Photoionization of the 3s23p3nd Rydberg series of Cl+ ion using the Screening constant by unit nuclear charge method

Radiation Physics and Chemistry 153 (2018) 111–119

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Photoionization of the 3s23p3nd Rydberg series of Cl+ ion using the Screening constant by unit nuclear charge method M.D. Baa, A. Dialloa, J.K. Badianea, M.T. Gninga, M. Sowb, I. Sakhoa, a b

T

⁎

Department of Physics, UFR Sciences and Technologies, University Assane Seck of Ziguinchor, Ziguinchor, Senegal Department of Physics, Faculty of Sciences and Techniques, University Cheikh Anta Diop of Dakar, Dakar, Senegal

ABSTRACT 3

3

° ° ) ]nd and [3s 23p (2P3/2 ) ]nd Rydberg series originating from the ground 3s 23p4 3P2 We report calculations of accurate high lying resonance energies of the [3s 23p (2D5/2

3

° ) ]nd Rydberg series originating from the 3s 23p4 1S0 metastable state of the Cl+ ion. and from the 3s 23p4 3P1,0 metastable states of Cl+ along with the [3s 23p (2P1/2 Calculations are performed using the Screening Constant by Unit Nuclear Charge (SCUNC) method up to n = 40. The results obtained are compared with the existing Dirac – Coulomb R-matrix (DCR) calculations (McLaughlin, 2017) and Advanced Light Source (ALS) measurements (Hernández et al., 2015). Analysis of the present results is achieved in the framework of the standard quantum-defect theory and of the SCUNC-procedure based on the calculation of the effective charge. It is shown that the SCUNC-method agree very well the ALS measurements up to n = 13. New high lying accurate resonance energies (n = 14–40) are tabulated as benchmarked data for the atomic physics community in connection with the modeling of plasma and astrophysical systems.

1. Introduction It is well known that visible matter in the Universe is almost in the plasma state and most observational data about the distant universe are conveyed through interstellar space by photons. Some of these photons are sufficiently energetic to induce the photoionization of atoms and ions according to the single process

hn + Xq+

X (q + m) + + me .

(1)

Photoionization of atoms and ions is then a fundamental process of importance in many astrophysical systems such as stars and nebulae. Of great important ions interesting to investigate are sulphur-like ions such as Cl+ in connection with their abundances in photoionized astrophysical objects. In the past, various studies have indicated the great importance of S-like Cl+ ion abundances for understanding extragalactic HII regions (Garnett, 1989). In addition, emission lines of S-like Cl+ ion have been observed in the spectra of the Io torus (Küppers and Schneider, 2000) and in the optical spectra of planetary nebulae NGC 6741 and IC 5117 (Keenan et al., 2003). The ground state configuration of the Cl+ ions is 3s23p4 3P in the LS coupling scheme. For the 3P term, the total spin S = 1 and the orbital momentum quantum number L = 1. As a result, the total angular momentum quantum number J takes the values |L + S| up to |L − S|, that means J = 2, 1 and 0. Due to spinorbit interaction, the Cl+ − 3s23p4 3P configuration splits into three subshells namely the 3s23p4 3P2 ground state and the 3s23p4 3P1,0 metastable states. In the works of Ralchenko et al. (2014), the energy

⁎

differences between the 3s23p4 3P2 and 3s23p4 3P1 states and between the 3s23p4 3P2 and 3s23p4 3P0 states are respectively equal to 86.3 meV and 123.5 meV. Then, according to the magnitude of the photon energy, the 3s23p4 3P2 ground state and the 3s23p4 3P1,0 metastable states of the Cl+ ion can be photoionized. So, two photoionization processes originating from the ground and metastable states of this sulphur-like ion can be observed. Using Eq. (1), the two possible photoexcitation processes are the following hν + Cl+ (3s23p4 3P2) → Cl+ [3s23p3 (3LJ)] nl· hν + Cl

+

2

4 3

(3s 3p

P1,0) → Cl

+

2

3

3

[3s 3p ( L′J′)] n′l’·

(2) (3)

These two processes can finally decay by single electron emission Cl+ [3s23p3 (3LJ)] nl → Cl2+ [3s23p3 (3LJ)] + e−·

(4)

Cl+ [3s23p3 (3L′J′)] n′l’ → Cl2+ [3s23p3 (3L′J′)] + e−·

(5)

In a very recent past, Hernández et al. (2015) measured at the Advanced Light Source at Lawrence Berkeley National Laboratory absolute photoionization cross-sections for the Cl+ ion in its ground and metastable states, 3s23p4 3P2,1,0 and 3s23p4 1D2, 1S0, using the merged beams photon–ion technique at a photon energy resolution of 15 meV in the energy range 19–28 eV. As stated by Covington et al. (2011), for comparison with high-resolution measurements such as from ALS experiments, state-of-the-art-theoretical methods are required using highly correlated wave functions. Relativistic effects are also required,

Corresponding author. E-mail address: [email protected] (I. Sakho).

https://doi.org/10.1016/j.radphyschem.2018.09.010 Received 16 June 2018; Received in revised form 4 September 2018; Accepted 6 September 2018 Available online 12 September 2018 0969-806X/ © 2018 Elsevier Ltd. All rights reserved.

Radiation Physics and Chemistry 153 (2018) 111–119

M.D. Ba et al.

formalism, the total energy of the (Nl, nl ) 2S + 1L excited states is expressed in the form (in Rydberg)

since fine-structure effects can be resolved. Using the Dirac–Coulomb Rmatrix (DCR) method, McLaughlin (2017) performed calculations in the energy range 19–28 eV as in the ALS experiments (Hernández et al., 2015) to assign and identify the resonance series in the ALS spectra of the Cl+ ion. The 3s23p3nd states have been identified in the Cl+ spectra as the prominent Rydberg series belonging to the 3p → nd transitions. The weaker Rydberg series 3s23p3ns, identified as 3p → ns transitions and window resonances 3s3p4 (4P) np features, due to 3s→np transitions, have also been found in the spectra (McLaughlin, 2017). In the present work, we focus our study on the 3s23p3nd prominent Rydberg series identified in the Cl+ spectra. The corresponding 3p → nd transitions are deduced from Eqs. (2) and (3) as follows,

E (Nl, nl ;

2S + 1

L )

1 1 + [1 N2 n2

Z2

=

(Nl, nl ;

2S + 1

2

L ; Z) ] . (10)

In this equation, the principal quantum numbers N and n are respectively for the inner and the outer electron of the helium-isoelectronic series. The β-parameters are screening constants by unit nuclear charge expanded in inverse powers of Z and given by q

2S + 1

(Nl nl ;

L ; Z)

=

fk k = 1

1 Z

k

.

