Photoionization study of the F2+ ion via the screening constant by unit nuclear charge method

Radiation Physics and Chemistry 96 (2014) 38–43

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Review

Photoionization study of the F2 þ ion via the screening constant by unit nuclear charge method M. Dieng a, M. Tine b, M. Sow a, B. Diop a, M. Guèye a, M. Faye a, I. Sakho b,n, M. Biaye c, A. Wagué a a

Department of Physics, Atoms Laser Laboratory, Faculty of Sciences and Technologies, University Cheikh Anta Diop, Dakar, Senegal Department of Physics, UFR of Sciences and Technologies, University Assane Seck of Ziguinchor, Ziguinchor, Senegal c Department of Physics and Chemistry, Faculty of Sciences and Technologies of Formation and Education, University Cheikh Anta Diop, Dakar, Senegal b

H I G H L I G H T S

    

Accurate energy positions of the 2s22p2 (1D)nd, 2s22p2 (1S)nd and 2s22p3 (3D)np Rydbergs states of F2 þ . Present results agree very well with Advanced Light Source data on F2 þ (Aguilar et al., 2005). Current quantum defects almost constants upto n ¼30. Effective charge decreases regularly toward the electric charge of the F3 þ core ion. Presented results may be useful guideline for photoionization studies on F2 þ ion.

art ic l e i nf o

a b s t r a c t

Article history: Received 25 March 2013 Accepted 12 August 2013 Available online 21 August 2013

In this paper, we have tabulated energy resonances of the 2s22p2 (1D)nd (2L), 2s22p2 (1S)nd (2L) and 2s22p3 (3D)np Rydberg series originating from the 2s22p3(2Po) and from the 2s22p3 (2Do) metastable states of F2 þ . In addition, energy resonances of the 2s2p3(5So)np (4P) Rydberg series originating from the 2s22p3 (4So) ground-state of F2 þ are also reported. Calculations are performed using the Screening constant by unit nuclear charge (SCUNC) method. Analysis of the present data is achieved by calculating the quantum defects and the effective nuclear charges for each series. The present results agree very well with the Advanced Light Source experiments on F2 þ (Aguilar et al., 2005). Upto n ¼30, the present quantum defects are almost constant and the effective charge decreases regularly toward the electric charge of the F3 þ core ion along each series. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Semi-empirical and empirical calculations Screening constant by unit nuclear charge Electron correlation calculations for atoms and ions Rydbergs states

1. Introduction One of the fundamental processes playing an important role in laboratory and astrophysical plasmas is the photoionization of atoms and ions. Therefore, it is an imperative task for physicists to provide accurate photoionization data for the modeling of astrophysical and laboratory plasmas. For such modeling, photoionization data have been generally provided by calculations because no experimental data were available (Kim and Manson, 2010). In the last decade, the use of merged beam facilities (Bizau et al., 2000; Covington et al., 2001) has enabled the measurements of total cross sections for many ions. Of great interest are the N-like ions for which photoabsorption from low-lying metastable states of open-shell nitrogen-like ions has been shown to be important in

n

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

0969-806X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.radphyschem.2013.08.006

the earth's upper atmosphere (Meier, 1991) as well as in astrophysical plasmas (Raju and Dwivedi, 1990). Recently, absolute photoionization measurements have been reported for admixtures of the ground and metastable states of F2 þ from 56.3 eV to 75.6 eV, and of Ne3 þ from 89.3 eV to 113.8 eV (Aguilar et al., 2005). These experiments were carried out at the Advanced Light Source (ALS) synchrotron radiation facility in Berkeley. For comparison with high-resolution measurements, state-ofthe-art-theoretical methods are required using highly correlated wave functions including relativistic effects (Covington et al., 2011). Among the ab initio methods applied in the photoionization studies of atoms and ions are the Hartree–Fock multi-configurationnal (MCDF) method (Bruneau, 1984; Simon et al., 2010), the Quantum Defect Theory (Dubau and Seaton., 1984), the R-matrix approach (Liang et al., 2013) widely used for international collaborations such as the Opacity Project (Seaton, 1987; Hofmann, 1990), the MultiConfiguration Dirac–Hartree–Fock method (MCDHF) (Grant, 2007), in the form of the grasp2k relativistic atomic structure package

