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Article

Multi-Configuration Dirac–Hartree–Fock (MCDHF) Calculations for B-Like Ions

by
Indu Khatri
1,*,
Arun Goyal
1,
Avnindra Kumar Singh
2 and
Man Mohan
1
1
Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
2
Department of Physics, D.D.U. College, University of Delhi, Delhi 110015, India
*
Author to whom correspondence should be addressed.
Atoms 2016, 4(2), 13; https://doi.org/10.3390/atoms4020013
Submission received: 9 December 2015 / Revised: 1 April 2016 / Accepted: 5 April 2016 / Published: 6 May 2016
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Abstract

:
Relativistic configuration interaction results are presented for several B-like ions (Ge XXVIII, Rb XXXIII, Sr XXXIV, Ru XL, Sn XLVI, and Ba LII) using the multi-configuration Dirac–Hartree–Fock (MCDHF) method. The calculations are carried out in the active space approximation with the inclusion of the Breit interaction, the finite nuclear size effect, and quantum electrodynamic corrections. Results for fine structure energy levels for 1s22s22p and 2s2p2 configurations relative to the ground state are reported. The transition wavelengths, transition probabilities, line strengths, and absorption oscillator strengths for 2s22p–2s2p2 electric dipole (E1) transitions are calculated. Both valence and core-valence correlation effects were accounted for through single-double multireference (SD-MR) expansions to increasing sets of active orbitals. Comparisons are made with the available data and good agreement is achieved. The values calculated using core–valence correlation are found to be very close to other theoretical and experimental values. The behavior of oscillator strengths as a function of nuclear charge is studied. We believe that our results can guide experimentalists in identifying the fine-structure levels in their future work.

1. Introduction

For ions with three electrons in the valence shell, it is quite simple to take all the configuration interactions between the states of the ground complex into consideration. Besides, three electron spectra are sufficiently complex to show unusual level anti-crossing effects. Due to increased availability of experimental data for highly-ionized systems obtained from beam-foil experiments and from astrophysical measurements, interest in transition rates and oscillator strengths in highly ionized atoms has increased. The calculated results are useful in the case of yet unobserved transitions and also for determining the density and temperature of the solar corona or in the diagnostic studies of thermonuclear plasmas. The X-ray spectra from L-shell ions lie in the wavelength region covered by the space observatories XMM-Newton and Chandra [1] and thus may be important for astrophysics. The spectral studies of boron-isoelectronic sequence are of great importance in diagnostics of solar, astrophysical, and fusion plasmas [2,3]. Transitions within n = 2 complex of ions in the boron isoelectronic sequences have been observed in tokamak [4,5] and astrophysical plasmas [6]. Transition within the 2s22p ground configuration are particularly useful for diagnostics of electron densities in the range 1012 to 1014 cm−3 [7,8]. Germanium is a useful element for plasma diagnostics [9].
Theoretically, many authors have contributed to the study of boron-like ions [10,11,12,13,14,15,16]. Bhatia et al. [17] calculated oscillator strengths, radiative decay rates, and collision strengths for many ions including Ge XXVIII. Energy levels and rates for electric dipole transitions in boron like ions between B I and Si IX were presented by Fischer and Tachiev [18,19] using multi-configuration Breit–Pauli wave functions. Fischer also calculated energy levels, transition rates, and lifetimes of boron-like ions using the multi-configuration Dirac–Hartree–Fock method [20]. Energy levels and transition rates were presented by Koc [21,22,23] based on the multireference relativistic configuration interaction method with the no-pair Dirac–Coulomb–Breit Hamiltonian. Corrégé and Hibbert [24] presented energy levels, oscillator strengths, and transition probabilities for C II, N III, and O IV using the CIV3 code. Energy levels, specific mass shift parameters, and transition probabilities for C II, N III, and O IV were presented by Jönsson et al. [25,26]. Hao and Jiang reported energy levels, transition rates, and line strengths for several ions along the B I isoelectronic sequence using multi-configuration Dirac–Hartree–Fock method [27]. Energies, transition rates, line strengths, oscillator strengths, and lifetimes were reported for boron-like ions between N III and Zn XXVI by Rynkun et al. [28]. Lifetime of the 2s22p 2P3/2 level and fine structure energy splitting between 2P3/2 and 2P1/2 levels in B-isoelectronic sequence were obtained by Marques et al. [29]. Energy, fine structure, hyperfine structure, and radiative transition rates of the high-lying multi-excited states for B-like Ne were obtained by Zhang et al. [30]. Chen [31] calculated energies, expectation values, fine structures, and hyperfine structures of the ground state and excited states for boron using the Rayleigh–Ritz variational method. An experimental study for transitions within n = 2 complex of Ba51+ has been made by Reader et al. [32].
In this paper, the MCDHF method is employed to determine fine structure energy levels, E1 wavelengths, transition probabilities, oscillator strengths, and line strengths between the states of 2s22p and 2s2p2 for Ge XXVIII, Rb XXXIII, Sr XXXIV, Ru XL, Sn XLVI, and Ba LII using the GRASP2K code. The valence–valence (VV) and core–valence (CV) correlation effects are taken into account in a systematic way using active space approximation. Breit interactions and quantum electrodynamics (QED) effects are added in subsequent relativistic configuration interaction calculations. The accuracy of wavefunctions and calculated eigenvalues are assessed from the analysis of the convergence patterns and from the comparison with the available data. The ratio of the length to velocity forms of the transition rates (Al/Av) are also used to estimate the accuracy of our calculations. The calculated data will be useful for identifying fine structure levels and transition lines in further investigations.

