Nowadays laser plasma is an excellent tool to simulate various magnetized supersonic and hydrodynamic flows.^1–3 Recently, for example, collimated long-scaled plasma streams were used in experiments to simulate jets in Young Stellar Objects (YSOs).⁴ In these laboratory experiments, the most interesting processes occur far away from the laser-irradiated surface of the source target, when the plasma is no longer heated by the laser. Therefore, the common steady-state models used to analyze x-ray spectroscopy observations are not relevant, and new models geared towards recombining plasmas⁵ are actively being developed. This new method of x-ray spectroscopy was applied in the investigation of magnetized supersonic flows and has shown remarkable results.⁶

The combination of external magnetic fields and laser-produced plasmas^6,7 allowed us to create a stable, collimated plasma stream which, in turn, let us investigate more complicated hydrodynamic phenomena. For example, this setup can be exploited to investigate the impact of a magnetized plasma column with an additional medium, e.g., a gas jet, solid obstacle, or another plasma. In particular, it has allowed us,⁸ by placing a solid obstacle in the path of the plasma column, as seen in Fig. 1, to simulate the astrophysical processes related to matter accretion onto a young star.^9,10

The purpose of this paper is to describe the new approach of x-ray diagnostics we used in such experiments, and that were recently applied successfully for the tasks of the laboratory simulation of magnetized accretion columns.⁸

The experiments dedicated to the study of accretion dynamics in Classical T Tauri Stars (CTTSs)¹¹ were performed at the nanosecond laser facility ELFIE at Ecole Polytechnique (France). The experimental platform was described in detail in Ref. 6 and is shown in Fig. 1. A narrow, ∼1-mm diameter, and centimeter-long collimated jet was created by the interaction of a laser beam with a solid thick target in the presence of a poloidal uniform large-scale magnetic field ∼20 T.^4,6 The plasma jet evolves parallel to the lines of the magnetic field and is made to interact with a solid obstacle, mimicking a stellar chromosphere (see Fig. 1).

The plasma flow strongly emits across a wide x-ray spectral range, and so, x-ray spectroscopy methods can be used for the investigation of plasma expansion as described in Refs. 5, 12, and 13. CF₂ (Teflon) targets with a low mean atomic number (Z_mean = 8) were chosen in respect of the given laser intensity on the target (∼10¹³ W/cm²), thus creating an almost fully ionized plasma. The focusing spectrometer (FSSR), which is based on a spherically bent mica crystal, with 2d = 19.9376 Å and a radius of curvature R = 150 mm, was used to analyze the x-ray emission stemming from the plasma. For this particular setup, the FSSR was implemented to measure the x-ray spectra of multicharged fluoride ions in the range 13–16 Å (800–950 eV) in the first order of reflection (see Fig. 2). The spectrometer was installed in the direction transverse to the jet expansion, providing a spatial resolution of about 350 µm along more than 12 mm of the jet.

The angles in the upwards (∼5°) and lateral (∼2°) directions were small enough for us to neglect skewing of the image. This arrangement allowed us to use the spectrometer in parallel with a Mach–Zehnder-type optical interferometer, which was used to provide complementary data about the plasma dynamics, observing the plasma electron density.⁸ The x-ray spectra were recorded on a fluorescent detector, a TR Fujifilm Image Plate.

Typical spectral images measured in the experiment are presented in Fig. 2. The emission is composed of Bremsstrahlung, and the following spectral components of fluoride ions: Helium-like ion emission starting with the transition 3p-1s (He_β line) and hydrogen-like ion emission with the Ly_α line and its dielectronic satellites. One can observe that the spectra change significantly as a function of the distance to the surface of the laser-irradiated target, as the plasma flow expands for many millimeters during tens of ns from the laser-interaction volume. We observed dielectronic satellites to the Ly_α line near the laser-irradiated target. Here the plasma temperature is sufficient to ionize inner ion shells. Further, i.e., at >1 mm from the target surface, the absolute intensities of the spectral lines decrease and the satellites disappear. In this remote region, the contribution of He-like transitions from the level with a larger main quantum number increases in the total spectrum, which is typical of a recombining plasma.¹⁴ Then, near the obstacle surface, the absolute line intensities increase again, and satellites of the Ly_α line are well distinguished, while the behavior of the He-like ion emission is preserved.

