2 ) . (1)

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

β ( N l n l ; L 2 S + 1 ; π Z ) = k = 1 q f k × ( 1 Z ) k . (2)

where f k = f k ( N l n l ; L 2 S + 1 π ) are parameters to be evaluated empirically.

2.2. Energies for the Ground State

For the ground state, Equations (1) and (2) give

E ( 1 s ; 2 S 1 0 ) = Z 2 ( 1 + { 1 f 1 Z f 2 Z 2 f 3 Z 3 } 2 ) . (3a)

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

E ( 1 s ; 2 S 1 0 ) = Z 2 ( 1 + { 1 0.625085938 Z 0.031315676 Z 2 0.059849712 Z 3 } 2 ) . (3b)

2.3. Spectral Lines of the 11S0 - 1s2p 1P1 Resonance Transition

During the 1s2 1S0 - 1s2p 1, 3P1 transitions, the energy of the system varies as

Δ E = h c λ = E ( 1 s 2 p ; P 1 , 3 1 ) E ( 1 s ; 2 S 1 0 ) . (4)

Using Equations (1) and (3b), we obtain from Equation (4)

For 2 £ Z £ 15

h c λ = Z 2 ( 1 + { 1 0.625085938 Z 0.031315676 Z 2 0.059849712 Z 3 } 2 ) Z 2 ( 1 + 1 4 { 1 f 1 Z f 2 × ( Z Z 0 ) Z 2 f 1 2 × ( Z Z 0 ) 2 × ( Z Z 0 ) Z 3 f 1 2 × ( Z Z 0 ) 2 × ( Z Z 0 ) 2 Z 4 f 1 × ( Z Z 0 ) 2 × ( Z Z 0 ) 2 Z 5 } 2 ) (5a)

In these equations, Z0 and Z 0 denote the nuclear charge of the helium-like systems used in the empirical determination of the f i —screening constants. On the basis of h = 6.626276 × 10 34 J s , c = 2.99792458 × 10 8 m / s and e = 1.602189 × 10 19 C and using for 1s2 1S0 - 1s2p3P1 the experimental wavelengths of He I (Z0 = 2) and that of Li II ( Z 0 = 3 ) respectively 584.3339 Å [19] and 199.280 Å [7], Equation (5a) gives f 1 = 1.004778731 and f 2 = 0.026277861 . We obtain then explicitly

1 λ = Z 2 ( 1 + { 1 0.625085938 Z 0.031315676 Z 2 0.059849712 Z 3 } 2 ) Z 2 ( 1 + 1 4 { 1 1.004778731 1 Z 0.026277861 Z 2 Z 2 0.000690525 ( Z 2 ) 2 × ( Z 3 ) Z 3 0.000690525 ( Z 2 ) 2 × ( Z 3 ) 2 Z 4 0.026277861 ( Z 2 ) 2 × ( Z 3 ) 2 Z 5 } 2 ) × 10973644.9 (5b)

In Equation (5b), wavelengths are expressed in meters (m) and the infinite rydberg energy 1 Ryd = 13.605698 eV is used along with 1 eV =1.602189 × 10−19 J. So Ryd/hc = 10973644.9 (m).

2.4. Spectral Lines of the 1s2 1S0 - 1s2p3P1 Intercombination Transition

Using Equations (1) and (3b), Equation (4) yields for the 1s2 1S0 - 1s2p 3P1 intercombination transition

h c λ = Z 2 ( 1 + { 1 0.625085938 Z 0.031315676 Z 2 0.059849712 Z 3 } 2 ) Z 2 ( 1 + 1 4 { 1 f 1 Z f 2 × ( Z Z 0 ) Z 2 f 2 2 × ( Z Z 0 ) × ( Z Z 0 ) 2 Z 3 f 2 2 × ( Z Z 0 ) 2 × ( Z Z 0 ) 2 Z 4 f 2 2 × ( Z Z 0 ) 2 × ( Z Z 0 ) 3 Z 5 } 2 ) (6a)

