The helium-like isoelectronic series emit strong X-ray wavelengths. The most intense lines of these systems are the resonance line designated by ω (also labelled r: 1s^{2 1}S₀ - 1s2p¹P₁), the intercombination lines (x + y) (or i: 1s^{2 1}S₀ - 1s2p³P_{2, 1}) and the forbidden line z (or f: 1s^{2 1}S₀ - 1s2s ³S₁). These three lines correspond to the transitions between the n = 2 excited shell and the n = 1 ground state shell. The determination of these lines is of great interest because the line ratios f/i and (f + i)/r provided respectively electrons density (n_e ~ 10⁸ - 10¹³ cm⁻³) and electrons temperature (T_e ~ 1 - 10 MK) as first shown by Gabriel and Jordan [1] and are widely used for collisional solar plasma diagnostics [1] [2] [3]. On the other hand, these line ratios enable also to determine the prevailed ionization processes (photo-ionization and/or collisional ionization) in the plasma [4] [5] [6]. Traditionally, He-like ions f/i line ratios have been used to derive electron densities of X-ray line-emitting regions since the populations of the 2³P level are controlled by collisional excitation from the 2³S level [1]. At low density, the n = 2 states are populated by electron excitation and then decay radiatively. Then, the relative intensities of the three lines are independent of the density [7]. Above the critical density (n_crtit): (n_crit C ~ A, C being the rate coefficient for collisional excitation 2³S → 2³P and A denotes the radiative transition probability of 2³S → 1¹S), the 2³S upper level of the forbidden line becomes to be depleted by collision to the 2³P upper levels of the intercombination line. As a result, when the electron density increases, the intensity of the forbidden line decreases strongly whereas that of the intercombination line increases. However, in the case of a strong UV radiation, the photo-excitation 2³S → 2³P becomes no negligible. Subsequently, the ratio (f/i) of the forbidden line on the intercombination line is no longer an electron density diagnostic. As concern the ratio (f + i)/r, it is sensitive to electron temperature as the dependence of the collisional excitation rates with the temperature for the resonance line is not the same for the forbidden and intercombination lines. In short, for plasma dominated by photo-ionization and recombination, the forbidden line (or the intercombination line at high density) becomes much stronger than the resonance line. In the case of plasma dominated by collisional ionization and excitation, the resonance line is stronger or comparable to the forbidden line and the intercombination line [4]. The following considerations indicate that the determination of the most intensive lines of Helium-like ions in the X-ray range is of great interest in laboratory and astrophysical plasma diagnostics.

On the experimental side, high-precision measurements of the energy difference between S and P levels in the helium isoelectronic series were made three decades ago. Robinson [7] presents measurements of the 1s^{2 1}S₀ - 1snp ¹P₁ series of the Helium isoelectronic sequence for Be III, B IV and C V. Since that time, many experiments have been improved. Twelve lines in the region 20 - 100 Å belonging to the resonance series of Be III, B IV, C V and O VII are remeasured by Svensson [8] using spectrograms. Beiersdorfer et al. [9] use the tokamak plasmas from the Princeton Large Torus (PLT) high-resolution Johann spectrometer to report the n = 2 → 1 X-ray transitions of Helium-like potassium, scandium, titanium, vanadium, chromium, and iron (Z = 19 - 26) along with wavelengths belonging to the 1s^{2 1}S₀ - 1snp ¹P₁ (n = 3 - 5) transitions. Furthermore, Engtröm and Litzén [10] generate spectra of C, N and O simultaneously by focusing 1 GW laser pulses on targets made of either ammonium hydrogen carbonate or ammonium oxalate and then determine the wavelengths of the 1s^{2 1}S - 1snp ¹P (n = 2 - 4) resonance lines in N VI and O VII (17 - 30 ÅÅ) with uncertainties ranging from 0.2 to 0.7 mÅÅ. Bartnik et al. [11] measure the wavelengths of the 1snp ¹P - 1s^{2 1}S (n = 4 - 10) transitions in He-like O VII in laser-produced gas puff plasmas with an accuracy measurement ranging between (1.5 - 3.0 mÅ).