The Doppler shift is negligible and is not observed in the experimental spectrum. Notably, since the plasma flow propagates laterally, with respect to the observer (Fig. 1), only a transverse Doppler shift could arise. This would yield a spectral shift of less than 10⁻⁴ Å for the plasma velocities measured in the experiment, which are in the range 100–1000 km/s.⁸ For the later stages of plasma expansion, when the longitudinal Doppler shift may affect spectral line broadening, the plasma velocity is estimated to be even lower, and so the total broadening is less than the instrumental line width.

Although the signal on the detector is integrated over time, in the case of a free plasma expansion (i.e., no obstacle), the flow moves with constant velocity (i.e., all acceleration processes are supposed to occur only close to the target surface) along the spatially resolved axis; reasonably good correspondence can be made between the spatial scale and temporal evolution of the plasma parameters. However, in the case when a solid obstacle is put in the way of a plasma plume, each spatial data point measured by the spectrometer should be considered as the sum of the emitted spectra: firstly, by the laser-initiated plasma, spreading towards the obstacle; and secondly, by the plasma formed after the impact of this plasma outflow on the obstacle surface. In the experiment, the ion emission, when the plasma is freely propagating (i.e., in the absence of an obstacle), is noticeably lower compared with the emission when the plasma impacts on the solid obstacle. This means that plasma emission occurs close to the obstacle, mainly following the impact of the plasma stream on the obstacle surface.

In addition, the plasma near the obstacle consists of ions generated from both the main and obstacle targets, obfuscating their respective ion emission. One way to circumvent this issue is to use different target materials for the main and obstacle targets, such that the ions from both materials emit in different spectral ranges, which allows the separation of their emissions. This was done in our experiment using CF₂ for one target and polyvinyl chloride [PVC (C₂H₃Cl)_n] for the other.⁸

The absolute and relative intensities of several spectral lines belonging to the different ionic charge states, such as those shown in Fig. 2, can be used to retrieve the plasma parameters (e.g., electron temperature and density) throughout the whole plasma expansion, and to unfold the whole scenario of the plasma-obstacle interaction.

First, one can observe that the most intense emission is located near the laser-irradiated target, since that plasma is characterized by the highest electron density, i.e., the critical density where the laser-matter interaction takes place, with N_e ∼ 10²¹ cm⁻³. The possibility of simultaneously observing the resonance lines of multiply charged K-ions and their satellites in this region is a consequence of the population and decay processes for the resonance and autoionizing levels. The satellites are initiated here by dielectronic recombination from the doubly excited states of ions which are situated very close to the corresponding resonance line (Ly_α in this case). They proved to be an effective method^15,16 of plasma diagnostics, since the satellite-to-resonance-line intensity ratio is essentially dependent on temperature, and in some cases on electron density. In a previous experiment conducted with identical parameters of laser and target,¹⁷ the temperature, ∼300 eV, was measured at the target surface using a steady-state plasma approach. Calculations to support inferring such a temperature were performed using the “zero-dimensional” program code FLYCHK,¹² which features a collisional-radiative model.

