Träbert, Beiersdorfer, Brickhouse and Golub35]. According to the wavelength and intensity data provided by the National Institute of Standards and Technology (NIST) database [Reference Kramida, Ralchenko and Reader36], the experimental lines, both well-resolved and some blended lines, were well-identified for the ion species and charge states. In our experiments, the spectral lines were mainly attributed to the transitions of 2s-np, 2p-ns, 2p-np, and 2p-nd (n = 2–6) for O III-VI and C IV-V ions, which are induced by L-shell ionization processes of carbon and oxygen. This indicates that the free plasma electrons originate mainly from the ionization processes of valence electrons.

The flat-field grating spectrometer was calibrated with the Hefei Synchrotron Radiation Light Source [Reference Zhongmou37, Reference Wei-Min, Lin and Guang-Yao38]. The wavelength accuracy is about 0.003 nm [Reference Du, Shen, Li, An, Shi and Wang39]. The wavelength resolution depends on the photon energy E, and we achieved E/ΔE∼300 at E = 62 eV or equivalently at a photon wavelength of 20 nm. There is an intensity uncertainty of about 11% according to the calibration of the sensitivity curve for the Typhoon FLA 7000 imaging plate scanner. This includes 10% uncertainties due to experimental conditions, such as the instability of the light source and scanner system, the spatial roughness of the imaging plate, and 5% uncertainties for the calculation of electron numbers [Reference Lu, Li and Wen40, Reference Williams, Maddox, Chen, Kojima and Millecchia41].

The relative intensity of emission lines with the same lower level is very sensitive to the electron temperature T_e and density n_e. Therefore, the intensity ratio of characteristic lines is a useful tool for diagnosing the plasma parameters. Assuming that the conditions for Boltzmann distribution of the upper levels are met, the radiation intensity I_mn when an electron transits from energy level m to level n, can be expressed as: $I_{mn}=N{A}_{m}h\nu \exp \left(-\Delta E_{mn}/k{T}_{ex}\right)$ , where N, g_m, A_m hν, $\Delta E_{mn}$ , and k, are the total number of atoms, statistical weight, transition probability, photon energy, excited level energy, and the Boltzmann constant, respectively. T_ex is the excitation temperature. This shows the obvious dependence on the temperature T_ex and the weak dependence on electron density. When the conditions of local thermodynamic equilibrium (LTE) are fulfilled (the excitation temperature is equal to the electron temperature), the relative emission intensity and excitation temperature satisfies [Reference Shaikh, Hafeez, Rashid, Mahmood and Baig42]:

(1) $\begin{array}{}\ln \frac{I_{mn}{\lambda }_{mn}}{A_m{g}_m}=-\frac{E_m}{k{T}_{ex}}+\ln \frac{N\left(T\right)}{U\left(T\right)},\end{array}$

where U (T) is the partition function and λ_mn is the transition wavelength.

This equation indicates that $\ln \left(I_{mn}{\lambda }_{mn}/A_m{g}_m\right)$ is proportional to the upper level energy E_m for the same atom or ion lines, and the slope is $-1/k{T}_{ex}$ , which contains the temperature information of the plasma. Therefore, the temperature can be obtained without detailed knowledge of the total density of atoms or the atomic species partition function.

In our experiments, the transition lines at 13.178 nm (2p-5p), 14.705 nm (2s−4p), 19.253 nm (2p-3d), 20.240 nm (2p-3d), 22.742 nm (2p-3s), and 23.130 nm (2p-3d) for O V ions and some spectral lines of O IV ions were used, respectively, for the determination of the electron temperature, they are labeled in Figure 4 along with the calibrated lines and main carbon transitions. These lines are well resolved, the transition probabilities used here are provided by the NIST database [Reference Kramida, Ralchenko and Reader36] with a well-known uncertainty. The wavelengths, degeneracies, and transition probabilities of these oxygen lines are summarized in Table 1.

Table 1: Spectroscopic parameters of the observed oxygen lines for plasma temperature determination.

Boltzmann plots were constructed from the relative intensity of oxygen lines as shown in Figure 5. The experimental data were fitted by least-squares regression. The linear coefficient of regression r turned out to be r > 0.9 on average. As a result, the plasma temperature is 16.6 and 17.2 eV obtained from O V and O IV ions, respectively.

