Abstract

We consider the 4s²4p⁴(³P₁)nd, 4s²4p⁴(¹D₂)nd and 4s²4p⁴(¹S₀)nd Rydberg series of Rb²⁺ originating from the 4s²4p⁵(²P⁰_1/2) ground state and 4s²4p⁵(²P⁰_3/2) metastable state of Rb²⁺ converging respectively to the 4s²4p⁴(³P₁), 4s²4p⁴(¹D₂) and 4s²4p⁴(¹S₀) series limit in Rb³⁺. The semi-empirical procedure of Modified Atomic Orbital Theory (MAOT) is applied to perform calculations in the present work from n = 6 up to n = 45. We note that the MAOT procedure gives accurate results compared to available Advanced Light Source (ALS) experimental data and other theoretical results. We tabulate new data from n = 22 - 45 for 4s²4p⁴(¹S₀)nd Rydberg series originating from the 4s²4p⁵(²P⁰_1/2) ground state and 4s²4p⁵(²P⁰_3/2) metastable state of Rb²⁺. The other Rydberg series 4s²4p⁴(³P₁)nd and 4s²4p⁴(¹D₂)nd are extended up to n = 45.

Keywords

Rydberg Series, MAOT, Photoionization, Ground State, Metastable State, Atomic Ions, Semi-Empirical Procedure

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1. Introduction

Photoionization of atomic ions is a fundamental process that occurs in the astrophysical environments like stars and nebulæ [1] [2] where the temperature is sufficient to ionize the matter. Most of the chemical heavier elements than iron are produced by slow neutron capture process or rapid neutron capture process and the Rubidium (Rb) is one of these elements [3] [4]. The provision of data obtained from the photoionization of Rubidium is a valuable aid to the simulation of astrophysical phenomena involving this element, particularly in the nucleosynthesis of heavy elements. Macaluso et al. [5] used a third-generation synchrotron radiation and the photo-ion, merged-beams technique at Advanced Light Source (ALS) at Lawrence Berkeley National Laboratory with a band pass resolution of 13.5 ± 2.5 meV full width half maximum (FWHM) to perform high-resolution photoionization cross section measurements of the for Rb²⁺ over the photon energy range 37.31 - 44.08 eV spanning the 4s²4p⁵(²P⁰_3/2) ground state and 4s²4p⁵(²P⁰_1/2) metastable state ionization thresholds. They identify three distinct Rydberg resonance series 4s²4p⁴(³P₁)nd, 4s²4p⁴(¹D₂)nd and 4s²4p⁴(¹S₀)nd originating from the 4s²4p⁵(²P⁰_3/2) ground state and 4s²4p⁵(²P⁰_1/2) metastable state. In tandem with the measurements, Breit-Pauli R-matrix calculations were performed in the intermediate coupling jK to facilitate the identification of several highly excited Auger Rydberg resonance states of the Rb²⁺ ions. Using a fully relativistic approach within the Dirac-Coulomb R-Matrix approximation (DARC), McLaughlin et al. [6] calculated the cross sections for the 4s²4p⁵(²P⁰_3/2) ground state and the corresponding 4s²4p⁵(²P⁰_1/2) metastable state. They used the high-resolution ALS photoionization cross sections to benchmark their PI cross sections and Auger Rydberg resonances. They obtained excellent results providing confidence in the theoretical data for astrophysical applications. Employing the Screen Constant by Unit Nuclear Charge (SCUNC), Sakho et al. [7] determined the energy resonances of the 4s²4p⁴(³P₁)nd and 4s²4p⁴(¹D₂)nd Rydberg series until n = 40 and reported precise high lying photoionization data of Rb²⁺ ions reducing energy deviations with respect to ALS data at a maximum of 0.001 eV. The Modified Atomic Orbital Theory (MAOT) has been applied successfully in the studies of trans-iron Photoionization by Diop et al. [8]. In this paper, we apply the MAOT formalism to investigate the 4s²4p⁴(³P₁)nd, 4s²4p⁴(¹D₂)nd and 4s²4p⁴(¹S₀)nd prominent Rydberg series in the photoionization spectra of Rb²⁺.

In Section 2, we present the theoretical procedure adopted in this work with a brief description of the MAOT formalism and the analytical expressions used in the calculations.

In Section 3, we present and discuss the results obtained along. The present results are compared with experimental data of Macaluso et al. [5] [9], the Dirac-Coulomb R-Matrix approximation (DARC) calculations of McLaughlin et al. [6] and SCUNC calculations of Sakho et al. [7].

