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Photoionization Study of Cl II, Ar II and Kr II Ions Using the Modified Atomic Orbital Theory ()

^{1}, Mamadou Diouldé Ba

^{1}, Jean Kouhissoré Badiane

^{1}, Momar Talla Gning

^{1}, Malick Sow

^{2}, Ibrahima Sakho

^{1*}

^{2}3p

^{3}(

^{2}D

_{5/2})]

*n*d and [3s

^{2}3p

^{3}(

^{2}P

_{3/2})]

*n*d, Ar II-3s

^{2}3p

^{4}(

^{1}D

_{2})

*ns*,

*n*d and of the Kr II [4s

^{2}4p

^{4}(

^{1}D

_{2})]

*ns*,

*n*d and 4s

^{2}4p

_{4}(

^{3}P

_{2},

^{3}P

_{1})]

*ns*, 4s

^{2}4p

^{4}(

^{3}D

_{2})]

*ns*,

*n*d and 4s

^{2}4p

^{4}(

^{3}D

_{2},

^{1}S

_{0})]

*ns*,

*n*d Rydberg series are reported. Natural widths of the Ar II-[3s

^{2}3p

^{4}(

^{1}D

_{2})]

*ns*,

*n*d series are also reported. Calculations are done in the framework of the Modified Atomic Orbital Theory (MAOT). Excellent agreements are obtained with available theoretical and experimental data. High lying accurate resonance energies up to

*n*= 40 are tabulated. The possibility to use the MAOT formalism report rapidly with an excellent accuracy the position of the excitation resonances as well as their width within simple analytical formulae is demonstrated.

Share and Cite:

*Journal of Modern Physics*,

**9**, 2594-2622. doi: 10.4236/jmp.2018.914162.

1. Introduction

In many astrophysical systems such as stars and nebulae, the main process governing light-atomic species interaction is Photoionization. Of great important ions interesting to investigate are Cl II (Cl^{+}), Ar II (Ar^{+}) and Kr II (Kr^{+}) ions. As far as Cl II is concerned, its interest is connected with it abundance in photoionized astrophysical objects. In the past, various studies have indicated the great importance of Cl II ions abundances for understanding extragalactic HII regions [1] . In addition, emission lines of Cl II ions have been observed in the spectra of the Io torus [2] and in the optical spectra of planetary nebulae NGC 6741 and IC 5117 [3] . In a very recent past, Hernández et al., [4] measured at the Advanced Light Source at Lawrence Berkeley National Laboratory absolute photoionization cross-sections for the of Cl II ions using the merged beams photon-ion technique at a photon energy resolution of 15 meV in the energy range 19 - 28 eV. Using the Dirac-Coulomb R-matrix (DCR) method, McLaughlin, [5] performed calculations in the same photon energy range that in the ALS experiments [4] to assign and identify the resonance series in the ALS spectra of the Cl II ions. In these experimental works, the 3s^{2}3p^{3}nd states have been identified in the Cl II spectra as the prominent Rydberg series belonging to the 3p → nd transitions. The weaker 3s^{2}3p^{3}ns Rydberg series, identified as 3p → ns transitions and window resonances 3s3p^{4}(^{4}P) np features, due to 3s → np transitions, have also been found in the spectra [5] . Besides, one of the important elements to study is Argon present in several astrophysical systems. An overabundance of the argon element in the spectra of X-rays of yellow supernovas was revealed by the satellite Chandra [6] dedicated to analyze the stellar object spectra. Furthermore, spectral rays of the argon element were observed in the optical spectra of planetary stars and nebulae [7] [8] . These few examples show the importance of the photoionization study of the argon element from the perspective of astrophysics. In a recent past, Covington et al., [9] performed the first experimental measurements of the photoionization cross-section of the Ar II ion. These authors also determined the resonance energies and natural widths related to the dominant Rydberg series 3s^{2}3p^{4}(^{1}D_{2})ns, nd and 3s^{2}3p^{4}(^{1}S_{0})ns, nd in the emission spectra of the Ar II ions. These energies were relatively measured at the metastable state Ar II (^{2}P°_{1/2}) and at the ground state Ar II (^{2}P°_{3/2}). As far as Krypton is concerned, it is also an element of major importance for diagnosing stellar plasmas such as stars and planetary nebulae as well as for diagnosing laboratory plasmas such as those obtained by inertial fusion. In a recent past, Hinojosa et al., [10] experimentally studied the photoionization of the Kr II ion at ALS at Berkley in the photonic energy range of 23 - 39 eV. In the photoionization spectra, these authors observed several Rydberg series, including the Kr II [4s^{2}4p^{4}(^{1}D_{2})] ns, nd series converging toward the excitation threshold Kr II [4s^{2}4p^{4}(^{1}D_{2})]. Very recently, Sakho, [11] applied the Screening constant by unit nuclear charge (SCUNC) formalism to report precise data belonging to various Rydberg series of Ar II and Kr II as observed in the works of Covington et al., [9] for Ar II and of Hinojosa et al., [10] for Kr II. In the present study, we use the Modified atomic orbital theory [12] [13] [14] [15] [16] to report accurate high lying resonance energies of the Cl I-[3s^{2}3p^{3}(^{2}D_{5/2})]nd and [3s^{2}3p^{3}(^{2}P_{3/2})]nd, Ar II-3s^{2}3p^{4}(^{1}D_{2})ns, nd and of the Kr II [4s^{2}4p^{4}(^{1}D_{2})]ns, nd and 4s^{2}4p^{4}(^{3}P_{2},^{3}P_{1})]ns, 4s^{2}4p^{4}(^{3}D_{2})]ns, nd and 4s^{2}4p^{4}(^{3}D_{2}, ^{1}S_{0})]ns, nd Rydberg series reported. Natural widths of the Ar II-[3s^{2}3p^{4}(^{1}D_{2})] ns, nd series are also reported. In Section 2, a brief description of the MAOT formalism is given. The results are present in Section 3. Section 4 concludes the study.

