Photoionization Study of the 2s²2p² (¹D)ns(²D), 2s²2p² (¹D)nd(²P), 2s²2p² (¹D)nd(²S), 2s²2p² (¹S)nd²D, and 2s²2p³ (³P)np(²D) Rydberg Series of O+ Ions via the Modified Atomic Orbital Theory

Malick Sow^*, Fatou Ndoye, Alassane Traoré, Abdou Diouf, Boubacar Sow, Youssou Gning, Papa Amadou Lamine Diagne

Department of Physics, Atoms Lasers Laboratory, Faculty of Sciences and Technologies, University Cheikh Anta Diop, Dakar, Senegal.

DOI: 10.4236/jmp.2021.1210086                

Abstract

We report in this paper energy positions of the 2P°_2s²2p²(¹D)nd ²P, 2P°_2s²2p²(¹D)nd ²S, 2P°_2s²2p²(¹D)ns ²D, 2P°_2s²2p²(¹S)nd ²D, and 2P°_2s²2p³(³P)np ²D Rydberg series in the photoionization spectra originating from 2P° metastable state of O+ ions. Calculations are performed up to n = 30 using the Modified Orbital Atomic Theory (MAOT). The present results are compared to the experimental data of Aguilar which are the only available values. The accurate data presented in this work may be a useful guideline for future experimental and other theoretical studies.

Keywords

Semiempirical Calculations, Modified Orbital Atomic Theory, Electron Correlation Calculations, Atoms and Ions, Rydberg Series, Quantum Defect

1. Introduction

The important role of studying Photoionization is a fundamental processes playing in laboratory and astrophysical systems such as nebulae plasmas [1], in inertial-confinement fusion experiments [2] and contributing to plasma opacity and radiation transfer inside plasmas. Thus, quantitative measurements of photoionization of ions provide precision data on ionic structure, and guidance to the development of theoretical approaches of multielectron interactions. Greatest attention has been concentrated on studying Rydberg series of O+ ions for which photoabsorption from low-lying metastable states of open-shell ions has been shown to be important in the earth’s upper atmosphere as well as in astrophysical plasmas. Formerly, studies on the O+ ion have been focused on ionization using the merged-beam technique. Thus, Aguilar et al. [3] performed the first experiment on the Absolute photoionization of O+ from 29.7 to 46.2 eV above the first ionization threshold, using a merged-beam line at the Advanced Light Source (ALS).

Therefore, it is an imperative task for physicists to provide accurate photoionization data for the modeling of astrophysical and laboratory plasmas.

The Opacity Project atomic database (at the Astronomic DataCenter of Strasbourg, France) was formed to re-estimate stellar envelope opacities in terms of atomic data computed by ab initio methods [4]. All these efforts led to the creation of several atomic databases widely used for astrophysical calculations [3].

In the present paper, we intend to provide accurate data on the photoionization of O+ ions that may be useful guideline for the physical atomic community. In addition, we aim to demonstrate the possibilities to use the Modified Atomic Orbital Theory of SOW et al. [5] [6] [7] [8] to reproduce excellently experimental data from merged beam facilities. For this purpose, we report calculations of energy resonances for the 2P˚_2s²2p²(¹D)nd ²P, 2P˚_2s²2p²(¹D)nd ²S, 2P˚_2s²2p²(¹D)ns ²D 2P˚_2s²2p²(¹S)nd ²D, and 2P˚_2s²2p³(³P)np ²D Rydberg series of O+ ions up to n = 30, via the MAOT procedure along with the quantum defect theory.

Energy resonances and quantum-defect are compared to the only available experimental data of ALS [3].

Section 2 gives MAOT theory with a brief description of the formalism and the analytical expressions used in the calculations. In Section 3, we present and discuss the results obtained, compared to available experimental. In Section 4, we summarize our study and draw conclusions.

2. Theory

2.1. Brief Description of the MAOT Formalism

In the framework of Modified Atomic Orbital Theory (MAOT), total energy of (νℓ)-given orbital is expressed in the form [8] [9].

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

For an atomic system of several electrons M, the total energy is given by (in Rydbergs):

$E=-{\displaystyle \underset{i=1}{\overset{M}{\sum }}\frac{{\left[Z-{\sigma }_i\left(\mathcal{l}\right)\right]}^2}{{\upsilon }_i^2}}$

With respect to the usual spectroscopic notation $\left(N\mathcal{l},N{\mathcal{l}}^{\prime }\right){}^{2S+1}L{}^{\pi }$, this equation becomes

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

In this formula (2), L characterizes the considered quantum state (S, P, D …) and the symbol π is the parity of the system.

