_{1}

^{*}

We report accurate wavelengths for the three most intense lines (resonance line: 1s
^{2}
^{1}S0 - 1s2p
^{1}P1, intercombination line: 1s
^{2 1}S0 - 1s2p
^{ 3}P1 and forbidden line: 1s
^{2}
^{1}S0 - 1s2s
^{3}S1) along with wavelengths for the 1s
^{2 1}S0 - 1s
*n*p
^{1}P
^{1} and
^{1}S0 - 1s
*n*p
^{3}P2 (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 0.004
Å
. A host of new data listed in this paper may be of interest in astrophysical and laboratory plasmas diagnostic.

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 [^{3}P level are controlled by collisional excitation from the 2^{3}S level [_{crtit}): (n_{crit} C ~ A, C being the rate coefficient for collisional excitation 2^{3}S → 2^{3}P and A denotes the radiative transition probability of 2^{3}S → 1^{1}S), the 2^{3}S upper level of the forbidden line becomes to be depleted by collision to the 2^{3}P 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 2^{3}S → 2^{3}P 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 [

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 [^{2} ^{1}S_{0} - 1snp ^{1}P_{1} 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 [^{2} ^{1}S_{0} - 1snp ^{1}P_{1} (n = 3 - 5) transitions. Furthermore, Engtröm and Litzén [^{2} ^{1}S - 1snp ^{1}P (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. [^{1}P - 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. [^{1}P - 1s^{21}S (n = 2 - 5) transitions. In addition, Safronova et al. [^{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 [^{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 [

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.

In the framework of Screening Constant per Unit Nuclear Charge formalism, total energy of ( N l , n l ′ ) L 2 S + 1 π excited states are expressed in the form (in rydberg units)

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

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 [

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

During the 1s^{2} ^{1}S_{0} - 1s2p ^{1, 3}P_{1} 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, Z_{0} 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 1s^{2} ^{1}S_{0} - 1s2p^{3}P_{1} the experimental wavelengths of He I (Z_{0} = 2) and that of Li II ( Z ′ 0 = 3 ) respectively 584.3339 Å [

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).

Using Equations (1) and (3b), Equation (4) yields for the 1s^{2} ^{1}S_{0} - 1s2p ^{3}P_{1} 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, Z_{0} and Z ′ 0 denote the nuclear charge of the helium-like systems used in the empirical determination of the f ″ i —parameters. For 1s^{2} ^{1}S_{0} - 1s2p^{3}P_{1}, the experimental wavelengths of He I (Z_{0} = 2) and that of B IV ( Z ′ 0 = 5 ) are respectively equal to 591.4121Å [

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)

For the 1s^{2} ^{1}S_{0} - 1s2s ^{3}S_{1} 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 1s^{2} ^{1}S_{0} - 1s2s ^{3}S_{1}, the experimental wavelengths from NIST [_{0} = 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)

Following the same reasoning above, we express from Equations (1) and (2) total energies belonging to the 1snp^{1}P_{1} 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 1s^{2} ^{1}S_{0} - 1snp^{1}P_{1} 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 1s^{2} ^{1}S_{0} - 1s3p ^{3}P_{1} and 1s^{2} ^{1}S_{0} - 1s4p ^{3}P_{1} transitions, the corresponding experimental wavelengths of Li II (Z_{0} = 3) are respectively equal to 178.014 Å and 171.575 Å [^{2} ^{1}S_{0} - 1s3p ^{3}P_{1} transition is 88.314 Å [^{2} ^{1}S_{0} - 1snp^{1}P_{1} 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 [

H = H 0 + W . (9)

In this expression, H_{0} denotes the nonrelativistic Hamiltonian and W is the sum of the perturbation operators which includes mainly correction to kinetic energy (W_{kin}), the Darwin term (W_{D}), mass polarization (W_{M}), spin-orbit corrections (W_{so}), spin-other orbit corrections (W_{soo}) and spin-spin corrections (W_{ss}). 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 Nl_{1}, nl_{2} configuration of He-like ions where N, n, and l_{1}, l_{2}, 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 E_{0}. 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.