2. Theory

2.1. Brief Description of the Modified Atomic Orbital Theory

In the framework of the modified atomic orbital theory (MAOT), the total energy of a (νl)-given orbital is expressed in the form in Rydberg units

$E\left(\nu l\right)=-\frac{{\left[Z-\sigma \left(l\right)\right]}^{2}}{{\nu}^{2}}$ (1)

In Equation (1), σ is the screening constant relative to the electron occupying the (νl)-orbital, l denotes the orbital quantum number, ν stands for the principal quantum number and Z represents the atomic number. In general, the doubly excited states (DES) in two electron systems are labelled as $\left(Nl,n{l}^{\prime}\right){}^{2S+1}L{}^{\pi}$ . In this notation, N and n denote respectively the principal quantum numbers of the inner and the outer electron, l and l’ are their respective orbital quantum numbers, S the total spin, L the total angular momentum and πthe parity of the system. For an atomic system of many M electrons, total energy is expressed as follows

$E=-{\displaystyle \underset{i=1}{\overset{M}{\sum}}\frac{{\left[Z-{\sigma}_{i}\left({}^{2S+1}L{}^{\pi}\right)\right]}^{2}}{{\nu}_{i}^{2}}}$ (2)

In the photoionisation study, energy resonances E_{n} are generally measured relatively to the E_{¥} converging limit of a given (^{2S+1}L_{J})nl-Rydberg series. For these states [12] [13] [14]

$\begin{array}{l}{E}_{n}={E}_{\infty}-\frac{1}{n{}^{2}}\{Z-{\sigma}_{1}\left({}^{2S+1}L{}_{J}\right)-{\sigma}_{2}\left({}^{2S+1}L{}_{J}\right)\times \frac{1}{n}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{{\sigma}_{2}^{\alpha}\left({}^{2}P{}_{3/2}^{0},{}^{1}D{}_{2}\right)\times \left(n-m\right)\times \left(n-q\right){\displaystyle \underset{k}{\sum}\frac{1}{{f}_{k}\left(n,m,q,s\right)}}\}}^{2}\end{array}$ (3)

In this equation, m and q (m < q) denote the principal quantum numbers of the (^{2S+1}L_{J})nl-Rydberg series of the considered atomic system used in the empirical determination of the
${\sigma}_{i}\left({}^{2S+1}L{}_{J}\right)$ -screening constants, s represents the spin of the nl-electron (s = 1/2), E_{¥} is the energy value of the series limit generally determined from NIST atomic database, and Z represents the nuclear charge of the considered element. The only problem that one may face by using the

MAOT formalism is linked to the determination of the $\underset{k}{\sum}\frac{1}{{f}_{k}\left(n,m,q,s\right)}$ -term.

The correct expression of this term is determined iteratively by imposing general Equation (3) to give accurate data with a constant quantum defect values along all the considered series. The value of α is fixed to 1 and or 2 during the iteration. The quantum defect is calculated from the standard formula

${E}_{n}={E}_{\infty}-\frac{R{Z}_{\text{core}}^{2}}{\left(n-\delta \right){}^{2}}$ (4)

In this equation, R is the Rydberg constant, E_{¥} denotes the converging limit, Z_{core} represents the electric charge of the core ion, and δ means the quantum defect. As far as the natural widths are concerned, they are given by (in Rydberg units)

$\begin{array}{l}{\Gamma}_{n}=\frac{1}{n{}^{2}}\{Z-{{\sigma}^{\prime}}_{1}\text{\hspace{0.05em}}\left({}^{2S+1}L{}_{J}\right)-{{\sigma}^{\prime}}_{2}\left({}^{2S+1}L{}_{J}\right)\times \frac{1}{n}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{{\sigma}_{2}^{\alpha}{}^{\prime}\left({}^{2}P{}_{3/2}^{0},{}^{1}D{}_{2}\right)\times \left(n-m\right)\times \left(n-q\right){\displaystyle \underset{k}{\sum}\frac{1}{{{f}^{\prime}}_{k}\left(n,m,q,s\right)}}\}}^{2}\end{array}$ (5)