In the photoionisation study, energy resonances are generally measured relatively to the E∞ converging limit of a given (^2S+1L_J) nl-Rydberg series. For these states, the general expression of the energy resonances is given by the formula of Sakho presented previously [10] (in Rydberg units):

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1\left({}^{2S+1}{L}_J\right)-{\sigma }_2\left({}^{2S+1}{L}_J\right)\times \frac{1}{n}\\ {-{\sigma }_2^{\mu }\left({}^{2S+1}{L}_J\right)\times \left(n-m\right)\times \left(n-q\right){\displaystyle \underset{k}{\sum }\frac{1}{f_k\left(n,m,q,s\right)}}\}}^2\end{array}$ (3)

In this equation m and q (m < q) denote the principal quantum numbers of the (^2S+1L_J)nl-Rydberg series of the considered atomic system used in the empirical determination of the σ_i(^2S+1L_J)-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 the NIST atomic database, E_n denotes the corresponding energy resonance, 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 ${\displaystyle \underset{k}{\sum }\frac{1}{f_k\left(n,m,q,s\right)}}$ 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 $\mu$ is fixed to 1 and 2 during the iteration. The quantum defect is calculated from the standard formula below

$E_n=E_{\infty }-\frac{R{Z}_{core}^2}{{\left(n-\delta \right)}^2}\Rightarrow \delta =n-Z_{core}\sqrt{\frac{R}{\left(E_{\infty }-E_n\right)}}$ (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 $\delta$ means the quantum defect.

2.2. Energy Resonances of the 2P˚_2s²2p²(¹D)nd(²P); 2P˚_2s²2p² (¹D)nd(²S); 2P˚_2s²2p² (¹D) ns (²D); 2P˚_2s²2p² (1S)nd(2D) and 2P˚_2s²2p³(³P)np(²D) Rydberg Series from 2P˚ Metastable State of O+

In the framework of the MAOT formalism, the energy positions of the 2P˚_2s²2p²(¹D)nd(²P); 2P˚_2s²2p² (¹D)nd(²S); 2P˚_2s²2p² (¹D) ns (²D); 2P˚_2s²2p² (1S)nd(2D) and 2P˚_2s²2p³(³P)np(²D) prominent Rydberg series from 2P˚ metastable state of O+ are given by (in Rydberg units)

· For 2P˚_2s²2p²(¹D)nd(²P) levels

$\begin{array}{l}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}+{\sigma }_2\times \left(n-m\right)\times \left(n-q\right)\times [\frac{1}{{\left(n+q-s\right)}^3}\\ {+\frac{1}{{\left(n+m-s\right)}^3}+\frac{1}{{\left(n+m+s\right)}^3}+\frac{1}{{\left(n+m-s\right)}^4}+\frac{1}{{\left(n+q-m+s\right)}^5}]\}}^2\end{array}$ (6)

Using the experimental data of ALS [3], we obtain (in eV) E₅ = 30.393 ± 0.15 (m = 5) and E₆ = 31.081 ± 0.15 (q = 6) respectively for the 2P˚_2s²2p²(¹D)5d ²P and 2P˚_2s²2p²(¹D)6d ²P levels. From NIST [11], we find E_∞ = 32.617 eV. Using these data, Equation (6) gives σ₁ = 6.012 ± 0.251 and σ₂ = −0.166 ± 0.009

· For 2P˚_2s²2p² (¹D)nd(²S) levels:

$E_n=E_{\infty }-\frac{1}{n^2}{\left\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}+{\sigma }_2\times \left(n-m\right)\times \left(n-q\right)\times \left[\frac{1}{{\left(n+m-s\right)}^3}+\frac{1}{{\left(n-s\right)}^4}\right]\right\}}^2$ (7)

For the 2P˚_2s²2p²(¹D)5d ²S and 2P˚_2s²2p²(¹D)6d ²S levels, we find using the experimental data of ALS et al. [3], E₅ = 30.213 ± 0.150 (m = 5) and E₆ = 30. 905 ± 0.150 (q = 6). From NIST [11], we find E_∞ = 32.617 eV Equation (7) provides then σ₁ = 6.061 ± 0.185 and σ₂ = −0.367 ± 0.092

· For 2P˚_2s²2p² (¹D) ns (²D) levels

$\begin{array}{l}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}+{\sigma }_2\times \left(n-m\right)\times \left(n-q\right)\times [\frac{1}{{\left(n+q-s\right)}^3}\\ +\frac{1}{{\left(n+q+s-m\right)}^4}+{\frac{1}{{\left(n+q-s\right)}^4}+\frac{1}{{\left(n+m-s\right)}^4}+\frac{1}{{\left(n+q-m+3s\right)}^5}]\}}^2\end{array}$ (8)

