 Research
 Open Access
 Published:
Rydberg energies and transition probabilities of Li I for np–ms (m ≤ 5) transitions
BeniSuef University Journal of Basic and Applied Sciences volume 11, Article number: 42 (2022)
Abstract
Background
Mathematical modeling provides grounds for understanding scientific systems theoretically. It serves as a guide for experimentalists in determining directions of investigation. Recently, the Covid19 pandemic has caused disturbances in almost every walk of life. Scientists have played their role and have continued research on the effects of the pandemic. Various mathematical models have been used in different branches of science (Djilali et al. in Phys Scr 96 12 124016, 2021; Math Biosci Eng 18(6):8245–8256, 2021; Zeb et al. in Alex Eng J 61(7):5649–5665). Wellestablished mathematical models give results close to those obtained by experiments. The Weakest Bound Electron Potential Model is one such model, which explains hydrogenlike atoms and ions. This model has been used extensively for hydrogenlike atoms and ions to calculate energies of Rydberg levels and ionization energies. This model has been used extensively for hydrogenlike atoms and ions to calculate energies of Rydberg levels and ionization energies.
Results
This paper presents the energies of the Rydberg series, 2s2ns, and 2s2np of Li I, calculated using WBEPM. The energies are used to calculate transition probabilities from np to 2s, 3s, 4s, and 5s levels. The transition probabilities are compared with corresponding values in published data where available. The agreement with known values is good; most of the transition probabilities calculated in this work are new. A computer program was developed to find the value of the dipole matrix element. The calculations were further verified by calculating the lifetimes of some lowlying levels.
Conclusions
Four series of Li I have been studied, and energies of the Rydberg levels in the series were calculated. The energies then are used to calculate transition probabilities from np to ms transitions, where m = 2, 3, 4, & 5 and n = 1–15. The results are compared where available. An excellent agreement with previously published data shows the reliability of calculations. Most of the transition probabilities are new.
1 Background
Atomic lithium is less abundant in the universe because of its nuclear instability; it is a part of compounds. Lithium is an ideal atom for the spectroscopic study of astronomical data.; its 6708 Å doublet is present in Ktype and cooler stars. The 6708 Å line is also among those with the maximum range of energy distribution. Furthermore, it is present in a part of the spectrum that has little effect from stellar atmospheric absorption or the crowding of lines [4]. More than 300 papers have been published on lithium describing various spectroscopic characteristics; however, only a few papers are on transition probabilities [5]. Heavens used central field approximation to calculate lifetime and transition probabilities between lower excited states of alkali atoms [6]. Fox and Dalgarno used a theoretical method and obtained eigenvalues and eigenfunctions for 1s^{2}2p ^{2}P^{o}, 1s2p^{2 2}P, and 1s2p^{2 2}D states of lithium sequence. With the help of eigenfunction, they also calculated transition probabilities for these states [7]. Lingard and Nielson performed extensive calculations based on the numerical approximation method to find dipole transition probabilities, oscillator strengths, lifetimes, and branching ratios of alkali isoelectronic sequences [8]. Kostelecky and Nieto presented a potential function that reproduces spectra in the limits of quantum defect theory. With the help of this potential function, they obtained analytical wavefunctions and calculated transition probabilities of some transitions in lithium and Sodium atoms [9]. Fischer et al. presented multiconfiguration BreitPauli energy levels, lifetimes, and transition probabilities for the lithium sequence up to Z = 8 [10]. Zheng et al. employed Weakest bound electron potential model WBEPM theory to calculate transition probabilities for lithium atoms and lithiumlike ions [11]. Celik used the weakest bound electron potential model and quantum defect theory to find atomic transition probabilities for the transitions in atomic lithium [12]. 2009 Wiese and Fuhr critically evaluated available literature sources and tabulated atomic transition probabilities for allowed and forbidden transitions in hydrogen, helium, and lithium [13]. Experimentally very few scientists have measured transition probabilities for the lithium atom. Most of the experimental work has measured transition probabilities of 2p and 3p levels of Li I. Anderson et al. measured lifetime and oscillator strength of 2p state of Li I throughbeam foil excitation method [14]. Gaupp et al. also found the lifetime and oscillator strength of the 2p state through laser excitation in an atomic beam experiment [15]. In this work, we calculated transition probabilities of 110 transitions using the semiempirical nonrelativistic method.
