3.1 Model 1: steady-state optically thin model

For the steady-state model, the rate equations used to calculate the charge state distribution are written in a shortened form,

(1)$$\begin{align}\frac{{\rm d}N_{i,j}}{{\rm d}t}&=\sum \limits_p\sum \limits_{m=i,i\pm 1}\sum \limits_n{N}_{m,n}{R}_{m,n\to i,j}^p\nonumber\\&\quad-{N}_{i,j}\sum \limits_d\sum \limits_{m=i,i\pm 1}\sum \limits_n{R}_{i,j\to m,n}^d\nonumber\\&=0,\end{align}$$

where ${N}_{i,j}$ and ${N}_{m,n}$ (cm⁻³) are the densities of the jth level of charge states $i$ and the $n$th level of charge states $m$, respectively, and ${R}_{m,n\to i,j}^p$ and ${R}_{i,j\to m,n}^d$ are the rate coefficients ($\mathrm{c}{\mathrm{m}}^3\cdot {\mathrm{s}}^{-1}$) of populating and depopulating processes. The first term of Equation (1) denotes the rates of the aforementioned processes that populate ${N}_{i,j}$ while the second term denotes the rates of the processes that depopulate this level. The rate equations are solved with the restrictions ${n}_i=\sum {N}_{i,j}$ and ${n}_{\rm e}=\sum {iN}_{i,j}$.

In this model the irradiated plasma and the BB source have the same values as in the experiment of Fujioka et al. The input parameters are radiation temperature ${T}_\mathrm{r}=500$ eV, dilution factor $a={10}^{-3}$, electron density ${n}_{\rm e}=5\times {10}^{19}$ cm⁻³ and electron temperature ${T}_{\rm e}=$ 26 eV, which are all in the experimental range. Under such conditions, photoionization dominates over electron impact ionization. Moreover, autoionization following photoexcitation also plays an important role^[Reference Han, Wang, Salzmann and Zhao^21]. The computed values of the ion fractions of Li-like, He-like and H-like ions are 38%, 54% and 2%, respectively. The simulated spectrum is shown in Figure 1. In a similar manner to the results of previous studies, the peaks at 1840 eV and 1864 eV are in agreement with the experimental ones, but the one at 1855 eV is still too weak.

In order to discover the reason for the disagreement between the experimental spectrum and the simulated one, we plotted the relative importance of the contributions of the various atomic processes to the population of four excited states that appear to be important in the spectral region of interest, as shown in Figure 3. These are the three He-like $1\mathrm{s}2{\mathrm{p}}\;{}^{1,3}{\mathrm{P}}_1$ or $1\mathrm{p}2\mathrm{s}$$\;^{1}{\mathrm{S}}_0$ levels and the doubly excited $1\mathrm{s}2\mathrm{s}2{\mathrm{p}}\;{}^2{\mathrm{P}}_{1/2}$ Li-like level, whose transitions to ground state generate the emission of the strongest spectral lines. Figure 3 shows the populating and depopulating contributions of every process for these levels. The populating contribution of process $p$ from level $i$ to level $j$ is calculated by

(2)$$\begin{align}{\mathcal{P}}_p=\frac{\sum \limits_{m=i,i\pm 1}\sum \limits_n{N}_{m,n}{R}_{m,n\to i,j}^p}{\sum \limits_q\sum \limits_{m=i,i\pm 1}\sum \limits_n{N}_{m,n}{R}_{m,n\to i,j}^q},\end{align}$$

and the depopulating contribution of process $d$ to ${N}_{i,j}$ is calculated by

(3)$$\begin{align}{\mathcal{D}}_d=\frac{\sum \limits_{m=i,i\pm 1}\sum \limits_n{R}_{i,j\to m,n}^d}{\sum \limits_q\sum \limits_{m=i,i\pm 1}\sum \limits_n{R}_{i,j\to m,n}^q}.\end{align}$$

Figure 3 Contributions of atomic processes under the conditions of the Fujioka et al. photoionization experiment for five selected levels. The atomic processes are listed on the x-axis. The blue processes are related to the radiation field, the dark-gray processes are controlled by collisions and the green processes are autoionization and dielectronic capture. Solid black lines represent the populating contributions for the levels, and the red dashed lines represent the depopulating contributions for the levels. The contribution friction of each process is also labeled.

In Figure 3, the $\rm 1s2s2{p}\ {}^2{P}_{1/2}$ and $\rm 1s2{p}\ {}^1{P}_1$ states, which account for the two strongest lines in the simulation, have similar behavior. These two levels are first populated by photoexcitation and then depopulated by radiative decay. As shown in Table 1, the A coefficients of the transitions of these two levels to ground state are larger than 3.0$\times$10¹³ s⁻¹, so the levels are rapidly depopulated by decay to ground state. In addition to these, there are several other Li-like and Be-like levels that have A coefficients around 3.0×10¹³ s⁻¹. These, also, are excited through photoexcitation and emit strong lines that contribute to the peak around 1845 eV.

