Multiply-excited states and their contribution to opacity in CO₂ laser-driven tin-plasma conditions

The first step in the present opacity calculations is the generation of all relevant atomic data (energy levels, oscillator strengths, ionisation potentials, etc). In Torretti et al [13], considerable effort was devoted to benchmarking the calculated atomic structures with available experimental data and therefore we have utilised these atomic structures in the present study. In brief, the Cats code [45] was employed in the calculation of level energies and absorption oscillator strengths. This code is based on Cowan's original atomic structure code [46, 47] and exhibits a favourable feature where one can scale the magnitude of the radial integrals in order to bring theoretical level structures into excellent agreement with experimental observations. This semi-empirical approach has proven to be an indispensable tool in atomic spectroscopy, especially for systems exhibiting complicated level structures [29, 48–51]. In the work of Torretti et al [13], optimum agreement between atomic structure calculations and previous experimental observations [17, 26, 27], was found by applying scaling factors of 0.87/0.75 to the radial integrals associated with singly-/multiply-excited energy levels. These two scaling factors were adopted for all charge states considered in this work. Extensive configuration sets were employed in the calculations through use of the so-called '2-mode' approach. In these calculations, configuration interaction (CI) was retained for a large number of configurations associated with Δn = 0 and Δn = 1 single, double and triple excitations from the n = 4 manifold of the ground-state configuration. Additional configurations were included using intermediate coupling (only interactions between levels of the same configuration are considered) [39]. This approach retains a significant level of detail provided by full CI calculations however at a much reduced computational cost.

Knowledge of opacities is crucial for understanding radiation transport in a host of plasma environments, such as those encountered in stellar interiors, fusion devices and short-wavelength plasma light sources [30, 52–55]. In this context, an opacity quantifies how photons, travelling in a plasma, are absorbed and scattered through interactions with the plasma constituents. For a given photon frequency ν, the opacity ${\kappa }_{\nu }^{\mathrm{T}}$ is defined as the contribution of four mechanisms:

$$\begin{equation}{\kappa }_{\nu }^{\mathrm{T}}={\kappa }_{\nu }^{\text{BB}}+{\kappa }_{\nu }^{\text{BF}}+{\kappa }_{\nu }^{\text{FF}}+{\kappa }_{\nu }^{\text{SCAT}}.\end{equation} \tag{ 1 }$$

The first three terms represent bound–bound (photoabsorption), bound–free (photoionisation) and free–free (inverse bremsstrahlung) contributions to the opacity and each contain a factor correcting for stimulated emission. The final term represents photon scattering. In the present work, we are mainly concerned with the bound–bound contribution to the opacity (since all other contributions are minor in the wavelength range of interest for these plasma conditions) and therefore we drop the superscript 'BB' notation in the following. The bound–bound opacity κ_ν is written

$$\begin{equation}{\kappa }_{\nu }={\rho }^{-1}\sum\limits _{i{< }j}{\sigma }_{\nu }^{ij}\left({n}_{i}-{g}_{i}{g}_{j}^{-1}{n}_{j}\right),\end{equation} \tag{ 2 }$$

where ρ is the mass density and the summation term is known as the absorptivity. Here, ${\sigma }_{\nu }^{ij}$ is the cross section for photoabsorption from level i to level j and n_i(j) are the population densities (simply called 'populations' in the following) of energy levels i(j) having statistical weights g_i(j). The sum runs over all transitions i→j associated with photon absorption at frequency ν. The second term in the summation is the aforementioned correction for stimulated emission. The cross section for photoabsorption ${\sigma }_{\nu }^{ij}$ is given by

$$\begin{equation}{\sigma }_{\nu }^{ij}=\frac{\pi {e}^{2}}{{m}_{\mathrm{e}}c}{f}_{ij}\phi \left(\nu \right),\end{equation} \tag{ 3 }$$

where e, m_e and c are the electron charge, electron mass and speed of light, respectively. The term f_ij is the absorption oscillator strength for a transition i → j and ϕ(ν) is the line profile function satisfying ${\int }_{0}^{\infty }\phi \left(\nu \right)\mathrm{d}\nu =1$.