2 Methods
In atoms, the outermost electrons are weakly bound to the nucleus as compared to the atomic core electrons. Lithium and other atoms have a single electron in their outermost shell; this electron is the weakest bound electron. Zheng et al. developed a theory, known as Weakest Bound Electron Potential Model (WBEPM), for the spectroscopic characteristics of such atoms [16]. The weakest bound electron model theory allows us to deal with manyelectron systems as a binary system; the weakest bound electron is considered an electron in a field created by a nucleus dressed up with an atomic core (consists of other electrons surrounding the nucleus). The weakest electron is excited to a higher energy level moves in an orbit that has a larger period; this reduces the coupling between core electrons and the weakest bound electron. Unlike Self consistent field (SCF) method in which each electron moves in average potential arising from other electrons, the WBEPM is based on the consideration of successive dynamic ionization [16]. The potential in which the weakest bound (WB) electron moves is given by
The radial part of the Schrodinger equation with this potential is as follows
The last two terms have the same denominator to be combined. Here \(A =  Z^{*}\) where \(Z^{*}\) is the effective charge of the nucleus,
here\({l}^{*}=l{\delta }_{n} , {n}^{*}=n{\delta }_{n}\), are effective orbital quantum number and principal quantum number, respectively, \({\delta }_{n}\) is the quantum defect in the quantum numbers and is given as a function of n.
\(\delta_{o}\) is the lowest quantum defect, & a, b, c, and d are found by fitting the first few energies of Rydberg levels, the energy (E) of the weakest bound electron is given by
The solution of Eq. (3) gives the radial wavefunction given by
The transition probability \((A_{fi} )\) of a transition for spontaneous emission between levels \(\left( {n_{f} ,l_{f} } \right) \& \left( {n_{i} ,l_{i} } \right)\), is given by
In above expression \(E_{f} > E_{i}\) and are the energies of upper and lower states, respectively, S represents electric dipole line strength. In lighter atoms, LS coupling dominates; therefore, line strength can be found by
The terms in the bracket contain two 6 J symbols and the matrix element \(P_{{l_{i} l_{f} }}^{\left( 1 \right)}\), which is given by
The integral in the above expression can be found by the formula based on WBEPM theory [11, 12].
in which \(S = {\text{min}}\left( {n_{f}^{*}  l_{f}^{*}  1  m_{1} , n_{i}^{*}  l_{i}^{*}  1  m_{2} } \right).\)
3 Results
Very few experimental results of transition probabilities and lifetimes are available for the transition studied in this work. In theory, only one study is found in which few transitions probabilities of lithium transitions are presented [8, 17]. Lindgren [8] used numerical coulomb approximation to calculate transition probabilities of various series for which n ≤ 12, l ≤ 4. The radiative transition probabilities of four Rydberg series in lithium are calculated in this work. This work employs a semiempirical nonrelativistic weakest bound electron potential model and consists of two parts. In the first part, Rydberg energies are calculated by finding corresponding quantum defects in principle quantum numbers of the Rydberg levels. The experimental values are used to calculate quantum defects and energies of the Rydberg levels and modified quantum numbers n* (Principal) and l* (orbital). The second part uses the energies and quantum defects to calculate transition probabilities for various transitions from np to 2s, 3s, 4s, and 5s levels. The quantum defects for Rydberg series 2s^{2}ns and 2s^{2}np are found using the following equations,
These quantum defects are then used to calculate the Rydberg energies of both series. WBEPM uses Eq. (10) for the calculations of matrix element, we developed a python program that calculates dipole matrix element using wavefunction given in Eq. (9). The python program's reliability is checked by calculating the radii of various ns and np levels, compared with the corresponding values in the literature. An excellent agreement was found between them. A generalized formula for each ns and np levels radii is generated and given in Eqs. (13–14) (which is a simple quadratic equation in 'n' (principal quantum number)). The energies and the values of matrix elements are used to calculate transition probabilities for the transitions from series np to state 2s, 3s, 4s, and 5s up to n = 30. The results of transition probabilities are given in Tables 1, 2, 3, and 4. The transition probability results are compared with the corresponding values listed in NIST line data [5] and previously published results [6, 8,9,10,11,12]. The experimental and theoretical data of transition probabilities is available only for a few transitions, especially for transitions from lower levels; therefore, most of the results are new. The calculation of transition probabilities using WBEPM requires modified values of principal and orbital quantum numbers; the modified values of n^{*} and l^{*} are determined by the quantum defects calculated in the first part. The transition probabilities are calculated using Eq. (7). The radii of ns and np series can be calculated by the simple quadratic equations given below:
3.1 Description of tables
The calculated transition probabilities for transitions np to 2s, 3s, 4s, 5s are shown in Tables 1, 2, 3, and 4. The first and second columns show the configurations of upper and lower levels of the transition, respectively. The third column shows the energies of the 1s^{2} np series calculated using quantum defect theory. The transition probabilities calculated using WBEPM for the transitions mentioned above are in the fourth column. In column five, corresponding probabilities published by other scientists are given.