Second, we note that the ion emission of the expanding plasma has a recombination behavior¹⁴ in the regions that are remote from the target surface. Here, at a distance >1 mm, ionization processes are negligible, and the relative intensities of the He-like spectral lines are changed due to the strong dependence of the population coefficients on the plasma parameters, as illustrated in Fig. 3(a). In general, this regime is time-dependent and not steady-state, but a quasi-stationary approach⁵ is commonly applied to measure plasma parameters. The approach is based on the calculation of the relative intensities of the spectral lines for ions having the same charge state [see Fig. 3(a)]. The method is sensitive to densities in the range 10¹⁶–10²⁰ cm⁻³, when the temperature ranges from 10 to 100 eV for ions with Z ∼ 10; note that here we assume that the ion mean charge is “frozen.” Here the relative intensities of the 1snp¹P₁ − 1s²¹S₀ transitions in the fluoride ions, where n = 3 − 7 (He_β, He_γ, He_δ, He_ε, He_ξ lines correspondingly), were considered. Despite a certain error in the measurement of the intensity ratios [quantified by the width of the bar in Fig. 3(a)], the utilization of several ratios inside the He-like ion series makes it possible to unambiguously determine the electron temperature and density at any point along the plasma expansion, as illustrated in Fig. 3(b). Finally, the retrieved plasma parameters for the plasma expanding from the main target are summarized in Figs. 3(c)–3(d).

Note that, in order to apply this quasi-stationary approach, the plasma must be optically thin under laboratory conditions. An estimation of the influence of the optical depth on the intensity of ion emission can be provided by means of a comparison of the relative intensities of the 1snp¹P₁ − 1s²¹S₀ transitions in the fluoride ions for optically thin and optically thick plasmas, while varying plasma size in the simulation, as illustrated in Fig. 4. In the optically thin case, the ratios of the corresponding spectral lines evidently do not depend on plasma thickness. Figure 4 gives the simulated ratio of the intensities of the spectral lines He_β and He_γ in a plasma at T_e ∼ 100 eV, with a variable density, corresponding to typical values recorded in the experiment, in the case of a recombining plasma, i.e., far from the main target [see Fig. 3(d)].¹⁷ These calculations are made for various lateral plasma sizes, using the FLYCHK code in the cases of optically thin (dashed lines) and thick plasmas (solid lines). Moreover, the experimentally measured range for the considered ratio is shown. In the experiment, the optical measurements allow^4,6 us to assess the lateral size of the recombining plasma to be about 1 mm for a density ∼ 10¹⁸ cm⁻³. As a consequence, what can be observed in Fig. 4 is that the measured ratio is quite consistent with the one calculated in the hypothesis of an optically thin plasma. Hence, this tends to give support to the optically thin plasma hypothesis for the considered recombining plasma.

In Figs. 3(c) and 3(d), one observes a fast decrease of T_e and N_e along the expansion axis in the case where no magnetic field is applied (blue dots). Conversely, in the case where we apply a magnetic field of 20 T strength (black dots), the initially heated plasma flow maintains itself far from the main target; i.e., at distances >2 mm, almost constant values of electron temperature and density exist, i.e., T_e ∼ 20 eV and N_e ∼ 4 × 10¹⁸ cm⁻³ respectively. When an obstacle is put in the way of the plasma propagation (green dots), we observe a substantial increase in the electron temperature and density, up to 50 eV and 2 × 10¹⁹ cm⁻³, respectively, in the region near the obstacle.

To retrieve the plasma parameters in the case when a solid unheated obstacle is put in the way of the recombining plasma, one has to carefully consider that the shock formation and plasma heating at the location of the obstacle can lead to an additional excitation of ions. Typically, this implies higher temperatures (T_e ≫ 100 eV, exceeding the ionization potential for the L-shell) and higher densities than in the free propagation case.^18,19 Here, the recombination approach alone is no longer applicable, and an additional contribution to the kinetic model by ionization should be taken into account. Figure 5(a) demonstrates the sudden appearance of a strong Ly_α line, as well as satellites near the obstacle position, both of which definitely cannot be described using solely a recombination model [as shown by the red lines in Fig. 5(a)]. The spectra shown here were integrated over 400 µm along the expansion axis to achieve a better signal-to-noise ratio. As can be seen in Fig. 5(b), a simulation performed in a steady-state collisional-radiative model with a single-temperature plasma fits the experimental ratio Ly_α/He_β, as well as describing the satellites nearby; however, the intensities of the other He-like lines, with a larger main quantum number n, are clearly underestimated. Thus, according to Fig. 5(a), a reasonable assumption is to assume the presence of an additional plasma component, in addition to the one modeled in Fig. 5(a), which can yield the observed strong Ly_α line. The observed x-ray emission stemming from near the obstacle surface can thus be envisioned as resulting from the overlap of two emissions originating from two different plasmas. The first emission originates from the recombining plasma coming from the main target which has propagated up to the obstacle, while the second emission, contributing to the observed higher charge states (Ly_α and satellites), comes from a hotter plasma generated at the obstacle surface, following the impact of the expanding jet. This latter emission can be described by a pure steady-state model.