2. Method of Calculation

2.1. Computational Procedure

The Grasp2K code [33] is based on the multi-configuration Dirac–Hartree–Fock (MCDHF) approach, taking relativistic and QED corrections into consideration. The MCDHF method has been described in detail by Grant [34]. We give a brief overview of the important features of the method.
The Dirac–Coulomb Hamiltonian is
H D C = ∑ i ( c α → i ⋅ p → i + ( β i − 1 ) c 2 + V i N ) + ∑ i > j 1 / r i j .
The first term denotes the one-body contribution for an electron due to kinetic energy and interaction with the nucleus in JJ coupling. Here, α and β are 4 × 4 Dirac matrices, c denotes the speed of light, and VN is the monopole part of the electron–nucleus coulomb interaction. The second term consists of the two-body Coulomb interactions between the electrons. The configuration state functions (CSFs) Φ(ΓαJP) are formed by symmetry-adapted linear combinations of Slater determinants of the Dirac orbitals. Atomic state functions are then constructed by a linear combination of these atomic state functions (ASFs).
Ψ i ( J P ) = ∑ α = 1 n c s f C i α Φ ( Γ α J P )
In the above equation, Ciα are mixing coefficients for the state i and ncsf denotes the number of CSFs used in the evaluation of ASFs. The one-electron and intermediate quantum numbers needed to define the CSFs are represented by Γα. The configuration mixing coefficients Ciα are obtained through diagonalization of the Dirac–Coulomb Hamiltonian given in Equation (1). The radial parts of the Dirac orbitals and the expansion coefficients are optimized self-consistently in the relativistic self-consistent field procedure. After this, relativistic configuration interaction (RCI) calculations [35] can be performed. The most important transverse photon interaction included in the Hamiltonian
H B r e i t = − ∑ i < j N [ α i ⋅ α j cos ( w i j r i j / c ) r i j + ( α i ⋅ ∇ i ) ( α j ⋅ ∇ j ) cos ( w i j r i j / c ) − 1 w i j 2 r i j / c 2 ]
The contributions from the Breit interaction, vacuum polarization, self-energy, and finite nuclear mass corrections are added as first-order perturbation correction. The spin-angular part of the matrix elements is calculated using the second quantization method in coupled tensorial form and quasispin technique [36].