Figure 5: Results of plasma temperature. (a, b) The O V characteristic lines (arrows) that were used to construct the Boltzmann plot. Intensities were obtained by line fitting procedure with Gaussian line profiles, and the O IV lines can be treated similarly. (c) Boltzmann plots for O IV and O V emission lines.

The validity of the local thermodynamic equilibrium assumption (LTE) was checked using McWhirter’s criterion [Reference Martino and D’Angelo43, Reference Mal, Junjuri, Gundawar and Khare44], which requires the electron number density to satisfy the conditions: $N_e\ge 1.6\times {10}^{12}{T}^{1/2}{\left(\Delta E\right)}^3$ , where $N_e\left({\text{cm}}^{-3}\right)$ is the electron number density, T(K) is the plasma temperature, and ΔE (eV) is the difference in the energies between the upper and lower states of all investigated transitions. With the evaluated temperature of 190 000 K in our experiment and $N_e=4.1\times {10}^{20}{\text{cm}}^{-3}$ , the maximum energy gap ΔE at 14.705 nm, and the LTE requirements are met. The determination of the electron density will be discussed later in this section.

Furthermore, the plasma is optically thin for most lines, as checked from the absorption coefficient due to inverse Bremsstrahlung [Reference Knudtson, Green and Sutton45] and the effect of self-absorption [Reference Shaikh, Rashid, Hafeez, Jamil and Baig46]. In our experiment, the maximum absorption coefficient for inverse Bremsstrahlung is 0.057 cm⁻¹ at 28.596 nm, since the plasma diameter is a millimeter, the weak absorption ( $1/K_{v\max }=17.5\text{cm}$ ) due to inverse Bremsstrahlung does not affect our quantitative spectroscopy. For the optically thick lines of transitions to the ground state 1s²2p², the intensities were modified according to the self-absorption coefficient.

For the uncertainty associated with the temperature obtained from the observed emission intensity, there are a number of factors that contribute to it. The main error of the Boltzmann-plot method arises from the uncertainty of the transition probability and the measurement error of the experimental spectral intensity. In our experiment, the transition probability accuracy of the spectral lines chosen for the evaluation is ≤10% [Reference Kramida, Ralchenko and Reader36], combined with the measured intensity error, the uncertainty caused by relative intensity, fitting, and transition probability is about 18%.

It should be noted that the temperature obtained by the Boltzmann-plot method is the time average temperature of the entire plasma [Reference Shaikh, Hafeez, Rashid, Mahmood and Baig42]. The spatial distribution and temporal evolution of the plasma density and temperature at different times can be obtained by hydrodynamic simulations [Reference Rosmej, Suslov and Martsovenko18]. With a similar experimental design, the plasma temperature variation is estimated to be about 25% during the 5 to 20 ns time span. These uncertainties, however, do not cause directly similar uncertainty in the temperature determination. The relative uncertainty in the electron temperature determination by the Boltzmann-plot method scales with the factor T/E_m, which is the ratio of the thermal energy of the plasma T and the level energy E_m. As a result, we have estimated that our plasma temperature relative error is about 7%.

Also from previous experiments [Reference Rosmej, Suslov and Martsovenko18], it is well documented that foam layers show high hydrodynamic stability during the heating process. The matter density is almost constant and stays close to the initial line density value. As a result of this, the radiative heat wave is supersonic. Thus, the radiative heat transfer proceeds at a higher speed than the thermal expansion.

Calculation of the ion charge state distributions using the collisional radiative code FLYCHK [Reference Lee, Chen, Morgan and Chung47] shows that the ionization degree of CHO foam is around ${\text{C}}_9^{3.80\pm 0.03+}{\text{H}}_{16}^{0.98\pm 0.003+}{\text{O}}_8^{4.33\pm 0.2+}$ for the 16.8 ± 1.1 eV electron temperature that we obtained from the Boltzmann-plot method and the corresponding free electron density of $4.0\pm 0.3\times {10}^{20}{\text{cm}}^{-3}$ . Here the relative error is 7%. The mean ion charge state $Z^{q+}$ is q = 2-3.