Hence, the experimental spectrum was simulated using the electron temperature and density, according to the experimental x-ray data: T₁ = 50 eV, N₁ = 2.3 × 10¹⁹ сm⁻³ for the shocked plasma, modeled by a recombining model; T₂ = (320–600) eV for the shocked plasma near the obstacle, using the simulation of Ly_α to satellites (see Fig. 6) in a steady-state model. The satellites have a low intensity in the experimental data, and so, the method based on analyzing their relative intensity to that of the Ly_α line produces significant error, represented in Fig. 6 by the rectangular hatching. Note that this uncertainty in the measurement also hampers us in determining reliable values for the density of the second plasma component, which is why it is not quoted here.

Notably, the plasma parameters inferred in this way for the two components of the plasma near the obstacle surface are very consistent with the simulated results using the GORGON²⁰ code; evidence is found of the formation of a hot plasma “shell” surrounding a central, colder “core” [see Figs. 7 and 3(d)]. Here the terms “hot” and “cold” refer to electron temperature only. The hot shell observed in the GORGON simulation is consistent with the plasma component that yields the strong Ly_α line and the associated satellites. The model used in the GORGON simulation describes a two-temperature, one-fluid resistive plasma which is optically thin. The model was initially developed for experiments in Z-pinches but is now widely used for laboratory astrophysics laser-plasma experiments.^21,22 The formation of the hot (in terms of electron plasma temperature) shell, as observed in the simulation performed using the GORGON code, results from the following scenario: after the laser-induced plasma stream has impacted the obstacle, the shocked plasma escapes radially but is recollimated on-axis by the external magnetic field. This recollimation of the plasma flow is accompanied by heating of the electrons within the peripheral regions, i.e., in the shell.⁸ Indeed, within the core, there is a decoupling of the ion and electron temperatures after the shock, the ions being the ones heated by the shock. Since the electrons can only be heated through collisions along their trajectories, given sufficient time, hot electrons are predominantly located in the shell.

Thus, the x-ray spectra shown in Fig. 5 describe two plasma components separated in space: one is a plasma shell and the other is a plasma core. Since we have already been able to derive the parameters of the cold core, we will now focus on ascertaining the density of the hot shell (the temperature of the latter was already inferred by the Ly_α/satellites ratio, see Fig. 6). For this, we will focus on the ratio between the Ly_α and He_β lines in the x-ray spectra, since the hydrogen and helium–like series for a given spectral range belong mostly to different plasma components. The He-like series of ion emission is emitted by the recombining plasma (i.e., the cold core), while the Ly_α line is mainly produced by the hot post-shocked zone (i.e., in the shell). The emissivity of the hot plasma can be evaluated for a wide range of electron densities (see Fig. 8) by a steady-state model in FLYCHK. The emissivity of the He-like emission, in contrast, is defined using a quasi-stationary approach.⁵ Here, the experimentally observed ratio can be expressed aswhere E is the emissivity of a spectral line, V the volume of a corresponding plasma component, and the emissivity, EHeβ, is calculated by a recombination model.⁵ Taking the volume ratio between the two plasma (shell/core) fractions retrieved from the interferometry data,⁸ it is possible to calculate the electron density of the hot plasma (i.e., the shell) component in the experiment as being up to ∼5 × 10¹⁸ cm⁻³.