3.1.1 Transitions 1s ^{2} np–1s ^{2} 2s
Table 1 gives the transition probabilities for the transitions 1s^{2} np to 1s^{2} 2s up to n = 30. Our transition probability results are close to corresponding NIST values; the average difference between is 3.6%, whereas the same difference between Lingard and NIST values is 6.3%. The results of this work are closer to NIST data compared to Zheng's results [11].
3.1.2 Transitions 1s ^{2} np–1s ^{2} 3s
Table 2 gives the transition probabilities for transitions 1s^{2} np–1s^{2} 3s. The results of this work are in good agreement with the NIST values, except for one transition probability for 1s^{2} 4p–1s^{2} 3s. Due to Cooper's minimum effect expected a large difference between the two values. Zheng’s results [11] for the transitions 1s^{2} 5p–1s^{2} 3s and 1s^{2} 6p–1s^{2} 3s have large deviations.
3.1.3 Transitions 1s ^{2} np–1s ^{2} 4s
Table 3 gives the transition probabilities for transitions 1s^{2} np–1s^{2} 4s. The NIST values for transition probability have uncertainties up to 10% in all transitions except 1s^{2} 4p–1s^{2} 4s. Only two transition results have large deviations from published data (1s^{2} 5p–1s^{2} 4s and 1s^{2} 6p–1s^{2} 4s). The transition probabilities for the transitions 1s^{2} 7p–1s^{2} 4s and 1s^{2} 8p to 1s^{2} 4s are closer to NIST data.
3.1.4 Transitions 1s ^{2} np–1s ^{2} 5s
Table 4 gives the transition probabilities for transitions 1s^{2} np–1s^{2} 5s. The transition probability for the transition 1s^{2} 6p–1s^{2} 5s, calculated in this work, differs from NIST values by 18%. The other values are in the acceptance range. Zheng presented transition probabilities for only two transitions; their results also agree with NIST data.
4 Discussion
The possible lifetimes of Rydberg levels are determined by using the relation \({\tau }_{i}={1 \mathord{\left/ {\vphantom {1 {\sum\nolimits_{j}A}_{ji}}} \right. \kern\nulldelimiterspace} {\sum\nolimits_{j}A}_{ji}}\) to check the reliability of the results. The lifetimes of 2p, 3s, 3p, and 4p Rydberg levels are determined and are shown in Table 5. The calculated values of lifetimes are compared with the experimental and theoretical results published in [6, 8, 17, 20,21,22,23,24,25]. An acceptable agreement is found between the lifetimes calculated in this work and the previously published results.
The dipole matrix elements are zero or have a minimum value for a particular set(s) of principal quantum numbers of upper and lower levels. It occurs due to the overlapping of positive and negative amplitudes of the wavefunctions of these levels. This situation is referred to as Cooper minimum. Due to this effect, the results of transition probabilities are not in good agreement for some of the transitions. Figure 1a–d show transition probabilities for the transition np–ms (m = 2, 3, 4, 5), respectively. A sharp drop can be seen in the transition probabilities in all figures. It shows that the probability of transitioning to neighboring levels is higher than those with larger principal quantum numbers. As principal quantum number increases, the value of transition probability decreases slower than the initial few levels. It results in large lifetimes for higher energy levels and occurs because the transition probability depends on oscillator strength. According to Fano and Cooper [19], the oscillator strength as a function of effective principal quantum number (n*) drops extremely rapidly towards a minimum, where a reversal of sign in the R integral occurs; this contributes to most of the strength. The rise from the minimum to the second maximum for higher n* is relatively slow. The second maximum has low oscillator strength and is usually not visible in the complete spectrum.