Transition Parameters

The transition parameters such as line strengths and rates for multipole transitions between two states ψα(PJM) and ψα(P′J′M′) can be expressed in terms of the transition matrix element:
〈 ψ α ( P J M ) ‖ Q k ( λ ) ‖ ψ α ( P ′ J ′ M ′ ) 〉 .
Here Q k ( λ ) denotes the corresponding transition operator of order k in Coulomb or Babushkin gauge [37]. Biorthogonal transformations of the atomic state functions were performed to compute the transition matrix element between two atomic state functions described by independently optimized orbital sets. Racah algebra techniques were used to evaluate the matrix element in the new representation.

2.2. Calculation Procedure

The extended optimal level (EOL) version of the MCDHF method is used to optimize the wave functions for all fine structure levels within a given term. In the EOL scheme [38], the optimization is on the weighted energy average of the states. The significant interactions between neighboring levels can be determined accurately in this method as simultaneous optimization of multiple levels with a specific J is performed in this method. We included different correlations in the calculation in a systematic approach; they are represented by the different constraints on the generation of CSFs included in Equation (2).The correlation between the valence electrons is defined as valence correlation (VV).In this, the core electrons are kept fixed and CSFs are generated by exciting valence electrons. The correlation between the valence electrons and core electrons is defined as core–valence correlation (CV), where one of the core electrons is excited to generate the CSFs. More than one core electron is allowed to excite in the core–core (CC) correlation, which is between the core electrons. We generated the CSFs using the active space approach [39,40]. This was done by exciting electrons from the reference configurations to a set of orbitals called the active set (AS).To generate configuration expansions for the fine structure terms belonging to the 2s22p ground configuration, single and double substitutions (SD) were performed from the {2s22p, 2p3} multireference set to an active set of orbitals. For the terms belonging to the 2s2p2 configuration, CSFs were generated by SD substitutions from the single reference configurations. By allowing excitations from a number of reference configurations to a set of relativistic orbitals, jj-coupled CSFs of particular parity and J symmetry were generated. We systematically enlarged the active sets to orbitals with principal quantum number n = 3, …, 7 and orbital quantum numbers l = 0, …, 4 (s,p,d,f,g) to observe the convergence.
AS1 = {n = 3, l = 0–2}
Then, the active set is increased in the way shown:
AS2 = AS1 + {n = 4, l = 0–3}
AS3 = AS2 + {n = 5, l = 0–4}
AS4 = AS3 + {n = 6, l = 0–4}
AS5 = AS4 + {n = 7, l = 0–4}.
Active set was increased in steps of orbital layers as orbitals with the same principal quantum number have similar energies. We optimized separately a set of orbitals for the even states and for the odd states.
In the present work, we included valence–valence (VV) and core–valence (CV) electron correlation effects to describe the inner properties. To reduce the processing time only the newly added orbitals were optimized. RCI (Relativistic Configuration Interaction) calculations including Breit interactions were performed to consider higher order correlation effects. Finally, the multireference sets for odd and even parity states were enlarged to include {2s22p, 2p3, 2s2p3d, 2p3d2} and {2s2p2, 2p23d, 2s23d, 2s3d2}, respectively. The configurations with largest weights in the preceding self-consistent field calculations were included in the multireference set. To the final RCI calculations, QED effects (vacuum polarization and self-energy) were added as perturbation. The mixing coefficients obtained in the block structure format using MCDHF and RCI orbital wave functions were then reformed into non-block format and the initial and final state orbital wave functions were redesigned to a new form in which the two orbitals are biorthonormal [41,42]. These biorthonormal wavefunctions were then used to evaluate the dipole transition rates.