2.2. Expressions of the Resonance Energies and of the Natural Widths

In the present work, for all the Rydberg series investigated for both Cl II, Ar II and Kr II, the resonance energies are given by the formula

$\begin{array}{c}{E}_{n}={E}_{\infty}-\frac{1}{{n}^{2}}\{Z-{\sigma}_{1}\left({}^{2S+1}L{}_{J}\right)-{\sigma}_{2}\left({}^{2S+1}L{}_{J}\right)\times \frac{1}{n}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\sigma}_{2}^{2}\left({}^{2S+1}L{}_{J}\right)\times {\left(n-m\right)}^{2}\left[\frac{1}{{\left(n+s+2\right)}^{4}}+\frac{1}{{\left(n-2s\right)}^{4}}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{{\sigma}_{2}\left({}^{2S+1}L{}_{J}\right)\times {\left(n-m\right)}^{2}\left[\frac{1}{{\left(n+m+s\right)}^{4}}+\frac{1}{{\left(n+m-s\right)}^{4}}\right]\}}^{2}\end{array}$ (6)

For the [3s^{2}3p^{4}(^{1}D_{2})]ns (j = 1/2) series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of Ar II ions, the natural widths are given by (in Rydberg units)

$\begin{array}{l}{\Gamma}_{n}=\frac{1}{{n}^{2}}\{Z-{\sigma}_{1}\left({}^{2S+1}L{}_{J}\right)-{\sigma}_{2}\left({}^{2S+1}L{}_{J}\right)\times \frac{1}{n}+{\sigma}_{2}^{2}\left({}^{2S+1}L{}_{J}\right)\times \left(n-m\right)\times \left(n-q\right)\\ {\text{\hspace{1em}}\text{\hspace{1em}}\times [\frac{1}{{\left(n+q-m+s+1\right)}^{4}}+\frac{1}{{\left(n+m+s\right)}^{4}}+\frac{1}{{\left(n+q+s\right)}^{4}}+\frac{1}{{\left(n+m-s\right)}^{4}}]\}}^{2}\end{array}$ (7)

The other expressions for the other series are of type Equation (7).^{ }

3. Results and Discussion

The σ_{1}-screening constants in Equations (6) and (7) are evaluated empirically using the data from Covington et al., [9] Hinojosa et al., [10] and from Hernández et al., [4] . The results obtained as indicated in the caption of the corresponding Table. As far as the σ_{2}-screening constant is concerned, it is evaluated theoretical from the simple equation σ_{2} = Z − Z_{core}. The electric charge of the core ion is deduced directly from the single Photoionization process for a given X^{p+}-plasma ion

$h\nu +{\text{X}}^{\text{p}+}\to {\text{X}}^{\left(\text{p}+\text{1}\right)}+{\text{e}}^{-}\Rightarrow {Z}_{\text{core}}=\left(\text{p}+\text{1}\right)$ (8)

So, for Cl II, Ar II and Kr II, we find respectively.

$h\nu +{\text{Cl}}^{+}\to {\text{Cl}}^{2+}+{\text{e}}^{-};{Z}_{\text{core}}=2\Rightarrow {\sigma}_{2}=15.00$

$h\nu +{\text{Ar}}^{+}\to {\text{Ar}}^{2+}+{\text{e}}^{-};{Z}_{\text{core}}=2\Rightarrow {\sigma}_{2}=16.00$

$h\nu +{\text{Kr}}^{+}\to {\text{Kr}}^{2+}+{\text{e}}^{-};{Z}_{\text{core}}=2\Rightarrow {\sigma}_{2}=34.00$