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 ^{1}S_{0} → np ^{1}P_{1} (1s^{2} ^{1}S_{0} → 1snp ^{1}P_{1}) transitions of helium-like ions up to Z = 8. The present SCUNC calculations values, are compared to the experimental data of Robinson [^{1}S_{0} → 2p ^{1}P_{1} transition, it is seen that the current SCUNC results compared very well to the experimental values. Here, the Δλ/λ percentage deviations with

1S-n ^{1}P | Li II λ | Be III λ | B IV λ | C V λ | N VI λ | O VII λ | F VIII λ | Ne IX λ | Na X λ | Mg XI λ | Al XII λ | Si XIII λ | P XIV λ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1S-3^{1}P 1S-4^{1}P 1S-5^{1}P 1S-6^{1}P 1S-7^{1}P 1S-8^{1}P 1S-9^{1}P 1S-10^{1}P 1S-11^{1}P 1S-12^{1}P 1S-13^{1}P 1S-14^{1}P 1S-15^{1}P 1S-16^{1}P 1S-17^{1}P 1S-18^{1}P 1S-19^{1}P 1S-20^{1}P 1S-21^{1}P 1S-22^{1}P 1S-23^{1}P 1S-24^{1}P 1S-25^{1}P | 178.0140 171.5750 168.7422 167.2401 166.3466 165.7714 165.3792 165.0997 164.8934 1647369 164.6153 164.5187 164.4410 164.3775 164.3248 164.2808 164.2435 164.2117 164.1843 164.1605 164.1399 164.1217 164.1057 | 88.3140 84.7502 83.1934 82.3706 81.8821 81.5679 81.3539 81.2015 81.0890 81.0037 80.9374 80.8849 80.8425 80.8079 80.7793 80.7553 80.7349 80.7176 80.7027 80.6898 80.6785 80.6686 80.6599 | 52.6852 50.4334 49.4536 48.9368 48.6302 48.4332 48.2990 48.2035 48.1331 48.0796 48.0381 48.0052 47.9787 47.9570 47.9391 47.9240 47.9113 47.9005 47.8911 47.8831 47.8760 47.8698 47.8644 | 34.9749 33.4271 32.7553 32.4014 32.1916 32.0568 31.9651 31.8997 31.8516 31.8150 31.7867 31.7642 31.7461 31.7312 31.7190 31.7087 31.7000 31.6926 31.6862 31.6807 31.6759 31.6717 31.6679 | 24.9012 23.7736 23.2850 23.0278 22.8754 22.7775 22.789 22.6635 22.6285 22.6020 22.5814 22.5651 22.5519 22.5412 22.5323 22.5248 22.5185 22.5131 22.5085 22.5045 22.5010 22.4979 22.4952 | 18.6283 17.7709 17.3999 17.2047 17.0891 17.0148 16.9643 16.9284 16.9018 16.8817 16.8661 16.8537 16.8438 16.8356 16.8289 16.8232 16.8184 16.8144 16.8109 16.8078 16.8052 16.8028 16.8008 | 14.4588 13.7853 13.4942 13.348 13.2504 13.1922 13.1525 13.1243 13.836 13.0878 13.0756 13.0659 13.0580 13.0517 13.0464 13.0419 13.0382 13.0350 13.0322 13.0299 13.0278 13.0260 13.0243 | 11.5474 11.5474 11.0046 10.7701 10.5738 10.5270 10.4951 10.4724 10.4557 10.4430 10.4331 10.4253 10.4190 10.4139 10.4096 10.4061 10.4030 10.4005 10.3983 10.3963 10.3947 10.3932 10.3919 | 9.4344 9.4344 8.9877 8.7949 8.6335 8.5950 8.5688 8.5501 8.5364 8.5259 8.5178 8.5114 8.5063 8.5020 8.4985 8.4956 8.4931 8.4910 8.4892 8.4876 8.4862 8.4850 8.4840 | 7.8524 7.8524 7.4785 7.3171 7.1821 7.1499 7.1280 7.1124 7.1009 7.0922 7.0854 7.0801 7.0757 7.0722 7.0693 7.0668 7.0648 7.0630 7.0615 7.0602 7.0590 7.0580 7.0571 | 6.6374 6.6374 6.3199 6.1829 6.0683 6.0409 6.0223 6.0091 5.9993 5.9919 5.9862 5.9816 5.9780 5.9750 5.9725 5.9704 5.9687 5.9672 5.9659 5.9648 5.9638 5.9629 5.9622 | 5.6841 5.6841 5.4110 5.2933 5.1948 5.1713 5.1553 5.1440 5.1356 5.1292 5.1243 5.1204 5.1172 5.1147 5.1125 5.1107 5.1092 5.1079 5.1068 5.1059 5.1050 5.1043 5.1036 | 4.9222 4.9222 4.6850 4.5827 4.4972 4.4768 4.4629 4.4531 4.4458 4.4403 4.4360 4.4326 4.4298 4.4276 4.4258 4.4242 4.4229 4.4218 4.4208 4.4200 4.4193 4.4186 4.4181 |