(11)

L ) are parameters to be evaluated empiriwhere fk = fk (Nl nl ; cally. For a given Rydberg series originating from a-2S+1LJ state, we obtain using (Sakho, 2018) 2S + 1

(6)

Z2 [1 n2

En = E

4 1

2

4 1

For the metastable states, 3s 3p S0 and 3s 3p D2 present in the parent ion beam of the ALS experiments, we obtain for the 3p → nd transitions. hν + Cl+ (3s23p4 1S0) → Cl+ [3s23p3 (2P1/2)] nd· hν + Cl

+

2

(3s 3p

4 1

+

D2) → Cl

2

3

2

[3s 3p ( D5/2)] nd·

(12)

In this equation, ν and µ (µ > ν) denote the principal quantum numbers of the (2S+1LJ) nl Rydberg series used in the empirical determination of the fi - screening constants, s represents the spin of the nlelectron (s = ½), E∞ is the energy value of the series limit, En denotes the resonance energy and Z stands for the atomic number. The βparameters are screening constants by unit nuclear charge expanded in inverse powers of Z and given by

(7) 2

(nl; s ; , ;2S + 1 L ; Z )]2 .

(8)

q

Z , 2S + 1LJ , n , s, µ ,

(9)

=

fk k = 1

1 Z

k

.

(13)

LJ , n, s , µ , ) are screening constants to be evalwhere fk = fk ( uated empirically. In Eq.(13), q stands for the number of terms in the expansion of the β–parameter. Generally, precise resonance energies are obtained for q < 5. The resonance energy are the in the form 2S + 1

In both the ALS measurements (Hernández et al., 2015) and the DCR calculations (McLaughlin, 2017), the studies have been limited to n = 13. In addition, comparison have been done between the DCR calculations and the ALS uncertain resonance energies for the 3p → nd transitions. Besides ALS measurements were absent for some transitions such as the 3s23p4 3P0 → [3s23p3 (2D5/2)] nd transitions where only the 24.219 eV precise data for the [3s23p3 (2D5/2)] 6d level and the uncertain value (25.000 eV) for the [3s23p3 (2D5/2)] 8d level have been quoted. In the energy range 19–28 eV, a huge of high lying resonance energies exist. These high lying Rydberg series are very useful data for the NIST database where many resonances are tabulated up to n = 60 for atomic systems such as Mg. The goal of the present study is to extend the previous ALS measurements (Hernández et al., 2015) and the DCR calculations (McLaughlin, 2017) to the high lying 3p → nd transitions with n = 6– 40 and to tabulated accurate data to be compared with the ALS uncertain and absent resonance energies for the 3p → nd transitions. Comparison with the DCR predictions is also aimed. For this purpose, we apply the Screening constant by unit nuclear charge (SCUNC) formalism (Goyal et al., 2016; Khatri et al., 2016; Sakho, 2016, 2017, 2018)very suitable in reproducing excellently experimental data. Analysis of the present results is achieved in the framework of the standard quantum-defect theory and of the SCUNC-procedure by calculating the effective charge. The present paper is organized as follows. Section 2 presents a brief outline of the theoretical part of the work. In Section 3, we present and discuss the results obtained compared with the existing ALS measurements (Hernández et al., 2015) and with the DCR calculations (McLaughlin, 2017). In Section 4, we summarize our study and draw conclusions.

f1 (2S+ 1LJ ) Z (n 1)

Z2 1 n2

En = E

f2 (2S + 1LJ ) ± Z

q

q

f1k F n, , , s ×

k =1 k =1

1 Z

k

2

.

(14) q

()

1 k

q

The quantity ± k = 1 k = 1 f1k F (n, , , s ) × Z is a corrective term introduced to stabilize the resonance energies with increasing the principal quantum number n. Besides, resonance energies are usually analyzed from the standard quantum-defect expansion formula 2 RZcore . (n )2

En = E

(15)

In this equation, R is the Rydberg constant, E∞ denotes the converging limit, Zcore represents the electric charge of the core ion, and δ means the quantum defect. In addition, theoretical and measured energy positions can be analyzed by calculating the Z*-effective charge in the framework of the SCUNC-procedure

Z *2 R. n2

En = E

(16) *

The relationship between Z and δ is in the form

Z *=

Zcore

(1

n

)

. (17)

According to this equation, each Rydberg series must satisfy the following conditions

2. Theory

Z* Zcore if Z* Zcore if lim Zn* = Zcore

2.1. Brief description of the SCUNC formalism In the framework of the Screening Constant by Unit Nuclear Charge 112

0 0. (18)

Radiation Physics and Chemistry 153 (2018) 111–119

M.D. Ba et al.

Besides, comparing Eq. (14) and Eq. (16), the effective charge is in the form

Z *=Z 1

f1 (2S + 1LJ ) Z (n

f2 (2S + 1LJ )

1)

Z

q

q

± k=1 k =1

1 Z

f1k F (n , , , s ) ×

k

Z2 n2

En = E

° ) f1 (2D5/2

1

Z (n

° f1 (2D5/2 )(n

.

Z2 (n

15.0 Z

1)

° f1 (2D5/2 )(n

)

+ 2s + 2)(n

Z3 (n

+ s + 1)

2

)

+ 2s )(n +

s

(23)

(19) Besides, the f2-parameter in Eq. (14) can be theoretically determined from Eq. (19)

f2 (2S + 1LJ )

lim Z * = Z 1

= Zcore .

Z

n

• For

Z2

En = E

° 5/2 ) ]nd

2

+

f1 ( Z 2 (n

f1 (

° D5/2 )

Z (n ° D5/2 )(n

3 2

tastable state

Z2 n2

En = E

° 5/2 ) ]nd

1

µ + 2s + 2) × (n µ + s + 1)

2

+

Z3 (n

° f1 (2P1/2 )(n

+ s + 3)

+

Z2 n2

En = E

3 2 ° 1/2 ) ]nd

µ + s + 1) × (n + µ

Z (n

+ 2s + 2)(n

f1 ( Z 3 (n

° D5/2 )(n

2

)

+ s + 1)(n

.