As one can see from equations (2) and (3), evaluation of the bound–bound opacity requires knowledge of both atomic structure (ν and f_ij ) and the level populations n_i(j). Evaluation of this latter quantity requires knowledge of the plasma environment in which the atomic species is embedded. Atomic, using as input the mass density ρ and electron temperature T_e, can calculate these level populations in LTE or non-LTE plasma conditions (the radiation temperature T_r can also be specified in non-LTE calculations). The LTE-mode of Atomic uses the ChemEOS equation-of-state (EOS) model to derive level populations and charge state distributions (CSDs) [56, 57]. This model is based on the free-energy-minimisation approach and the occupation probability formalism (this latter procedure allows plasma density effects to be included in a consistent manner). Importantly, non-ideal plasma effects, such as plasma microfields and strong coupling are included in the free energy term. This leads to modified Boltzmann and Saha relations from which level populations and CSDs are derived.

Atomic also has the capability of modelling plasmas under non-LTE conditions [58, 59]. Unlike LTE plasmas, the calculation of level populations in non-LTE conditions requires a microscopic viewpoint in which the rates of all relevant atomic populating and depopulating processes are calculated. The most general definition of a non-LTE plasma is one which exhibits any characteristic departure from LTE conditions, such as the existence of a non-Maxwellian electron energy distribution function (EEDF) or a non-Planckian radiation field. As such, there are a variety of ways that a plasma can be defined as under non-LTE conditions. Consider an atomic state (energy level, configuration or superconfiguration) i associated with some charge state ζ in the plasma. In non-LTE plasmas, the population of this state evolves according to a collisional-radiative (CR) equation of the form [60, 61]

$$\begin{equation}\frac{\mathrm{d}{n}_{i\zeta }}{\mathrm{d}t}=\sum\limits _{j{\zeta }^{\prime }}{n}_{j{\zeta }^{\prime }}{R}_{j{\zeta }^{\prime }\to i\zeta }-{n}_{i\zeta }\sum\limits _{j{\zeta }^{\prime }}{R}_{i\zeta \to j{\zeta }^{\prime }}.\end{equation} \tag{ 4 }$$

Here, ${R}_{j{\zeta }^{\prime }\to i\zeta }={R}_{j{\zeta }^{\prime }\to i\zeta }^{\mathrm{C}}+{R}_{j{\zeta }^{\prime }\to i\zeta }^{\mathrm{R}}$ denotes the rate of collisional (${R}_{j{\zeta }^{\prime }\to i\zeta }^{\mathrm{C}}$) and radiative (${R}_{j{\zeta }^{\prime }\to i\zeta }^{\mathrm{R}}$) processes connecting j (associated with some charge state ζ') to i. In a similar fashion, rates describing i → j are denoted R_iζ→jζ . The first term in equation (4) represents population influx into i and the second term describes population outflux from i. A CR equation of the form equation (4) exists for all atomic states i in each charge state ζ present in the plasma. Coupling these states yields the so-called rate equation

$$\begin{equation}\frac{\mathrm{d}\mathbf{N}}{\mathrm{d}t}=A\mathbf{N}.\end{equation} \tag{ 5 }$$

Here, N is a vector of atomic state populations (n_iζ ∈ N) and A is the so-called rate matrix containing all relevant rates ${R}_{j{\zeta }^{\prime }\to i\zeta ,i\zeta \to j{\zeta }^{\prime }}^{\mathrm{C},\mathrm{R}}$.

The choice of atomic processes to be included in the rate matrix A is highly case-dependent; in coronal plasmas, for example, one can neglect the role of three-body recombination. Such simplifications are not possible in the current case and therefore we must consider a wide range of atomic processes (see section 3 for more details). Fortunately, Atomic has the capability of modelling a variety of (de-)populating processes such as electron-impact excitation/ionisation, photoexcitation, photoionisation, autoionisation (AI), etc. Cross sections for electron-impact excitation were calculated in the plane-wave Born (PWB) approximation using the CATS code. Electron-impact ionisation cross-sections were obtained from the GIPPER code (photoionisation cross sections can also be obtained if an external radiation field is included in the non-LTE model) [62]. Atomic uses these cross sections to calculate the rates which form the entries of A. Invoking a conservation equation of the form

$$\begin{equation}\sum\limits _{\zeta =0}^{Z}\sum\limits _{i=0}^{{N}_{i}-1}{n}_{i\zeta }={n}_{\text{ion}},\end{equation} \tag{ 6 }$$

where N_i is the number of atomic states and n_ion is the ion density allows one to solve equation (5) for the atomic state populations n_iζ . These populations can then used to produce opacities [equation (2)] in a similar fashion as in the LTE mode of Atomic.

Finally, it should be emphasized that a consistent atomic structure has been employed in all stages of the calculation, ranging from the evaluation of collisional cross-sections to the derivation of atomic state populations and level-resolved opacities. This procedure ensures self-consistency in all aspects of the calculation.