3. Results and Discussion

A very efficient way to ensure the convergence of atomic property within a certain correlation model is to use the active set approach to enlarge the configuration expansion systematically. We optimized the states of 2s22p, 2s2p2 configurations layer by layer. In order to consider VV correlations, calculations were performed with CSFs generated by single and double excitations from the 2s and 2p shells of the reference configurations 2s22p and 2s2p2 to the active set. For CV calculations, we allow excitations from the 1s orbital also. Table 1 displays our computed level energies for 10 levels belonging to 2s22p and 2s2p2 configurations of Ge XXVIII, Rb XXXIII, Sr XXXIV, Ru XL, Sn XLVI, and Ba LII as functions of the increasing active sets for VV and CV correlations. Energy contributions from Breit interaction and QED corrections are included in the calculations. For the odd parity states of Ge XXVIII, our RCI calculations included 182,470 and 6055 CSFs distributed over J = 1/2, 3/2 angular symmetries for the n = 7 results in the CV and VV correlations, respectively. For the even parity states, there were 277,127 and 10,546 CSFs distributed over J = 1/2, 3/2, and 5/2. Comparing our VV and CV calculations with other available data, we observe an improvement in the agreement when core orbital excitations are included. The computed energies for Rb XXXIII, Sr XXXIV, and Ba LII from CV correlation results agree well with the experimental values. The largest discrepancy between our computed energies from n = 7 CV calculations and National Institute of Standards and Technology (NIST) [43] values is 0.57% for 2s2p2 2P1/2 level of Rb XXXIII. Further, we find our calculated energies are in good agreement with Koc [22] energy values. As Z increases, level ordering changes for some levels as seen from Table 1. For Ru XL, Sn XLVI and Ba LII, 2s2p2 4P1/2 lies above 2s22p 2P°3/2 and for Sn XLVI and Ba LII, the levels 2s2p2 4P5/2 and 2s2p2 2D5/2 are interchanged.
The zero-order Dirac–Fock wave functions given by the reference configuration in the absence of electron correlation include limited number of configurations and hence are insufficient to represent the occupied orbitals. Therefore, more configurations must be added to represent electron correlations. The CSFs generated from these configurations must have same angular momentum and parity as the occupied orbital. In Table 2, we have presented the mixing coefficients for the wave functions of our calculated levels. For instance, the configuration mixed wave function for the 2s2p2 4P1/2 level for Ge XXVIII is represented as
2s2p2 4P1/2 = 0.94 2s2p2(4P1/2) + 0.30 2s2p2(2S1/2) + 0.13 2s2p2 (2P1/2),
where 0.94, 0.30 and 0.13 are the configuration mixing coefficients. The maximum contribution to the total wave function of a given level is from the same configuration. The contribution from each level is also listed in the table.
In Table 3, the radiative data for its 2s22p 2P°1/2–2s2p2 2D3/2 transition in Ge XXVIII are shown as functions of increasing active sets in VV and CV correlations. In both correlation calculations, the convergence of the results can be clearly seen as n increases. A good agreement between Coulomb and Babushkin gauges is found and this agreement improves with increasing n.
In Table 4, we have presented transition wavelengths as well as radiative rates, oscillator strengths, and line strengths for E1 (electric dipole) transitions from the ground state and first excited state for CV correlations. As CV results are better and converged fully, we have performed the calculations including CV correlation with n = 7. Results are provided in both Coulomb and Babushkin gauges. Both forms agree well as can be seen from Table 4, which indicates the accuracy of our results. A comparison between our computed transition wavelengths with NIST values [44] has been provided wherever possible and a good agreement achieved. The values of δT, which represents the deviation of ratio of length and velocity form of line strengths from unity and thus is an accuracy indicator, have also been tabulated. The maximum value of δT is 0.24, which confirms the accuracy of our results. In Figure 1 we have plotted transition wavelengths from ground state 2s22p2 P°1/2 to 2s2p2 4P1/2, 2s2p2 4P3/2, 2s2p2 2D3/2, 2s2p2 2P1/2, 2s2p2 2P3/2 and 2s2p2 2S1/2 levels as a function of Z. It is observed that wavelength decreases with increasing Z. The Al/Av values of E1 transitions from B-like ions for various Z are plotted in Figure 2. The ratios range from 0.90 to 1.01.The small discrepancy in the Al and Av values may be taken as a measure of the reliability of the computed rates. In Figure 3 and Figure 4 we display the weighted oscillator strengths (gf) in length form for various E1 transitions from ground state as length form is considered more stable. For an allowed transition, the Z dependence depends on the Δj value. The jumping electron in such transition is either of type 2s1/2–2p1/2 (Δj = 0) or 2s1/2–2p3/2 (Δj = 1). For the B I sequence, the f value for an allowed transition (Δj = 1) increases slowly with Z while for transitions (Δj = 0) the f value decrease slowly with Z [45]. However, in the intermediate-Z region anti-crossings of the energy levels occur between two levels of same configuration, with the same J value having the same parity due to strong mixing in the corresponding wavefunctions. These states are nearly degenerate at a well-defined Z value. These anti-crossings of energy levels have a significant influence on the f value of the corresponding lines and can account for the anomalies in the systematic trends of the oscillator strengths involving the corresponding states.