+ s)

.

3

µ

2 ° D5/2)

Z (n 2 ° D5/2)(n

s ) × (n

15.0 Z

1) )2

µ + s + 1)

+

° f1 (2P1/2 ,

2 ° D5/2)(n

Z3 (n

)(n

2s )(n

s

+ 1) 1)2

2

.

Let us first precise the sign of the quantum defect δ using the SCUNC analysis conditions (18) by considering the lowest resonance corresponding to the first entry for the 3p → nd. For this purpose, we focus 3 ° ) ]nd and the demonstration on the particular cases of the [3s 23p (2D5/2 3

° ) ]nd Rydberg series. The calculations are of similar for the [3s 23p (2P3/2 other series.

• for ° f1 (2D5/2 )(n

) + s + 1)

1)

(25)

15.0 Z

1)

s

series originating from the 3s 23p4 1S0 me-

° f1 (2P1/2 ,

1

° f1 (2P1/2 ,

Z2 (n

series originating from the 3s 23p4 3P1 me-

° f1 (2D5/2 )

2

)

° ) ]nd series originating from tastable state and for the [3s 23p (2D5/2 the 3s 23p4 1D2 metastable state

Z 3 (n

2

)

+ 2s )(n + s + 1)

3

° ) ]nd series originating from the 3s 23p4 3P2 the [3s 23p (2D5/2 ground state, the lowest resonance corresponds to the 3p → 6d (µlow = 6). From Table 1, we pull f1 (2D5/2) = − 0.642. From Eq. (21), we * as follows deduce the expression of the effective charge Zmax

.

(22)

•

)2

3. Results and discussion 2

)

° f1 (2D5/2 )(n

Z 2 (n

° f1 (2P1/2 )(n

• For the [3s 3p ( P

(21) 2

15.0 Z

Z (n 1)

(24)

15.0 Z

1)

+ s + 1)(n

• For the [3s 3p ( D

° f1 (2P3/2 )

1

from the 3s 23p4 3P2 ground state 2

1

n2

Z2 n2

Z2 (n

Using Eq. (14), we obtain the following energy positions (in Rydberg) 2

4 3

En = E

(20)

2.2. Energy resonances of the 3p→ nd transitions

• For the [3s 3p ( D

3

° ) ]nd series originating from the ground the [3s 23p (2P3/2 3s 3p P2 state and from the 3s 23p4 3P1,0 metastable states

2

We get then f2 = Z – Zcore, where Zcore is deduced from the photoionization process of the considered atomic Xp+ system, Xp+ + hν → X(p+1)+ + e−. We find then Zcore = p + 1. So, for the Cl+ ion, Zcore = 2 and f2 = (17 − 2) = 15.0. The remaining f1-parameter is to be evaluated empirically using the ALS measurements (Hernández et al., 2015) for a given (2S+1LJ) µl level with ν = 0 in Eq. (14). The results obtained are quoted in Table 1.

3 2

.

1)

* =Z Zmax

3

° ) ]nd series originating from the 3s 23p4 3P0 For the [3s 23p (2D5/2 metastable state

= 17

1+ 1+

° f1 (2D5/2 )

Z(

low

0.642 7(6 1)

15.0 Z

1)

15.0 17

= 2.128.

(26)

* = 2.128 > Zcore. The quantum defect δ is As Zcore = 2.0, then Zmax Table 1 Screening constants evaluated from Eq. (14). The experimental energy resonances En (in eV) are taken from Hernández et al. (2015) and the energy limits E∞ (in eV) are taken from the NIST tabulations (Ralchenko et al., 2014). Transitions

Levels

µ

En (eV)

E∞ (eV)

f1

2

6

24.348 ± 0.013

26.060

− 0.642 ± 0.040

6

24.259 ± 0.013

25.974

− 0.651 ± 0.040

6

24.219 ± 0.013

25.936

− 0.657 ± 0.040

7

26.246 ± 0.013

27.522

− 0.862 ± 0.065

7

26.157 ± 0.013

27.435

− 0.872 ± 0.060

7

26.108 ± 0.013

27.398

− 0.932 ± 0.065

4

20.426 ± 0.013

24.054

− 0.197 ± 0.011

6

22.979 ± 0.013

24.615

− 0.403 ± 0.041

2 2 2

2 2

P2

° (3D5/2 ) nd

° (3D5/2 )6d

P1

° (3D5/2 ) nd 3 ° ( D5/2) nd ° (3P3/2 ) nd 3 ° ( P3/2) nd ° (3P3/2 ) nd 3 ° ( P1/2) nd ° (3P1/2 ) nd

° (3D5/2 )6d

P0 P2

P1 P0

1

S0

1

D2

° (3D5/2 )6d 3 ° ( P3/2)7d ° (3P3/2 )7d 3 ° ( P3/2)7d ° (3P1/2 )4d ° (3D5/2 )6d

113

Radiation Physics and Chemistry 153 (2018) 111–119

M.D. Ba et al.

Table 2 Resonance energies (E), quantum defect (δ) and effective charge (Z*) of the 3 ° ) ]nd Rydberg series originating from the 3s 23p4 3P2° ground state of [3s 23p (2D5/2 3 ° ) series limit of Cl2+. The present the Cl+ ions converging to the 3s 23p (2D5/2 SCUNC results are compared with the ALS measurements (Hernández et al., 2015) and with the Dirac – Coulomb R-matrix (DCR) calculations (McLaughlin, 2017). The experimental resonance energies are calibrated to ± 0.013 eV. The energy limits (E∞ = 26.060 eV) is taken from the NIST tabulations (Ralchenko et al., 2014). n

SCUNC E

DCR E

ALS E

|ΔΕ|a

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 … ∞

24.348 24.832 25.134 25.336 25.479 25.583 25.661 25.722 25.769 25.808 25.839 25.865 25.886 25.904 25.920 25.933 25.944 25.954 25.963 25.971 25.978 25.984 25.989 25.994 25.998 26.002 26.006 26.009 26.012 26.015 26.017 26.020 26.022 26.024 26.025 … 26.060

24.353 24.846 25.130 25.334 25.476 25.584

24.348 24.829 25.128 25.335 25.479 25.583

0.000 0.003 0.006 0.001 0.000 0.000

… 26.060

… 26.060

Table 3 Resonance energies (E), quantum defect (δ) and effective charge (Z*) of the 3 ° ) ]nd Rydberg series originating from the 3s 23p4 3P1° metastable [3s 23p (2D5/2 3 ° ) series limit of Cl2+. The state of the Cl+ ions converging to the 3s 23p (2D5/2 present SCUNC results are compared with the ALS experimental data (Hernández et al., 2015) and with the Dirac – Coulomb R-matrix (DCR) calculations (McLaughlin, 2017). The ALS experimental resonance energies are calibrated to ± 0.013 eV. The energy limits (E∞ = 25.974 eV) is taken from the NIST tabulations (Ralchenko et al., 2014).