In order to circumvent computational intractabilities associated with level-resolved, non-LTE opacity calculations involving multiply-excited configurations, we have employed the ionisation temperature method of Busquet [64, 65] to effectively 'mimick' non-LTE opacities using LTE computations. The key observation underlying this method is that non-LTE CSDs (performed at some T_e) often resemble LTE CSDs undertaken at a temperature T_Z different to T_e [66]. The first step in Busquet's approach is to identify the LTE temperature T_Z which yields optimum agreement in average charge state ⟨Z⟩ between the non-LTE and LTE calculations, i.e.

$$\begin{equation}\langle Z{\left({T}_{\mathrm{e}}\right)\rangle }_{\text{non}-\text{LTE}}=\langle Z{\left({T}_{Z}\right)\rangle }_{\text{LTE}}.\end{equation} \tag{ 7 }$$

In the current context, T_Z is known as the 'ionisation temperature'. Once T_Z is determined, one then performs detailed, level-resolved LTE opacity calculations at T_Z . Higher-order moments of the CSD, such as the width, are not considered in this method. Although this approach provides a numerically cost-efficient means of incorporating non-LTE opacities (and EOS quantities) into hydrodynamic codes, Klapisch and Bar-Shalom have stressed that it should only be applied to plasmas not far from pure LTE conditions [66].

We wish to test the validity this method in the context of the current problem. Guided by previous modelling efforts of Sasaki and co-workers [37, 40, 41], we have undertaken a non-LTE configuration-averaged (CA) Atomic calculation at ρ = 2 × 10⁻⁴ g cm⁻³ and T_e = 36 eV. The electron density of this plasma is n_e = 1.27 × 10¹⁹ cm⁻³, close to the critical electron density for CO₂ laser light (${n}_{\text{crit},{\mathrm{C}\mathrm{O}}_{\mathrm{2}}}=1.1{\times}1{0}^{19}\enspace {\mathrm{c}\mathrm{m}}^{-3}$). The CA approach generates appropriately averaged energies, oscillator strengths and PWB electron-impact excitation cross sections from which configuration-to-configuration rates are derived [59]. This approach is employed in the present context to ensure computational tractability. Configurations sets employed in the CA CATS calculations for Sn¹²⁺ and Sn¹⁴⁺ are provided in table 1. The calculations were performed under steady-state conditions, i.e. dN/dt = 0. This approximation is valid for plasmas where the time taken to generate a given CSD (through ionisation and recombination processes) occurs on timescales shorter than the laser pulse duration. This is certainly true for industrial LPP EUV sources, which are typically driven by multi-10 nanosecond-long CO₂ pulses. We calculated the rates of electron-impact excitation/ionisation and AI (as well as their inverse processes) assuming a Maxwell–Boltzmann EEDF at T_e = 36 eV. The calculations were performed without an external radiation field, i.e. photoexcitation and photoionisation processes were not considered (see subsection 3.2 below for further discussion). The process of spontaneous emission was included in the model.

Table 1. Configuration sets adopted for Sn¹²⁺ and Sn¹⁴⁺ in the non-LTE/LTE CA calculations (see main text). Configurations are grouped according to excitation degree. Here, angular momenta l = 0 – 4 (s − g), l' = 1–4 (p − g) and $\tilde {l}=2\enspace --\enspace 4\enspace \left(d-g\right)$, respectively.

Sn12+	Sn14+
Ground configuration	Ground configuration
4s24p64d2	4s24p6
Singly-excited configurations	Singly-excited configurations
4s24p64d + {4f, 5l}	4s24p5 + {4d, 4f, 5l}
4s24p5 + {4d3, 4d24f, 4d25l}	4s14p6 + {4d, 4f, 5l}
4s4p6 + {4d3, 4d24f, 4d25l}
Doubly-excited configurations	Doubly-excited configurations
4s24p6 + $\left\{4{f}^{2},4f5l,5s5l,5p5{l}^{\prime },5d5\tilde {l},5f5g\right\}$	4s24p4 + {4d2, 4d4f, 4f2, 4d5l, 4f5l, 5s5l, 5p5l', 5d5f, 5d5g, 5f5g}
4s24p5 + {4d4f2, 4d5s2, 4d4f5l, 4d5s5p, 4d5s5d}	4s14p5 + {4d2, 4d4f, 4f2, 4d5l, 4f5l, 5s5l, 5p5l', 5d5f, 5d5g, 5f5g}
4s24p4 + {4d4, 4d34f, 4d35l, 4d24f2}	4p6 + {4d2, 4d4f, 4d5l, 4f5l}
4s4p54d4
Triply-excited configuration	Triply-excited configurations
4s24p34d5	4s24p34d3, 4s24p34d24f, 4s4p44d3, 4s4p44d24f