4. Conclusions

In this work, we have provided a detailed and systematic study of fine-structure energy levels, wavelengths, transition rates, and line strengths for transitions among levels belonging to 2s22p and 2s2p2 configurations of Ge XXVIII, Rb XXXIII, Sr XXXIV, Ru XL, Sn XLVI, and Ba LII. The MCDHF method has been adopted for the calculations. The calculations were performed for valence–valence and core–valence correlations through large configuration expansions in a systematic way using active set approach. The self-consistent field approximation, Breit interaction, and QED effects are included to improve the atomic state functions and the corresponding energies. Results from our present calculations are in good agreement with other available theoretical and experimental results. Nearly equal values of length and velocity forms indicate the accuracy of our results. It is clear that the relativistic and configuration interaction effects are important in the accurate evaluation of atomic data. It is clear that for all Z ions, the MCDHF method including core–valence correlation is an accurate approach for the whole sequence. We hope that these results will be useful for analyzing data from fusion devices and from astrophysical sources and in the modeling and characterization of plasmas.

Acknowledgments

Indu Khatri and Arun Goyal are thankful to U.G.C., India for their Junior Research Fellowship. M.M. Man Mohan is thankful to U.G.C. for Basic Science Research Fellowship.

Author Contributions

Indu Khatri and Arun Goyal proposed the main idea of the paper and performed the calculations. Man Mohan and Avnindra Kumar Singh provided the required computer hardware, were involved in the discussion and in writing the manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

