1 Introduction
The concept of photoionizing plasmas[Reference Liedahl1–Reference Behar and Netzer5] was first introduced by Tarter, Tucker and Salpeter (1969)[Reference Tarter, Tucker and Salpeter6] to explain X-ray spectra from highly ionized but low-temperature celestial objects. Since then, more astronomical sources with similar spectral properties have been observed[Reference Schulz, Canizares, Lee and Sako7–Reference van Kerkwijk, van Paradijs, Zuiderwijk, Hammerschlag-Hensberge, Kaper and Sterken9]. These are either diffuse sources (interstellar gas, supernova remnants, etc.) or high-mass X-ray binary systems (HMXBs) around compact objects (black holes, neutron stars, white dwarfs)[Reference Schulz, Canizares, Lee and Sako7–Reference Wang, Salzmann, Zhao and Takabe11].
In recent years there have also been a few attempts to produce photoionizing plasmas in the laboratory under controlled experimental conditions[Reference Foord, Heeter, van Hoof, Thoe, Bailey, Cuneo, Chung, Liedahl, Fournier, Chandler, Jonauskas, Kisielius, Mix, Ramsbottom, Springer, Keenan, Rose and Goldstein12, Reference Fujioka, Takabe, Yamamoto, Salzmann, Wang, Nishimura, Li, Dong, Wang, Zhang, Rhee, Lee, Han, Tanabe, Fujiwara, Nakabayashi, Zhao, Zhang and Mima13]. Several problems make such experiments extremely difficult. The first one is, of course, the generation of a high-intensity X-ray beam that can ionize high-Z material to He- and H-like species. This requires extreme technology such as the Z machine[Reference Foord, Heeter, van Hoof, Thoe, Bailey, Cuneo, Chung, Liedahl, Fournier, Chandler, Jonauskas, Kisielius, Mix, Ramsbottom, Springer, Keenan, Rose and Goldstein12] or high-intensity lasers[Reference Fujioka, Takabe, Yamamoto, Salzmann, Wang, Nishimura, Li, Dong, Wang, Zhang, Rhee, Lee, Han, Tanabe, Fujiwara, Nakabayashi, Zhao, Zhang and Mima13, Reference Fujioka, Yamamoto, Salzmann, Wang, Li, Dong, Wang, Zhang, Rhee, Lee, Han, Kwon, Zhong, Zhao, Tanabe, Fujiwara, Nakabayashi, Zhang, Nishimura, Takabe and Mima14]. A second difficulty is the analysis of such experiments: while astrophysical photoionizing plasmas are stable, time-independent objects, laboratory plasmas vary on timescales of nanoseconds and spatial extents of micrometers. Nevertheless, both kinds of plasmas produce He-
$\mathrm{\alpha}$ triplet lines: the central diagnostic tool in the analysis of photoionizing plasmas[Reference Porquet, Dubau and Grosso15].
The main aim of this paper is a new analysis of the experiment of Fujioka et al., which was carried out on the GEKKO-XII laser facility. In this experiment a black-body (BB) radiation source with temperature of 
${T}_{\rm r}=480\pm 20$ eV was generated by converging 12 high-intensity laser beams on a CH2 pellet, with the experimental dilution factor 
$a=\left(6.5\pm 3.5\right)\times {10}^{-4}$. The Planck radiation from this source was used to irradiate, through a narrow slit, a low-temperature silicon plasma having electron density 
${n}_{\rm e}=\left(0.75\pm 0.25\right)\times {10}^{20}$ cm−3 [Reference Fujioka, Takabe, Yamamoto, Salzmann, Wang, Nishimura, Li, Dong, Wang, Zhang, Rhee, Lee, Han, Tanabe, Fujiwara, Nakabayashi, Zhao, Zhang and Mima13], which was generated simultaneously. The central measurement of the experiment is the He-
$\mathrm{\alpha}$ triplet spectrum from the silicon plasma in the range of 1820–1865 eV, as well as the electron temperature, which was measured to be 
${T}_{\rm e}=27.5\pm 1.5$ eV. A photoionized silicon plasma was produced, whose ionization parameter, 
$\xi =16{\pi}^2J/{n}_{\rm e}$, reached (5.9
$\pm$3.8) erg·cm·s−1 (
$J$ is the radiation intensity and 
${n}_{\rm e}$ is the electron density).
