The Influence of External Radiation on the Emission Properties of H- and He-like Argon Ions in High Temperature Plasma

Abstract

In the present work, the influence of external X-ray radiation on the kinetics of multicharged ions in high-temperature plasma is investigated. A generalized diagnostic approach is proposed for the electron density and temperature measurements of photo-pumped plasma based on the relative intensity of the H-like ion resonance line and its dielectronic satellites. Based on detailed kinetic calculations performed for argon plasma, the conditions under which these techniques can be applied without modification to the photo-pumped plasma are determined, and the relative intensities of these lines are calculated for cases where the external influence significantly alters the kinetics of their excitation. The development of such diagnostic methods is of particular importance for the experiments with powerful X-ray free-electron lasers and thermonuclear laser plasma.

1. Introduction

Since the middle of the last century, X-ray spectroscopy of multicharged ions has been used to study the rarefied high-temperature plasma of the solar corona or the plasma generated in facilities with magnetic confinement. This led to the development of X-ray spectroscopy methods for plasma diagnostics, mainly based on measurement of the relative intensities of different spectral lines or on measurement of the profiles of their contours. The further development of laser technology and other methods of plasma generation led to the emergence of short-lived dense plasmas. In contrast to astrophysical and tokamak plasmas, laser plasma had a high density, and its study required the development and generalization of already developed methods [1,2,3]. Such short-lived plasmas are used today for a variety of physical investigations [4,5,6,7,8,9,10,11,12,13,14] and can be generated by pico- and femtosecond laser pulses or in ultrafast electrical discharges of the X-pinch type. The typical lifetime of such plasma does not exceed a few tens of picoseconds.

Its unique properties, such as a short lifetime, ultra-high density—close to that of the solid state—non-equilibrium energy distribution functions of electrons and ions, and excitation of strong oscillating electric fields, require further development and modification of the traditional methods of X-ray spectroscopy. These include the excitation of hollow ion states [15,16], the influence of an external magnetic field on the ionization state of plasma [17,18,19,20], the excitation of plasma satellites [21,22], the generation of hot electrons [23,24,25], the energy transfer into the depth of solid-state targets [26], and the non-stationarity of generated plasma [27,28,29]. Based on a pioneering study [30], the ratio of intensities of resonance lines and their dielectronic satellites is a practical “thermometer” that can be used to measure the electron temperature of hot plasmas, and the intensity ratios within the satellite structure enable measurements of density in laser or X-pinch plasmas.

In recent years, laser plasmas have been exposed to strong external short-wavelength radiation, which has led to the emergence of new processes that did not play a role in the previously studied plasmas. The intensity of this external radiation can exceed the intensity of the radiation generated in the plasma itself by orders of magnitude [31,32]. This increases the probabilities of the corresponding processes of photoexcitation and photoionization to such an extent that they begin to play a decisive role in plasma kinetics [32]. Such new plasma objects are, for example, laser plasmas generated by indirect heating of hohlraum-type targets or laser plasmas exposed to radiation from high-power X-ray free-electron lasers (XFELs) [30,33,34,35,36,37,38,39,40,41,42]. It should be emphasized that XFELs can already reach an X-ray intensity in the range of 10¹⁸–10²² W/cm² [41,42,43,44,45]. Under such conditions, the absorption of external radiation is mainly due to photoionization of electrons from the lower shells of atoms, leading to the formation of exotic, highly non-equilibrium states of matter: atoms or ions with empty inner and filled outer electron shells, so-called “hollow” atoms or ions [15,16].

X-ray spectroscopic methods for diagnosing such plasmas have not really been developed yet. Also, the possibility of using already developed techniques in this case has not yet been investigated. Recently, we considered [46] the possibility of diagnosing such laser plasmas with external photon pumping based on the relative intensities of the resonance and intercombination lines of multicharged He-like ions [1,47,48]. Based on detailed kinetic calculations in [46], the conditions under which this technique can be applied without modification were determined and calculations of the relative intensities of these lines were performed for cases where the external influence significantly changes the kinetics of their excitation.