The resonance energies of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ ]nd Rydberg series originating from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{3}\text{P}{}_{2}^{\xb0}$ ground state and from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{3}\text{P}{}_{1}^{\xb0}$ ,
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{3}\text{P}{}_{0}^{\xb0}$ ,
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{1}\text{S}{}_{0}^{\xb0}$ and
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{1}\text{D}{}_{2}$ metastable states of Cl II ions are listed in Tables 1-8. Comparisons of the present MAOT calculations are done with the available Dirac-Coulomb R-matrix (DCR) calculations [5] and with the ALS experimental data [4] . For both the DCR [5] and ALS [4] studies, the determination of the resonances energies have been limited to n = 13 (see Table 4). In general due to interaction configuration and other electron-electron effects, the peaks of the cross section overlap involving difficulty for the identification of lines in the atomic spectra with increasing n. But, it can be seen that, the MAOT formulas are enough stable so that very high lying resonances can be tabulate up to n = 40 with a quantum defect practically constant along all the series investigated. For many resonances, the uncertain experimental entries in parenthesis are enlightened. In Table 2, the resonance energy of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ ]11d level are equal to 25.493 eV (MAOT), 25.493 eV (DCR) and (25.492 eV) for the ALS uncertain experimental data. The excellent agreement between theories indicate that the ALS data can be stated as accurate at 25.492 eV. The same conclusion can be drawn for the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ ]8d state quoted in Table 3 where the MAOT prediction at 25.001 eV agree very well with both the DCR value at 24.999 eV [5] and the uncertain ALS measurement [4] equal to (25.000 eV). For this level the ALS data must be considered as precise at 25.000 eV. Table 4 lists resonance energies of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{3/2}^{\xb0}\right)$ ]nd Rydberg series originating from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{3}\text{P}{}_{2}^{\xb0}$ ground state of the Cl^{+} ions. In this table, two uncertain ALS values are quoted for the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{3/2}^{\xb0}\right)$ ]6d and [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{3/2}^{\xb0}\right)$ ]12d levels respectively at (25.745 eV) and (27.114 eV) to be compared to the MAOT predictions at 25.749 eV and 27.113 eV and to the DCR data [5] respectively equal to 25.755 eV and 27.115 eV. For the n = 11 and 13, the MAOT calculations respectively at 27.031 eV and 27.176 eV are seen to agree very well with the ALS measurements [4] at 27.031 eV and 27.175 eV. Subsequently the DCR data at 25.755 eV (n = 6) and at 27.178 eV (n = 13) are probably greater than the accurate data. The ALS experimental entries in parenthesis can be considered as certain at 25.745 eV and 27.114 eV. Besides, the MAOT data at 25.659 eV quoted in Table 5 is seen to agree very well with the uncertain ALS measurement [4] at (25.660 eV). A slight discrepancy is observed when comparing with the corresponding DCR calculation [5] equal to 25.668 eV. Comparison indicates clearly that the ALS value [4] is correct at 25.660 eV. In Table 6 and Table 7, all the ALS data [4] are certain. In general, good agreements are obtained between theory and experiment. In Table 8, the uncertain ALS data [4] at (24.152 eV) for the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ ]11d level is difficult to enlighten. For this level, the MAOT prediction at 24.146 eV compared fairly well with the DCR calculations [5] equal to 24.138 eV. A new measurement or calculation is needed to clarify this uncertain ALS value [4] . Overall, for the entire data quoted in Tables 2-8, comparisons indicate that the MAOT formula reproduces with a very good accuracy the ALS measurements [4] via a simple formalism without using computational codes in contrast with the DCR formalism [5] . Tables 9-11 list resonance energies of the

Table 1. Resonance energies of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ ]nd Rydberg series originating from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{\text{3}}\text{P}{}_{2}^{\xb0}$ ground state of the Cl^{+} ions converging to the
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ threshold of Cl^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Dirac-Coulomb R-matrix (DCR) calculations of McLaughlin [5] and with the ALS experimental data of Hernández et al., [4] . The ALS experimental resonance energies are calibrated to ±0.013 eV. The energy limits is taken from the NIST tabulations of Ralchenko et al., [19] . σ_{1}(^{2}D_{5/2}) = -0.770 ± 0.048; σ_{2}(^{2}D_{5/2}) = 15.00.

Table 2. Resonance energies (E) and quantum defect (δ) of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ ]nd Rydberg series originating from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{\text{3}}\text{P}{}_{1}^{\xb0}$ metastable state of the Cl^{+} ions converging to the
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ threshold of Cl^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Dirac-Coulomb R-matrix (DCR) calculations of McLaughlin [5] and with the ALS experimental data of Hernández et al., [4] . The ALS experimental resonance energies are calibrated to ±0.013 eV. The energy limits is taken from the NIST tabulations of Ralchenko et al., [19] . σ_{1}(^{2}D_{5/2}) = −0.781 ± 0.048; σ_{2}(^{2}D_{5/2}) = 15.00.

Table 3. Resonance energies (E) and quantum defect (δ) of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ ]nd Rydberg series originating from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{\text{3}}\text{P}{}_{0}^{\xb0}$ metastable state of the Cl^{+} ions converging to the
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ threshold of Cl^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Dirac-Coulomb R-matrix (DCR) calculations of McLaughlin [5] and with the ALS experimental data of Hernández et al., [4] . The ALS experimental resonance energies are calibrated to ±0.013 eV. The energy limits is taken from the NIST tabulations of Ralchenko et al., [19] . σ_{1}(^{2}D_{5/2}) = −0.789 ± 0.048; σ_{2}(^{2}D_{5/2}) = 15.00.