The experimental spectrum is shown in Figure 1, and it contains the three He-
$\mathrm{\alpha}$ triplet peaks. The peak at 1864 eV is the resonance line (1s2 1S0-1s2p 1P2) and the peak at 1855 eV is the intercombination line (1s2 1S0-1s2p 3P2 and 1s2 1S0-1s2p 3P1), whose reproduction was the aim of the aforementioned papers and this paper as well[Reference Fujioka, Takabe, Yamamoto, Salzmann, Wang, Nishimura, Li, Dong, Wang, Zhang, Rhee, Lee, Han, Tanabe, Fujiwara, Nakabayashi, Zhao, Zhang and Mima13, Reference Wang, Salzmann, Zhao, Takabe, Fujioka, Yamamoto, Nishimura and Zhang16–Reference Wu, Duan, Li and Yan20]. The forbidden line (1s2 1S0-1s2s 3S1) is expected to show up around 1840 eV and is supposed to be negligibly low. Wang et al. employed a relatively simple time-dependent model, which included only photoionization and radiative recombination as the main atomic processes. Using this model they could reproduce correctly the average plasma properties, such as the plasma temperature and the average ionization degree, 
$\overline{Z},$ and their behavior as a function of time. Their simulated spectrum, however, could not reproduce correctly the experimental He-
$\mathrm{\alpha}$ spectrum and in particular the ratio between the intercombination and the resonance peaks.
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 Mima13]. Blue line: the theoretical result of an optically thin model. f, Li, i and r denote the position of the forbidden line, satellite lines, the intercombination line and the resonance line, respectively.
Several studies tried to overcome the mismatch between the simulated and measured spectra. Hill and Rose used a detailed configuration for recording atomic data in a time-dependent model. They also added opacity effects into the simulation. They correctly predicted the resonance line, but the time-dependent relative intensities were always different from those in the experimental results. Later, Bao et al. simulated the spectrum with a steady-state model. While their simulated peaks around 1864 eV and 1840 eV fitted the experimental peaks, the peak at 1855 eV was weaker than the experimental value. Wu et al. developed a time-dependent model, but, similarly, their central peak was still too low. As a matter of fact, all these papers report results that are not significantly different from those of Wang et al. in Figure 1.
In this study we reanalyzed the results obtained by Fujioka et al.[Reference Fujioka, Takabe, Yamamoto, Salzmann, Wang, Nishimura, Li, Dong, Wang, Zhang, Rhee, Lee, Han, Tanabe, Fujiwara, Nakabayashi, Zhao, Zhang and Mima13, Reference Fujioka, Yamamoto, Salzmann, Wang, Li, Dong, Wang, Zhang, Rhee, Lee, Han, Kwon, Zhong, Zhao, Tanabe, Fujiwara, Nakabayashi, Zhang, Nishimura, Takabe and Mima14]. Based on assumed very high accuracy atomic data, this work focuses on the detailed contributions of every atomic process to investigate the line emission mechanism under experimental conditions. Our analysis also takes the opacity effect into account. The line emission mechanism is investigated with three models: (i) a steady-state optically thin model; (ii) a time-dependent optically thin model; and finally (iii) a steady-state optically thick model. With all these elements in our simulations, we finally succeeded to obtain reasonably good agreement between our third model and the experiment.
A brief description of the theoretical model and comparisons of atomic data are given in Section 2. In Section 3, the line emission mechanism is investigated under experimental conditions, and the reason for the weak central peak in previous studies is explained. Finally, a summary is given in Section 4.
2 Theoretical model
In the computations presented in this paper, we use radiative-collisional code based on FAC (RCF)[Reference Han, Wang, Salzmann and Zhao21, Reference Han, Wang, Zhong, Liang, Wei, Yuan, Zhu, Li, Liu, Li, Zhao, Zhang, Wang, Xiong, Jia, Hua, Zhu, Li, Zhao and Zhang22] to calculate the spectrum emitted from a photoionizing plasma. RCF is collisional-radiative code that simulates plasma under nonlocal thermodynamic equilibrium conditions. RCF takes into account ten atomic processes in the rate equations to calculate the charge state distribution, the level distribution and the corresponding spectrum. The atomic processes are divided into five mutually inverse groups[Reference Han, Wang, Salzmann and Zhao21, Reference Salzmann23]. Two of these groups are (i) free electron collisional ionization (CI) and three-body recombination (TR); and (ii) electron collisional excitation (CE) and collisional deexcitation (CD). The other three groups include the photon-induced processes that take place when a plasma is irradiated by a strong radiation field, which may turn out to become the dominant processes: (iii) photoionization (PI) and radiative recombination (RR); (iv) photoexcitation (PE) of ionic levels and subsequent spontaneous radiative decay (A); and (v) autoionization (AI) and dielectronic capture (DC) – doubly excited states can ionize spontaneously by AI or be produced through DC, which eventually decay to ground state through emission of the corresponding satellite lines.