In the present work, the influence of external photon pumping on the dielectronic satellites of the resonance lines of H-like ions is investigated for the first time. Based on detailed kinetic calculations, the conditions under which external photopumping does not affect the kinetics of autoionization levels were determined, and the relative intensities of the spectral lines were calculated for cases where the external influence significantly changes the kinetics of their excitation. In numerical calculations, the case of argon plasma is considered, since it is a bright representative of middle-Z elements frequently used for the interactions of high-power lasers with matter [49].

2. Influence of Photon Pumping on the Excitation Kinetics of Autoionization States for He-like Ions

The relative intensities of the resonance lines of H-like multiply charged ions and their dielectronic satellites emitted during the radiative decay of the autoionization states of He-like ions are known to be sensitive to the electron temperature $T_e$ and electron density $N_e$ of the plasma and can therefore be used for measurements. Moreover, if the plasma density is relatively low and the ion kinetics are adequately described by coronal approximation, then the ratio of the intensities of the dielectronic satellites and the resonance line itself depends only on the value of $T_e$, making it a very convenient “thermometer”. Such a method for temperature diagnostics was proposed in [48,50] and actively used to study the solar corona. Note, when we talk about multiply charged ions with nuclear charges > 10, coronal approximation is applicable for both the resonance level and the autoionization levels, even for very dense plasmas with $N_e<$ 10²¹ cm⁻³, which made it possible to use this technique to diagnose the temperature of laser plasmas heated with nanosecond laser pulses (see e.g., review articles [1,3]).

With a further increase in plasma density, coronal approximation becomes inapplicable, and the considered intensity ratio starts to depend not only on $T_e$ but also on $N_e$. This undoubtedly complicates the diagnostic application, since the plasma density must also be known in order to be able to measure the temperature. The above-mentioned technique described in [47,51] can be used to determine the density, but it is much more convenient to use the intensity ratio of the satellite groups, which is caused by the decay of the 2s2p ³P and 2p^{2 3}P states and which, as shown, e.g., in [51,52], is mainly sensitive to the plasma density and depends only weakly on the temperature. Thus, by simultaneously using two pairs of intensity ratios between the resonance line and the satellite and between two groups of satellites, both plasma parameters $T_e$ and $N_e$ can be measured. This approach can be described as a generalization of the classical method [48,50] for the case of dense plasma.

If the plasma is exposed to an external flow of strong short-wavelength radiation (we call this photopumping for simplicity), then the considered intensities of spectral lines tend to depend on another parameter, namely the photopumping intensity, starting from a certain value I_ph. If the intensity I_ph is known, then the diagnostic approach described above remains applicable, but requires the use of calculations of relative intensities that consider the effect of photopumping. When the intensity I_ph is not known, a different additional ratio of satellite intensities can be used (for example, the ratio of the intensities for the satellite 2p^{2 1}D₂–1s2p ¹P₁ and the satellite group 2s2p ³P–1s2s ¹S₀), and so the parameters $T_e$,$N_e$, and I_ph can be measured using these three ratios simultaneously. It is reasonable to describe this diagnostic approach as a generalization of the classical method for the case of a dense plasma with external photopumping.

The possibility of determining the temperature from the intensity ratio of resonance lines and their dielectronic satellites is based on a combination of several features, the most important of which are the following [50]:

(1) The main channels for populating both resonance and autoionization levels are processes involving the ground state of the same ion. For example, if we consider the resonance line of the H-like ion and its dielectronic satellites, this state is the level 1s, whose excitation by electron collisions generates the state 2p, and the dielectronic capture produces the autoionization levels 2l₁2l₂, whose radiative decay causes the emission of the dielectronic satellites. It follows that the intensity ratio of these lines does not depend on the ionization state of the plasma, at least to a first approximation.

