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

Resonance energies of the Cl II-[3s3p(D5/2)]nd and [3s3p(P3/2)]nd, Ar II-3s3p(D2)ns, nd and of the Kr II [4s4p(D2)]ns, nd and 4s4p(P2,P1)]ns, 4s4p(D2)]ns, nd and 4s4p(D2, S0)]ns, nd Rydberg series are reported. Natural widths of the Ar II-[3s3p(D2)]ns, nd 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.


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 A. Diallo et al.
energy of a (νl)-given orbital is expressed in the form in Rydberg units ( ) ( ) 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 ( ) , S Nl nl L π + ′ . 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 ( ) 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] ( ) ( ) , , , 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 ( ) -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 ( ) 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 ( ) 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)

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 For the [3s 2 3p 4 ( 1 D 2 )]ns (j = 1/2) series originating from the 3s 3p P ° me- tastable state of Ar II ions, the natural widths are given by (in Rydberg units) The other expressions for the other series are of type Equation (7).

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 So, for Cl II, Ar II and Kr II, we find respectively.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 3s 3p D °]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.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 [ ( ) 3s 3p D °]11d level is difficult to enlighten.For this level, the MAOT pre- diction 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  [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].[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  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.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 3s 3p P ° 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.3s 3p P ° 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].( 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 For the same levels, the MAOT and SCUNC calculations [11]   [11] are compared with the ALS experimental data of Hinojoha et al., [10].
°]12d levels re- spectively 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.
° ground state of the Ar II ions.For both Tables9-12, 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.
metastable states of Cl II ions are listed in Tables 1-8.Comparisons of the present MAOT calculations are done with the available

Table 2
, the resonance energy of the [

Table 4
° ground state of the Cl + ions.In this table, two uncertain ALS values are quoted for the [ ( )

Table 1 .
Resonance energies of the [

Table 4 .
Resonance energies (E) and quantum defect (δ) of the [ ° 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
Table 13 and Table 14 quote respec-

Table 15
[9]round 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
*This line is not well identified.

Table 19 .
[11]nance energies (E) and quantum defect (δ) of the 4s 2 4p 4 ( 3 P 2 , 3 P 1 )]ns series ° 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 Hinojo- 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.Table21presents resonance energies and quantum defect for the

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 [10] 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.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. 4s