For the 2P˚_2s²2p² (¹D) 6s (²D) and 2P˚_2s²2p² (¹D) 7s (²D) levels the experimental energy positions ALS et al. [3] are, E₆ = 30.578 ± 0.15 (m = 6) and E₇ = 31. 188 ± 0.15 (q = 7). From NIST [11], we find E_∞ = 32.617 eV. In that case, we find using Equation (8) σ₁ = 6.056 ± 0.322 and σ₂ = −2.274 ± 0.413

· For 2P˚_2s²2p² (1S)nd(²D) levels

$\begin{array}{l}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}+{\sigma }_2\times \left(n-m\right)\times \left(n-q\right)\times [\frac{1}{\left(n+q-m+s\right){\left(n-s\right)}^2}\\ +\frac{1}{{\left(n+2m-q\right)}^3}+{\frac{1}{{\left(n+q+s-m\right)}^4}]\}}^2\end{array}$ (9)

From ALS of Aguilar et al. [3], we obtain for the 2P˚_2s²2p² (1S)4d(2D) and 2P˚_2s²2p² (1S)5d(²D) E₄ = 31.924 ± 0.15 (m = 4) and E₅ = 33. 217 ± 0.15 (q = 5). From NIST [11], we find E_∞ = 35.458 eV. We find then using Equation (9) σ₁ = 6.008 ± 0.167 and σ₂ = −0.187 ± 0.05.

· For 2s²2p³(³P)np(²D) levels

$\begin{array}{l}{E}_n=E_{\infty }-\frac{1}{n^2}\{Z-{\sigma }_1-\frac{{\sigma }_2}{n}+{\sigma }_2\times \left(n-m\right)\times \left(n-q\right)\times [\frac{1}{{\left(n-s\right)}^2}\\ +\frac{1}{{\left(n+s-m\right)}^2}-\frac{1}{{\left(n+s-m\right)}^3}{+\frac{1}{{\left(n+s-m\right)}^4}]\}}^2\end{array}$ (10)

From ALS et al. [3], we obtain for the 2P˚_2s²2p³(³P)3p(²D) and 2P˚_2s²2p³(³P)4p(²D) E₃ = 39.478 ± 0.15 (m = 3) and E₄ = 43.115 ± 0.15 (q = 4). From NIST [11], we find E_∞ = 47.527eV. We find then using Equation (10) σ₁ = 5.411 ± 0.411 and σ₂ = −0.844 ± 0.022

3. Results and Discussions

The results obtained in the present paper are listed in Tables 1-5 and compared with the Advanced Light Source experimental data of Aguilar et al. [3].

In Table 1, we quote the present MAOT results for energy resonances (E) and quantum defect (δ) of the 2P˚_2s²2p²(¹D)nd(²P) Rydberg series relatively to the 2P˚_metastable state of O+ ion. The current energy positions are calculated from equations (6) along with Z = 8, m = 5, and q = 6, σ₁ = 6.012 ± 0.251 and σ₂ = −0.166 ± 0.009. All these screening constant are evaluated using the Advanced Light Source (ALS) experimental results of Aguilar et al. [3], and take from NIST [11] the E_∞ energy limits which is 32.617 eV. Then our results are converted into eV for direct comparison by using the infinite Rydberg (1 Ry = 0.5 a.u = 13.605698 eV). It is seen that the data obtained compared very well to the experimental data of Aguilar et al. [3].

Up to n = 11, the maximum energy differences relative to the experimental data is less than 0.006 eV. In addition, the present quantum defect is almost constant up to n = 30. This may expect our results for n > 11 to be accurate.

In Table 2, we compare the present MAOT energy resonances (E) and quantum defect (δ) of the 2P˚_2s²2p²(¹D)nd(²S) Rydberg series relatively to the 2P˚_metastable state of O+ ion to experimental data [3]. All our energy values are obtained empirically using Equation (7) and converted into (eV) for direct comparison. Here again, the agreements are seen to be very good. Along the series, the present quantum defect is almost constant.

In Table 3, we show a comparison of the energy resonances (E) and quantum defect (δ) of the 2P˚_2s²2p²(¹D)ns (²D) Rydberg states relatively to the 2P˚_metastable state of O+ ion. The current energy positions are calculated from equations (8) along with Z = 8, m = 6, and q = 7, σ₁ = 6.056 ± 0.322 and σ₂ = −2.274 ± 0.413. The agreements between the studies are seen to be very good and the quantum defect is almost constant along the series. The agreements between the MOAT results and experimental data are seen to be very good. Along all the series investigated, the quantum defect is practically constant. This may expect our results up to n = 30 to be accurate.