SCUNC δ

DCR δ

ALS δ

SCUNC Za

n

SCUNC E

DCR E

ALS E

|ΔΕ|a

0.36 0.34 0.33 0.33 0.33 0.32 0.32 0.32 0.32 0.32 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31

0.30 0.35 0.30 0.34 0.35 0.30

0.38 0.35 0.36 0.34 0.32 0.32

24.259 24.735 25.039 25.243 25.387 25.493 25.572 25.633 25.681 25.720 25.752 25.777 25.799 25.817 25.833 25.846 25.858 25.868 25.877 25.884 25.891 25.897 25.903 25.908 25.912 25.916 25.920 25.923 25.926 25.929 25.931 25.933 25.936 25.938 25.939 … 25.974

24.259 24.750 25.036 25.238 25.384 (25.492)

0.000 0.015 0.003 0.005 0.003 0.001

…

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 … ∞

24.264 24.762 25.039 25.237 25.385 25.493

…

2.128 2.103 2.087 2.076 2.067 2.060 2.055 2.050 2.046 2.043 2.040 2.038 2.035 2.033 2.032 2.030 2.029 2.027 2.026 2.025 2.024 2.023 2.022 2.021 2.021 2.020 2.019 2.019 2.018 2.018 2.017 2.017 2.016 2.016 2.015 … 2.000

… 25.974

… 25.974

SCUNC δ

DCR δ

ALS δ

SCUNC Za

0.37 0.37 0.37 0.37 0.37 0.37 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 …

0.36 0.30 0.37 0.40 0.40 0.36

0.37 0.33 0.38 0.40 0.42 (0.37)

…

…

2.130 2.112 2.098 2.086 2.076 2.069 2.063 2.058 2.053 2.049 2.046 2.043 2.041 2.038 2.036 2.035 2.033 2.031 2.030 2.029 2.028 2.027 2.026 2.025 2.024 2.023 2.022 2.022 2.021 2.020 2.020 2.019 2.019 2.018 2.018 … 2.000

a energy difference with respect to the ALS measurements (Hernández et al., 2015).

a energy difference with respect to the ALS measurements (Hernández et al., 2015).

therefore positive according to the SCUNC analysis conditions (18). 2 4 1 2 3 2 ° S0 me1/2 ) ]nd series originating from the 3s 3p tastable state, the lowest resonance corresponds to the 3p → 4d (µlow = 4).

(McLaughlin, 2017). The data are quoted in Tables 2–9. Table 2 lists resonance energies, quantum defect and effective charge of the 3 ° ) ]nd Rydberg series originating from the 3s 23p4 3P2° ground [3s 23p (2D5/2

• for the [3s 3p ( P

3

° ) series limit of Cl2+. state of the Cl+ ion converging to the 3s 23p (2D5/2 The present SCUNC results are compared with the ALS measurements (Hernández et al., 2015) and with the DCR calculations (McLaughlin, 2017). Up to n = 11, it can be seen the very good agreement between the quoted data for both resonance energies and quantum defects. As far as resonance energies are concerned, the maximum energy difference between the SCUNC predictions and the ALS measurements is equal to 0.006 eV for the 8d level. For this level, the present SCUNC value is at 25.134 eV to be compared to the ALS data at 25.128 eV along with the DCR prediction equal to 25.130 eV. It should be underlined the excellent reproduction of the ALS measurements by the SCUNC formulae (21) for the two last quoted ALS data belonging to 10d and 11d levels. This may give credit to the accuracy of the present calculations for the high lying resonances n = 12–40 with a constant quantum defect. Table 3 lists resonance energies, quantum defect and effective 3 ° ) ]nd Rydberg series originating from the charge of the [3s 23p (2D5/2 2 4 3 ° 3s 3p P1 metastable state of the Cl+ ion converging to the

From Table 1 we pull f1 (2D5/2) = − 0.197. From Eq. (25), we de* as follows duce the expression of the effective charge Zmax

* =Z Zmax = 17

1+ 1+

° f1 (2P1/2 )

Z(

low

0.197 7(4 1)

1)

15.0 Z 15.0 17

= 2, 159.

(27)

* = 2.159 > Zcore. The quantum defect δ is As Zcore = 2.0, then Zmax consequently positive according to the SCUNC analysis conditions (18). It can be seen in all the Tables listed in this work, that both the ALS and DCR quantum defects are positive in agreement with the SCUNC analysis conditions (18). Let us now move on comparing the present predictions for the resonance energies with the only existing ALS measurements (Hernández et al., 2015) and with the DCR calculations 114

Radiation Physics and Chemistry 153 (2018) 111–119

M.D. Ba et al.

Table 4 Resonance energies (E), quantum defect (δ) and effective charge (Z*) of the 3 ° ) ]nd Rydberg series originating from the 3s 23p4 3P0° metastable [3s 23p (2D5/2 3 ° ) series limit of Cl2+. The state of the Cl+ ions converging to the 3s 23p (2D5/2 present SCUNC results are compared with the ALS measurements (Hernández et al., 2015) and with the Dirac – Coulomb R-matrix (DCR) calculations (McLaughlin, 2017). The experimental resonance energies are calibrated to ± 0.013 eV. The energy limits (E∞ = 25.936 eV) is taken from the NIST tabulations (Ralchenko et al., 2014). n

SCUNC E

DCR E

ALS E

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 … ∞

24.219 24.696 25.000 25.205 25.349 25.454 25.534 25.595 25.643 25.682 25.713 25.739 25.761 25.779 25.795 25.808 25.820 25.830 25.839 25.846 25.853 25.859 25.865 25.870 25.874 25.878 25.882 25.885 25.888 25.891 25.893 25.895 25.898 25.900 25.901 … 25.936

24.223 24.733 24.999 25.198 25.345

24.219 (25.000)

… 25.936

… 25.936

|ΔΕ|a 0.000 (0.000)

Table 5 Resonance energies (E), quantum defect (δ) and effective charge (Z*) of the 3 ° ) ]nd Rydberg series originating from the 3s 23p4 3P2° ground state of [3s 23p (2P3/2 3 ° ) series limit of Cl2+. The present the Cl+ ions converging to the 3s 23p (2P3/2 SCUNC results are compared with the ALS measurements (Hernández et al., 2015) and with the Dirac – Coulomb R-matrix (DCR) calculations (McLaughlin, 2017). The experimental resonance energies are calibrated to ± 0.013 eV. The energy limits (E∞ = 27.522 eV) is taken from the NIST tabulations (Ralchenko et al., 2014).