Table 4. Resonance energies (E) and quantum defect (δ) of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{3/2}^{\xb0}\right)$ ]nd Rydberg series originating from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{\text{3}}\text{P}{}_{2}^{\xb0}$ ground state of the Cl^{+} ions converging to the
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{3/2}^{\xb0}\right)$ threshold of Cl^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Dirac-Coulomb R-matrix (DCR) calculations of McLaughlin [5] and with the ALS experimental data of Hernández et al., [4] . The ALS experimental resonance energies are calibrated to ±0.013 eV. The energy limits is taken from the NIST tabulations of Ralchenko et al., [19] . σ_{1}(^{2}P_{3/2}) = −1.006 ± 0.077; σ_{2}(^{2}P_{3/2}) = 15.00.

Table 5. Resonance energies (E, eV), quantum defect (δ) of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{3/2}^{\xb0}\right)$ ]nd Rydberg series originating from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{\text{3}}\text{P}{}_{1}^{\xb0}$ metastable state of the Cl^{+} ions converging to the
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{3/2}^{\xb0}\right)$ threshold of Cl^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Dirac-Coulomb R-matrix (DCR) calculations of McLaughlin [5] and with the ALS experimental data of Hernández et al., [4] . The ALS experimental resonance energies are calibrated to ±0.013 eV. The energy limits is taken from the NIST tabulations of Ralchenko et al., [19] . σ_{1}(^{2}P_{3/2}) = −1.018 ± 0.076; σ_{2}(^{2}P_{3/2}) = 15.00.

Table 6. Resonance energies (E) and quantum defect (δ) of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{3/2}^{\xb0}\right)$ ]nd Rydberg series originating from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{\text{3}}\text{P}{}_{0}^{\xb0}$ metastable state of the Cl^{+} ions converging to the
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{3/2}^{\xb0}\right)$ threshold of Cl^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Dirac-Coulomb R-matrix (DCR) calculations of McLaughlin [5] and with the ALS experimental data of Hernández et al., [4] . The ALS experimental resonance energies are calibrated to ±0.013 eV. The energy limits is taken from the NIST tabulations of Ralchenko et al., [19] . σ_{1}(^{2}P_{3/2}) = −1.088 ± 0.076; σ_{2}(^{2}P_{32}) = 15.00.

Table 7. Resonance energies (E, eV), quantum defect (δ) of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{1/2}^{\xb0}\right)$ ]nd Rydberg series originating from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{\text{1}}\text{S}{}_{0}^{\xb0}$ metastable state of the Cl^{+} ions converging to the
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{P}{}_{1/2}^{\xb0}\right)$ threshold of Cl^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Dirac-Coulomb R-matrix (DCR) calculations of McLaughlin [5] and with the ALS experimental data of Hernández et al., [4] . The ALS experimental resonance energies are calibrated to ±0.013 eV. The energy limits is taken from the NIST tabulations of Ralchenko et al., [19] . σ_{1}(^{2}P_{1/2}) = -0.262 ± 0.015; σ_{2}(^{2}P_{1/2}) = 15.00.

Table 8. Resonance energies (E) and quantum defect (δ) of the [
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ ]nd series originating from the
$3\text{s}{}^{2}3{\text{p}}^{4}{}^{\text{1}}\text{D}{}_{2}$ metastable state of the Cl^{+} ions converging to the
$3\text{s}{}^{2}3{\text{p}}^{3}\left({}^{2}\text{D}{}_{5/2}^{\xb0}\right)$ threshold of Cl^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Dirac-Coulomb R-matrix (DCR) calculations of McLaughlin [5] and with the ALS experimental data of Hernández et al., [4] . The ALS experimental resonance energies are calibrated to ±0.013 eV. The energy limits is taken from the NIST tabulations of Ralchenko et al., [19] . σ_{1}(^{2}D_{5/2}) = -0.483 ± 0.050; σ_{2}(^{2}D_{5/2}) = 15.00.

Table 9. Resonance energies (E) and quantum defect (δ) of the 3s^{2}3p^{4}(^{1}D_{2})ns (j = 1/2 series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of the Ar^{+} ions converging to the ^{1}D_{2} threshold of Ar^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the QB data of Covington et al., [9] . The energy limits eV) is taken from the NIST tabulations of Ralchenko et al., [17] . σ_{1}(^{1}D_{2}) = −4.227 ± 0.224; σ_{2}(^{1}D_{2}) = 16.00.

Table 10. Resonance energies (E) and quantum defect (δ) of the 3s^{2}3p^{4} (^{1}D_{2})nd (j = 1/2 series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of the Ar^{+} ions converging to the 3s^{2}3p^{4}(^{1}D_{2}) threshold of Ar^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the QB data of Covington et al., [9] . The energy limits eV) is taken from the NIST tabulations of Ralchenko et al., [17] . σ_{1}(^{1}D_{2}) = −1.159 ± 0.265; σ_{2}(^{1}D_{2}) = 16.00.