The CSD resulting from this calculation is shown in black in figure 1. Importantly, the relevant EUV-emitting Sn¹¹⁺–Sn¹⁴⁺ ions are produced under such conditions, with Sn¹²⁺ and Sn¹³⁺ clearly the dominant species. The average charge state of this plasma is ⟨Z⟩_non-LTE = 12.6. Also illustrated in figure 1 is an LTE CA calculation (shown in red) performed at ρ = 2 × 10⁻⁴ g cm⁻³ and T_e = 26 eV. These calculations yield ⟨Z⟩_LTE = 12.6, identical to the non-LTE calculation. Also, we note that the CSDs are very nearly identical. An ionisation temperature of T_Z = 26 eV is thus established. We note that the ratio T_e/T_Z = 36/26 ≈ 1.4 is larger than that obtained by the scaling law ${T}_{\mathrm{e}}/{T}_{Z}\approx {\left(1+\left(5.36{\times}1{0}^{13}\right){T}_{\mathrm{e}}^{7/2}{n}_{\mathrm{e}}^{-1}\right)}^{0.215}\approx 1.2$ obtained by Busquet et al [65]. Much better agreement with this scaling law is obtained in the Nd:YAG-relevant case considered in [13] (n_e = 1.27 × 10²⁰ cm⁻³, T_Z = 32 eV, T_e = 34.5 eV), where T_e/T_Z = 34.5/32 ≈ 1.08 and the scaling law yields T_e/T_Z ≈ 1.02.

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It is interesting to consider why optimum agreement in CSDs is obtained for T_e > T_Z . Exact reproduction of an LTE CSD is possible using a T_e = 26 eV non-LTE calculation, however one must include an external radiation field in the calculation characterised by T_r = T_e. In this way, all atomic processes are in detailed balance. However, as detailed above, our non-LTE calculations do not include an external radiation field. In order to match our present calculations with a T_Z = 26 eV LTE calculation, we must therefore compensate for the missing (de)excitation and ionisation channels provided by photoexcitation and photoionisation by increasing T_e above T_Z . The difference T_e − T_Z is effectively a gauge for how close the system is to LTE conditions; in the work of Torretti et al [13], for example, a value T_e − T_Z = 2.5 eV (where T_e = 34.5 eV and T_Z = 32 eV) highlights the dominance of collisional rates over radiative rates at near-LTE conditions (n_e = 1.27 × 10²⁰ cm⁻³). The same cannot be said for the CO₂-relevant conditions, where a value T_e − T_Z = 10 eV is indicative of truly non-LTE conditions, i.e. collisional rates do not significantly outweigh radiative rates.

The replication of a non-LTE CSD using LTE computations, such as that shown in figure 1, is key for justifying the use of LTE opacities as an approximation for non-LTE opacities. An even stronger argument in favour of this approximation is to establish equivalence between the non-LTE and LTE-computed configuration populations (the cumulative sum of which yields the CSD—see equation (6)). To this end, we present in figure 2 non-LTE- and LTE-computed configuration populations (P_non-LTE, P_LTE) as a function of CA energy (E_av) for Sn¹²⁺ (upper panel) and Sn¹⁴⁺ (lower panel). The non-LTE configuration population calculations are shown in figure 2(a) (Sn¹²⁺) and figure 2(d) (Sn¹⁴⁺). The corresponding LTE calculations are shown in figures 2(b) and (e). Configurations associated with single-, double- and triple-electron excitations are shown as blue, orange and green circles, respectively. The ground-state configurations, fixed at E_av = 0 eV, are shown as grey circles. Perhaps the most striking feature of these calculations is the near-coincidence in non-LTE- and LTE-computed configuration populations. This is quantified in figures 2(c) and (f), where, for a given configuration C, we plot the non-LTE-normalised Configuration Population Difference (CPD) defined as (P_C,non−LTE − P_C,LTE)/P_C,non−LTE. For the majority of configurations, this quantity takes on values below 1%. This indicates a near-coincidence in non-LTE and LTE-computed configuration populations. The few configurations where the non-LTE-normalised CPD values are above 1% all have LTE-computed populations greater than the corresponding non-LTE populations.