M. Dieng et al. / Radiation Physics and Chemistry 96 (2014) 38–43

(Jönsson et al., 2007). The MCDHF has been used to compute with high precision the 2 3Pl–23Po separation energy, including relativistic contributions to electron–electron correlations and radiative corrections (Indelicato et al., 1989). In addition, recent excitation energies and lifetimes for Zn-like sequence from Z¼48 to 54 (Feng et al., 2011) and spectral properties of Sb IV (Jönsson et al., 2012) have been reported from MCDHF calculations. In the works of Aguilar et al. (2005), the R-matrix method has been used in the calculations for single photoionization of the N-like ions investigated. In the present paper, we intend to provide accurate data on the photoionization of N-like ions that may be useful guideline for the physical atomic community. In addition, we aim to demonstrate the possibilities to use the Screening constant by unit nuclear charge method (Sakho, 2012; Sakho et al., 2013; Faye et al., 2013; Diop et al., 2013) to reproduce excellently experimental data from merged beam facilities. For this purpose, we report calculations of energy resonances of the 2s22p2 (1D)nd (2L), 2s22p2 (1S)nd (2L), and 2s22p3 (3D)np Rydberg series originating from the 2s22p3(2Po) and 2s22p3 (2Do) metastable states of F2 þ . In addition, energy resonances of the 2s2p3(5S0)np (4P) Rydberg series originating from the 2s22p3 (4So) ground state of F2 þ are also reported. Calculations are performed using the SCUNCmethod and analysis of the data tabulated is achieved via the SCUNC procedure along with the quantum defect theory. Section 2 presents the theoretical procedure adopted in this work. In Section 3, we present and discuss the results obtained along with comparison with the only available experimental data (Aguilar et al., 2005).

39

2.2. Energy resonances of the 2s22p2 (1D)nd (2L) and 2s22p2 (1S)nd (2L) Rydberg series originating from the 2s22p3(2Po) metastable state Using Eqs. (1) and (2), the energy resonances of the 2s22p2 (1D) nd (2L) and 2s22p2 (1S)nd (2L) Rydberg series originating from the 2s22p3(2Po) metastable state of F2 þ are given by

 For the 2s22p3(2Po)-2s22p2 (1D)nd (2L) transitions ( Z 20 f ð1 DÞ f 2 ð1 DÞ f 1 ð1 DÞ En ¼ E1  2 1 1   Z 0 ðn1Þ Z0 n Z 20 ðnνÞ  ðnmÞ f ð1 DÞ þ 1 3  ðnνs þ 2Þðnμs þ 2Þ Z0 2 ðnνÞ  ðnmÞ  ðnνs þ 2Þðnμs þ 2Þ

ð3Þ

From Aguilar et al. (2005), we get (in eV) E7 ¼56.914 70.015 (ν ¼ 7) and E8 ¼ 57.509 7 0.015 (m ¼8) and E1 ¼59.445. Using these data, Eq.(3) gives f1 ¼  0.058804866; f2 ¼ 5.990656816. Taking into account the uncertainties in the energy positions, we obtain Δf1 ¼0.070 and Δf2 ¼0.009. Then we get finally f1 ¼  0.05970.070 and f2 ¼5.991 70.009.

 For the 2s22p3(2Po)-2s22p2 (1S)nd (2L) transitions

( Z 20 f ð1 SÞ f 2 ð1 SÞ f 1 ð1 SÞ  1 1 þ  ðnνÞ Z 0 ðn1Þ Z0 n2 Z 20   1 1 ðnmÞ þ ðnmsþ 3Þðn þ νmsÞ ðnν2s þ3Þðnms þ3Þ )2 f 1 ð1 SÞ ðnνÞ  ðnmÞ þ  : ð4Þ ðnνs þ 3Þðnms þ 3Þ Z 30

En ¼ E1 

2. Theory 2.1. Brief description of the SCUNC formalism In the framework of the Screening Constant by Unit Nuclear π Charge formalism, total energy of the ðNl; nl′Þ2S þ 1 L excited states is expressed in the form (in Rydberg) π

EðNl; nl′; 2S þ 1 L Þ ¼ Z 2




 1 1 2S þ 1 π 2 : þ ½1βðNl; nl′; L ; ZÞ N 2 n2

In this equation, the principal quantum numbers N and n are respectively for the inner and the outer electron of the heliumisoelectronic series. For a given Rydberg series originating from a-2S þ 1LJ state, we obtain for the energy resonance En the general following expression (Sakho, 2012) Z2 En ¼ E1  02 f1βðZ 0 ; 2S þ 1 LJ ; n; s; μ; νÞg2 : n