On the theoretical side, many techniques are presented. Acaad et al. [12] construct wave function expanded in a triple series of Laguerre polynomials of the perimertric coordinates to study the S and P states of the helium isoelectronic sequence and report nonrelativistic wavelengths and total wavelengths including mass polarization relativistic and, the Lamb shift corrections for Z = 2 - 9 belonging to the 1snp ¹P - 1s²¹S (n = 2 - 5) transitions. In addition, Safronova et al. [13] apply the MZ code through a perturbation theory based on hydrogen-like functions to compute wavelengths of highly charged He-like ions (Z = 6 - 54) for both satellite lines (1s2l’nl - 1s²n’l’, n, n’ = 2, 3) and (1snp ^{1, 3}P - 1s², n = 2, 3 and 1s2s ^{1, 3}S - 1s²) transitions. Additionally, the plasma simulation code CLOUDY is used by Porter [14] to present wavelengths of the UV, intercombination, forbidden, and resonance transitions oh He-like ions for Z = 6 - 14 and for Z = 16, 18, 20, and 26. But, as far as we know, the wavelengths cannot be directly determined within a single analytical formula for a whole members of He-like ions using one of the preceding method or one of the other existing computational techniques. Then, analytical spectral lines in two-electron systems such as the Balmer or the Lyman spectral lines of the hydrogen-like systems are not yet established. In this paper, we intend to present analytical spectral lines belonging to the resonance line: 1s^{2 1}S₀ - 1s2p ¹P₁ and intercombination line: 1s^{2 1}S₀ - 1s2p ³P_{2, 1} along the 1s^{2 1}S₀ - 1snp ¹P₁ (n £ 10) transitions in the helium isoelectronic sequence. In our study, we use the Screening Constant per Unit Nuclear Charge (SCUNC) method suitable in the analysis of atomic spectra [15] [16]. All the results obtained in the present work compared very well to the available experimental and theoretical literature data. A host of data listed in this paper may be of interest in astrophysical and laboratory plasmas diagnostic.

In section 2, we present the theoretical procedure adopted in this work. In section 3, the presentation and the discussion of the results are made. A comparison of our results with available experimental and theoretical results is also made.

The present SCUNC wavelengths predictions for the wavelengths belonging to the 1s^{2 1}S₀ → 1snp ¹P1 (3 ≤ n ≤ 13) transitions in He-like (Z = 3 - 38) ions are quoted in Table 1. Table 2 Presents a comparison between theoretical and experimental wavelengths of the 1 ¹S₀ → np ¹P₁ (1s^{2 1}S₀ → 1snp ¹P₁) transitions of helium-like ions up to Z = 8. The present SCUNC calculations values, are compared to the experimental data of Robinson [7], Svensson [8], Bartnik et al. [11] and to the experimental data of Engtröm and Litzén [10]. For the resonance 1 ¹S₀ → 2p ¹P₁ transition, it is seen that the current SCUNC results compared very well to the experimental values. Here, the Δλ/λ percentage deviations with

Here, λ^p denotes the present SCUNC calculations values, λexp represents the experimental values and Δλ/λ stands for the percentage deviations with respect to the experimental value of the corresponding system. (a), experimental data of Robinson [7]; (b), experimental data of Svensson [8]; (c), experimental data of Bartnik et al. [11]; (d), experimental data of Engtröm and Litzén [10]. Wavelengths are in angstroms.

respect to the experimental values of the corresponding system are less than 0.009%. The slight discrepancies can be explained by the fact that the present formalism disregards explicitly mass polarization, relativistic and QED corrections. For the transitions 1 ¹S₀ → np ¹P₁ (n ≥ 3), comparison with the quoted experimental data indicates again good agreements. For these levels, the percentage deviations with respect to the experimental value of the corresponding system are less than 0.05%. Here, the discrepancies may be imputed mainly to mass polarization corrections which are not taken into account in the present calculations. In fact, and as well mentioned by Beiersdorfer et al. [9], the n ≥ 3 levels are less affected by electron-electron interactions, relativistic and QED corrections. Then, for n ≥ 3 states, the ratio m/M (m and M respectively the electron and nuclear masses) becomes important while increasing the Z-charge number. Nevertheless, the present SCUNC semi-empirical formulas may be considered as good representative of experimental data when electron-electron interactions, relativistic and QED corrections are disregarded. In Table 3, the SCUNC predictions for the wavelengths belonging to the 1s^{2 1}S₀ → 1s2p ^1,3P₁ transitions in He-like ions are compared to the ab initio calculations of Acaad et al., [12] using wave function expanded in a triple series of Laguerre polynomials of the perimertric coordinates, the computational results of Safronova et al., [13] applying the MZ code through a perturbation theory based on hydrogen-like functions and with the data of Porter [14] using the plasma simulation code CLOUDY. The overall agreement between the calculations is reasonably gratifying. Here, the |Δλ_theo| differences in wavelengths between the present calculations and the theoretical literature data [12] [13] [15] have never overrun 0.003 Å for the 1s^{2 1}S₀ → 1s2p ¹P₁ resonance line and 0.008 Å for the 1s^{2 1}S₀ → 1s2p ³P₁ intercombination line up to Z = 22. This may point out