SCUNC δ

DCR δ

ALS δ

SCUNC Za

n

SCUNC E

DCR E

ALS E

|ΔΕ|a

0.37 0.38 0.38 0.37 0.37 0.37 0.37 0.37 0.37 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.35 …

0.36 0.27 0.38 0.40 0.40

0.37 (0.38)

25.757 26.246 26.561 26.773 26.922 27.031 27.112 27.175 27.225 27.264 27.296 27.323 27.345 27.364 27.379 27.393 27.405 27.415 27.424 27.432 27.439 27.445 27.450 27.455 27.460 27.464 27.467 27.471 27.474 27.476 27.479 27.481 27.483 27.485 27.487 … 27.522

(25.745) 26.246 26.564 26.778 26.928 27.031 (27.114) 27.175

(0.012) 0.000 0.003 0.005 0.006 0.000 (0.002) 0.000

…

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 … ∞

25.755 26.253 26.576 26.780 26.927 27.035 27.115 27.178

…

2.054 2.050 2.047 2.044 2.041 2.039 2.037 2.035 2.033 2.032 2.030 2.029 2.028 2.027 2.026 2.025 2.024 2.023 2.023 2.022 2.021 2.021 2.020 2.019 2.019 2.018 2.018 2.054 2.050 2.047 2.044 2.041 2.039 2.037 2.035 … 2.000

… 27.522

… 27.522

a energy difference with respect to the ALS measurements (Hernández et al., 2015).

SCUNC δ

DCR δ

ALS δ

SCUNC Za

0.45 0.47 0.47 0.48 0.48 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 …

0.46 0.45 0.45 0.44 0.44 0.43 0.44 0.42

(0.47) 0.47 0.46 0.45 0.43 0.47 (0.45) 0.48

…

…

2.065 2.061 2.057 2.054 2.051 2.048 2.046 2.043 2.041 2.040 2.038 2.037 2.035 2.034 2.033 2.032 2.030 2.029 2.029 2.028 2.027 2.026 2.025 2.025 2.024 2.023 2.065 2.061 2.057 2.054 2.051 2.048 2.046 2.043 2.041 … 2.000

a energy difference with respect to the ALS measurements (Hernández et al., 2015).

3

(Hernández et al., 2015) corresponding values are respectively equal to 24.129 eV and (25.000 eV). The DCR calculations (McLaughlin, 2017) have been performed up to n = 10. Comparison between theory and experiments shows that the uncertain ALS data at (25.000 eV) agrees excellently with the SCUNC prediction at 25.000 eV and with the DCR value at 24.999 eV. This may indicate the accuracy of the ALS measurement. For the remaining resonances relative to the 7d, 9d and 10d levels, the SCUNC predictions respectively at 24.696 eV, 25.205 eV and 25.349 eV compare well with the corresponding DCR predictions respectively at 24.733 eV, 25.198 eV and 25.345 eV. The energy differences between the calculations are equal to 0.037 eV, 0.007 eV and 0.004 eV respectively. These agreements may indicate that the SCUNC data are good representative for the missing ALS measurements and for the entire high lying n = 10–40 resonances for which the SCUNC quantum defect is seen to be constant along the series. Table 5 lists resonance energies, quantum defect and effective charge of the 3 ° ) ]nd Rydberg series originating from the 3s 23p4 3P2° ground [3s 23p (2P3/2

° 3s 23p (2D5/2 ) series limit of Cl2+. Here again, the agreement between theory and experiment is very good. For the 7d level, the SCUNC data at 24.735 eV differs to 0.015 eV to the ALS data at 24.750 eV. For this same level the energy difference between the DCR prediction at 24.762 eV and the ALS data is equal to 0.012 eV. Except for the 7d level, the maximum energy difference between the SCUNC predictions and the ALS measurements is less than 0.006 eV up to n = 11. For this last quoted data, it should be underlined the excellent agreement between the SCUNC and DCR predictions equal both to 25.493 eV compared to the uncertain ALS data at (24.492 eV). Subsequently, the ALS measurement for the 11d level may considered as accurate and equal to 24.492 eV with a quantum defect at 0.37 agreeing very well with both the SCUNC and DCR predictions respectively at 0.37 and 0.36. In addition, it can also be mentioned the constant SCUNC quantum defect with an average of 0.36 up to n = 40. Table 4 presents resonance en3 ° ) ]nd ergies, quantum defect and effective charge of the [3s 23p (2D5/2 2 4 3 ° Rydberg series originating from the 3s 3p P0 metastable state of the

3

° ) series limit of Cl2+. state of the Cl+ ion converging to the 3s 23p (2P3/2 The present SCUNC results are compared with the ALS measurements (Hernández et al., 2015) and with the DCR calculations (McLaughlin,

3

° ) series limit of Cl2+. For this Cl+ ion converging to the 3s 23p (2D5/2 Rydberg series, only two experimental data relative to the 6d and 8d levels have been measured in the Cl+ spectra. The ALS measurements

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Table 6 Resonance energies (E), quantum defect (δ) and effective charge (Z*) of the 3 ° ) ]nd Rydberg series originating from the 3s 23p4 3P1° metastable [3s 23p (2P3/2 3 ° ) series limit of Cl2+. The state of the Cl+ ions converging to the 3s 23p (2P3/2 present SCUNC results are compared with the ALS measurements (Hernández et al., 2015) and with the Dirac – Coulomb R-matrix (DCR) calculations (McLaughlin, 2017). The experimental resonance energies are calibrated to ± 0.013 eV. The energy limits (E∞ = 27.435 eV) is taken from the NIST tabulations (Ralchenko et al., 2014). n