ð1Þ

In Eq. (1), ν and m (m4 ν) denote the principal quantum numbers of the (2S þ 1LJ)-nl Rydberg series used in the empirical determination of the fi-screening constants, s represents the spin of the nl-electron (s¼ ½), E1 is the energy value of the series limit, and Z0 stands for the nuclear charge. The β-parameters are screening constants by unit nuclear charge expanded in inverse powers of Z0 and given by (Sakho, 2012) q

βðZ 0 ; 2S þ 1 LJ ; n; s; μ; νÞ ¼ ∑ f k k¼1




1 Z0

where f k ¼ f k ð2S þ 1 LJ ; n; s; μ; νÞ evaluated empirically.

k ð2Þ are screening constants to be

From Aguilar et al. (2005), we obtain (in eV) E5 ¼57.871 70.015 (ν ¼5) and E6 ¼59.437 70.015 (m¼ 6) and E1 ¼ 62.954. Taking into account these data, Eq.(4) provides f1 ¼  0.111 70.070 and f2 ¼ 5.972 70.009. 2.3. Energy resonances of the 2s22p2(1D)nd (2L) and 2s22p3 (3D)np Rydberg series originating from the 2s22p3 (2Do) metastable state For the 2s22p2(1D)nd (2L) and 2s22p3 (3D)np Rydberg series originating from the 2s22p3 (2Do) metastable state, the energy resonance is given by

 For the 2s22p3 (2Do)-2s22p2(1D)nd (2L) transitions En ¼ E1 

( Z 20 f ð1 DÞ f 2 ð1 DÞ f 1 ð1 DÞ ðnνÞ  ðnmÞ  1 1  2 Z 0 ðn1Þ Z0 Z 0 ðn þ νmÞðn þ νmsÞ n

f 1 ð1 DÞ ðnνÞ  ðnmÞ Z 20 ðn þ νmsÞðn þ νms1Þ 2

þ

f ð1 DÞ ðnνÞ  ðnmÞ  1 3  ðn þ νmsÞðn þ νms1Þ Z0 2

)2 :

ð5Þ

Using the experimental data (in eV) of Aguilar et al. (2005) E8 ¼59.6617 0.015 (ν ¼8) and E9 ¼ 60.074 70.015 (m¼ 9) along with E1 ¼ 61.609, we obtain from Eq. (5) f1 ¼  0.23 70.06 and f2 ¼6.007 0.01.

 For the 2s22p3 (2Do)-2s2p3 (3D)np transitions En ¼ E1 

( Z 20 f ð3 DÞ f 2 ð3 DÞ  1 1 2 Z 0 ðn1Þ Z0 n

40

M. Dieng et al. / Radiation Physics and Chemistry 96 (2014) 38–43

f ð3 DÞ ðnvÞ  ðnμÞ f 1 ð3 DÞ ðnvÞ  ðnμÞ  1 2 þ  ðn þ vμ2sÞ2 Z 0 ðn þ vμs1Þ3 Z 30 )2 f ð3 DÞ ðnvÞ  ðnμÞ þ 1 4  ðn þ vμs3Þ3 Z0

standard quantum-defect expansion formula expressed as follows:

ð6Þ

Using (in eV) E4 ¼67.2197 0.015 (ν ¼ 4) and E5 ¼70.98470.015 (m ¼5) and E1 ¼76.815 (Aguilar et al., 2005), Eq.(6) provides f1 ¼  1.032 70.070 and f2 ¼5.985 70.009. 2.4. Energy resonances of the 2s2p3(5S0)np (4P) Rydberg series originating from the 2s22p3 (4So) ground state For the 2s2p3(5S0)np (4P) Rydberg series originating from the 2s22p3 (4So) ground state of F2 þ , the energy resonance is given by ( Z 20 f ð5 SÞ f 2 ð5 SÞ f 1 ð5 SÞ  En ¼ E1  2 1 1 þ  ðnνÞ Z 0 ðn1Þ Z0 n Z 20  1 ðnmÞ ðn þ νmsÞðn þ νms2Þ ) f ð5 SÞ ðnνÞ  ðnmÞ3 þ 1 4  2: ð7Þ ðn þ νm2s1Þðn þ 2ν þ 2ms2Þ2 Z0 From Aguilar et al. (2005), we get (in eV) E5 ¼66.319 70.015 (ν ¼5) and E6 ¼68.062 70.015 (m ¼6) and E1 ¼ 71.907. Using these data, Eq.(7) gives f1 ¼  0.798 70.060 and f2 ¼5.700 70.010.