Here, λ^p denotes the present SCUNC calculations, λtheo represents the theoretical values and |Δλtheo| stands for the difference in wavelengths between the present calculations and the other theoretical ones (λtheo^a or λtheo^b). (a): calculations of Accad et al., [12], (b): calculations of Safronova et al. [13]; (c): calculations of Porter [14]. Wavelengths are in angstroms.

the good agreement between the calculations. The discrepancies with respect to the accurate ab initio computations are due to the present none-relativistic formalism. Table 4, shows a comparison of the present wavelengths for the forbidden 1s^{2 1}S₀ → 1s2s ³S₁ transitions of He-like systems (Z = 2 - 15) with the NIST compiled data. Excellent agreement is obtained between the SCUNC predictions and the NIST data. Except for Z = 8, the maximum shift in wavelengths with respect to the NIST values is at 0.003 Å. In Table 5, the present theoretical wavelengths for the 1snp ¹P1 → 1s^{2 1}S0 (2 ≤ n ≤ 5) transitions of the helium-like ions up to Z = 9 are compared to the λnrel—nonrelativistic wavelengths values and to the λ_tot—total wavelengths (including mass polarization, relativistic corrections and the Lamb-shift correction for the 1 ¹S level) computed by Accad et al. [12]. For the 1s^{2 1}S₀ → 1s2p ¹P₁ resonance line, the uncertainties between the present calculations and the λ_tot—total wavelengths

*|Δλ| = |λ^SCUNC − |λ^NIST|.

results [12] are less than 0.003 Å. As far as comparison with the λ_nrel—non-relativistic wavelengths values are concerned, it is seen that the uncertainties are about 0.01 Å for Z = 5 - 9. This points out that, the present SCUNC results are most accurate than the λ_nrel—nonrelativistic wavelengths obtained by Accad et al. [12] when increasing the nuclear charge. For n ≥ 3 states, it can also be seen that the present SCUNC wavelengths values are most accurate than that of Accad et al. [12]. Here, the uncertainties with respect to the λ_tot—total wavelengths are less than 0.005 Å for all the entire series considered (Z = 2 - 9) whereas the uncertainties with respect to the λ_nrel—nonrelativistic wavelengths increase up to 0.01 Å for Z = 9. This may point out again that, in the SCUNC formalism, relativistic effects are implicitly incorporated in the fi—screening constants evaluated from experimental data. Besides, it should be mentioned that the λ_tot—total wavelengths equal to 88.3075 Å for the 1s^{2 1}S₀ → 1s3p ¹P₁ transition of Be III may be probably lower as the corresponding high precision measurement is at 88.3140 Å [7] to be compared to the present prediction at 88.3140 Å.

The Screening Constant per Unit Nuclear Charge method has been applied to inaugurate the first spectral lines for the three most intense lines (resonance line 1s^{2 1}S₀ - 1s2p¹P₁ intercombination line 1s^{2 1}S₀ - 1s2p ³P₁ and forbidden line 1s^{2 1}S₀ - 1s2s ³S₁ and for the 1s^{2 1}S₀ - 1snp¹P₁ transitions in the helium isoelectronic sequence. In our knowledge, only the spectral lines of the Hydrogen-like ions have determined empirically in the past. At present hour, the possibilities to calculate easily the most intense lines of helium-like systems in the X-ray range in connection with plasma diagnostic are demonstrated in this work. All the results obtained in the present paper compared very well to various experimental and theoretical literature data. It should be underlined the merit of the SCUNC formalism providing accurate results via simple analytical formulas without needing to use codes of simulation. The accurate results obtained in this work point out the possibilities to investigate highly charged He-positive like ions in the framework of the SCUNC method.

The author declares no conflicts of interest regarding the publication of this paper.

Sakho, I. (2020) Most Intense X-Ray Lines of the Helium Isoelectronic Sequence for Plasmas Diagnostic. Journal of Modern Physics, 11, 487-501. https://doi.org/10.4236/jmp.2020.114031