In Table 4, we list the present energy resonances (E) and quantum defect (δ) for the 2P˚_2s²2p²(¹S)nd (²D) Rydberg states relatively to the 2P˚_metastable state of O+ ion compared to the experimental data [3]. The current energy positions are calculated from equations (9) along with Z = 8, m = 4, and q = 5, E_∞ = 35.458 eV; σ₁ = 6.008 ± 0.167 and σ₂ = −0.187 ± 0.05. Comparison shows that the maximum energy deviation is at 0.006 up to n = 14. This indicates the very good accuracy between the results. For n ≥ 15 it should be underlined that, since the MAOT formalism reproduces excellently the experimental measurements [3], the present results quoted in Table 4 for the 2P˚_2s²2p²(¹S)nd (²D) levels may be a very good representative of the nonexistent experimental data.

Table 1. Energy resonances (E) and quantum defect (δ) of the 2P˚_2s²2p² (¹D)nd (²P) Rydberg series observed in the photoionization spectra originating from the 2P˚ metastable states of O⁺. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [2]. The results are expressed in eV. The energy uncertainties in the present calculations and in the experimental data are indicated into parenthesis.

^aNIST atomic database [11]. |ΔE|: energy differences relative to the experimental data.

Table 2. Energy resonances (E) and quantum defect (δ) of the 2P˚_2s²2p² (¹D)nd (²S) Rydberg series observed in the photoionization spectra originating from the 2P˚ metastable states of O⁺. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [2]. The results are expressed in eV. The energy uncertainties in the present calculations and in the experimental data are indicated into parenthesis.

^aNIST atomic database [11]. |ΔE|: energy differences relative to the experimental data.

Table 3. Energy resonances (E) and quantum defect (δ) of the 2P˚_2s²2p² (¹D)ns (²D) Rydberg series observed in the photoionization spectra originating from the 2P˚ metastable states of O⁺. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [2]. The results are expressed in eV. The energy uncertainties in the present calculations and in the experimental data are indicated into parenthesis.

^aNIST atomic database [11]. |ΔE|: energy differences relative to the experimental data.

Table 4. Energy resonances (E) and quantum defect (δ) of the 2P˚_2s²2p² (¹S)nd (²D) Rydberg series observed in the photoionization spectra originating from the 2P˚ metastable states of O⁺. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [2]. The results are expressed in eV. The energy uncertainties in the present calculations and in the experimental data are indicated into parenthesis.

^aNIST atomic database [11]. |ΔE|: energy differences relative to the experimental data.

Table 5. Energy resonances (E) and quantum defect (δ) of the 2P˚_2s²2p³ (3P˚)nd (²D) Rydberg series observed in the photoionization spectra originating from the 2P˚ metastable states of O⁺. The present results (MAOT) are compared to the Advanced Light Source (ALS) of Aguilar et al. [2]. The results are expressed in eV. The energy uncertainties in the present calculations and in the experimental data are indicated into parenthesis.

^aNIST atomic database [11]. |ΔE|: energy differences relative to the experimental data.

In Table 5, we compare the present MAOT energy resonances (E) and quantum defect (δ) of the 2P˚_2s²2p²(¹D)nd(²S) Rydberg series relatively to the 2P˚_metastable state of O+ ion to experimental data [3]. Our current energy positions are calculated from Equations (10) with Z = 8 along with m = 3, and q = 4, E_∞ = 47.527 eV, σ₁ = 5.411 ± 0.411 and σ₂ = −0.844 ± 0.022. Here again, the agreements are seen to be very good. Along the series, the present quantum defect is almost constant. In a few series where discrepancies are observed, the maximum energy difference relative to the experimental data is at 0.001 eV. This indicates the excellent agreements between the present calculations and the experimental measurements for energy positions.

For all the Rydberg series investigated, the slight discrepancies between the present calculations and experiment may be explain by the simplicity of the MAOT formalism which does not include explicitly any relativistic corrections.

4. Summary and Conclusion

In this paper, energy resonances of the 2s²2p²(¹D)ns(²D), 2s²2p²(¹D)nd(²P), 2s²2p²(¹D)nd(²S), 2s²2p²(¹S)nd (²D), and 2s²2p³(³P)np (²D) Rydberg series in the photoionization spectra originating from 2P metastable state of O+ ions are reported in this paper using the Modified Orbital Atomic Theory (MAOT). Over the entire Rydberg series investigated, it is shown that the present MOAT results agree very well with the only available experimental data of ALS [3]. A host of accurate data up to n = 30 are quoted in the recent work. The very good result obtained is this work points out the possibilities to use the MAOT formalism in the investigation of high lying Rydberg series of ions containing several electrons in the framework of a soft procedure. This work may be of interest for future experimental and theoretical studies in the photoabsorption spectrum of O⁺.

Acknowledgements

The authors are grateful to the Orsay Institute of Molecular Sciences (OIMS), Paris, France and the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy.

Conflicts of Interest

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

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