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  45. Farrag, A.; Luc-Koenig, E.; Sinzelle, J. Systematic trends of the relativistic f values for electric dipole transitions within the ground complex of B-like ions. J. Phys. B At. Mol. Opt. Phys. 1980, 13, 3939–3955. [Google Scholar] [CrossRef]
Atoms 04 00013 g001 1024
Figure 1. Transition wavelengths from ground state 2s22p 2P°1/2 as a function of Z where curves a, b, c, d, e, f represent transitions to 2s2p2 4P1/2, 2s2p2 4P3/2, 2s2p2 2D3/2, 2s2p2 2P1/2, 2s2p2 2P3/2, and 2s2p2 2S1/2 levels, respectively.
Figure 1. Transition wavelengths from ground state 2s22p 2P°1/2 as a function of Z where curves a, b, c, d, e, f represent transitions to 2s2p2 4P1/2, 2s2p2 4P3/2, 2s2p2 2D3/2, 2s2p2 2P1/2, 2s2p2 2P3/2, and 2s2p2 2S1/2 levels, respectively.
Atoms 04 00013 g001
Atoms 04 00013 g002 1024
Figure 2. Ratios of length to velocity forms of rates for various transitions from 2s22p 2P°1/2 state. Here 1, 2, 3, 4, 5, 6 represent transitions to 2s2p2 4P1/2, 2s2p2 4P3/2, 2s2p2 2D3/2, 2s2p2 2P1/2, 2s2p2 2P3/2, and 2s2p2 2S1/2 state, respectively.
Figure 2. Ratios of length to velocity forms of rates for various transitions from 2s22p 2P°1/2 state. Here 1, 2, 3, 4, 5, 6 represent transitions to 2s2p2 4P1/2, 2s2p2 4P3/2, 2s2p2 2D3/2, 2s2p2 2P1/2, 2s2p2 2P3/2, and 2s2p2 2S1/2 state, respectively.
Atoms 04 00013 g002
Atoms 04 00013 g003 1024
Figure 3. Weighted oscillator strengths (gf) as a function of Z. The curves a, b, c, d correspond to transitions to 2s2p2 4P1/2, 2s2p2 2P3/2, 2s2p2 4P3/2, 2s2p2 2S1/2 from 2s22p 2P°1/2 state, respectively
Figure 3. Weighted oscillator strengths (gf) as a function of Z. The curves a, b, c, d correspond to transitions to 2s2p2 4P1/2, 2s2p2 2P3/2, 2s2p2 4P3/2, 2s2p2 2S1/2 from 2s22p 2P°1/2 state, respectively
Atoms 04 00013 g003
Atoms 04 00013 g004 1024
Figure 4. Weighted oscillator strengths (gf) as a function of Z. The curves e, and f correspond to transitions to 2s2p2 2D3/2, and 2s2p2 2P1/2 from 2s22p 2P°1/2 state, respectively.
Figure 4. Weighted oscillator strengths (gf) as a function of Z. The curves e, and f correspond to transitions to 2s2p2 2D3/2, and 2s2p2 2P1/2 from 2s22p 2P°1/2 state, respectively.
Atoms 04 00013 g004
Table
Table 1. Energies (in cm−1) of fine-structure relativistic levels of Ge XXVIII, Rb XXXIII, Sr XXXIV, Ru XL, Sn XLVI, and Ba LII relative to ground state of B-like ions.
Table 1. Energies (in cm−1) of fine-structure relativistic levels of Ge XXVIII, Rb XXXIII, Sr XXXIV, Ru XL, Sn XLVI, and Ba LII relative to ground state of B-like ions.
Ge
ConfigurationVVCV
n = 4n = 5n = 6n = 7n = 4n = 5n = 6n = 7Ref. [17]Δ%
2s2 2p 2P° 1/20.000.000.000.000.000.000.000.000.000.00
2s2 2p 2P° 3/2294,359.19294,335.58294,358.61294,371.64294,445.85294,544.84294,700.43294,824.39294,5390.09
294,658a0.06
294,668b0.05
294,550c0.09
294,614d0.07
2s2p2 4P1/2572,810.09573,172.37573,274.34573,321.06572,252.78572,596.33572,794.37572,943.86574,5590.28
573,139a0.03
572,858b0.01
2s2p2 4P3/2740,529.66740,925.26741,020.59741,066.48740,080.93740,426.04740,591.13740,721.62742,5980.25
2s2p2 4P5/2846,036.05846,212.79846,272.93846,303.89845,542.28845,510.69845,513.03845,531.07847,4060.22
2s2p2 2D3/21,103,153.701,102,793.771,102,766.601,102,760.551,102,276.491,101,807.991,101,836.131,101,909.391,102,1110.02