1S-n ^{1}P | Li II | Be III | B IV | ||||||
---|---|---|---|---|---|---|---|---|---|

λ^{p} | λexp^{(a)} | Δλ/λ | λ^{p} | λexp^{(a) } | Δλ/λ | λ^{p} | λexp^{(a) } | Δλ/λ | |

1S-2^{1}P | 199.2800 | 199.280 | 0.0000% | 80.2522 | 80.254 | 0.0018% | 60.390 | 60.313 | 0.0033% |

1S-3^{1}P | 178.0140 | 178.014 | 0.0000% | 88.3140 | 88.314 | 0.0000% | 52.6852 | 52.679 | 0.098% |

1S-4^{1}P | 171.5750 | 171.575 | 0.0000% | 84.7502 | 84.758 | 0.0092% | 50.4334 | 50.435 | 0.0032% |

1S-5^{1}P | 168.7421 | 83.1934 | 83.202 | 0.083% | 49.4536 | 49.456 | 0.0048% | ||

1S-6^{1}P | 167.2401 | 82.3706 | 82.377 | 0.0198% | 48.9368 | ||||

1S-7^{1}P | 166.3466 | 81.8821 | 81.891 | 0.089% | 48.6302 | ||||

1S-8^{1}P | 165.7714 | 81.5679 | 48.4332 | ||||||

1S-9^{1}P | 165.3792 | 81.3539 | 48.2990 | ||||||

1S-8^{1}P | 165.0997 | 81.2015 | 48.2035 |

1S-n ^{1}P | CV | NVI | OVII | ||||||
---|---|---|---|---|---|---|---|---|---|

λnr^{p } | λexp^{(a, b) } | Δλ/λ* | λnr^{p } | λexp^{(d) } | Δλ/λ | λnr^{p } | λexp^{(a, c)} | ||

1S-2^{1}P | 40.2647 | 40.268^{b} | 0.0082% | 28.7857 | 28.787 | 0.0045% | 21.6021 | 21.602^{a } | 0.0005% |

1S-3^{1}P | 34.9749 | 34.973^{a, b } | 0.0054% | 24.9012 | 24.898 | 0.0128% | 18.6283 | ^{ } | |

1S-4^{1}P | 33.4271 | 33.426^{a, b } | 0.0033% | 23.7736 | 23.771 | 0.089% | 17.7709 | ^{ } | |

1S-5^{1}P | 32.7553 | 32.754^{a, b } | 0.0039% | 23.2850 | 23.281 | 0.0172% | 17.3999 | ^{ } | |

1S-6^{1}P | 32.4014 | 32.399^{b} | 0.0074% | 23.0278 | 23.024 | 0.0165% | 17.2047 | 17.199^{c } | 0.0331% |

1S-7^{1}P | 32.1916 | 22.8754 | 17.0891 | 17.083^{c } | 0.0357% | ||||

1S-8^{1}P | 32.0568 | 22.7775 | 17.0148 | 17.008^{c } | 0.0399% | ||||

1S-9^{1}P | 31.9651 | 22.789 | 16.9643 | 16.957^{c } | 0.0431% | ||||

1S-8^{1}P | 31.8997 | 22.6635 | 16.9284 | 16.924^{c } | 0.0230% |

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 [

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. [^{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., [_{theo}| differences in wavelengths between the present calculations and the theoretical literature data [^{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