3. Results and discussions The results obtained in this work are listed in Tables 1–5 and comparison is done with the only available literature data (Aguilar et al., 2005). The energy resonances are analyzed using the Table 1 Energy resonances (E), quantum defect (δ) and effective nuclear charge (Z*) for the 2s22p2(1D)nd (2L) Rydberg series originating from the 2s22p3(2Po) metastable state of F2 þ . The present results (SCUNC) are compared to the Advanced Light Source (ALS) experiments of Aguilar et al. (2005). SCUNC

ALS

n

E (eV)

E (eV)

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 … 1

56.914 (10) 57.509 (8) 57.918 (7) 58.210 (6) 58.425 (6) 58.589 (5) 58.716 (4) 58.816 (4) 58.898 (3) 58.964 (3) 59.019 (3) 59.065 (2) 59.104 (2) 59.138 (2) 59.166 (2) 59.191 (2) 59.213 (2) 59.232 (1) 59.248 (1) 59.263 (1) 59.276 (1) 59.288 (1) 59.299 (1) 59.308 (1) … 59.445

56.914 57.509 57.908 58.210 58.427 58.592 58.722 58.822 58.900

(15) (15) (15) (15) (15) (15) (15) (15) (15)

RZ 2 En ¼ E1  core2 ðnδÞ

ð8Þ

In this equation, R, E1, Zcore and δ are respectively the Rydberg constant, the converging limit, the electric charge of the core ion and the quantum defect. For all the Rydberg series investigated, the present energy resonances quoted in Tables 1–5 agree very well with the experimental data (Aguilar et al., 2005) up to n ¼17. If the present quantum defects are almost constant along the series, it is seen that the experimental quantum defects vary notably with a minimum of  0.022 for n ¼14 (Table 1) and  0.048 for n ¼17 (Table 3). Let us move on analyzing the present data in the framework of the SCUNC formalism by calculating the effective nuclear charge Zn using the formula n

Z2 En ¼ E1  2 : n

ð9Þ

In this equation, the effective nuclear charge is given by Z n ¼ Z 0 f1F½f i ð1 L2 ; 2 P j Þ; n; ν; μ; s; Z 0 g:

ð10Þ

Eq. (10) is defined for each series. For instance, for the 2s22p3(2Po)-2s22p2 (1D)nd (2L) transitions, the effective nuclear charge is in the form using Eq. (3) ( f ð1 DÞ f 2 ð1 DÞ f 1 ð1 DÞ ðnνÞ  ðnmÞ    Z n ¼ Z 0 1 1 Z 0 ðn1Þ Z0 ðnνs þ 2Þðnμs þ 2Þ Z 20 2 f ð1 DÞ ðnνÞ  ðnmÞ þ 1 3  : ð11Þ ðnνs þ 2Þðnμs þ2Þ Z0 Table 2 Energy resonances (E), quantum defect (δ) and effective nuclear charge (Z*) for the 2s22p2(1S)nd (2L) Rydberg series originating from the 2s22p3(2Po) metastable state of F2 þ . The present results (SCUNC) are compared to the Advanced Light Source (ALS) experiments of Aguilar et al. (2005). SCUNC

ALS

SCUNC

ALS

SCUNC

ALS

n

E (eV)

E (eV)

|ΔE|

δ

δ

Zn

Zn

0.000 0.000 0.010 0.000 0.002 0.003 0.006 0.006 0.002

0.044 0.047 0.045 0.044 0.043 0.042 0.042 0.042 0.042 0.043 0.043 0.043 0.044 0.044 0.045 0.045 0.046 0.046 0.047 0.047 0.048 0.049 0.049 0.050

0.042 0.046 0.075 0.044 0.035 0.018  0.016  0.022 0.012

3.019 3.018 3.015 3.013 3.012 3.011 3.010 3.009 3.009 3.008 3.008 3.007 3.007 3.007 3.006 3.006 3.006 3.006 3.006 3.005 3.005 3.005 3.005 3.005