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In addition to validating the use of Busquet's approach in the current context, the equivalence in configuration populations reveals an interesting characteristic of the non-LTE-computed populations in that they appear to follow a single Boltzmann-esque temperature law of the form

$$\begin{equation}{P}_{C,\mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{L}\mathrm{T}\mathrm{E}}\propto {g}_{C}\enspace \mathrm{exp}\left(-\frac{E}{{T}_{\text{eff}}}\right),\end{equation} \tag{ 8 }$$

where ${g}_{C}={\sum }_{{J}_{C}}\left(2{J}_{C}+1\right)$ is the total statistical weight of configuration C (J_C is an energy level associated with C) and T_eff is a so-called effective temperature [67] (the Boltzmann constant k_B typically appearing in equations of the form equation (8) is here 'absorbed' into the definition of T_eff).

In order to evaluate T_eff, we plot in figure 3 the natural logarithm of the ratio P_C,non−LTE/g_C as a function of E_av for (a) Sn¹²⁺ and (b) Sn¹⁴⁺. First-order polynomial fits to the data are shown in red. Excellent fits to the data are characterised by R² values of 0.99 attained in both cases. The effective temperatures T_eff are obtained from the inverse-negative of the slopes of the lines of best fit. As indicated in the plots, the fits yield values of T_eff = 26.6 (Sn¹²⁺) and 26.9 eV (Sn¹⁴⁺), clearly very similar to the LTE ionisation temperature T_Z = 26 eV. This close agreement between T_eff and T_Z is perhaps not too surprizing given the near-coincidence in non-LTE and LTE-computed configuration populations.

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This observation, that non-LTE-computed configuration populations follow Boltzmann-like temperature laws has been studied extensively in the works of Bauche, Bauche-Arnoult and others [67–72]. The emergence of temperature laws is attributed to the complex interplay of atomic processes defining the steady-state CR model. More specifically, the combination of (i) the microreversibility nature of collisional processes (which drives the population distributions towards LTE conditions) and (ii) correlation laws associated with spontaneous emission (high/low-lying upper states typically decay towards high/low-lying lower states) tend to generate Boltzmann-type population distributions [71]. This is in fact a general phenomenon of non-LTE hot plasmas [68]. Finally, it should be noted that the approach of Bauche et al [70], does not ascribe a single effective temperature T_eff to a population distribution. Rather, configurations are grouped according to superconfigurations and equation (8) is used to derive superconfiguration-specific temperatures. These temperatures may vary significantly from superconfiguration-to-superconfiguration (see figure 1 of [70]). Although this approach differs to that presented in figure 3 (where we have performed a 'global' fit to all configurations), the non-LTE-computed population distributions are clearly well-described by a single-temperature Boltzmann-type law.

Finally, we wish to comment on the importance of a non-zero radiation field in our non-LTE computations. As outlined in section 3, all non-LTE calculations presented thus far have not included an external radiation field in the underlying atomic kinetics. Indeed, identifying an external radiation field appropriate for the current study is rather difficult as it is a purely non-local quantity, i.e. it depends on the radiative characteristics of the plasma surrounding the studied (ρ, T_e) plasma. Proper evaluation of an external radiation field requires complex simulations involving the coupling of non-LTE atomic kinetics with a model for radiation transport in a multi-(ρ, T_e, T_r) plasma. Such an endeavour is outside the scope of the present work. Instead, we have performed an additional non-LTE calculation where we have included a Planckian radiation field (characterised by a radiation field T_r). We fixed T_r and tuned T_e until equation (7) is established with a T_Z = 26 eV LTE plasma. We find that a T_e = 33 eV and T_r = 21 eV plasma, for example, reproduces both equation (7) and the LTE CSD. Moreover, the configuration populations associated with Sn¹²⁺ and Sn¹⁴⁺ are also found to be in excellent agreement with the LTE-computed configuration populations. This is demonstrated in figure 4, where, in a similar vein to figure 3, we plot the natural logarithm of the ratio P_C,non−LTE/g_C as a function of E_av for (a) Sn¹²⁺ and (b) Sn¹⁴⁺. Effective temperatures obtained from the slopes of the lines of best fit (shown in red) are T_eff = 26.6 eV for both Sn¹²⁺ and Sn¹⁴⁺, clearly in excellent agreement with the data presented in figure 3. The introduction of a non-zero radiation field in these computations therefore does not break the observed Boltzmann-esque temperature law once equivalence between the non-LTE and LTE-computed CSDs is established.

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