At the electron temperature of ${T}_{\mathrm{e}}=26$ eV the collisional excitation from ground state to the $\rm 1s2{l}$ He-like levels (whose energies are about 1840–1860 eV above ground state) is negligibly small. Nevertheless, collisional processes dominate the redistribution of population among these excited levels, because the electrons’ energy is close to the energy intervals between the $1\mathrm{s}2\mathrm{l}$ levels. As 1s2s¹S₀ and 1s2p³P₀ are metastable states, the redistribution generates an accumulation of He-like ions in these metastable states and, as a consequence, they become overpopulated relative to the other $1\mathrm{s}2\mathrm{l}$ levels. Collisional processes then move the population from these metastable states to the allowed ones. Our computations indicate that about 12% of the $1\mathrm{s}2{\mathrm{p}}\;{}^1{\mathrm{P}}_1$ population is excited from the $1\mathrm{s}2{\mathrm{s}}\;{}^1{\mathrm{S}}_0$ and $1\mathrm{s}2{\mathrm{s}}\;{}^3{\mathrm{S}}_1$ metastable states by such collisions.

Altogether, this simulation shows that the main excitation of the He-like $1\mathrm{s}2\mathrm{l}$ levels is carried out through photoexcitation, but the collisional processes also have an important contribution through the redistribution of the excited states.

3.2 Model 2: time-dependent optically thin model

Next we tried to investigate the influence of a time-varying radiation pulse on the emission spectrum. For this purpose BB radiation having a Gaussian temporal profile was used,

(4)$$\begin{align}G(t)=\exp \left[-\frac{{\left(t-{t}_0\right)}^2}{2{\sigma}^2}\right],\end{align}$$

where ${t}_0=160$ ps and $\sigma \sim 80$ ps^[Reference Fujioka, Takabe, Yamamoto, Salzmann, Wang, Nishimura, Li, Dong, Wang, Zhang, Rhee, Lee, Han, Tanabe, Fujiwara, Nakabayashi, Zhao, Zhang and Mima^13, Reference Wang, Salzmann, Zhao, Takabe, Fujioka, Yamamoto, Nishimura and Zhang^16, Reference Wu, Duan, Li and Yan^20]. The rate equations in Model 2 take the form

(5)$$\begin{align}&{N}_{i,j}\left(t+\Delta t\right)={N}_{i,j}(t)+\Delta {N}_{i,j}(t)\nonumber\\&\quad{}={N}_{i,j}(t)+\Delta t\cdot \Big[\sum \limits_p\sum \limits_{m=i,i\pm 1}\sum \limits_n{N}_{m,n}(t){R}_{m,n\to i,j}^p(t)\nonumber\\&\qquad-{N}_{i,j}(t)\sum \limits_d\sum \limits_{m=i,i\pm 1}\sum \limits_n{R}_{i,j\to m,n}^d(t)\Big],\end{align}$$

where ${N}_{i,j}\left(t+\Delta t\right)$ is the level density at $t+\Delta t$, ${N}_{i,j}(t)$ and ${N}_{m,n}(t)$ are the level densities at the previous timestep, and ${R}_{m,n\to i,j}^p(t)$ and ${R}_{i,j\to m,n}^d(t)$ are the reaction rates at $t$. The electron temperature and density are assumed to be constant throughout the simulation^[Reference Wang, Salzmann, Zhao, Takabe, Fujioka, Yamamoto, Nishimura and Zhang^16, Reference Hill and Rose^18, Reference Wu, Duan, Li and Yan^20]. To compensate for the lower incident energy in this model compared to Model 1, the dilution factor was increased to $a=1\times {10}^{-2}$.

Figure 4 shows the temporal evolution of charge states up to 1 ns. The average charge state, $\overline{Z}$, reaches its maximum at the end of the radiation pulse and decreases slowly as the plasma cools. Figure 5 shows the contributions of the various atomic processes to the evolution of the $1\mathrm{s}2{\mathrm{p}}\;{}^1{\mathrm{P}}_1$ level. This reflects also the behavior of the resonance line as a function of time. Figure 5 indicates that as the radiation pulse decreases, the level population is dominated by the collisional processes (CE and CD). In a similar manner to Model 1, the Li-like satellite lines might have similar behaviors. In Figure 6, the green line is the time-integrated spectrum. The resonance line and Li-like satellite lines are close to the experimental spectrum, but the peak around 1855 eV is still weak. In conclusion, the time-dependent optically thin model cannot reproduce the experimental spectrum.