Table 11. Resonance energies (E) and quantum defect (δ) of the 3s^{2}3p^{4}(^{1}D_{2})nd (j = 3/2 series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of the Ar^{+} ions converging to the 3s^{2}3p^{4}(^{1}D_{2}) threshold of Ar^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the QB data of Covington et al., [9] . The energy limits eV) is taken from the NIST tabulations of Ralchenko et al., [17] . σ_{1}(^{1}D_{2}) = -1.159 ± 0.265; σ_{2}(^{1}D_{2}) = 16.00.

3s^{2}3p^{4}(^{1}D_{2})ns, nd (j = 1/2) and of the 3s^{2}3p^{4}(^{1}D_{2})nd (j = 3/2) series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of the Ar II ions. Comparison indicate an excellent agreements between the present results from the Modified atomic orbital theory (MAOT) and both the Screening constant by unit nuclear charge (SCUNC) results [11] and the multichannel R-matrix QB technique which defines matrices Q and B in terms of asymptotic solutions [9] . These very good agreements are also observed comparing the resonance energies 3s^{2}3p^{4}(^{1}D_{2})ns, nd (j = 1/2) series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{3/2}^{\xb0}$ ground state of the Ar II ions. For both Tables 9-12, the QB data are limited to n = 16 and the SCUNC values to n = 30. High lying MAOT data are tabulated up to n = 40 with a constant quantum defect along each series. Table 13 and Table 14 quote respectively resonance energies of the 3s^{2}3p^{4}(^{1}D_{2})nd (j = 1/2)and of the 3s^{2}3p^{4}(^{1}D_{2})ns (j = 3/2) series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{3/2}^{\xb0}$ ground state of the Ar II ions. Here again, the agreements between the present MAOT results and both SCUNC of Sakho [11] and QB results of Covington et al., [9] are very good. Table 15 presents resonance energies of the 3s^{2}3p^{4} (^{1}D_{2})nd (j = 3/2) series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{3/2}^{\xb0}$ ground state of the Ar II ions. In this Table, an uncertain QB data [9] is quoted at (28.774 eV) for the 3s^{2}3p^{4}(^{1}D_{2})10d level. For this level, the MOAT prediction is at 28.735 eV to be compared to the SCUNC forecast at 28.734 eV. Subsequently, the QB data at (28.774 eV) associated with a quantum defect equal to (0.422) is less precise. For this level, both the MAOT and SCUNC [11] predictions associated with the quantum defects 0.721 and 0.724 are preferable. Therefore, it should be underlined that the SCUNC calculations are more precise than the MOAT calculations. This is due mainly to the fact that, the SCUNC formalism is a development of 1/Z taking implicitly into account more relativistic effects than the MAOT formalism witch is a simple development on 1/n. In addition, in the SCUNC formalism, great accuracy are obtained when performing the analytical formula for each atomic system [11] . In the present work, the same Formula (6) is used for both Cl II, Ar II and Kr II in contrast with the work of Sakho [11] where the resonance energy expression for the Ar II ions is different to that of the Kr II ions. Table 16 and Table 17 list natural widths of the [3s^{2}3p^{4}(^{1}D_{2})]ns, nd (j = 1/2) (Table 16) and of the [3s^{2}3p^{4}(^{1}D_{2})]ns, nd (j = 3/2) (Table 17) series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of Ar^{+} ions. It can be seen that the present MAOT data agree well with both the SCUNC results [11] and QB data [9] . It should be mentioned again that the SCUNC calculations are more precise than the MOAT calculations for the reason explained above. Table 18 lists resonance energies and quantum defect of the [4s^{2}4p^{4} (^{1}D_{2})]nd series originating from the
$4\text{s}{}^{2}4{\text{p}}^{5}{}^{2}\text{P}{}_{3/2}^{\xb0}$ ground state of the Kr II ions. Comparisons indicate very good agreements between the present results from the MAOT formalism and both the SCUNC calculations [11] up to n = 30 and ALS measurements of [10] up to n = 13. In this iable, two uncertain ALS data are quoted for the [4s^{2}4p^{4}(^{1}D_{2})]12d and [4s^{2}4p^{4}(^{1}D_{2})]13d levels respectively at (25.880 eV) and (25.926 eV). The

Table 12. Resonance energies (E) and quantum defect (δ) of the 3s^{2}3p^{4}(^{1}D_{2})ns (j = 1/2 series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{3/2}^{\xb0}$ ground state of the Ar^{+} ions converging to the 3s^{2}3p^{4}(^{1}D_{2}) threshold of Ar^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the QB data of Covington et al., [9] . The energy limits eV) is taken from the NIST tabulations of Ralchenko et al., [17] . σ_{1}(^{1}D_{2}) = -4.234 ± 0.224; σ_{2}(^{1}D_{2}) = 16.00.