SCUNC E

DCR E

ALS E

|ΔΕ|a

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 … ∞

25.667 26.157 26.473 26.685 26.834 26.943 27.025 27.088 27.137 27.177 27.209 27.236 27.258 27.276 27.292 27.306 27.318 27.328 27.337 27.345 27.351 27.358 27.363 27.368 27.373 27.377 27.380 27.384 27.387 27.389 27.392 27.394 27.396 27.398 27.400 … 27.435

25.668 26.162 26.480 26.695 26.842

(25.660) 26.157 26.475 26.687 26.833

(0.007) 0.000 0.002 0.002 0.001

… 27.435

… 27.435

Table 7 Resonance energies (E), quantum defect (δ) and effective charge (Z*) of the 3 ° ) ]nd Rydberg series originating from the 3s 23p4 3P0° metastable [3s 23p (2P3/2 3 ° ) series limit of Cl2+. The state of the Cl+ ions converging to the 3s 23p (2P3/2 present SCUNC results are compared with the ALS measurements (Hernández et al., 2015) and with the Dirac – Coulomb R-matrix (DCR) calculations (McLaughlin, 2017). The experimental resonance energies are calibrated to ± 0.013 eV. The energy limits (E∞ = 27.398 eV) is taken from the NIST tabulations (Ralchenko et al., 2014).

SCUNC δ

DCR δ

ALS δ

SCUNC Za

n

SCUNC E

DCR E

ALS E

|ΔΕ|a

0.45 0.47 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 …

0.45 0.46 0.45 0.43 0.42

(0.45) 0.47 0.47 0.46 0.49

25.612 26.108 26.428 26.643 26.793 26.903 26.986 27.049 27.099 27.139 27.171 27.198 27.220 27.239 27.255 27.268 27.280 27.290 27.299 27.307 27.314 27.320 27.326 27.331 27.335 27.339 27.343 27.346 27.350 27.352 27.355 27.357 27.359 27.361 27.363 … 27.398

26.108 26.430 26.648

0.000 0.002 0.005

…

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 … ∞

25.635 26.117 26.434 26.650 26.797

…

2.163 2.145 2.128 2.113 2.101 2.091 2.083 2.076 2.071 2.066 2.061 2.058 2.054 2.051 2.049 2.046 2.044 2.042 2.040 2.038 2.037 2.036 2.034 2.033 2.032 2.031 2.030 2.029 2.028 2.027 2.026 2.026 2.025 2.024 2.024 … 2.000

… 27.398

… 27.398

SCUNC δ

DCR δ

ALS δ

0.48 0.50 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 …

0.44 0.48 0.49 0.47 0.48

0.51 0.50 0.49

…

…

SCUNC Za 2.174 2.155 2.136 2.121 2.108 2.098 2.089 2.082 2.076 2.070 2.066 2.062 2.058 2.055 2.052 2.049 2.047 2.045 2.043 2.041 2.040 2.038 2.037 2.035 2.034 2.033 2.032 2.031 2.030 2.029 2.028 2.027 2.027 2.026 2.025 … 2.000

a energy difference with respect to the ALS measurements (Hernández et al., 2015).

a energy difference with respect to the ALS measurements (Hernández et al., 2015).

2017). Here, both the ALS measurements and the DCR calculations have been performed up to n = 13. For this Rydberg series, the f1 (2P3/ 2) screening constant has been evaluated from the first certain ALS data 26.246 eV corresponding to the 7d level. The SCUNC prediction for the upper uncertain 6d ALS resonance at (25.745 eV) is equal to 25.757 eV to be compared to the DCR result at 25.755 eV. The energy difference between the calculations is equal to 0.002 eV. The ALS data still then uncertain and the SCUNC prediction may be used as the precise experimental representative data. For the n = 8–13 levels, the agreements between theory and experiments are seen to be very good. Here, the maximum energy difference between the SCUNC predictions and the ALS measurements is equal to 0.006 eV for the 10d level. For the 11d uncertain ALS resonance at (27.114 eV), the predictions from the SCUNC and DCR calculations are respectively equal to 27.112 eV and 27.115 eV. But for the 11d level and the 13d level, the SCUNC data and the ALS measurements are both equal to 27.031 eV and 27.175 eV. This may indicate that for the intermediate 12d level, the uncertain ALS resonance at (27.114 eV) may be equal to 27.112 eV as predicted by the SCUNC calculations. The high lying n = 14–40 resonances are then expected to be very precise with an average constant quantum defect at

0.4676 ≈ 0.47 agreeing excellently with the 0.47 ALS measurements for the first certain ALS data at 26.246 eV used to evaluate the f1 (2P3/2) screening constant. Table 6 quotes resonance energies, quantum defect 3 ° ) ]nd Rydberg series originating and effective charge of the [3s 23p (2P3/2 2 4 3 ° from the 3s 3p P1 metastable state of the Cl+ ion converging to the 3

° 3s 23p (2P3/2 ) series limit of Cl2+. Both the ALS measurements (Hernández et al., 2015) and the DCR calculations (McLaughlin, 2017) have been performed up to n = 10. The agreements between experiments and theory is very good for the n = 7–10. Here again, the f1 (2P3/ 2) screening constant has been evaluated from the first certain ALS data 26.157 eV corresponding to the 7d level. The SCUNC prediction for the upper uncertain 6d ALS resonance at (25.660 eV) is equal to 25.667 eV compared to the DCR data at 25.668 eV. The energy difference between the calculations is equal to 0.001 eV. The ALS data still then uncertain and the SCUNC prediction may be used as the precise experimental representative data. For the n = 8–10 levels, the agreements between theory and experiments are seen to be very good. Here, the maximum energy difference between the SCUNC predictions and the ALS measurements is equal to 0.002 eV for the 9d level with an energy difference at 0.001 eV for the last 10d state. The high lying n = 11–40

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Table 8 Resonance energies (E), quantum defect (δ) and effective charge (Z*) of the 3 ° ) ]nd Rydberg series originating from the 3s 23p4 1S0° metastable [3s 23p (2P1/2 3 ° ) series limit of Cl2+. The state of the Cl+ ions converging to the 3s 23p (2P1/2 present SCUNC results are compared with the ALS measurements (Hernández et al., 2015) and with the Dirac – Coulomb R-matrix (DCR) calculations (McLaughlin, 2017). The experimental resonance energies are calibrated to ± 0.013 eV. The energy limits (E∞ = 24.054 eV) is taken from the NIST tabulations (Ralchenko et al., 2014). n