Figure 4 Evolution of fractions of charge states from C-like ion to bare nuclei in the time-dependent model with a Gaussian radiation pulse (red line). The radiation pulse is adapted as a Gaussian distribution with FWHM of 160 ps and $\sigma$=80 ps^[Reference Fujioka, Takabe, Yamamoto, Salzmann, Wang, Nishimura, Li, Dong, Wang, Zhang, Rhee, Lee, Han, Tanabe, Fujiwara, Nakabayashi, Zhao, Zhang and Mima^13, Reference Wang, Salzmann, Zhao, Takabe, Fujioka, Yamamoto, Nishimura and Zhang^16, Reference Wu, Duan, Li and Yan^20].

Figure 5 Evolution of process contributions to 1s2p ¹P₁. Upper panel: populating contributions. Lower panel: depopulating contributions. The radiation pulse is also plotted.

Figure 6 Black line: the experimental spectrum of Fujioka et al.^[Reference Fujioka, Takabe, Yamamoto, Salzmann, Wang, Nishimura, Li, Dong, Wang, Zhang, Rhee, Lee, Han, Tanabe, Fujiwara, Nakabayashi, Zhao, Zhang and Mima^13]. Green line: theoretical spectrum of time-dependent model. Red line: theoretical spectrum of optically thick model.

3.3 Model 3: steady-state optically thick model

Finally, the opacity effects were added into Model 3. Recently, in the experiment of Loisel et al. on photoionized silicon plasma, optical depth up to 60 was measured for the He-like resonance line. This indicates that reabsorption along the line between the point of emission and the measuring device in the photoionization experiment has to be taken into account. As the assumption of time dependence of the incident pulse could not account for the difference between the experimental and simulated spectra, in Model 3 a steady-state plasma was used, as in Model 1.

To account for the opacity effect, i.e., photon reabsorption (PR) on the way to the spectrometer, we used the escape probability^[Reference Kallman and Bautista^31] method, i.e.,

(6)$$\begin{align}{I}_{\mathrm{abs}}\left(\lambda \right)={I}_0\left(\lambda \right)T\left(\lambda \right),\end{align}$$

where $\lambda$ is the photon wavelength, ${I}_\mathrm{abs}\left(\lambda \right)$ and ${I}_0\left(\lambda \right)$ are the absorbed and unabsorbed line intensities, respectively and $T\left(\lambda \right)$ is the transmittance of the absorbing medium,

(7)$$\begin{align}T\left(\lambda \right)=\frac{1-{e}^{-\tau \left(\lambda \right)}}{\tau \left(\lambda \right)}.\end{align}$$

$\tau \left(\lambda \right)$ is the optical depth^[Reference Mehdipour, Kaastra and Raassen^32],

(8)$$\begin{align}\tau \left(\lambda \right)={\tau}_0\phi \left(\lambda \right),\end{align}$$

where ${\tau}_0$ is the optical depth at the line center, and $\phi \left(\lambda \right)$ is the line profile, which is assumed to have Gaussian shape. The line center optical depth is calculated by

(9)$$\begin{align}{\tau}_0=\frac{\sqrt{\pi }{e}^2{\lambda}_{\mathrm{c}}{f}_{\lambda }}{m_{\mathrm{e}}\kern0.1em c}{\left(\frac{M_{\mathrm{i}}}{2{T}_{\mathrm{i}}}\right)}^{1/2}{N}_{\mathrm{l}},\end{align}$$

where ${\lambda}_{\mathrm{c}}$ is the line center wavelength, ${f}_{\lambda }$ is the oscillator strength, ${m}_{\mathrm{e}}$ is the electron mass, ${N}_{\mathrm{l}}=\int {n}_{\mathrm{l}}\kern0.1em {\rm d}\mathrm{l}$ is the column density of the lower level from the point of emission to the spectrometer, ${M}_{\mathrm{i}}$ is the ion mass and ion temperature ${T}_{\mathrm{i}}={T}_{\mathrm{e}}$^[Reference Pradhan and Nahar^33].

As the Einstein A coefficient is proportional to ${f}_{\lambda }$, lines which have strong transition probability have a greater probability of being absorbed as well. This is especially true for the He-$\mathrm{\alpha}$ resonance line, while the intercombination and forbidden lines’ absorption is much lower, owing to the lower A coefficient. In fact, the difference of six orders of magnitude between the A coefficients of the resonance line ($A=3.76\times {10}^{13}\kern0.1em \hbox{s}^{-1}$) and the intercombination line ($A=1.04\times {10}^7\kern0.1em \mathrm{s}^{-1}$) indicates that the resonance line has reached its BB limit^[Reference Salzmann^23], while the intercombination line goes through a transparent medium. This explains the greatly reduced value of the resonance line relative to the other He-$\mathrm{\alpha}$ ones. Regarding the satellite lines, some of these have large A coefficients. For instance, the A coefficient of the $\rm 1s\kern0.1em 2{p}^{2}\ {}^{2}{P}_{1/2}\to 1{s}^22{p}\ {}^2{P}_{1/2}$ transition is $A=3.31\times {10}^{13}\kern0.1em \mathrm{s}^{-1}$. Several other transitions, also have A coefficients of the order of ${10}^{13}\kern0.1em \mathrm{s}^{-1}.$ These lines are also greatly reduced for the same reason as for the resonance line.