5 Conclusions
Lithium is a lighter atom, so a nonrelativistic approach is suitable to calculate various spectroscopic properties of such atoms. Weakest bound electron potential theory (a semiempirical method was employed in this work to calculate transition probabilities of various transitions from series 1s^{2} np–1s^{2} ms (\(2\le n\le 30, 2\le m\le 5\)). We present 110 transition probabilities, while Heavens [6], Lingard and Nelson, Kostelecky & Nieto [9], Fischer [10], Zheng et al. [11], and Celik reported 14, 48, 6, 1, 15, 4, and 35 transition probabilities, respectively. Meanwhile, NIST data shows 26 transition probabilities of these series. The results of this work are in good agreement except for four (1s^{2} 4p–1s^{2} 3s, 1s^{2} 5p–1s^{2} 4s, 1s^{2} 6p–1s^{2} 4s, 1s^{2} 6p–1s^{2} 5s) of the 110 transitions, and most of the results are new. Most of the results of transition probabilities in the literature focus on measurement of transition from 2p to 2s (experimental), and very few results of transition probabilities of lithium are reported yet. Lindgard calculated transition probabilities (theoretically) for some series of transitions in lithium for n ≤ 12. we extended the calculation up to n ≤ 30. These values of transition probabilities are helpful in the measurement of oscillator strengths and polarizabilities of Li I.
The graphs of transition probabilities show a sharp decrease towards a minimum, followed by a slight rise and eventually a slight decrease. It was observed that the dipole matrix elements have negative values for a few lower transitions that affect oscillator strengths and transition probabilities. It causes transition probabilities to have a minimum known as Cooper minimum. Cooper's minimum effect was first observed in a study of photoionization crosssection of alkali atoms. The calculations are based on the weakest bound electron potential model, which treats lithium as hydrogenlike, but hydrogen does not show zero dipole moment in photoionization. The Cooper minimum reminds us that Weakest bound electron potential model is an approximation; even though lithium is approximately hydrogenlike, it has two additional electrons in the inner shell.
Quantum defect theory was used to calculate quantum defects in principal quantum numbers, which are then used to calculate energies of Rydberg series 1s^{2} np, and 1s^{2} ns. A generalized formula of quantum defect for each of the series is developed. A computer program in python was developed and used to find the values of the matrix element; this part of the calculation makes our method slightly different from WBEPM theory. A quadratic equation can calculate the radii of the levels in ns and np series in 'n', which was developed to check the program's reliability. The matrix element, the energies, and quantum defects are used to calculate transition probabilities. For further verification of the results of this work, the lifetimes of some of the Rydberg levels are determined by using calculated transition probabilities. The results are compared with the previously published data, and a good agreement between the two results indicates the reliability of this work.
Availability of data and materials
Not applicable.
Abbreviations
 Li:

Lithium
 SCF:

Self consistent field
 WB:

Weakest bound
 WBEPM:

Weakest bound electron potential model
References
Djilali S, Bentout S, Ghanbari B, Kumar S (2021) Modeling escaping behavior from the herd in different ecological interactions. Phys Scr 96(12):124016
Djillali S, Bentout S, Touaoula TM, Tridane A (2021) Global dynamics of alcoholism epidemic model with distributed delays. Math Biosci Eng 18(6):8245–8256
Zeb A, Atangana A, Khan ZA, Djillali S (2022) A robust study of a piecewise fractional order COVID19 mathematical model. Alex Eng J 61(7):5649–5665
Bonsack WK (1959) The abundance of lithium and convective mixing in stars of type. Astrophys J 130:843
Heavens OS (1961) Radiative transition probabilities of the lower excited states of the alkali metals. JOSA 51(10):1058–1061