(2) The autoionization levels have a very fast collisionless decay channel. This leads to the fact that the populations of the autoionization levels are well described by the coronal approximation. The intensity ratio of the resonance line and its satellites therefore does not depend on the plasma density, since both population channels of these spectral lines are proportional to the electron density. Of course, the latter only applies to not-too-dense-plasmas, because with increasing density, the coronal approximation becomes inapplicable. The applicability of coronal approximation is violated for satellite lines at higher densities than, for example, for the intercombination line. Moreover, if the ionization state of a dense plasma is strongly transient, then there is an additional dependence of the 2p-level population on the fraction ratio for H-like ions and nuclei. As a result, it is possible to diagnose not only moderate-density plasmas, but also ultra-dense ones characterized by solid densities [51] due to the relative intensities of satellites.

In the analysis of plasma density, we are primarily interested in the relative intensities of the dielectronic satellites themselves. As we have already established, in the satellites related to the resonance line of the H-like ions, the populations of levels 2l₁2l₂ are mainly determined by dielectronic trapping from the ground state 1s ²S_1/2 and due to radiative transitions in the He-like ions. Other population mechanisms related to the collisional excitation from the ground state of the He-like ion are insignificant for ions with $Z\ge 10$ because the effective cross sections of the two-electron excitation of 1s²–2l₁2l₂ are negligible. To qualitatively determine the dependence of the populations and intensities of the 2l₁2l₂–1s2l₂ satellite lines on the plasma density and temperature, we can use the kinetic model [1] described by the following system of equations:

$$N_i\left({\sum }_m{A}_{im}+{\mathsf{Г}}_i+N_e{\sum }_m<v{\sigma }_{im}>\right)=q_i+N_i{N}_e{\sum }_m<v{\sigma }_{mi}>,$$

(1)

$$q_i=N_H\left(1s\right)<v{\sigma }_0^{i,d}>N_e,$$

(2)

where $q_i$—population rate of the satellite states, $N_H\left(1s\right)$—ground state population of the H-like ion, $<v{\sigma }_0^{i,d}>—$rate of dielectronic capture associated with the autoionization probability ${\mathsf{Г}}_i$ through the detailed equilibrium relation, $<v{\sigma }_{im}>$—rate of collisional transition from i to m, and $A_{im}$—spontaneous radiative transition probability i-m.

As follows from Equation (1), in the coronal equilibrium, when $\sum _m{A}_{im}+{\mathsf{Г}}_i\gg max(N_e<v{\sigma }_{im}>)$, the population of autoionization levels and satellite intensities are proportional to the electron density and the factor ${\displaystyle \frac{\mathsf{Г}}{\mathsf{Г}+{\sum }_m{A}_m}}$. In the case of the high-density limit, the local thermodynamic equilibrium (LTE) is realized; the level populations are determined by the Boltzmann distribution and do not depend on the electron density. It is significant that in the coronal and LTE limits, the intensity ratio of any two satellites is determined by different atomic constants (${\Gamma }_i$ and $A_{im}$ in the coronal limit and only $A_{im}$ in the LTE limit) and therefore has a different value. This means that there is a range in which the ratio of their intensities is a function of density and can be used to measure it.

In the case of two groups of closely spaced triplet levels, (1) 2p² (transitions 2p^{2 3}P₂–1s2p ³P₂, 2p^{2 3}P₂–1s2p ³P₁, 2p^{2 3}P₁–1s2p ³P₁, 2p^{2 3}P₁–1s2p ³P₂) and (2) 2s2p (transitions 2s2p ³P₂–1s2s ³S₁, 2s2p ³P₁–1s2s ³S₁, 2s2p ³P₀–1s2s ³S₁), the possibility of determining the plasma density by the intensity ratio $R=I_1/I_2$ is due to the fact that the dielectronic capture rate for the 2p² states is much smaller than for the 2s2p levels. In the coronal limit, the states of the second group are predominantly populated, while the populations of the first group remain insignificant. As the electron density increases, the 2p² states are populated due to collisional transitions from the 2s2p levels.

When plasma is exposed to strong external short-wavelength radiation, its influence on the kinetics of He-like ions consists, on the one hand, of the emergence of an additional channel for the decay of excited states and, on the other hand, of the possible transition of the plasma into the recombination regime. We note that the transition to the recombination mode is possible if the energy of the incident photon is sufficient for the photoionization of the ion ground state [46].