Model 3 uses the same input parameters as Model 1. The red line in Figure 6 is the simulation result of Model 3. Compared with the optically thin model, the resonance line and the satellite line of the Li-like ion are reduced by PR so much that they are comparable to the intercombination line. It turns out that the result of Model 3 is much closer to the experimental spectrum than the previous results.

In Model 3, the column density of the He-like ion is ${N}_{{\rm Si}^{12+}}=0.74\times {10}^{17}$ cm⁻², but ${N}_{{\rm Si}^{12+}}$ is always smaller than $0.6\times {10}^{17}$ cm⁻² in Model 2. Even if PR was added into the time-dependent model, it would not have sufficient absorption. For a bigger value of ${N}_{{\rm Si}^{12+}}$ in the time-dependent model, it again needs a larger dilution factor, which was not tried in this study. Considering the present result is much closer than the previous results and the time-dependent model consumes much more CPU time than the steady-state model, we think that a steady-state optically thick model is sufficient to simulate the Fujioka et al. photoionization experiment.

3.4 Line ratios

Now that the experimental spectrum is successfully explained by an optically thick model, the line ratios of He-$\mathrm{\alpha}$ triplet lines need to be rediscussed. There are two important line ratios for the He-$\mathrm{\alpha}$ triplet,

(10)$$\begin{align}G=\frac{I_{\mathrm{F}}+{I}_{\mathrm{I}}}{I_{\mathrm{R}}},\end{align}$$

and

(11)$$\begin{align}R=\frac{I_\mathrm{F}}{I_\mathrm{I}},\end{align}$$

where ${I}_{\mathrm{R}}$, ${I}_{\mathrm{I}}$ and ${I}_{\mathrm{F}}$ are the intensities of the resonance, intercombination and forbidden lines, respectively. Gabriel and Jordan proposed that G is sensitive to the electron temperature and R is relative to the electron density, and these two ratios have been widely applied to diagnose the electron temperature and electron density of photoionizing plasmas^[Reference Porquet and Dubau^34–Reference Ji, Schulz, Nowak, Marshall and Kallman^37]. According to the above results, the line ratios are influenced by the satellite lines and the PR effect. The He-$\mathrm{\alpha}$ lines are surrounded by a lot of intense satellite lines, especially the forbidden line^[Reference Han, Wang, Liang and Zhao^19, Reference Wang, Han, Salzmann and Zhao^25, Reference Wang, Han, Jin, Salzmann, Liang, Wei, Zhong, Zhao and Li^38], and these can hardly be distinguished from each other. Thus, under some low resolution conditions, the influence of satellite lines needs to be taken into account in the calculation of line intensities. We denote by ${I}_{\mathrm{R}+}$, ${I}_{\mathrm{I}+}$ and ${I}_{\mathrm{F}+}$ the intensities of the spectrum near three He-$\mathrm{\alpha}$ lines, and the corresponding line ratios by R ₊ and G ₊^[Reference Han, Wang, Zhong, Liang, Wei, Yuan, Zhu, Li, Liu, Li, Zhao, Zhang, Wang, Xiong, Jia, Hua, Zhu, Li, Zhao and Zhang^22, Reference Wang, Han, Salzmann and Zhao^25]. Specified to the Fujioka et al. photoionization experiment, the forbidden line is invisible among the surrounding satellite lines, and it is impossible to get an exact ${I}_{\mathrm{F}}$. It is easy to take the left peak as a unity and use the new line ratios to diagnose the electron density and temperature. Consequently, the experimental line ratios are ${G}_{\rm e+}\sim 0.94$ and ${R}_{\rm e+}\sim 2.09$. The optically thin model gives the traditional line ratios of ${G}_1=3.16\times {10}^{-2}$ and ${R}_1=1.61\times {10}^{-6}$, which have similar values of A coefficient and are hardly measurable in the experimental environment. The deduced ratios in Model 1 are ${R}_{1+}$=5.55 and ${G}_{1+}$=0.39, and in Model 3 are ${R}_{3+}$=1.02 and ${G}_{3+}=1.48$.