Here again, Z0 and Z 0 denote the nuclear charge of the helium-like systems used in the empirical determination of the f i —parameters. For 1s2 1S0 - 1s2p3P1, the experimental wavelengths of He I (Z0 = 2) and that of B IV ( Z 0 = 5 ) are respectively equal to 591.4121Å [19] and 61.0880 Å [9] as quoted in Ref. [12], we obtain from Equation (6a) f 1 = 0.967951498 and f 2 = 0.06781546 . Equation (6a) becomes then

1 λ = Z 2 ( 1 + { 1 0.625085938 Z 0.031315676 Z 2 0.059849712 Z 3 } 2 ) Z 2 ( 1 + 1 4 { 1 0.967951498 1 Z + 0.06781546 Z 2 Z 2 0.004598936 ( Z 2 ) × ( Z 5 ) 2 Z 3 0.004598936 ( Z 2 ) 2 × ( Z 5 ) 2 Z 4 0.004598936 ( Z 2 ) 2 × ( Z 5 ) 3 Z 5 } 2 ) × 10973644.9 (6b)

2.5. Spectral Lines of the 1s2 1S0 - 1s2s 3S1 Forbidden Transitions

For the 1s2 1S0 - 1s2s 3S1 forbidden transitions, the spectral lines are given by

h c λ = Z 2 ( 1 + { 1 0.625085938 Z 0.031315676 Z 2 0.059849712 Z 3 } 2 ) Z 2 ( 1 + 1 4 { 1 f 1 Z f 2 × ( Z Z 0 ) Z 2 f 2 2 × ( Z Z 0 ) × ( Z Z 0 ) Z 3 f 2 2 × ( Z Z 0 ) 2 × ( Z Z 0 ) Z 4 f 2 2 × ( Z Z 0 ) 2 × ( Z Z 0 ) 3 Z 5 f 2 2 × ( Z Z 0 ) × ( Z Z 0 ) 3 Z 6 } 2 ) (7a)

For 1s2 1S0 - 1s2s 3S1, the experimental wavelengths from NIST [20] for He I (Z0 = 2) and for Li II ( Z 0 = 3 ) are respectively equal to 625.563 Å and 210.069 Å. Equation (7a) provides then f 1 = 0.816109425 and f 2 = 0.079252785 . Equation (7a) becomes explicitly

h c λ = Z 2 ( 1 + { 1 0.625085938 Z 0.031315676 Z 2 0.059849712 Z 3 } 2 ) Z 2 ( 1 + 1 4 { 1 0.816109425 Z 0.079252785 × ( Z 2 ) Z 2 0.006281003 × ( Z 2 ) × ( Z 3 ) Z 3 0.006281003 × ( Z 2 ) × ( Z 3 ) Z 3 0.006281003 × ( Z 2 ) 2 × ( Z 3 ) 2 Z 4 0.006281003 × ( Z 2 ) 2 × ( Z 3 ) 3 Z 5 0.006281003 × ( Z 2 ) × ( Z 3 ) 3 Z 6 } 2 ) × 10973644.9 (7b)

2.6. Spectral Lines of the 1s2 1S0 - 1snp1P1 Transitions

Following the same reasoning above, we express from Equations (1) and (2) total energies belonging to the 1snp1P1 levels

E ( 1 s n p ; P 1 1 ) = Z 2 ( 1 + 1 n 2 { 1 f 1 Z ( n 1 ) f 2 Z f 3 × ( Z Z 0 ) Z n 2 2 f 3 × ( Z Z 0 ) 2 × ( Z Z 0 ) Z 3 f 3 × ( Z Z 0 ) 2 × ( Z Z 0 ) 2 Z 4 } 2 ) (8a)