Table 13. Resonance energies (E) and quantum defect (δ) of the 3s^{2}3p^{4}(^{1}D_{2})nd (j = 1/2 series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{3/2}^{\xb0}$ ground state of the Ar^{+} ions converging to the 3s^{2}3p^{4}(^{1}D_{2}) threshold of Ar^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the QB data of Covington et al., [9] . The energy limits eV) is taken from the NIST tabulations of Ralchenko et al., [17] . σ_{1}(^{1}D_{2}) = -1.159 ± 0.265; σ_{2}(^{1}D_{2}) = 16.00.

Table 14. Resonance energies (E) and quantum defect (δ) of the 3s^{2}3p^{4}(^{1}D_{2})ns (j = 3/2 series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{3/2}^{\xb0}$ ground state of the Ar^{+} ions converging to the 3s^{2}3p^{4}(^{1}D_{2}) threshold of Ar^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the QB data of Covington et al., [9] . The energy limits eV) is taken from the NIST tabulations of Ralchenko et al., [17] . σ_{1}(^{1}D_{2}) = -4.294 ± 0.224; σ_{2}(^{1}D_{2}) = 16.00.

Table 15. Resonance energies (E, eV) and quantum defect (δ) of the 3s^{2}3p^{4}(^{1}D_{2})nd (j = 3/2 series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{3/2}^{\xb0}$ ground state of the Ar^{+} ions converging to the 3s^{2}3p^{4}(^{1}D_{2}) threshold of Ar^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the QB data of Covington et al., [9] . The energy limits eV) is taken from the NIST tabulations of Ralchenko et al., [17] . σ_{1}(^{1}D_{2}) = -1.575 ± 0.265; σ_{2}(^{1}D_{2}) = 16.00.

*This line is not well identified.

Table 16. Natural widths (G, meV) of the [3s^{2}3p^{4}(^{1}D_{2})]ns, nd (j = 1/2) series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of Ar^{+} ions. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the QB data of Covington et al., [9] . For: [4s^{2}4p^{4}(^{1}D_{2})] ns: σ_{1}(^{1}D_{2}) = -5.494 ± 0.010; σ_{2}(^{1}D_{2}) = 17.989 ± 0.010. For: [4s^{2}4p^{4}(^{1}D_{2})]nd: σ_{1}(^{1}D_{2}) = -0.168 ± 0.010; σ_{2}(^{1}D_{2}) = 17.998 ± 0.010.

Table 17. Natural widths (G, meV) of the [3s^{2}3p^{4}(^{1}D_{2})]ns, nd (j = 3/2) series originating from the
$3\text{s}{}^{2}3{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of Ar^{+} ions. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the QB data of Covington et al., [9] . For: [4s^{2}4p^{4}(^{1}D_{2})] ns: σ_{1}(^{1}D_{2}) = -0.670 ± 0.010; σ_{2}(^{1}D_{2}) = 18.038 ± 0.010. For: [4s^{2}4p^{4} (^{1}D_{2})] nd: σ_{1}(^{1}D_{2}) = -1.399 ± 0.010; σ_{2}(^{1}D_{2}) = 17.827 ± 0.010.

Table 18. Resonance energies (E) and quantum defect (δ) of the [4s^{2}4p^{4}(^{1}D_{2})]nd series originating from the
$4\text{s}{}^{2}4{\text{p}}^{5}{}^{2}\text{P}{}_{3/2}^{\xb0}$ ground state of the Kr^{+} ions converging to the [4s^{2}4p^{4} (^{1}D_{2})] threshold of Kr^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the ALS experimental data of Hinojoha et al., [10] . The ALS resonance energies are calibrated to ±30 meV and quantum defects are estimated to within an error of 20%. The energy limits is taken from the NIST tabulations of Ralchenko et al., [18] . Here σ_{1}(^{1}D_{2}) = −0.719 ± 0.116; σ_{2}(^{1}D_{2}) = 34.00.