Z | 1s^{2} ^{1}S_{0} → 1s2p ^{1}P_{1} (resonance line: r) | 1s^{2} ^{1}S_{0} → 1s2p ^{3}P_{1} (intercombinaison line: i) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

λp | λtheo^{a, b } | λtheo^{c } | |Δλtheo|^{a, b } | |Δλtheo|^{c } | λp | λtheo^{a, b } | λtheo^{c } | |Δλtheo|^{a, b } | |Δλtheo|^{c } | |

2 | 584.3339 | 584.3343^{a } | 0.0004 | 591.4121 | 591.499^{a } | 0.0002 | ||||

3 | 199.2800 | 199.2791^{a } | 0.0009 | 202.2252 | ||||||

4 | 80.2522 | 80.2535^{a } | 0.0013 | 81.6677 | ||||||

5 | 60.390 | 60.3135^{a } | 0.0020 | 61.0880 | 61.0882^{a } | 0.0002 | ||||

6 | 40.2647 | 40.2671^{a } | 40.2680 | 0.0024 | 0.0033 | 40.7302 | 40.7299^{a } | 40.7310 | 0.0003 | 0.0008 |

7 | 28.7857 | 28.7867^{a } | 28.7870 | 0.0010 | 0.0013 | 29.0818 | 29.0840^{a } | 29.0840 | 0.0022 | 0.0022 |

8 | 21.6021 | 21.6012^{a } | 21.6020 | 0.0009 | 0.0001 | 21.7988 | 21.8033^{a } | 21.8070 | 0.0008 | 0.0045 |

9 | 16.8088 | 16.8061^{a } | 16.8070 | 0.0027 | 0.0018 | 16.9438 | 16.9496^{a } | 16.9470 | 0.0045 | 0.0082 |

8 | 13.4514 | 13.4470 | 0.0044 | 13.5464 | 13.5530 | 0.0066 | ||||

9 | 9.0050 | 9.0030 | 0.0020 | 9.0880 | 9.0830 | 0.0060 | ||||

12 | 9.1689 | 9.1688 | 0.0001 | 9.2310 | 9.2312 | 0.0062 | ||||

13 | 7.7568 | 7.7573 | 0.0005 | 7.8044 | 7.8070 | 0.0026 | ||||

14 | 6.6475 | 6.6480 | 0.0005 | 6.6847 | 6.6883 | 0.0036 | ||||

15 | 5.7701 | 5.0386^{b } | 5.7898 | 5.0667^{b } | ||||||

16 | 5.0387 | 5.0387 | 0.0002 | 0.0000 | 5.0667 | 5.0665 | 0.0000 | 0.0002 | ||

17 | 4.4445 | 4.4445^{b } | 0.0002 | 4.4682 | 4.4681^{b } | 0.0001 | ||||

18 | 3.9491 | 3.9492^{b } | 3.9488 | 0.0001 | 0.0003 | 3.9694 | 3.9695^{b } | 3.9691 | 0.0001 | 0.0003 |

19 | 3.5318 | 3.5319^{b } | 0.0002 | 3.5493 | ||||||

20 | 3.1771 | 3.1772^{b } | 3.1772 | 0.0001 | 0.0001 | 3.1924 | 3.1928^{b } | 3.1928 | 0.0004 | 0.0004 |

21 | 2.8731 | 2.8731^{b } | 0.0000 | 2.8866 | 2.8871^{b } | 0.0005 | ||||

22 | 2.684 | 2.685^{b } | 0.0001 | 2.6226 | 2.6230^{b } | 0.0004 |

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., [

the good agreement between the calculations. The discrepancies with respect to the accurate ab initio computations are due to the present none-relativistic formalism. ^{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 ^{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. [^{2} ^{1}S_{0} → 1s2p ^{1}P_{1} resonance line, the uncertainties between the present calculations and the λ_{tot}—total wavelengths

