Most Intense X-Ray Lines of the Helium Isoelectronic Sequence for Plasmas Diagnostic

We report accurate wavelengths for the three most intense lines (resonance line: 1s2 1S0 - 1s2p 1P1, intercombination line: 1s2 1S0 - 1s2p 3P1 and forbidden line: 1s2 1S0 - 1s2s 3S1) along with wavelengths for the 1s2 1S0 - 1snp1P1 and 1S0 - 1snp3P2 (2 ≤ n ≤ 25) transitions in He-like systems (Z = 2 – 13). The first spectral lines that belong to the above transitions are established in the framework of the Screening Constant per Unit Nuclear Charge method. The results obtained agree excellently with various experimental and theoretical literature data. The uncertainties in wavelengths between the present calculations and the available literature data are less than 0.004A. A host of new data listed in this paper may be of interest in astrophysical and laboratory plasmas diagnostic.


Introduction
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 S 0 -1s2p 1 P 1 ), the intercombination lines (x + y) (or i: 1s 2 1 S 0 -1s2p 3 P 2, 1 ) and the forbidden line z (or f: 1s 2 1 S 0 -1s2s 3 S 1 ). 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 ~ 10 8 -10 13 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 ioni-struct 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 1 P -1s 21 S (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 -1s 2 n'l', n, n' = 2, 3) and (1snp 1, 3 P -1s 2 , n = 2, 3 and 1s2s 1, 3 S -1s 2 ) 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 S 0 -1s2p 1 P 1 and intercombination line: 1s 2 1 S 0 -1s2p 3 P 2, 1 along the 1s 2 1 S 0 -1snp 1 P 1 (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.

Brief Description of the SCUNC Formalism
In the framework of Screening Constant per Unit Nuclear Charge formalism, total energy of ( ) 2 1 , S N n L π + ′   excited states are expressed in the form (in rydberg units) In this equation, the principal quantum numbers N and n, are respectively for the inner and the outer electron of He-isoelectronic series. In this equation, the β-parameters are screening constant by unit nuclear charge expanded in inverse powers of Z and given by ( ) ; where ( )   are parameters to be evaluated empirically. 1 ; Using the experimental total energy of He I, Li II and Be III respectively (in eV) −79.01 [17], −198.09 [18] and −371.60 [18], the screening constants in Equation (4) are evaluated by use of the infinite rydberg energy 1 Ryd = 13.605698 eV. We find then ( )
Before presenting and discussing the results obtained in this work, let us first move on explaining how electron-electrons and relativistic effects are accounted in the present SCUNC formalism. As mentioned previously [16] In these expressions, α denotes the fine structure constant and M is the nuclear mass of the Q-electron systems. The energy value of the Hamiltonian (9a) is in the form This equation can be expressed in the same shape than Equation (9b) Using (9c) and the last equation in (10) Equation (11)

Results and Discussions
The present SCUNC wavelengths predictions for the wavelengths belonging to the 1s 2 1 S 0 → 1snp 1 P1 (3 ≤ n ≤ 13) transitions in He-like (Z = 3 -38) ions are quoted in Table 1. 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 1 S 0 → np 1 P 1 (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 S 0 → 1s2p 1,3 P 1 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 S 0 → 1s2p 1 P 1 resonance line and 0.008 Å for the 1s 2 1 S 0 → 1s2p 3 P 1 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 S 0 → 1s2s 3 S 1 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 1 P1 → 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 1 S level) computed by Accad et al. [12]. For the 1s 2 1 S 0 → 1s2p 1 P 1 resonance line, the uncertainties between the present calculations and the λ tot -total wavelengths  results [12] are less than 0.003 Å. As far as comparison with the λ nrel -nonrelativistic 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 S 0 → 1s3p 1 P 1 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.

Conclusion
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 S 0 -1s2p 1 P 1 intercombination line 1s 2 1 S 0 -1s2p 3 P 1 and forbidden line 1s 2 1 S 0 -1s2s 3 S 1 and for the 1s 2 1 S 0 -1snp 1 P 1 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.

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