Fox JL, Dalgarno A (1977) Radiative transition probabilities of the 1s 2p^{2} ^{2}P and 1s 2p^{2} ^{2}D states of the lithium isoelectronic sequence. Phys Rev A 16(1):283
Lindgård A, Nielsen SE (1977) Transition probabilities for the alkali isoelectronic sequences Li I, Na I, KI, Rb I, Cs I, Fr I. At Data Nucl Data Tables 19(6):533–633
Kostelecký VA, Nieto MM (1985) Analytical wave functions for atomic quantumdefect theory. Phys Rev A 32(6):3243
Fischer CF, Saparov M, Gaigalas G, Godefroid M (1998) BreitPauli energies, transition probabilities, and lifetimes for 2s, 2p, 3s, 3p, 3d, 4s^{2 } levels of the lithium sequence, Z= 3–8. Atom Data Nuclear Data Tables 70(1):119–134
Zheng NW, Sun YJ, Wang T, Ma DX, Zhang Y, Su W (2000) Transition probability of lithium atom and lithiumlike ions with weakest bound electron wave functions and coupled equations. Int J Quantum Chem 76(1):51–61
Çelik G (2007) The calculation of transition probabilities between individual lines for atomic lithium. J Quant Spectrosc Radiat Transf 103(3):578–587
Wiese WL, Fuhr JR (2009) Accurate atomic transition probabilities for hydrogen, helium, and lithium. J Phys Chem Ref Data 38(3):565–720
Andersen T, Jessen KA, Sørensen G (1969) Mean life measurements of 2p ^{2}p^{0} levels in the lithium sequence. Phys Lett A 29(7):384–385
Gaupp A, Kuske P, Andrä HJ (1982) Accurate lifetime measurements of the lowest ^{2}P_{1/2} states in neutral lithium and sodium. Phys Rev A 26(6):3351
Zheng NW, Wang T, Ma DX, Zhou T, Fan J (2004) Weakest bound electron potential model theory. Int J Quantum Chem 98(3):281–290
Theodosiou CE (1984) Lifetimes of alkalimetal—atom Rydberg states. Phys Rev A 30(6):2881
McAlexander WI, Abraham ERI, Hulet RG (1996) Radiative lifetime of the 2 P state of lithium. Phys Rev A 54(1):R5
Fano U, Cooper JW (1968) Spectral distribution of atomic oscillator strengths. Rev Mod Phys 40(3):441
Carlsson J, Sturesson L (1989) Accurate timeresolved laser spectroscopy on lithium atoms. Zeitschrift für Physik D Atoms, Mol Clust 14(4):281–287
McAlexander WI, Abraham ERI, Ritchie NWM, Williams CJ, Stoof HTC, Hulet RG (1995) Precise atomic radiative lifetime via photoassociative spectroscopy of ultracold lithium. Phys Rev A 51(2):R871
Volz U, Schmoranzer H (1996) Precision lifetime measurements on alkali atoms and on helium by beam–gas–laser spectroscopy. Phys Scr 1996(T65):48
Heldt J, Leuchs G (1979) Lifetime measurements of Lithium Dstates. Zeitschrift für Physik A Atoms and Nuclei 291(1):11–13
Hansen W (1983) Radiative lifetimes of excited Li I states. J Phys B Atom Mol Phys 16(6):933
Pipin J, Bishop DM (1992) Accurate variational calculations of energies of the 2 ^{2}S, 2 ^{2}P, and 3 ^{2}D states and the dipole, quadrupole, and dipolequadrupole polarizabilities and hyperpolarizability of the lithium atom. Phys Rev A 45(5):2736
Ashby R, van Wijngaarden WA (2003) Lifetimes and polarizabilities of low lying lithium S, P and D states. J Quant Spectrosc Radiat Transf 76(3–4):467–473
Nagourney W, Happer W, Lurio A (1978) Levelcrossing study of the hyperfine structure of lithium. Phys Rev A 17(4):1394
Isler RC, Marcus S, Novick R (1969) Hyperfine structure of the 3^{2}P and 4^{2}P states of lithium and lifetime of the 3^{2}P state. Phys Rev 187(1):76
Acknowledgements
Not applicable.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
M N Hameed, M H Zaheer, M B Khan contributed to making computer program for calculations, R Siddiq wrote the manuscript and did calculations, Z Uddin supervised this work. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Siddiq, R., Hameed, M.N., Zaheer, M.H. et al. Rydberg energies and transition probabilities of Li I for np–ms (m ≤ 5) transitions. BeniSuef Univ J Basic Appl Sci 11, 42 (2022). https://doi.org/10.1186/s43088022002240
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s43088022002240
Keywords
 Rydberg atom
 Rydberg levels
 Transition probability
 Lifetime of atoms
 Lithium