For the 1s2 1S0 - 1snp1P1 transitions, we get

h c λ = Z 2 ( 1 + { 1 0.625085938 Z 0.031315676 Z 2 0.059849712 Z 3 } 2 ) Z 2 ( 1 + 1 n 2 { 1 f 1 Z ( n 1 ) f 2 Z f 3 × ( Z Z 0 ) Z n 2 2 f 3 × ( Z Z 0 ) 2 × ( Z Z 0 ) Z 3 f 3 × ( Z Z 0 ) 2 × ( Z Z 0 ) 2 Z 4 } 2 ) (8b)

For 1s2 1S0 - 1s3p 3P1 and 1s2 1S0 - 1s4p 3P1 transitions, the corresponding experimental wavelengths of Li II (Z0 = 3) are respectively equal to 178.014 Å and 171.575 Å [7]. In addition, for Be III ( Z 0 = 4 ), the wavelength for to the 1s2 1S0 - 1s3p 3P1 transition is 88.314 Å [7]. Using these wavelengths, we get from Equation (8b) f 1 = 0.011679205 , f 2 = 1.003675341 , and f 3 = 0.008177868 . The spectral lines belonging to the 1s2 1S0 - 1snp1P1 transitions is then in the shape.

1 λ = Z 2 ( 1 + { 1 0.625085938 Z 0.031315676 Z 2 0.059849712 Z 3 } 2 ) Z 2 ( 1 + 1 n 2 { 1 0.011679205 1 Z ( n 1 ) 1.003675341 1 Z 0.008177868 Z 3 Z n 2 2 0.008177868 ( Z 3 ) 2 × ( Z 4 ) Z 3 0.008177868 ( Z 3 ) 2 × ( Z 4 ) 2 Z 4 } 2 ) × 10973644.9 (8c)

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 the framework of the SCUNC formalism, all the relativistic corrections in many electron systems are incorporated in the β-parameters. To enlighten this point, let us move on considering the main relativistic terms in the Hamiltonian operator of Q-electron systems. For Q-electron systems, the Hamiltonian can be expressed as follows

H = H 0 + W . (9)

In this expression, H0 denotes the nonrelativistic Hamiltonian and W is the sum of the perturbation operators which includes mainly correction to kinetic energy (Wkin), the Darwin term (WD), mass polarization (WM), spin-orbit corrections (Wso), spin-other orbit corrections (Wsoo) and spin-spin corrections (Wss). For Q-electron systems, the non-relativistic Hamiltonian and the perturbation operators are explicitly the following

H 0 = i = 1 Q [ 1 2 i 2 Z r i ] + i , j = 1 i j Q 1 r i j ; W kin = α 2 8 i = 1 Q p i 4 ; W D = 3 π α 2 2 i = 1 Q δ ( r i ) .

W M = 1 M i , j = 1 i j Q i j ; W so = Z 2 c 2 i = 1 Q l i s i r i 3 ;

W soo = 1 2 c 2 i , j = 1 i j Q [ 1 r i j 3 ( r i r j ) × p i ] ( s i + 2 s j ) .

W ss = 1 c 2 i , j = 1 j > i Q 1 r i j 3 [ s i s j 3 ( s i r i j ) ( s j r i j ) r i j 2 ] .

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

E = E 0 + w . (9b)

with

w = W kin + W D + W M + W so + W soo + W ss . (9c)

For a-given Nl1, nl2 configuration of He-like ions where N, n, and l1, l2, are respectively principal and orbital quantum numbers, the total energy is given by

E = Z 2 N 2 Z 2 n 2 [ 1 β ( N l 1 n l 2 ; L 2 S + 1 ; π Z ) ] 2 . (9d)

Developing Equation (9d), we obtain

E = Z 2 N 2 Z 2 n 2 + Z 2 n 2 β ( N l 1 n l 2 ; L 2 S + 1 ; π Z ) [ 2 β ( N l 1 n l 2 ; L 2 S + 1 ; π Z ) ] . (9e)

Equation (9e) can be rewritten in the form

E = Z 2 N 2 Z 2 n 2 + i = 1 2 Z 2 ν i 2 β i × [ 2 β i ] .