Within the framework of a simplified kinetic model considering only six levels, 2s2p ³P_J and 2p^{2 3}P_J, an expression of the intensity ratio $R=I_1/I_2$ was obtained in [51] for the satellite groups due to radiative transitions from the levels 2p^{2 3}P_j (denoted by indices 1–3) and the levels 2s2p ³P_j (denoted by indices 4–6), respectively:

$$R=a\alpha +{\displaystyle \frac{x^2\left[a(1-\alpha )+b\right]+x\left[a(1-\alpha +\beta )+b\right]}{{(1+x)}^2}},$$

(3)

where $a=5/9$ [$A$(2p^{2 3}P₁−1s2p ³P₁) $+A$(2p^{2 3}P₁−1s2p ³P₂)]/$A$(2s2p ³P₁−1s2p ³S₁), $b=$ 1/3 [$A$(2p2 ³P₂−1s2p ³P₂) $+A$(2p2 ³P₂−1s2p ³P₁)]/$A$(2s2p ³P₁−1s2p ³S₁), $\alpha ={\mathsf{Г}}_3{K}_4/{\mathsf{Г}}_4{K}_3$, $\beta ={\mathsf{Г}}_3/{\mathsf{Г}}_4$, $K_i={\mathsf{Г}}_i+{\sum }_m{A}_{im}$, and $\chi =N_e<v{\sigma }_{42}>/K_3$. Equation (3) describes both the coronal and Boltzmann distributions between triplet states in the absence of external pumping.

Within the same model, a similar formula can be obtained if one considers the ionization of excited states at photon energies that are not sufficient for the ionization of the ground states of H- and He-like ions. In this case, the photoionization process should be added to the probability of collisionless ${\mathrm{K}}_{\mathrm{i}}$ decay of the autoionization state i:

$${K\prime }_i={\mathsf{Г}}_i+{\sum }_m{A}_{im}+I{F}_i,$$

(4)

where $F_i—$photoionization cross section. As follows from (1), photon pumping affects the ratio R at intensities as follows:

$$I_{cr}>({\mathsf{Г}}_i+{\sum }_m{A}_{im})/F_i,$$

(5)

In fact, the effect of photoionization is reduced in this case, first by shifting the density of LTE onset between the 2l₁2l₂ levels to higher values and second by changing the asymptotes in the coronal limit.

At a photon energy sufficient to ionize the ground states of H- and He-like ions, the dependence of the intensity ratio R on the pump intensity is more complex and may depend strongly on the temperature. At temperatures $T_e<<Z^{2}2R_y/4$ ($Z$ is the spectroscopic symbol of the He-like ion), the number of H-like ions is insignificant, and the presence of external pumping can strongly change the charge state of the plasma and lead to the appearance of bright dielectronic satellites. The satellite states can be populated by both dielectronic recombination and collisional processes of excited states of He-like ions. At temperatures of the order of $Z^{2}2R_y/4$, a significant number of H-like ions are present in the plasma, and the influence of external short-wavelength radiation is mainly reduced to the depopulation of the excited states.

The relation between the intensities of the satellites and the intensity of the Ly_α resonance line is of particular interest for the measurement of plasma temperature. Let us see how photopumping affects this ratio. In the absence of photon pumping, since the 2l₁2l₂ and 2p states are populated from the same ground state 1s, the intensity ratio $Y\left(T_e\right)=I_S/I_{Ly}\sim (A_S/A_{2p}) [{\mathsf{Г}}_S/(A_S+{\mathsf{Г}}_S\left)\right] (A_{2p}/W_{2p})exp(-(E_S-E_{2p})/T_e)$ does not depend on the density. Here the collisional excitation rate of H-like ions for the transitions from ground states to 2p states is indicated as ${<v\sigma >}_{2p}=W_{2p}exp(-E_{2p}/T_e)$ to emphasize an exponential dependence of $Y\left(T_e\right)$. External radiation with a photon energy sufficient for the ionization of excited states of He-like ions creates an additional decay channel:

$$Y\left(T_e\right)=I_S/I_{Ly}\sim (A_S/A_{2p}) [{\mathsf{Г}}_S/(A_S+{\mathsf{Г}}_S+I{F}_S\left)\right](A_{2p}+{IF}_{2p})/W_{2p}exp(-(E_S-E_{2p})/T_e),$$

(6)

If the photopumping energy is sufficient for the ionization of the ground state 1s, this can lead to the transition of the plasma into the recombination regime when the Ly_α line is populated by recombination processes, and the dependence $Y\left(T_e\right)$ changes dramatically.

