Calculation of the Rydberg Energy Levels for Francium Atom

Abstract

Based on the weakest bound electron potential model theory, the Rydberg energy levels and quantum defects of 𝑛𝑠²S_1/2(𝑛=8-50), 𝑛𝑑²D_3/2(𝑛=6-50), and 𝑛𝑑²D_5/2(𝑛=6-50) spectrum series for francium atom are calculated. The calculated results are in excellent agreement with the 74 known experimentally measured levels (the absolute difference is less than 0.03 cm^-1) and 58 energy levels for highly excited states are predicted.

1. Introduction

Much work [1–19] has been carried out to investigate the energy structure in excited states of francium atom, the heaviest alkali metal atom. The experimental investigations of the francium spectrum were pioneered by Liberman et al. in 1978 [1]. Later on, the energy levels of 7p²P⁰_1/2,3/2 and 8p²P⁰_1/2,3/2 were investigated by Doppler-free laser spectroscopy techniques in 1986 and 1987 [2, 3]. The 𝑛𝑠 (𝑛=23, 25-27, 29-31) and 𝑛𝑑 (𝑛=22-33) Rydberg levels were observed by two-step laser excitation and detection techniques also in 1987 [4], the 𝑛𝑠 (𝑛=10-22) and 𝑛𝑑 (𝑛=8-20) series were determined by stepwise laser excitation in 1989 and 1990 [5, 6], and the 9s level was observed in 1996 [7]. Meanwhile, the ionization potential was determined in 1988 [8] and refined in 1989 [5]. More recently, an interferometric method was used to improve the accuracy of the 7S-7P transition frequency of francium by 1 order of magnitude in 2009 [9]. The theoretical investigation of the energy structure of francium could be outlined as follows. In 1995, many-body perturbation theory in screened Coulomb interaction was used to calculate energy levels, 𝐸1 transition amplitudes, and the parity-nonconserving 𝐸1 amplitude of the 7s-8s transition in francium by Dzuba et al. [10]. In 1998, Ritz formalism was utilized to predict the francium Rydberg 𝑛𝑠, 𝑛𝑝, and 𝑛𝑑 levels up to 𝑛=30 [12]. In 1999, removal energies of the 𝑛=7-10 states and hyperfine constants of the 𝑛=7 and 8 states in francium were calculated by Safronova et al. [11], the calculations were based on the relativistic single-double approximation in which single and double excitations of Dirac-Hartree-Fock wave functions are included to all orders in perturbation theory. In 2004, pseudorelativistic Hartree-Fock method including core-polarization effects was used to investigate the radiative parameters for electric dipole transitions in the ions Ra II, Ac III, Th IV, and U VI along the francium isoelectronic sequence by Biémont et al. [13]. Also in 2004, relativistic Hartree-Fock method with several correlations was used to perform ab initio calculation of isotope shifts for isotopes of francium (from 𝐴=207 to 𝐴=228) by Dzuba et al. [14]. In 2005, radiative corrections to 𝐸1 matrix elements for 𝑛𝑠-𝑛𝑝 transitions in the alkali-metal atoms lithium through francium are evaluated by Sapirstein and Cheng [15]. In 2006, relativistic third-order and all-order methods were used to calculate the scalar and tensor polarizabilities for the 5𝑓_5/2 ground state in francium-like ion Th³⁺ by Safronova et al. [16]. In 2007, some 𝐸1 transitions in the francium isoelectronic sequence were computed in the “Dirac-Fock + core-polarization” approximation by Migdalek and Glowacz-Proszkiewicz [17], in which the core-valence electron correlation was treated in a semiclassical picture. Also in 2007, relativistic many-body perturbation theory was applied to study energies of the 7𝑠, 7𝑝, 6𝑑, and 5𝑓 states of francium-like ions with nuclear charges 𝑍=87-100, and also the transition rates, oscillator strengths, and lifetimes of francium-like ions with 𝑍=87-92 by Safronova et al. [18]. Most of the energy levels for neutral Francium have been compiled by Sansonetti in 2007 [19]

In all these theoretical works; however, Rydberg energy levels for highly excited states (𝑛>31) of francium-like atoms have not been efficiently studied. The reason is that no matter what approximation method (including many-body perturbation theory, relativistic single-double approximation, pseudo-relativistic Hartree-Fock method, relativistic third-order and all-order methods, Dirac-Fock + core-polarization’ approximation method, and so on ) is used, it is hard to investigate the highly excited states for such a complicated system that the configuration of the ground state is 1𝑠²2𝑠²2𝑝⁶3𝑠²3𝑝⁶3𝑑¹⁰4𝑠²4𝑝⁶4𝑑¹⁰4𝑓¹⁴5𝑠²5𝑝⁶5𝑑¹⁰6𝑠²6𝑝⁶7𝑠. Apossible approach to solve this problem is to utilize model potential method. In a previous work [20], we have successfully calculated the Rydberg energy levels and the quantum defects of 1𝑠²2𝑠²2𝑝𝑛𝑠³𝑃⁰_2,1,0 (𝑛=3-50) and 1𝑠²2𝑠²2𝑝𝑛𝑑³𝐹⁰_3,2 (𝑛=3-50) spectrum series for Carbon I atom by the weakest bound electron potential model theory (WBEPMT) [21–23]. In this paper, the WBEPMT is utilized to investigate the Rydberg energy levels and the quantum defects of 𝑛𝑠²S_1/2 (𝑛=8-50), 𝑛𝑑²D_3/2 (𝑛=6-50) and 𝑛𝑑²D_5/2 (𝑛=6-50) spectrum series for francium atom. The calculated results are in excellent agreement with the 74 known experimentally measured levels [19] (the absolute difference is less than 0.03 cm⁻¹), and 58 energy levels for highly excited states are predicted, which might be significant for experiment guidance.