SCUNC E

DCR E

ALS E

|ΔΕ|a

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 … ∞

20.426 21.748 22.475 22.903 23.178 23.364 23.497 23.595 23.669 23.726 23.772 23.808 23.838 23.863 23.884 23.901 23.916 23.929 23.940 23.950 23.959 23.966 23.973 23.979 23.984 23.989 23.993 23.997 24.000 24.004 24.007 24.009 24.012 24.014 24.016 24.018 24.020 … 24.054

20.420 21.734 22.459 22.890 23.170 23.361

20.426 21.742 22.463 22.900 23.178

0.000 0.006 0.012 0.003 0.000

… 24.054

… 24.054

SCUNC δ

DCR δ

ALS δ

SCUNC Za

0.13 0.14 0.13 0.12 0.12 0.12 0.12 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 …

0.13 0.16 0.16 0.16 0.15 0.14

0.13 0.15 0.15 0.13 0.12

2.066 2.062 2.045 2.036 2.031 2.027 2.024 2.021 2.019 2.018 2.016 2.015 2.014 2.013 2.012 2.012 2.011 2.011 2.010 2.010 2.009 2.009 2.008 2.008 2.008 2.008 2.007 2.007 2.007 2.007 2.006 2.006 2.006 2.006 2.006 2.006 2.005 … 2.000

…

…

Table 9 Resonance energies (E), quantum defect (δ) and effective charge (Z*) of the 3 ° ) ]nd series originating from the 3s 23p4 1D2 metastable state of the [3s 23p (2D5/2 3 ° ) series limit of Cl2+. The present SCUNC Cl+ ions converging to the 3s 23p (2D5/2 results are compared with the ALS measurements (Hernández et al., 2015) and with the Dirac – Coulomb R-matrix (DCR) calculations (McLaughlin, 2017). The experimental resonance energies are calibrated to ± 0.013 eV. The energy limits (E∞ = 24.615 eV) is taken from the NIST tabulations (Ralchenko et al., 2014). n

SCUNC E

DCR E

ALS E

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 … ∞

22.979 23.426 23.713 23.907 24.045 24.146 24.222 24.281 24.328 24.366 24.396 24.422 24.443 24.461 24.476 24.489 24.500 24.510 24.519 24.526 24.533 24.539 24.544 24.549 24.554 24.558 24.561 24.564 24.567 24.570 24.572 24.575 24.577 24.579 24.581 … 24.615

22.969 23.392 23.719 23.907 24.039 24.138

22.979

… 24.615

… 24.615

|ΔΕ|a

23.718 23.907 24.040 (24.152)

0.000 0.005 0.000 0.005 (0.006)

SCUNC δ

DCR δ

ALS δ

SCUNC Za

0.23 0.23 0.23 0.23 0.23 0.23 0.23 0.23 0.23 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 …

0.25 0.32 0.21 0.23 0.28 0.32

0.27

…

…

2.081 2.069 2.060 2.053 2.047 2.042 2.039 2.035 2.033 2.030 2.028 2.027 2.025 2.024 2.022 2.021 2.020 2.019 2.018 2.018 2.017 2.016 2.016 2.015 2.015 2.014 2.014 2.013 2.013 2.013 2.012 2.012 2.011 2.011 2.011 … 2.000

0.21 0.23 0.27 (0.27)

a energy difference with respect to the ALS measurements (Hernández et al., 2015).

a

energy difference with respect to the ALS measurements (Hernández et al., 2015).

3

° ) ]nd Rydberg series originating from effective charge of the [3s 23p (2P1/2 2 4 1 ° the 3s 3p S0 metastable state of the Cl+ ion converging to the

resonances are then expected to be very precise with an average constant quantum defect at 0.4729 ≈ 0.47 agreeing excellently with the 0.47 ALS measurements for the first certain ALS data at 26.157 eV used to evaluate the f1 (2P3/2) screening constant. Table 7 presents resonance 3 ° ) ]nd energies, quantum defect and effective charge of the [3s 23p (2P3/2 Rydberg series originating from the 3s 23p4 3P0° metastable state of the

3

° 3s 23p (2P1/2 ) series limit of Cl2+. In general, the agreements between theory and experiments are very good for n = 4 – 8. It should be underlined the large discrepancy between the SCUNC prediction and ALS data equal to 0.012 eV for the 6d state. But, for the last state 8d, both the SCUNC predictions and the ALS data are at 23.178 eV to be compared to the DCR calculations equal to 23.170 eV. Table 9 lists resonance energies, quantum defect and effective charge of the 3 ° ) ]nd series originating from the 3s 23p4 1D2 metastable state [3s 23p (2D5/2

3

° ) series limit of Cl2+. The present Cl+ ion converging to the 3s 23p (2P3/2 SCUNC results are compared with the ALS measurements (Hernández et al., 2015) and with the DCR calculations (McLaughlin, 2017). Here, only three resonances have been measured for n = 7–10. The maximum energy difference between the SCUNC predictions and the ALS measurements is equal to 0.005 eV for the 9d resonance. The high lying n = 11–40 resonances are then expected to be very precise with an average constant quantum defect at 0.503 ≈ 0.50 agreeing very well with the 0.51 ALS measurements for the first certain ALS data at 26.108 eV used to evaluate the f1 (2P3/2) screening constant. As far as comparison between theories is concerned, the agreements are seen to be very good. Table 8 lists resonance energies, quantum defect and

3

° ) series limit of Cl2+. The of the Cl+ ion converging to the 3s 23p (2D5/2 present SCUNC results are compared with the ALS measurements (Hernández et al., 2015) and with DCR calculations (McLaughlin, 2017) for the resonances n = 6–10. For this series, the maximum energy difference between the SCUNC predictions and the ALS measurements is equal to (0.006 eV) for the last 11d level. For this resonance the uncertain ALS resonance at (24.152 eV) be compared to the SCUNC and DCR results at 24.146 eV and 24.138 eV. The ALS data may be precise. For the high lying n = 12–40 resonances, the quantum defect is

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practically constant and equal to 0.22. AS far as the Z*- effective charge is concerned, for all the resonances investigated, it can be seen that Z* tends toward the electric charge of the core ion Zcore = 2.000 with increasing n accordingly to the SCUNC analysis conditions (9).

by Unit Nuclear Charge (SCUNC) method up to n = 40. Excellent agreements are obtained between the present predictions and previous studies from Advanced Light Source experiments at Lawrence Berkeley National Laboratory (Hernández et al., 2015) and calculations from Dirac–Coulomb R-matrix (McLaughlin, 2017). The very good results obtained in this work show that the SCUNC-method can be used to assist the sophisticated R-matrix-method for locating and determining the properties of atomic resonances. Finally, our predicted data up to n = 40 may be of great importance for the atomic physics community in connection with the determination of accurate abundances for sulphurlike chlorine for understanding extragalactic HII regions, optical spectra of planetary nebulae NGC 6741 and IC 5117 and spectra of the Io torus.