In Section 4, we summarize and conclude the present work.

2. Theory

2.1. Brief Description of the MOAT Formalism

In the framework of Modified Atomic Orbital Theory (MAOT), total energy of (νl)-given orbital is expressed in the form [10]-[13]

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

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

$E=-{\displaystyle \sum _{i=1}^N\frac{{\left[Z-{\sigma }_i\left(l\right)\right]}^2}{{\nu }_i^2}}$

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

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

In the photoionization of atoms and ions, energy resonances are generally measured relatively to the E∞ converging limit of a given $\left({}^{2S+1}L{}_J\right)nl$ -Rydberg series. For these states, the general expression of the energy resonances En is given by the formula of Sakho presented previously [8] (in Rydberg units)

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

In this equation, m and q (m < q) denote the principal quantum numbers of the $\left({}^{2S+1}L{}_J\right)nl$ -Rydberg series of the considered atomic system used in the empirical determination of the ${\sigma }_i\left({}^{2S+1}L{}_J\right)$ -screening constants, s represents the spin of the nl-electron (s = 1/2), E_∞ is the energy value of the series limit generally determined from NIST atomic database, En 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 ${\displaystyle \sum _k\frac{1}{f_k\left(n,m,q,s\right)}}$ term.

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

$E_n=E_{\infty }-\frac{R{Z}_{core}^2}{{\left(n-\delta \right)}^2}\Rightarrow \delta =n-Z_{core}\sqrt{\frac{R}{E_{\infty }-E_n}}$ (4)

In this equation, R is the Rydberg constant, E∞ is the converging limit, Zcore represents the electric charge of the core ion, and δ is the quantum defect.

Zcore is directly obtained by the photoionization process from an atomic $X^{p+}$ system:

$X^{p+}+h\nu \to X^{\left(p+1\right)+}+{\text{e}}^{-}$ . We find $Z_{core}=p+1$ .

In addition, theoretical and measured energy positions can be analyzed by calculating the Z effective charge in relationship with the quantum defect (δ).

The relationship between $Z^{\ast }$ and δ is in this form below:

$Z^{\ast }=\frac{Z_{core}}{1-\frac{\delta }{n}}$ (5)

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

$\{\begin{array}{l}{Z}^{\ast }\ge Z_{core}\text{if}\delta \ge 0\\ Z^{\ast }\prec Z_{core}\text{if}\delta \prec 0\\ \underset{n\to \infty }{\lim }{Z}^{\ast }=Z_{core}\end{array}$ (6)

2.2. Energy Resonances of the 4s²4p⁴(³P₁)nd, 4s²4p⁴(¹D₂)nd and 4s²4p⁴(¹S₀)nd Rydberg Series of Rb²⁺

Using Equation (3), the energy resonances of the 4s²4p⁴(³P₁)nd, 4s²4p⁴(¹D₂)nd and 4s²4p⁴(¹S₀)nd are given below (in Rydberg units).

• For the 4s²4p⁵(²P⁰_1/2) → 4s²4p⁴(³P₁)nd transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}[Z-{\sigma }_1-{\sigma }_2\times \frac{1}{n}+{\sigma }_2\left(n-m\right)\left(n-q\right)(\frac{1}{{\left(n+m-3s\right)}^2}\\ -\frac{1}{{\left(n+q-8s\right)}^2}+{\frac{1}{{\left(n+q-m+7s\right)}^3})]}^2\end{array}$ (7)

• For the 4s²4p⁵(²P⁰_3/2) → 4s²4p⁴(³P₁)nd transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}[Z-{\sigma }_1-{\sigma }_2\times \frac{1}{n}+{\sigma }_2\left(n-m\right)\left(n-q\right)(\frac{1}{{\left(n+m+s\right)}^2}\\ -\frac{1}{{\left(n+q-s\right)}^2}+{\frac{1}{{\left(n+q-m+9s\right)}^3})]}^2\end{array}$ (8)

• For the 4s²4p⁵(²P⁰_1/2) → 4s²4p⁴(¹D₂)nd transition

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

• For the 4s²4p⁵(²P⁰_3/2) → 4s²4p⁴(¹D₂)nd transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}[Z-{\sigma }_1-{\sigma }_2\times \frac{1}{n}-{\sigma }_2\left(n-m\right)\left(n-q\right)(\frac{1}{{\left(n+m-q+8s\right)}^2}\\ +\frac{1}{{\left(n+q-m+2s\right)}^2}+{\frac{1}{{\left(n+m-q+5s\right)}^3})]}^2\end{array}$ (10)