2s2p2 2P1/21,199,600.451,199,379.881,199,412.071,199,425.531197969.031197600.771,197,678.131,197,770.151,197,3230.04
2s2p2 2D5/21,211,205.851,210,750.861,210,686.991,210,664.841,210,505.671,209,856.501,209,748.391,209,723.041,211,6080.15
2s2p2 2S1/21,513,917.871,513,582.641,513,577.791,513,573.671,512,905.951,512,529.921,512,614.731,512,712.551,505,5790.47
2s2p2 2P3/21,527,073.631,526,714.161,526,697.231,526,695.121525292.771524737.121,524,740.201,524,799.941,523,4910.08
Rb
ConfigurationVVCV
n = 4n = 5n = 6n = 7n = 4n = 5n = 6n = 7OtherΔ%
2s22p 2P° 1/20.000.000.000.000.000.000.000.000.000.00
2s22p 2P° 3/2554,879.16554,870.27554,891.60554,907.79554,994.57555,105.27555,259.44555,379.85554,700c0.12
555,329a0.01
555,338b0.01
2s2p2 4P1/2730,395.31730,657.17730,740.17730,781.53729,723.78729,940.10730,109.73730,241.58726,900c0.46
730,429a0.02
730,041b0.03
2s2p2 4P3/21,095,432.101,095,811.161,095,900.171,095,946.771,094,951.961,095,471.351,095,401.411,095,517.751,089,900c0.51
2s2p2 4P5/21,245,199.881,245,245.181,245,275.421,245,296.701,244,612.541,244,431.321,244,402.221,244,404.331,242,100c0.18
2s2p2 2D3/21,527,957.941,527,589.891,527,553.991,527,548.251,527,778.221,526,581.581,526,417.031,526,473.271,532,900c0.42
2s2p2 2P1/21,597,246.421,597,079.831,597,124.781,597,147.921,595,466.011,595,131.271,595,213.201,595,303.301,604,400c0.57
2s2p2 2D5/21,814,677.321,814,403.751,814,366.821,814,358.661,814,002.121,813,535.331,813,459.031,813,444.101,814,700c0.07
2s2p2 2S1/22,187,867.672,187,546.022,187,542.462,187,542.522,186,881.542,186,524.702,186,614.522,186,711.392,191,200c0.20
2s2p2 2P3/22,199,518.572,199,195.732,199,184.272,199,188.432,198,582.202,197,311.712,197,213.852,197,269.662,207,900c0.48
Sr
ConfigurationVVCV
n = 4n = 5n = 6n = 7n = 4n = 5n = 6n = 7OtherΔ%
2s22p 2P° 1/20.000.000.000.000.000.000.000.000.000.00
2s22p 2P° 3/2623,284.03623,275.74623,301.64623,316.67623,405.36623,519.82623,675.76623,379.36623,100c0.04
623,739a0.06
623,747b0.06
2s2p2 4P1/2763,024.73763,262.18763,345.64763,384.59762,324.87762,514.86762,679.97762,807.35759,800c0.39
762,976a0.02
762,565b0.03
2s2p2 4P3/21,183,539.381,183,909.041,184,000.991,184,046.381,183,036.481,183,318.821,183,448.771,183,551.331,178,000c0.47
2s2p2 4P5/21,341,468.231,341,487.091,341,515.581,341,534.531,340,860.771,340,652.951,340,618.811,340,617.831,338,600c0.15
2s2p2 2D3/21,629,937.451,629,540.421,629,503.961,629,495.581,628,859.191,628,304.401,628,288.421,628,331.201,634,900c0.40
2s2p2 2P1/21,693,721.641,693,562.921,693,614.611,693,638.491,691,914.661,691,588.561,691,673.261,691,762.371,700,800c0.53
2s2p2 2D5/21,969,084.811,968,840.721,968,812.811,968,807.121,968,414.671,967,981.541,967,912.631,967,900.111,968,800c0.04
2s2p2 2S1/22,355,943.752,355,623.322,355,624.882,355,624.752,354,959.942,354,609.242,354,702.022,354,798.152,359,100c0.18
2s2p2 2P3/22,367,421.822,367,072.082,367,061.002,367,063.132,365,622.032,365,064.972,365,055.702,365,100.132,375,500c0.44
Ru
ConfigurationVVCV
n = 4n = 5n = 6n = 7n = 4n = 5n = 6n = 7Ref. [23]Δ%
2s22p 2P° 1/20.000.000.000.000.000.000.000.000.00
2s2p2 4P1/2963,735.96963,847.87963,910.53963,929.75962,833.63962,867.47962,993.35963,105.03963,285a0.02
962,734b0.04
2s22p 2P° 3/21,180,758.611,180,767.781,180,797.791,180,804.101,180,916.591,181,063.551,181,219.581,181,344.531,181,453a0.01