3.019 3.018 3.025 3.013 3.009 3.005 2.997 2.996 3.002

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 … 1

57.871 (14) 59.433 (12) 60.382 (11) 60.992 (9) 61.408 (8) 61.705 (7) 61.923 (6) 62.089 (5) 62.218 (5) 62.320 (4) 62.402 (4) 62.470 (3) 62.525 (3) 62.572 (3) 62.611 (3) 62.645 (2) 62.674 (2) 62.699 (2) 62.720 (2) 62.740 (2) 62.756 (2) 62.771 (1) 62.785 (1) 62.797 (1) 62.807 (1) 62.817 (1) … 62.954

57.871 59.437 60.371 60.988 61.411 61.921

SCUNC

(15) (15) (15) (15) (15) (15)

ALS

SCUNC n

|ΔE|

δ

δ

Z

0.000 0.004 0.011 0.004 0.003 0.216

0.092 0.099 0.100 0.100 0.100 0.099 0.099 0.099 0.099 0.100 0.100 0.100 0.100 0.101 0.101 0.102 0.102 0.103 0.103 0.104 0.104 0.105 0.105 0.106 0.106 0.107

0.092 0.100 0.115 0.107 0.100 0.114

3.056 3.051 3.044 3.038 3.034 3.030 3.027 3.025 3.023 3.021 3.020 3.019 3.018 3.017 3.016 3.015 3.015 3.014 3.014 3.013 3.013 3.012 3.012 3.011 3.011 3.011

ALS Zn 3.055 3.050 3.050 3.041 3.031 2.755

M. Dieng et al. / Radiation Physics and Chemistry 96 (2014) 38–43

Table 3 Energy resonances (E), quantum defect (δ) and effective nuclear charge (Z*) for the 2s22p2(1D)nd (2L) Rydberg series originating from the 2s22p3(2Do) metastable state of F2 þ . The present results (SCUNC) are compared to the Advanced Light Source (ALS) experiments of Aguilar et al. (2005). SCUNC

ALS

n

E (eV)

E (eV)

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 … 1

59.661 (8) 60.074 (7) 60.368 (6) 60.584 (5) 60.749 (4) 60.877 (3) 60.978 (3) 61.059 (3) 61.126 (2) 61.182 (2) 61.228 (2) 61.267 (2) 61.301 (2) 61.329 (1) 61.354 (1) 61.376 (1) 61.395 (1) 61.412 (1) 61.427 (1) 61.440 (1) 61.452 (1) 61.463 (1) 61.472 (1)

59.661 60.074 60.371 60.586 60.751 60.877 60.988 61.064 61.129 61.188

SCUNC

(15) (15) (15) (15) (15) (15) (15) (15) (15) (15)

ALS

SCUNC n

|ΔE|

δ

δ

Z

0.000 0.000 0.003 0.002 0.002 0.000 0.010 0.005 0.003 0.006

0.072 0.068 0.068 0.068 0.069 0.070 0.071 0.072 0.073 0.074 0.074 0.074 0.074 0.074 0.074 0.074 0.073 0.073 0.072 0.072 0.071 0.070 0.071

0.072 0.070 0.056 0.062 0.055 0.064  0.046 0.011 0.020  0.048

3.027 3.023 3.021 3.019 3.017 3.016 3.015 3.015 3.014 3.013 3.012 3.012 3.011 3.011 3.010 3.010 3.009 3.009 3.008 3.008 3.008 3.007 3.007

ALS Z

n

3.028 3.023 3.016 3.016 3.013 3.015 2.981 3.002 3.005 2.990

61.609

Table 4 Energy resonances (E), quantum defect (δ) and effective nuclear charge (Z*) for the 2s22p3(3D)np Rydberg series originating from the 2s22p3(2Do) metastable state of F2 þ . The present results (SCUNC) are compared to the Advanced Light Source (ALS) experiments of Aguilar et al. (2005). SCUNC

ALS

SCUNC

ALS

SCUNC

From Eqs. (8) and (9), we see that the Zn is linked to the quantum defect δ by (Sakho, 2012) Zn ¼

Z core 1ðδ=nÞ

ð12Þ

This equation indicates that, in the framework of the SCUNCformalism each Rydberg series must satisfy the following conditions (Sakho, 2012) ( n Z Z Z core if δ Z 0 Z n r Z core

if

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 … 1

E (eV) 67.219 (15) 70.984 (15) 72.885 (12) 73.988 (10) 74.685 (8) 75.152 (7) 75.481 (6) 75.722 (5) 75.902 (4) 76.042 (4) 76.151 (3) 76.239 (3) 76.311 (3) 76.370 (2) 76.419 (2) 76.460 (2) 76.496 (2) 76.526 (2) 76.552 (1) 76.575 (1) 76.595 (1) 76.612 (1) 76.628 (1) 76.642 (1) 76.654 (1) 76.665 (1) 76.675 (1) … 76.815