• For the 4s²4p⁵(²P⁰_1/2) → 4s²4p⁴(¹S₀)nd transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}[Z-{\sigma }_1-{\sigma }_2\times \frac{1}{n}+{\sigma }_2\left(n-m\right)\left(n-q\right)(\frac{1}{{\left(n+2q+5s\right)}^2}\\ +\frac{1}{\left(n+m\right){\left(n+q+3s\right)}^2}-{\frac{1}{{\left(n+m+q+6s\right)}^2})]}^2\end{array}$ (11)

• For the 4s²4p⁵(²P⁰_3/2) → 4s²4p⁴(¹S₀)nd transition

$\begin{array}{c}{E}_n=E_{\infty }-\frac{1}{n^2}[Z-{\sigma }_1-{\sigma }_2\times \frac{1}{n}-{\sigma }_2\left(n-m\right)\left(n-q\right)(\frac{1}{{\left(n+q+s\right)}^2}\\ +\frac{1}{{\left(n+m+s\right)}^2}+{\frac{1}{{\left(n-m+q+s\right)}^3})]}^2\end{array}$ (12)

To evaluate the σ_i-screen constants, we use the E_exp(nl) experimental energy resonances and the E_∞ energy limits obtained by Macaluso et al. [5] [9]. We find:

• For the 4s²4p⁵(²P⁰_1/2) → 4s²4p⁴(³P₁)nd transition

Using E_∞ = 39.109 eV; E_exp(13d) = 38.352 eV (m = 13); E_exp(14d) = 38.459 eV (q = 14). Equation (7) gives σ₁ = 34.023 and σ₂ = −1.164

• For the 4s²4p⁵(²P⁰_3/2) → 4s²4p⁴(³P₁)nd transition

Using E_∞ = 40.023 eV; E_exp(13d) = 39.268 eV, E_exp(14d) = 39.374eV (m = 13 and q = 14). Equation (8) gives σ₁ = 34.003 and σ₂ = −0.855

• For the 4s²4p⁵(²P⁰_1/2) → 4s²4p⁴(¹D₂)nd transition

Using E_∞ = 40.485 eV; E_exp(8d) = 38.446 eV, E_exp(9d) = 38.885 eV (m = 13 and q = 14). Equation (9) gives σ₁ = 33.999 and σ₂ = −0.767

• For the 4s²4p⁵(²P⁰_3/2) → 4s²4p⁴(³P₁)nd transition

Using E_∞ = 41.399 eV; E_exp(8d) = 39.347 eV, E_exp(8d) = 39.790 eV (m = 13 and q = 14). Equation (10) gives σ₁ = 34.000 and σ₂ = −0.852

• For the 4s²4p⁵(²P⁰_1/2) → 4s²4p⁴(¹S₀)nd transition

Using E_∞ = 43.104 eV; E_exp(6d) = 39.332 eV, E_exp(7d) = 40.374 eV (m = 13 and q = 14). Equation (11) gives σ₁ = 34.006 and σ₂ = −0.992

• For the 4s²4p⁵(²P⁰_3/2) → 4s²4p⁴(¹S₀)nd transition

Using E_∞ = 44.018 eV; E_exp(6d) = 40.242 eV, E_exp(7d) = 41.286 eV (m = 13 and q = 14). Equation (12) gives σ₁ = 34.008 and σ₂ = −1.014

3. Results and Discussions

Our present results are listed in Tables 1-6 and compared with the ALS experimental data of Macaluso et al. [5], the SCUNC calculations of Sakho et al. [7] and DARC calculations of McLaughlin and Babb [6]. Our results are expressed into eV (1 Ry = 0.5 a.u = 13.605698 eV) for direct comparison. The effective nuclear charge $Z^{\ast }$ is calculated to validate the MAOT’s analysis conditions in Equation (6). The analysis of the nuclear effective charge values indicates that the quantum defect values obtained along the calculations and quoted in Tables 1-6 are also in agreement with the MAOT’s conditions of Equation (6).