Z | λSCUNC | λNIST | |Δλ|* |
---|---|---|---|

2 | 625.563 | 625.563 | 0.000 |

3 | 210.069 | 210.069 | 0.000 |

4 | 104.547 | 104.548 | 0.001 |

5 | 62.439 | 62.440 | 0.001 |

6 | 41.469 | 41.472 | 0.003 |

7 | 29.531 | 29.534 | 0.003 |

8 | 22.094 | 22.101 | 0.007 |

9 | 17.149 | - | |

10 | 13.696 | 13.699 | 0.003 |

11 | 11.190 | 11.192 | 0.002 |

12 | 9.313 | - | |

13 | 7.872 | - | |

14 | 6.741 | 6.740 | 0.001 |

15 | 5.838 | - |

*|Δλ| = |λ^{SCUNC} − |λ^{NIST}|.

System | Transition | Theory | Comparison | |||
---|---|---|---|---|---|---|

Present λ | Accad et al. λnrel | Accad et al. λtot | |λ – λnrel| | |λ – λtot| | ||

1S-2^{1}P | 199.2800 | 199.2813 | 199.2791 | 0.0013 | 0.0009 | |

Li II | 1S-3^{1}P | 178.0140 | 178.0162 | 178.0143 | 0.0022 | 0.0003 |

1S-4^{1}P | 171.5750 | 171.5776 | 171.5757 | 0.0026 | 0.0007 | |

1S-5^{1}P | 168.7421 | |||||

1S-2^{1}P | 80.2522 | 80.2600 | 80.2535 | 0.0078 | 0.0013 | |

Be III | 1S-3^{1}P | 88.3140 | 88.3134 | 88.3075 | 0.0006 | 0.0065 |

1S-4^{1}P | 84.7502 | 84.7588 | 84.7532 | 0.0086 | 0.0030 | |

1S-5^{1}P | 83.1934 | 83.2044 | 83.1989 | 0.090 | 0.0055 | |

1S-2^{1}P | 60.390 | 60.3224 | 60.3135 | 0.094 | 0.0025 | |

B IV | 1S-3^{1}P | 52.6852 | 52.6876 | 52.6800 | 0.0024 | 0.0052 |

1S-4^{1}P | 50.4334 | 50.4408 | 50.4335 | 0.0074 | 0.0001 | |

1S-5^{1}P | 49.4536 | 49.4621 | 49.4549 | 0.0085 | 0.0013 | |

1S-2^{1}P | 40.2647 | 40.2774 | 40.2671 | 0.0127 | 0.0024 |

C V | 1S-3^{1}P | 34.9749 | 34.9811 | 34.9723 | 0.0062 | 0.0026 |
---|---|---|---|---|---|---|

1S-4^{1}P | 33.4271 | 33.4343 | 33.4259 | 0.0072 | 0.0012 | |

1S-5^{1}P | 32.7553 | 32.7622 | 32.7540 | 0.0069 | 0.0013 | |

1S-2^{1}P | 28.7857 | 28.7980 | 28.7867 | 0.0123 | 0.0010 | |

N VI | 1S-3^{1}P | 24.9012 | 24.9098 | 24.9002 | 0.0086 | 0.0010 |

1S-4^{1}P | 23.7736 | 23.7806 | 23.7714 | 0.0070 | 0.0022 | |

1S-5^{1}P | 23.2850 | |||||

1S-2^{1}P 1S-3^{1}P | ||||||

21.6021 | 21.6133 | 21.6012 | 0.092 | 0.0009 | ||

O VII | 18.6283 | 18.6381 | 18.6280 | 0.0098 | 0.0003 | |

1S-4^{1}P | 17.7709 | 17.7777 | 17.7680 | 0.0068 | 0.0029 | |

1S-5^{1}P | 17.3999 | 17.4051 | 17.3957 | 0.0052 | 0.0042 | |

1S-2^{1}P | 16.8088 | 16.8188 | 16.8061 | 0.080 | 0.0027 | |

F VIII | 1S-3^{1}P | 14.4588 | 14.4690 | 14.4584 | 0.082 | 0.0004 |

1S-4^{1}P | 13.7853 | |||||

1S-5^{1}P | 13.494 2 |

results [_{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. [_{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 [

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.

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