This equation can be expressed in the same shape than Equation (9b)

E = E 0 + w .

where

{ E 0 = Z 2 N 2 Z 2 n 2 w = i = 1 2 Z 2 ν i 2 β i × [ 2 β i ] (10)

Using (9c) and the last equation in (10), we find

i = 1 2 Z 2 ν i 2 β i × [ 2 β i ] = W kin + W D + W M + W so + W soo + W ss . (11)

Equation (11) indicates clearly that, in the framework of the SCUNC-formalism, all the relativistic corrections are incorporated in the β-screening constants per unit nuclear charge. In the structure of the independent particles model disregarding all the relativistic effects, total energy is given by E0. Subsequently w = 0. This involves automatically β = 0. Then, all relativistic effects are accounted implicitly in general Equation (1) via the β-parameters expanded in inverse powers of Z as shown by Equation (2) where the f k = f k ( N l n l ; L 2 S + 1 π ) —screening constants are evaluated empirically using experimental data incorporating all the relativistic effects and all electrons-electrons effects in many electron systems.

3. Results and Discussions

The present SCUNC wavelengths predictions for the wavelengths belonging to the 1s2 1S0 → 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 (1s2 1S0 → 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

Table 1. Present wavelengths (λ, in Å) of the 1s2 1S0 → 1snp 1P1 transitions in He-like (Z = 3 - 15) ions.

Table 2. Theoretical and experimental wavelengths of the 1 1S0 → np 1P1 (1s2 1S0 → 1snp 1P1) transitions of helium-like ions up to Z = 8.

Here, λp 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 1s2 1S0 → 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 |Δλtheo| differences in wavelengths between the present calculations and the theoretical literature data [12] [13] [15] have never overrun 0.003 Å for the 1s2 1S0 → 1s2p 1P1 resonance line and 0.008 Å for the 1s2 1S0 → 1s2p 3P1 intercombination line up to Z = 22. This may point out

Table 3. Theoretical wavelengths for 1s2 1S0 → 1s2p 1,3P1 for He-like ions (2 ≤ Z ≤ 22).

Here, λp 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 (λtheoa or λtheob). (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 1s2 1S0 → 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 → 1s2 1S0 (2 ≤ n ≤ 5) transitions of the helium-like ions up to Z = 9 are compared to the λnrel—nonrelativistic wavelengths values and to the λtot—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 1s2 1S0 → 1s2p 1P1 resonance line, the uncertainties between the present calculations and the λtot—total wavelengths

Table 4. Comparison of the SCUNC predictions with the NIST data the wavelengths belonging to the forbidden 1s2 1S0 → 1s2s 3S1 transitions in He-like (Z = 2 - 15) systems. Wavelengths are in angstroms.

*|Δλ| = |λSCUNC − |λNIST|.

Table 5. Theoretical wavelengths for the 1s2 1S0 → 1snp 1P1 (2 ≤ n ≤ 5) transitions in He-like (Z = 3 - 9) ions. Here, λ denotes the present SCUNC calculations, λnrel denotes the nonrelativistic wavelengths and λtot the theoretical wavelengths of Accad et al. [12] including mass polarization, relativistic corrections and the Lamb-shift correction for the 1S level. Wavelengths are in angstroms.

results [12] are less than 0.003 Å. As far as comparison with the λnrel—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 λnrel—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 λtot—total wavelengths are less than 0.005 Å for all the entire series considered (Z = 2 - 9) whereas the uncertainties with respect to the λnrel—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 λtot—total wavelengths equal to 88.3075 Å for the 1s2 1S0 → 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.

4. 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 1s2 1S0 - 1s2p1P1 intercombination line 1s2 1S0 - 1s2p 3P1 and forbidden line 1s2 1S0 - 1s2s 3S1 and for the 1s2 1S0 - 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.

Cite 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. doi: 10.4236/jmp.2020.114031.
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