Table 1. Resonance Energy (E_n, eV), quantum defect (δ) and effective nuclear charge $Z^{\ast }$ of the 4s²4p⁴(³P₁)nd Rydberg series converging to the Rb³⁺ 4s²4p⁴(³P₁) threshold origin Rb²⁺ 4s²4p⁵(²P⁰_1/2) metastable state. The present MAOT calculations are compared to the SCUNC calculations of Sakho et al. [7], DARC calculations of McLaughlin and Babb [6] and the ALS experimental measurements of Macaluso et al. [5]. |ΔE| denotes the energy difference between the MAOT calculations and the ALS measurements.

Table 2. Resonance Energy (E_n, eV), quantum defect (δ) and effective nuclear charge $Z^{\ast }$ of the 4s²4p⁴(³P₁)nd Rydberg series converging to the Rb³⁺ 4s²4p⁴(³P₁) threshold origin Rb²⁺ 4s²4p⁵(²P⁰_3/2) ground state. The present MAOT calculations are compared to the SCUNC calculations of Sakho et al. [7], DARC calculations of McLaughlin and Babb [6] and the ALS experimental measurements of Macaluso et al. [9]. |ΔE| denotes the energy difference between the MAOT calculations and the ALS measurements.

Table 3. Resonance Energy (E_n, eV), quantum defect (δ) and effective nuclear charge $Z^{\ast }$ of the 4s²4p⁴(¹D₂)nd Rydberg series converging to the Rb³⁺ 4s²4p⁴(¹D₂) threshold origin Rb²⁺ 4s²4p⁵(²P⁰_1/2) metastable state. The present MAOT calculations are compared to the SCUNC calculations of Sakho et al. [7], DARC calculations of McLaughlin and Babb [6] and the ALS experimental measurements of Macaluso et al. [9]. |ΔE| denotes the energy difference between the MAOT calculations and the ALS measurements.

Table 4. Resonance Energy (E_n, eV), quantum defect (δ) and effective nuclear charge $Z^{\ast }$ of the 4s²4p⁴(¹D₂)nd Rydberg series converging to the Rb³⁺ 4s²4p⁴(¹D₂) threshold origin Rb²⁺ 4s²4p⁵(²P⁰_3/2) ground state. The present MAOT calculations are compared to the SCUNC calculations of Sakho et al. [7], DARC calculations of McLaughlin and Babb [6] and the ALS experimental measurements of Macaluso et al. [9]. |ΔE| denotes the energy difference between the MAOT calculations and the ALS measurements.

Table 5. Resonance Energy (E_n, eV), quantum defect (δ) and effective nuclear charge $Z^{\ast }$ of the 4s²4p⁴(¹S₀)nd Rydberg series converging to the Rb³⁺ 4s²4p⁴(¹S₀) threshold origin Rb²⁺ 4s²4p⁵(²P⁰_1/2) metastable state. The present MAOT alculations are compared to the DARC calculations of McLaughlin and Babb [6] and the ALS experimental measurements of Macaluso et al. [9]. |ΔE| denotes the energy difference between the MAOT calculations and the ALS measurements.

Table 6. Resonance Energy (E_n, eV) and quantum defect (δ) and effective nuclear charge $Z^{\ast }$ of the 4s²4p⁴(¹S₀)nd Rydberg series converging to the Rb³⁺ 4s²4p⁴(¹S₀) threshold originating from Rb²⁺ 4s²4p⁵(²P⁰_3/2) ground state. The present MAOT calculations are compared to the DARC calculations of McLaughlin and Babb [6] and the ALS experimental measurements of Macaluso et al. [9]. |ΔE| denotes the energy difference between the MAOT calculations and the ALS measurements.

In Table 1, we quote the present MAOT results for resonance energies (E_n in eV), quantum defect (δ) and effective nuclear charge $Z^{\ast }$ of the 4s²4p⁴(³P₁)nd Rydberg series relatively to the 4s²4p⁵(²P⁰_1/2) metastable state of Rb²⁺. The results presented in this table are calculated by using the Equation (7). Thus, the maximum energy difference |ΔE| between the present calculations and the ALS experimental data of Macaluso et al. [5] is 0.001 eV. For n = 13 up to n = 21, we note a good agreement between results. For n ≥ 22 up to n = 40, our results are compared with those of the SCUNC calculations of Sakho et al. [7]. For n = 40 our value at 39.032 eV is in good agreement with the SCUNC value at 39.031 eV. We also note that the nuclear effective charge $Z^{\ast }$ and the quantum defect δ are in good agreement with the MAOT analysis conditions via Equation (6). This good agreement allows us to expect our results up to n = 45 to be accurate.