E (eV) 67.219 70.984 72.892 73.991 74.695 75.164

(15) (15) (15) (15) (15) (15)

|ΔE|

δ

δ

Z

0.000 0.000 0.007 0.003 0.010 0.012

0.428 0.417 0.418 0.418 0.418 0.418 0.418 0.418 0.417 0.417 0.417 0.417 0.417 0.417 0.417 0.417 0.417 0.417 0.417 0.417 0.418 0.418 0.418 0.418 0.418 0.419 0.419

0.428 0.418 0.413 0.415 0.401 0.390

3.359 3.273 3.225 3.191 3.166 3.146 3.131 3.118 3.108 3.099 3.092 3.086 3.080 3.075 3.071 3.067 3.064 3.061 3.058 3.055 3.053 3.051 3.049 3.047 3.046 3.044 3.042

δr0

n

limZ n-1 ¼ Z core :

ð13Þ

From Tables 1–5, it is clearly seen that the conditions in (13) are well satisfied for both theory and experiments. As a result, analysis of experimental data and theoretical calculations can also be done in the framework of the SCUNC formalism taking into account (13). It should be mentioned the particular ALS value Zn ¼2.755 for n ¼10 corresponding to the experimental energy position (in eV) at 61.921. In general, the experimental value of Zn evaluated using Eq. (9) is about 3.05 as shown in Tables 1–5. This may indicate that the experimental data at 61.921 is probably greater that the accurate value. Our prediction at 61.705 may be more accurate. On the other hand, for the Rydberg states considered, the present quantum defects are almost constant along all the series up to n ¼30. This may emphasize the accuracy of our predicted results where no literature data are available to date for direct comparison. Besides, it is well known that the quantum defect is being zero for a pure hydrogenic state (Hinojoha et al., 2012). Subsequently, for a given Rydberg series, calculations or measurements must be stopped when δ is equal to zero or become negative where positive values are only allowed. In addition, calculations

Table 5 Energy resonances (E), quantum defect (δ) and effective nuclear charge (Z*) for the 2s2p3(5S0)np (4P) Rydberg series originating from the 2s22p3(4So) ground state of F2 þ . The present results (SCUNC) are compared to the Advanced Light Source(ALS) experiments of Aguilar et al. (2005).

ALS SCUNC

n

n

41

ALS

SCUNC

ALS

SCUNC

ALS

Zn 3.359 3.273 3.222 3.189 3.158 3.135

n

E (eV)

E (eV)

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 … 1

66.295 (16) 68.062 (12) 69.129 (10) 69.809 (8) 70.267 (7) 70.590 (6) 70.826 (5) 71.003 (4) 71.141 (4) 71.249 (3) 71.336 (3) 71.406 (3) 71.465 (2) 71.513 (2) 71.554 (2) 71.589 (2) 71.619 (2) 71.645 (1) 71.668 (1) 71.688 (1) 71.705 (1) 71.721 (1) 71.734 (1) 71.747 (1) 71.758 (1) 71.777 (1)

66.319 68.062 69.129 69.808 70.266 70.588 70.824

71.907

(15) (15) (15) (15) (15) (15) (15)

n

|ΔE|

δ

δ

Z

0.024 0.000 0.000 0.001 0.001 0.002 0.002

0.329 0.357 0.361 0.360 0.359 0.359 0.359 0.359 0.360 0.361 0.362 0.363 0.364 0.365 0.366 0.366 0.366 0.366 0.366 0.365 0.364 0.362 0.360 0.357 0.354 0.350

0.319 0.357 0.361 0.363 0.363 0.366 0.371

3.211 3.190 3.163 3.142 3.125 3.112 3.101 3.093 3.085 3.079 3.074 3.070 3.066 3.062 3.059 3.056 3.053 3.051 3.048 3.046 3.044 3.042 3.040 3.039 3.037 3.035