3. Application of the Method to Ar XVII and XVIII Ions

The above considerations can only claim a qualitative description of the excitation kinetics of the spectra of He-like ions, since they do not consider different configurations of satellite states n₁l₁n₂l₂, stepwise excitation processes, and ionization from singly excited states. In the present work, a kinetic calculation for Ar XVII and XVIII ions was performed using the collisional–radiative code iPRAX [49]. It was designed for modelling the radiation spectra of the plasma of multicharged ions in a wide range of temperatures and densities and offers the possibility of performing calculations within the framework of stationary and non-stationary models. The software takes into account the most important elementary processes: collisional excitation and deexcitation, spontaneous and stimulated emission, collisional ionization and triple recombination, photoionization and photorecombination, and autoionization and dielectronic capture. The database, containing information about the cross sections and rates of these quantities and the energy structure of ion levels was calculated using the code cFAC [52]. To account for cascade processes, configurations with a principal quantum number n ≤ 10 and multiply excited states were considered. The flux of external photons was quasi-monochromatic, with a spectral width of 3 eV, which is typical for experiments with the European free-electron laser.

Examples of calculations are shown in Figure 1 for the photon energy $E_{ph}$ = 1200 eV, which is sufficient for the ionization of states 2l₁2l₂ and the excited states of Ar XVIII, although it is lower than the ionization potential of the ground state 1s. Figure 1a demonstrates that photoionization at low density leads to an increase in the R-value, as also follows from Equation (2). For example, at $I={10}^{19}$ W/cm² and an electron density of 10²¹ cm⁻³, the calculated value is $R=2.31$, which is consistent with the value of aα (See Equation (3)). Note, a slight discrepancy between analytical and numerical calculations is possible and related to the disruption of some conditions supposed for Equation (3) or error bars of data used for the radiative–collisional codes. As the density increases, the collisional processes between 2l₁2l₂ states become important and it leads to an LTE equilibrium within the configuration. In Figure 1b, condition (5) is satisfied at $I={10}^{16}$ W/cm², which corresponds to the beginning of the change in the intensity ratio.

Examples of results calculated for the pump photon energy $E_{ph}$ = 4500 eV are shown in Figure 2. In the range of moderate densities, the change in the R-ratio is associated with the transition of plasma into the recombination regime. The photopumping mechanism affects the ionization state of the plasma and consequently the populations of excited states at photopumping intensity:

$$I>N_e{C}^i\left(1s^2-1s\right)/F\left(1s^2\right),$$

(7)

where $C^i\left(1s^2-1s\right)$—collisional ionization rate of the 1s² state. With increasing pump intensity, the 1s2l states are populated by recombination from H-like ions, and their role in excitation transfer to the 2l₁2l₂ states increases. With a further increase in intensity, plasma becomes fully ionized and the population of 2l₁2l₂ occurs mostly by recombination from 2l states. Figure 2b shows that the R-ratio is sensitive to the electron temperature due to the dependence of the collisional ionization and photorecombination rates.

Equation (5) also applies to the ratio of satellite lines to Ly_α in the case that the photopumping energy is sufficient for the ionization of excited states. The results for the photon energy $E_{ph}$ = 1200 eV are shown in Figure 3. The transition 2p^{2 1}D_2–1s2p ¹P₁ was chosen as the most intense, but the functional dependence of all satellites 2l₁2l₂ or their sum remains the same. For the case of $T_e$ = 1200 eV in Figure 3b, condition (5) is satisfied at $I>{2·10}^{16}$ W/cm² when the ratio Y is changed. By increasing the external pump intensity, the ratio Y shifts to higher values, while the functional form of temperature dependence remains the same, which is confirmed by Figure 3a.