The final parameters of the plasma components for the magnetic field strengths used in the experiment, 20 T and 6 T, are depicted in Table I. Note that, in the latter case, the analysis was performed according to what has been detailed above. Here, in line with the GORGON simulations and the optical interferometry data,⁸ we have assumed that the hot plasma component is a shell enveloping a cold core. We observe a significant difference between the 20 T and 6 T cases, in terms of electron densities, for the core plasma, up to ∼10 times; note that such a difference is fully consistent with optical interferometry data. Moreover, the electron temperature is similar for both cases. We associate this effect with the additional ionization of plasma and collimation of plasma flow when the magnetic field is applied. This leads to the increase in plasma density and the faster recombination rate, making it possible to measure the electron density using our technique.

For the shell, its optical depth can be evaluated in FLYCHK, in a similar way to what was done for the recombining plasma (i.e., the core), as presented in Fig. 4. The value of the optical depth is found here to be τ ∼ 0.3, validating, a posteriori, the possibility of using the Ly_α to He_β ratio for evaluating the shell plasma parameters. Even though we note that the shell revealed in the experiment is optically thin, we have to note that this is not necessarily so in the astrophysical case; i.e., the transport of x-ray emission is not scalable from the laboratory to the astrophysical case. For example, strong emission in the x-ray and UV ranges is suggested to take place following the impact of accretion columns on YSOs, due to the high temperature induced in the shocked plasma.²³ However, recent astrophysical observations^24,25 have shown that the mass-accretion rates are different for calculations based on x-ray and optical ranges of radiation. The effect of the optical depth in the x-ray range is given as one possible reason. The explanation can be also found in Ref. 26 where the idea of an “accretion-fed corona” is presented. In that case, the x-ray emission originates from three plasma components: a hot corona; a high-density, post-shock region close to the shock front; and a cold, less-dense, post-shock cooling region. We stress here that in the astrophysical case,²⁶ the situation, in terms of electron temperature, is the reverse of the situation observed in the laboratory, namely one of a cold shell surrounding a hot core. This is due to the very short equilibration time between ions and electrons in the astrophysical case compared with the time-scale of hydrodynamic-shell formation; in contrast, in the laboratory experiment, shell formation takes place on shorter time scales than the ions’ and electrons’ equilibration. Hence, the formation of a cold-shell plasma, highlighted here in laboratory experiments, could, in the astrophysical case, potentially lead to x-ray absorption of the emission originating from the hot core, directly influencing the calculated mass-accretion rate.

We developed a combined approach to characterize the dynamics of a magnetized plasma flow when it collides with a solid obstacle, mimicking the formation of accretion columns in young stars. The approach is based on the simulation of the relative intensities of x-ray spectral lines emitted by differently charged ionic states. On top of the plasma flow colliding onto the obstacle, a shock develops in which the ions are first heated, and hence, the electrons stay relatively cold. We assume the generation of an additional, hotter plasma component that results from the expansion into vacuum of the plasma escaping from the shock region. That second plasma component induces a significant x-ray emission from higher charge states, i.e., the Ly_α line and its dielectronic satellites. The ratio between the emitted Ly_α and He_β lines in the x-ray region was successfully used to derive the localization of these two plasma components, since these two lines are predominantly emitted by the hotter (post-shock) and colder (shocked) plasma components, respectively. Overall, this leads to evidence for the formation of a hot plasma shell with T_e > 320 eV, enveloping a cold, dense core with T_e ∼ 50 eV and N_e ∼ 2 × 10¹⁹ cm⁻³. These plasma parameters are consistent with numeric simulations performed by the magnetohydrodynamic code GORGON and with optical interferometry measurements of the same plasma.⁸ The formation of shell plasma in the astrophysical case could lead to substantial absorption of x-ray emission of accretion columns generated in YSOs.⁸