Zn 3.204 3.190 3.163 3.142 3.126 3.113 3.103

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are accurate if at least the quantum defect is constant with increasing the principal quantum number n of the nl-electron valence. Otherwise δ may decrease slowly. Therefore, calculations should be limited when the quantum defect exhibits a monotonic increasing when n increases. Let us move now on speculating upon the advantages, disadvantages and utility of the SCUNC formalism. As well known, to compare with high resolution measurements, state-of-the-art-theoretical methods are required using highly correlated wave functions including relativistic effects. Such ab initio methods are generally based on a complex mathematical formalism. In addition, a computer program via specific codes is required to tabulate accurate resonance parameters. For instance, calculations from the MCDF method are performed using the code developed by Bruneau (Bruneau, 1984) and the widely used R-matrix method is based on the DiracAtomic-R-matrix-Codes (Wang et al., 2010; Berrington et al., 1995). In addition, calculations from the relativistic MCDHF method are performed using the GRASP2K code (Jönsson et al., 2007). The main advantage of the SCUNC method is the possibilities to overcome these mathematical complexities in the framework of very soft empirical procedure without using any code. In fact, as shown by Eq.(2) the SCUNC-method is based on screening constant parameters derived empirically using experimental results including relativistic effects. This indicates that relativistic effects are implicitly taken into account in Eq. (1) via the parameter β and why the SCUNC method has been able to reproduce excellently high-resolution measurements (Sakho, 2012; Sakho et al., 2013; Faye et al., 2013; Diop et al., 2013). In addition, it should be mentioned the simplicity of the present formalism where a single stable formula is used to calculate accurate energy resonances and quantum defects for any Rydberg series up to very high n states. In general, measurements and calculations from sophisticated ab initio methods are limited to n ¼30 due probably to interaction between series inducing the overlapping of the cross section's peaks or due to limitations in the experimental resolution used. But, it should be mentioned the main inconvenient of the SCUNC formalism. Calculations need experimental data as input to evaluate empirically the screening constants. Consequently, the method is not applicable ahead of experiments. But fortunately, high measurements are continuously performed using synchrotron radiations such as ASTRID (Kjeldsen et al., 1999), SOLEIL (Bizau et al., 2000, 2011), ALS (Aguilar et al., 2005) and Spring 8 (Oura et al., 2000). The development of these synchrotron light sources provides high accurate experimental data for benchmarking state-of-the-art theoretical quantum mechanical methods. Consequently, the SCUNC method can continuously be used to provide useful photoionization data for the physical community.

4. Summary and conclusion In this paper, we have calculated accurate energy resonances, quantum defects and effective nuclear charges of the 2s22p2(1D)nd (2L), and 2s22p2(1S)nd (2L) Rydberg series originating from the 2s22p3(2Po) metastable state of F2 þ and of the 2s22p2(1D)nd (2L), and 2s22p3(3D)np Rydberg series from the 2s22p3(2Do) metastable state of the F2 þ ion. Accurate energy resonances, quantum defects and effective nuclear charges of the 2s2p3(5S0)np (4P) Rydberg series from the 2s22p3 (4So) ground state of F2 þ are also calculated. These resonance parameters are obtained using the Screening Constant by Unit Nuclear (SCUNC) Charge method up to n¼ 30. For the lowest members (n r17), the present results are in good agreement with the only available experimental data (Aguilar et al., 2005). Ours predicted results expected to be accurate for the high lying states 18 rn r30 for which no other experimental and theoretical values are available as far as we know, may be

useful guideline for investigators focusing on the photoionization spectra of F2 þ . The good agreement between the present theory and the ALS experimental data of Aguilar et al. (2005) is explained arguing that the SCUNC-method includes relativistic effects in the calculations via the screening constant parameters evaluated from experimental data. It should be mentioned the simplicity of the present formalism where no highly correlated wave functions and codes are required to succeed on obtaining very accurate results up to high n. References Aguilar, A., Emmons, E.D., Gharaibeh, M.F., Covington, A.M., Bozek, J.D., Ackerman, G., Canton, S., Rude, B., Schlachter, A.S., Hinojosa, G., Álvarez, I., Cisneros, C., McLaughlin, B.M., Phaneuf, R.A., 2005. Photoionization of ions of the nitrogen isoelectronic sequence: experiment and theory for F2 þ and Ne3 þ . Journal of Physics B: Atomic, Molecular and Optical Physics 38, 343. Berrington, K.A., Eissner, W.B., Norrington, P.H., 1995. 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