4. Summary and conclusion The second calculations of resonance energies and quantum defects 3 3 ° ° ) ]nd Rydberg series origi) ]nd and [3s 23p (2P3/2 of the [3s 23p (2D5/2 2 4 3 nating from the ground 3s 3p P2 and from the 3s 23p4 3P1,0 metastable 3

° ) ]nd Rydberg series originating states of Cl+ along with the [3s 23p (2P1/2 2 4 1 from the 3s 3p S0 metastable state of the Cl+ have been investigated. Calculations are performed in the framework of the Screening Constant

Appendix. Detailed processes to evaluate empirically the screening constants fk In the framework of the Screening constant by unit nuclear (SCUNC) method, the screening constants fk are evaluated from experimental values. They are then determined empirically with a certain absolute error linked to the experimental measurement errors. We move on explaining in detail the principle for determining the absolute values, Δfk. Within the framework of the SCUNC formalism, the screening constants, fk, are presented as fk = fkexp ± Δfk. The absolute errors, Δfk are given by Sakho (2018)

(fk

fk =

f k+ )2 + (fk

f k )2

.

2

(28)

In general, the experimental resonance energies are expressed in the form En = Eexp ± ΔE, with ΔE the absolute error on the resonance energies. The f k± screening constants are evaluated using the experimental resonance energies for n = ν and n = µ for the nl 2S+1LJ Rydberg series considered. As a result, the corrective term in Eq. (14) is equal to zero and we obtain (in Rydberg)

f1 (2S + 1LJ )

Z2 1 n2

En = E

Z (n

2

f2 (2S + 1LJ )

1)

.

Z

(29)

In the present work, only f1 is to be evaluated as f2 = 15.0. In this case, one equation is required to find the value of f1 in Eq. (29) using the following relations for n = µ

E + = Eexp + E

;

E = Eexp

(30)

E. 3 2

Let us then apply Eq. (30) to evaluate both f1 and Δf1 considering for example the [3s 3p ( ground state. For these states, the resonance energies are given by Eq. (21) reminded below 2

Z2 n2 2

En = E

° f1 (2D5/2 )

1

Z (n

15.0 + Z Z 2 (n

1)

° f1 (2D5/2 )(n

)

+ s + 1)(n

+ s + 3)

+

° D5/2 ) ]nd

Rydberg series originating from the 3s 3p

° f1 (2D5/2 )(n

Z 3 (n

2

P2

2

)

+ s + 1)(n

4 3

+ s)

.

3

° ° ) . The first entry for the [3s 23p (2D5/2 ) ]nd is n = µ = 6. As Z = 17 the Cl+ ion, Eq. (21) above takes For sake of simplification we put f1 = f1 (2D5/2 the form

172 1 62

E6 = E

f1 17 × (5

15.0 17

1)

2

× 13, 60569 = E

172 1 62

f1 68

15.0 17

2

× 13, 60569.

(31)

In Table 1 we pull the experimental resonance energy E6 = (24.348 ± 0.013) eV n = µ = 6 from Hernández et al. (2015) along with the energy limits E∞ = 26.060 eV taken from the NIST tabulations (Ralchenko et al., 2014). Using Eqs. (30) and (31), we find 172 62

24.348 = 26.060

{1

}

15.0 2 17

f1 68

24.348 + 0.013 = 26.060

172 62

{1

68

24.348

172 62

{1

68

0.013 = 26.060

f1+ f1

× 13, 60569 15.0 17

} × 13, 60569. 2

}

15.0 2 17

× 13, 60569

Simplifying these equations, we get f1

1

68 f1+

1 1

68 f1 68

15.0 17

= 0.125197

15.0 17 15.0 17

= 0.124720755

= 0.12567144

f1 = f1+ = f1 =

0.641745 0.601264175 . 0.6820724

(32)

Using the results (32) and Eq. (28), the absolute error, Δf1 is equal to

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M.D. Ba et al.

f1 =

(

0.641745 + 0.601264175)2 + ( 2

0.641745 + 0.6820724) 2

= 0.040404184.

The empirical screening constant, f1 = − 0.641745, is then presented as f1 = − 0.642 ± 0.040. The other absolute errors, Δfk for the remaining series are evaluated similarly. One find then the data quod in Table 1.

Küppers, M.E., Schneider, N.M., 2000. Geophys. Res. Lett. 25, 513. McLaughlin, B.M., 2017. MNRAS 464, 1990. Ralchenko, Y., Kramida, A.E., Reader, J., 2014. NIST Atomic Spectra Database (Version 5.2). National Institute of Standards and Technology, Gaithersburg, MD. Available at: 〈http://physics.nist.gov/〉. Sakho, I., 2016. At. Data Nucl. Data Tables 108, 57. Sakho, I., 2017. J. Electron Spectrosc. Relat. Phenom. 222, 40. Sakho, I., 2018. The Screening Constant by Unit Nuclear Charge Method. Description & application to the photoionization of atomic systems ISTE Science Publishing Ltd, London and John Wiley & Sons, Inc, USA.

References Covington, A.M., et al., 2011. Phys. Rev. A 84, 013413. Garnett, 1989. Astrophys. J. 345, 771. Goyal, A., Khatri, I., Sow, M., Sakho, I., Aggarwal, S., Singh, A.K., Mohan, M., 2016. Radiat. Phys. Chem. 125, 50. Hernández, et al., 2015. J. Quant. Spectrosc. Radiat. Transf. 151, 217. Keenan, et al., 2003. Astrophys. J. 543, 385. Khatri, I., Goyal, A.1, Ba, M.D., Faye, M., Sow, M., Sakho, I., Aggarwal, S., Singh, A.K., Mohan, M., Wagué, A., 2016. Radiat. Phys. Chem. 130, 208.

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