The situation is different with photon energy, which is sufficient for photoionization of the ground state of Ar XVII. Since the ionization potentials of He- and H-like ions are close to each other, the ionization temperature of the plasma deviates from the electron temperature $T_e$ when external pumping is considered, and condition (7) is satisfied. The populations of the 2p_1/2 and 2p_3/2 states are determined by photorecombination at low densities and by three-body recombination of Ar XIX ions at high densities. As can be seen from Figure 4b, the curves for different temperatures practically merge when $I>{10}^{12}$ W/cm² (condition (7)). In this case, the population of Ly_α from excited states dominates over the collisional excitation from the ground state of Ar XVIII, while the 2l₁2l₂ states are still populated by dielectronic recombination, leading to a decrease in the Y ratio. The overlaps of the diagrams for different temperatures are since the cascade recombination processes leading to Ly_α line emission depend in a complex way on pump intensity, temperature, and density.

Figure 4a shows the dependence $Y\left(T_e\right)$, which is traditionally used to determine the plasma temperature. The dashed curve is no longer applicable at a photopumping intensity of the order of 10¹¹ W/cm² and the temperature measurements should be based on calculations considering external X-ray radiation. Note that condition (7) depends on the plasma temperature as $\sim T_e^{-3/2}exp(-\mathsf{\Delta }E/T_e)$ and that the curves for fixed and not-too-large external pump intensities turn into the usual asymptotics, corresponding to the absence of the recombination mode. However, as the intensity increases, plasma becomes fully ionized, leading to a different asymptote.

It should be noted that the results calculated with the iPRAX code (dashed curves in Figure 1, Figure 2, Figure 3 and Figure 4) are in a good agreement—within 10%—with the results obtained by using the FLYCHK code in the absence of photopumping [53].

4. Conclusions

In our work, it is shown that external pumping with a short-wavelength source can significantly modify the excitation kinetics of the autoionization states of He-like ions. On the one hand, it changes the lifetime of the excited states due to the emergence of an excitation-free channel for their decay; on the other hand, it changes the charge composition of the plasma and the dynamics of population channels due to the transfer of excitation from other excited states.

Our approach, which takes these effects into account, opens up the investigation of a dense photopumped plasma in experimental studies based on the relative intensity of the H-like ion resonance line and its dielectronic satellites. In particular, these results can be used in the interpretation of the results of experiments at XFEL facilities to determine the plasma parameters. In addition, we have presented calculations of the intensities for the satellite groups 2s2p and 2p² to Ly_α for the Ar plasma, which is typically used as a gas or cryogenic target in many laser–matter experiments including XFEL ([54,55]). It is shown that for photon energies sufficient to ionize the excited states of He- and H-like argon ions, the effect of external radiation must be taken into account at intensities of the order of 10¹⁶ W/cm². In cases where the photon energy exceeds the ionization potential of the ground state, modification may be required at much lower intensities of incident radiation, the corresponding threshold being a function of plasma density. For example, for $N_e ~$ 10²¹ cm⁻³, a modification of the kinetic equations is already required at intensities of more than $7·{10}^{12}$W/cm². Similar values were obtained for the temperature measurement using the ratio of the intensities of the dielectronic satellite lines to Ly_α.

It should be noted that the results presented here can be used if the lifetime of the plasma object is of the order of the photopumping pulse duration. In the case of the interaction of an XFEL pulse with a duration of up to 100 fs with a cluster argon target, this leads to a limitation of the cluster size, which should not exceed 100 nm, since the plasma lifetime of clusters with a larger size exceeds 100 fs. If the plasma is generated by the indirect drive heating scheme using hohlraum-type targets, the duration of the photopumping pulses can be in the subnano- and nanosecond range and even coincide with the plasma lifetime. In this case, X-ray spectra measured with a picosecond time resolution allow the dynamics of target compression to be studied.