The relevant atomic data, such as the energy levels, cross-sections and rate coefficients of these processes, are calculated by flexible atomic code (FAC)[Reference Gu24]. FAC solves the relativistic Dirac equation for the given atom/ion in a j–j coupling scheme. Charge states from bare nuclei down to C-like and levels up to 
$n=8$ are included in the present calculations[Reference Wang, Han, Salzmann and Zhao25]. The accuracy of atomic data has been checked for several conditions in our previous publications[Reference Han, Wang, Salzmann and Zhao21, Reference Wang, Han, Salzmann and Zhao25]. Comparisons of the atomic data in this study with the data in others are shown in Table 1 and Figure 2. In Table 1, the energies and Einstein A coefficients of the 29 strongest lines are listed. They show good agreement within 0.05% and 10%, respectively, when compared with the data from Palmeri et al. Comparisons of the collisional excitation cross-sections for transitions among the 
$1\mathrm{s}2{\mathrm{l}}$ levels of silicon with data from the NIFS database[Reference Badnell26–Reference Sampson and Clark29] are shown in Figure 2, which also show good agreement. The Stewart–Pyatt continuum lowering formula[Reference Stewart and Pyatt30] is taken into account for the high-density conditions, above 
${n}_{\rm e}\sim {10}^{20}\ {\rm cm}^{-3}$. More details about the calculation of the various atomic processes are given in Ref. [Reference Han, Wang, Salzmann and Zhao21]. The simulated emission lines were broadened with a Gaussian profile having FWHM of 7 eV.
Transition energies and Einstein A coefficients of some intense silicon lines between 1820 eV and 1865 eV. The resonance, intercombination and forbidden lines are marked as 
$R/w$, 
$I/\left(x+y\right)$ and 
$F/z$, respectively.
Comparison of the collisional excitation cross-section with the results of Refs. [Reference Badnell26–Reference Sampson and Clark29]. The transitions include 1s2s 3S
${}_1\to$ 1s2p 1P1, 1s2s 3S
${}_1\to$ 1s2s 1S1, 1s2p 3P
${}_1\to$ 1s2p 1P1 and 1s2s 1S
${}_1\to$ 1s2p 1P1 of He-like Si.
3 Results and discussion
Three models were used in our study.
Model 1: The plasma is assumed to be in a steady state, having constant density and temperature throughout the period of the irradiation. It is assumed also to be optically thin.
Model 2: The time dependence of the charge state distribution in the plasma is taken into account, but the model still assumes an optically thin plasma.
Model 3: A steady-state but optically thick plasma is assumed that accounts for the reabsorption of the emerging radiation inside the irradiated plasma.
The line emission mechanism is investigated with all three models, and the differences between the results, in particular the opacity effects, are explained and discussed.
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,
$$\begin{align}\frac{{\rm d}N_{i,j}}{{\rm d}t}&=\sum \limits_p\sum \limits_{m=i,i\pm 1}\sum \limits_n{N}_{m,n}{R}_{m,n\to i,j}^p\nonumber\\&\quad-{N}_{i,j}\sum \limits_d\sum \limits_{m=i,i\pm 1}\sum \limits_n{R}_{i,j\to m,n}^d\nonumber\\&=0,\end{align}$$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 Zhao21]. 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
$$\begin{align}{\mathcal{P}}_p=\frac{\sum \limits_{m=i,i\pm 1}\sum \limits_n{N}_{m,n}{R}_{m,n\to i,j}^p}{\sum \limits_q\sum \limits_{m=i,i\pm 1}\sum \limits_n{N}_{m,n}{R}_{m,n\to i,j}^q},\end{align}$$and the depopulating contribution of process 
$d$ to 
${N}_{i,j}$ is calculated by
$$\begin{align}{\mathcal{D}}_d=\frac{\sum \limits_{m=i,i\pm 1}\sum \limits_n{R}_{i,j\to m,n}^d}{\sum \limits_q\sum \limits_{m=i,i\pm 1}\sum \limits_n{R}_{i,j\to m,n}^q}.\end{align}$$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,
$$\begin{align}G(t)=\exp \left[-\frac{{\left(t-{t}_0\right)}^2}{2{\sigma}^2}\right],\end{align}$$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 Mima13, Reference Wang, Salzmann, Zhao, Takabe, Fujioka, Yamamoto, Nishimura and Zhang16, Reference Wu, Duan, Li and Yan20]. The rate equations in Model 2 take the form
$$\begin{align}&{N}_{i,j}\left(t+\Delta t\right)={N}_{i,j}(t)+\Delta {N}_{i,j}(t)\nonumber\\&\quad{}={N}_{i,j}(t)+\Delta t\cdot \Big[\sum \limits_p\sum \limits_{m=i,i\pm 1}\sum \limits_n{N}_{m,n}(t){R}_{m,n\to i,j}^p(t)\nonumber\\&\qquad-{N}_{i,j}(t)\sum \limits_d\sum \limits_{m=i,i\pm 1}\sum \limits_n{R}_{i,j\to m,n}^d(t)\Big],\end{align}$$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 Zhang16, Reference Hill and Rose18, Reference Wu, Duan, Li and Yan20]. 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.