2. Theory of WBEPM

In the weakest bound electron potential model theory (WBEPMT) [21–23], the term energy of an atom system is expressed as𝑇=𝑇_limit+𝐸, (1) where 𝑇_limit is the ionization limit corresponding to a given spectrum series, and 𝐸 is the energy of the weakest bound electron (WBE), which is determined by the radial equation (in atom unit)12𝑑²𝑅𝑑𝑟²+1𝑟𝑑𝑅+(𝑟)−ℓ()𝑑𝑟𝐸−𝑉ℓ+12𝑟²𝑅=0. (2) The potential that suffered by the WBE is modeled as [23]𝑉(𝑟)=−𝑍′𝑟+𝑘(𝑘+1)+2𝑘ℓ2𝑟², (3) in which 𝑍′ stands for the number of effective nuclear charge, and 𝑘 an adjustable parameter and is not necessarily an integer. The first term of (3) represents the Coulomb potential and the second term represents the dipole potential that originated from polarization.

Inserting (3) into (2), the radial equation becomes12𝑑²𝑅𝑑𝑟²+1𝑟𝑑𝑅+𝑑𝑟𝐸+𝑍′𝑟−ℓ^′ℓ^′+12𝑟²𝑅=0, (4) withℓ^′=ℓ+𝑘. (5)

The solution to (4) is known as𝑅_𝑛ℓ(𝑟)=𝐴exp−𝑍′𝑟𝑟𝑛′^ℓ′𝐿^2ℓ′+1_{𝑛−ℓ−1}2𝑍′𝑟,𝑛′𝐸=−𝑅𝑍′𝑛′², (6) where 𝑛′=𝑛+𝑘, 𝐴 is the normalization constant, and 𝑅 the Rydberg constant. Let𝑍′=𝑍𝑛′𝑛−𝛿^net_𝑛, (7) where 𝑍_net is the net charge number of the atomic kernel and 𝑍_net=1 for atoms, 𝛿_𝑛 is the quantum defect, the term energy described by (1) is turned into𝑇=𝑇_limit+𝐸=𝑇_limit𝑍−𝑅^net𝑛−𝛿_𝑛². (8)

According to Martin’s formula about quantum defect [24], 𝛿_𝑛 can be expressed as𝛿_𝑛=³_𝑖=0𝑎_𝑖𝑛−𝛿₀^2𝑖, (9) where 𝛿₀ is the quantum defect of the lowest energy level in a given series, 𝑎_𝑖 (𝑖=0,1,2,3) are parameters that can all be obtained through the least-square fitting of (8) with the first few experimental data in a given spectrum series.

3. Results and Discussion

By using the experimental data of the first four lowest energy levels of a given spectrum series, parameters 𝑎_𝑖 are fitted by least-square method, the results are listed in Table 1.

Table 1 

Spectral coefficients of the four energy series for francium atom by fitting the experimental values.

The Rydberg energy levels of the spectrum series of 𝑛𝑠²S_1/2 (𝑛=8-50), 𝑛𝑑²D_3/2 (𝑛=6-50), and 𝑛𝑑²D_5/2 (𝑛=6-50) for francium atom are calculated by (9). The calculated energy levels, quantum defects, and the differences between calculated and measured energy levels [19] are shown in Table 2 to Table 4, respectively. The unit of energy levels is expressed in cm⁻¹.

Table 2 

Comparison of the calculated and measured values of 𝑛𝑠²S_1/2 (𝑛=8-50) energy levels series for francium atoms.

From Tables 2, 3, and 4, it can be seen that the calculated results are in excellent agreement with the known experimental data since the absolute difference is less than 0.03 cm⁻¹.

Table 3 

Comparison of the calculated and measured values of 𝑛𝑑²D_3/2 (𝑛=6-50) energy levels series for francium atoms.

Table 4 

Comparison of the calculated and measured values of 𝑛𝑑²D_5/2 (𝑛=6-50) energy levels series for francium atoms.

Therefore, by using the experimental data of the first four lowest energy levels for a given spectrum series, the Rydberg energy levels and the quantum defects of 𝑛𝑠²S_1/2 (𝑛=8-50), 𝑛𝑑²D_3/2 (𝑛=6-50), and 𝑛𝑑²D_5/2 (𝑛=6-50) spectrum series for francium atom have been calculated by virtue of the WBEPMT, the calculated results are in excellent agreement with the 74 known experimentally measured levels, one exception (22𝑑²D_5/2) is noticed, and 58 energy levels for highly excited states are predicted, which might be significant for experiment guidance.

This implies that the WBEPMT is not only reliable to investigate the Rydberg energy levels of CI, but also dependable to probe the Rydberg energy levels of FrI. Compared with many-body perturbation theory, relativistic single-double approximation, pseudo-relativistic Hartree-Fock method, relativistic third-order and all-order methods, and Dirac-Fock + core-polarization’ approximation method, the advantage of WBEPMT is that the program code, which can be easily written by the Mathematica language, is simpler and the running time is much shorter. The disadvantage of WBEPMT; however, is that the wave functions of the Rydberg states cannot be obtained, which might be settled by combining the WBEPMT with other theories.

Acknowledgments

This work was supported by the Scientific Research Foundation of the State Human Resource Ministry for Returned Chinese Scholars, China (Grant No. 2005LXAH06) and the Research Foundation of Education Bureau of Anhui Province, China (Grant Nos. KJ2008A145 and 2002HBL05).