associated ALS quantum defects are equal to (−0.671) and (−0.663) respectively. For the same levels, the MAOT and SCUNC calculations [11] are respectively at 25.863 eV and 25.861 eV for the [4s^{2}4p^{4}(^{1}D_{2})]12d state and at 25.913 eV and 25.912 eV for the [4s^{2}4p^{4}(^{1}D_{2})]13d state. Constant quantum defects are tabulated for the MAOT and SCUNC predictions [11] respectively at −0.31 and −0.37. So the MAOT and SCUNC estimations can be considered as the accurate data for the [4s^{2}4p^{4}(^{1}D_{2})]12d and [4s^{2}4p^{4}(^{1}D_{2})]13d resonances. Table 19 quotes resonance energies and quantum defect of the 4s^{2}4p^{4}(^{3}P_{2},^{3}P_{1})]ns and 4s^{2}4p^{4}(^{3}P_{1})]ns series originating from the
$4\text{s}{}^{2}4{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of the Kr II ions. For the 4s^{2}4p^{4}(^{3}P_{2})]ns series, comparisons indicate good agreements between theory and experiments. It should be underlined the very good agreements for n = 13 - 20. This may enlighten the accuracy of the uncertain ALS measurement [10] listed into parenthesis. For the 4s^{2}4p^{4}(^{3}P_{1})]ns series only one ALS data at 23.996 eV is quoted. New MAOT values from n = 14 to 40 are tabulated with a constant quantum defect about 0.42. In Table 20, resonance energies and quantum defect of the 4s^{2}4p^{4}(^{3}D_{2})]ns, ndseries originating from the
$4\text{s}{}^{2}4{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of the Kr II ions are listed. Here again, the MAOT data agree very well with the ALS data of Hinojoha et al., [10] . It should be underlined the excellent agreements between theory and experiments for the 4s^{2}4p^{4}(^{3}D_{2})]8s and 4s^{2}4p^{4}(^{3}D_{2})]9s levels. For these states, both the MAOT and ALS work provide the same values respectively equal to 24.650 eV and 24.842 eV. The excellent agreement for the quantum defects can also be mentioned, 0.20 for both theory and experiments. Table 21 presents resonance energies and quantum defect for the

Table 19. Resonance energies (E) and quantum defect (δ) of the 4s^{2}4p^{4}(^{3}P_{2},^{3}P_{1})]ns series originating from the
$4\text{s}{}^{2}4{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of the Kr^{+} ions converging to the [4s^{2}4p^{4}(^{1}D_{2})] threshold of Kr^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the ALS experimental data of Hinojoha et al., [10] . The ALS resonance energies are calibrated to ±30 meV and quantum defects are estimated to within an error of 20%.The energy limits is taken from the NIST tabulations of Ralchenko et al., [18] . σ_{1}(^{3}P_{2}) = −1.438 ± 0.104; σ_{2}(^{3}P_{2}) = 34.00; σ_{1}(^{3}P_{1}) = −0.863 ± 0.138; σ_{2}(^{3}P_{1}) = 34.00.

Table 20. Resonance energies (E) and quantum defect (δ) of the 4s^{2}4p^{4}(^{3}D_{2})]ns, nd series originating from the
$4\text{s}{}^{2}4{\text{p}}^{5}{}^{2}\text{P}{}_{1/2}^{\xb0}$ metastable state of the Kr^{+} ions converging to the [4s^{2}4p^{4}(^{1}D_{2})] threshold of Kr^{2+}. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the ALS experimental data of Hinojoha et al., [10] . The ALS resonance energies are calibrated to ±30 meV and quantum defects are estimated to within an error of 20%. The energy limits is taken from the NIST tabulations of Ralchenko et al., [18] . σ_{1}(^{1}D_{2}) = −0.414 ± 0.116; σ_{2}(^{1}D_{2}) = 34.00 for 4s^{2}4p^{4}(^{3}D_{2})]ns. σ_{1}(^{1}D_{2}) = −0.698 ± 0.116; σ_{2}(^{1}D_{2}) = 34.00 for 4s^{2}4p^{4}(^{3}D_{2})]nd.

Table 21. Resonance energies (E) and quantum defect (δ) of the 4s^{2}4p^{4}(^{3}D_{2}, ^{1}S_{0})]ns, nd series of the Kr^{+} ions. The present results from the Modified atomic orbital theory (MAOT) are compared with the Screening constant by unit nuclear charge (SCUNC) results of Sakho [11] are compared with the ALS experimental data of Hinojoha et al., [10] . The ALS resonance energies are calibrated to ±30 meV and quantum defects are estimated to within an error of 20%. σ_{1}(^{1}D_{2}) = -0.785 ± 0.116; σ_{2}(^{1}D_{2}) = 34.00; σ_{1}(^{1}S_{0}) = -0.811 ± 0.116; σ_{2}(^{1}S_{0}) = 34.00.

4s^{2}4p^{4}(^{3}D_{2}, ^{1}S_{0})]ns, nd series of the Kr II ions. Here again, the good agreements between the present MAOT and ALS data [10] may enlighten the accuracy of the uncertain ALS measurement listed into parenthesis. Overall, for both the Cl II, Ar II and Kr II ions, it is demonstrated in this paper the possibilities to reproduce excellently high ALS measurements from single MAOT analytical expression. This is the main strength of the present work.

4. Conclusion

Accurate high lying resonance energy up to n = 40 is reported applying the Modified atomic orbital theory. For both the Cl II, Ar II and Kr II ions investigated, a single formula has been established to reproduce with a very good accuracy high experimental measurements such as those performed at the Advanced Light Source at Lawrence Berkeley National Laboratory. A huge number of results are tabulated as useful reference data for interpreting atomic spectra from astrophysical objects containing chlorine, argon and krypton elements.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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