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 Mima13, Reference Wang, Salzmann, Zhao, Takabe, Fujioka, Yamamoto, Nishimura and Zhang16, Reference Wu, Duan, Li and Yan20].
Evolution of process contributions to 1s2p 1P1. Upper panel: populating contributions. Lower panel: depopulating contributions. The radiation pulse is also plotted.
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 Mima13]. 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 Bautista31] method, i.e.,
$$\begin{align}{I}_{\mathrm{abs}}\left(\lambda \right)={I}_0\left(\lambda \right)T\left(\lambda \right),\end{align}$$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,
$$\begin{align}T\left(\lambda \right)=\frac{1-{e}^{-\tau \left(\lambda \right)}}{\tau \left(\lambda \right)}.\end{align}$$
$\tau \left(\lambda \right)$ is the optical depth[Reference Mehdipour, Kaastra and Raassen32],
$$\begin{align}\tau \left(\lambda \right)={\tau}_0\phi \left(\lambda \right),\end{align}$$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
$$\begin{align}{\tau}_0=\frac{\sqrt{\pi }{e}^2{\lambda}_{\mathrm{c}}{f}_{\lambda }}{m_{\mathrm{e}}\kern0.1em c}{\left(\frac{M_{\mathrm{i}}}{2{T}_{\mathrm{i}}}\right)}^{1/2}{N}_{\mathrm{l}},\end{align}$$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 Nahar33].
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 Salzmann23], 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,
$$\begin{align}G=\frac{I_{\mathrm{F}}+{I}_{\mathrm{I}}}{I_{\mathrm{R}}},\end{align}$$and
$$\begin{align}R=\frac{I_\mathrm{F}}{I_\mathrm{I}},\end{align}$$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 Dubau34–Reference Ji, Schulz, Nowak, Marshall and Kallman37]. 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 Zhao19, Reference Wang, Han, Salzmann and Zhao25, Reference Wang, Han, Jin, Salzmann, Liang, Wei, Zhong, Zhao and Li38], 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 Zhang22, Reference Wang, Han, Salzmann and Zhao25]. 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 
${G}_{3+}=1.48$.
4 Conclusion
In this paper, the X-ray spectrum of photoionized silicon plasmas under the conditions of the Fujioka et al. photoionization experiment was investigated with a steady-state optically thin model, a time-dependent optically thin model and a steady-state optically thick model, respectively.
The following was found.
Photoexcitation produces a strong resonance line and some nearby Li-like lines because of large photoexcitation rates and lower-level densities. Consequently, these lines are much stronger than the forbidden line and intercombination line in the optically thin models (Model 1 and Model 2).
The resonance line and some satellite lines are easily absorbed as the photons travel to the detector. As a result, the experimental spectrum is successfully simulated by an optically thick model, where the intercombination lines become more visible than in previous studies.
The
$G$ and $R$ ratios of He-$\mathrm{\alpha}$ lines are influenced by the satellite lines, optical effect and plasma ionization structure. Therefore, one needs to be careful when using these line ratios in plasma diagnosis.
Acknowledgments
This work was supported by the Science Challenge Project (No. TZ2016005), the National Natural Science Foundation of China (Nos. 11903006, 12073043 and U1930108), and the Strategic Priority Research Program of Chinese Academy of Sciences (No. XDA25030700).
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