1,181,450b0.01
2s2p2 4P3/21,865,365.701,865,718.291,865,804.501,865,836.541,864,874.291,865,056.331,865,163.521,865,257.38
2s2p2 4P5/22,066,640.942,066,553.942,066,559.152,066,556.662,065,913.402,065,586.262,065,510.672,065,498.15
2s2p2 2D3/22,393,724.802,393,327.092,393,285.002,393,262.222,392,484.982,391,858.982,391,815.832,391,847.82
2s2p2 2P1/22,428,031.862,427,931.612,427,998.542,428,019.482,426,075.452,425,798.572,425,886.932,425,980.87
2s2p2 2D5/23,199,253.573,199,167.003,199,170.593,199,166.723,198,607.593,198,345.463,198,301.123,198,306.13
2s2p2 2S1/23,665,385.673,665,095.713,665,106.493,665,100.753,664,395.963,465,296.953,465,029.123,464,957.16
2s2p2 2P3/23,676,012.003,675,708.303,675,706.903,675,701.933,674,204.063,664,086.233,664,186.223,664,289.99
Sn
ConfigurationVVCV
n = 4n = 5n = 6n = 7n = 4n = 5n = 6n = 7Ref. [23]Δ%
2s22p 2P° 1/20.000.000.000.000.000.000.000.000.00
2s2p2 4P1/21,172,932.391,172,931.391,172,977.251,173,002.491,171,793.521,171,681.511,171,782.281,171,874.841,172,053a0.01
1,171,358b0.04
2s22p 2P° 3/22,063,472.212,063,518.862,063,547.862,063,572.302,063,683.872,063,868.262,064,045.182,064,175.012,064,478a0.01
2,064,448b0.01
2s2p2 4P3/22,884,412.962,884,756.242,884,839.772,884,884.612,883,794.522,884,027.382,884,133.522,884,220.49
2s2p2 2D5/23,121,078.423,120,936.003,120,931.843,120,935.403,120,228.303,119,821.853,119,731.043,119,706.80
2s2p2 2D3/23,493,429.543,493,073.933,493,038.563,493,033.783,491,945.093,491,391.153,491,352.023,491,378.27
2s2p2 2P1/23,505,894.953,505,837.963,505,919.223,505,960.423,503,775.073,503,534.713,503,638.303,503,732.41
2s2p2 4P5/25,096,696.595,096,720.505,096,752.035,096,771.525,096,062.174,761,571.794,761,262.074,761,169.15
2s2p2 2S1/25,640,312.355,640,057.955,640,081.855,640,096.005,639,304.585,095,918.025,095,910.465,095,927.34
2s2p2 2P3/25,650,330.145,650,123.495,650,145.575,650,165.525,648,454.695,639,044.375,639,168.235,639,277.45
Ba
ConfigurationVVCV
n = 4n = 5n = 6n = 7n = 4n = 5n = 6n = 7Ref. [23]Δ%
2s22p 2P° 1/20.000.000.000.000.000.000.000.00
2s2p2 4P1/21,394,480.971,394,373.831,394,398.451,394,422.041,393,070.221,392,820.601,392,896.061,392,944.14
1,393,276a0.02
1,392,459b0.03
2s22p 2P° 3/23,393,333.193,393,412.403,393,454.033,393,489.633,393,607.703,393,850.703,394,049.823,394,191.00
3,394,759a0.02
3,394,676b0.01
2s2p2 4P3/24,365,092.434,365,428.444,365,506.064,365,555.284,364,405.034,364,626.734,364,728.874,364,816.11
2s2p2 2D5/24,631,483.734,631,314.924,631,300.474,631,309.824,630,674.884,630,045.474,629,940.554,629,909.23
2s2p2 2P1/25,052,120.835,052,094.245,052,183.325,052,236.955,049,800.115,049,589.515,049,709.125,049,720.16
2s2p2 2D3/25,054,465.275,054,148.705,054,121.675,054,125.235,052,767.665,052,245.405,052,210.605,052,239.37
2s2p2 4P5/27,905,439.737,905,551.787,905,601.747,905,639.217,904,940.536,535,783.596,535,433.146,535,318.45
2s2p2 2S1/28,528,078.138,527,865.228,527,899.718,527,927.198,527,043.947,904,769.397,904,793.467,904,826.22
2s2p2 2P3/28,537,396.208,537,283.198,537,328.528,537,364.668,535,451.458,526,851.598,527,006.468,527,195.97
Theory – DHFa: a QED corrections estimated from hydrogenic self-energy [23];
Theory – DHF b: b QED corrections from screened model [23]
c: NIST [43]; Δ% : | M C D H F − O t h e r s | O t h e r s × 100
d: Ref. [29]
Table
Table 2. The configuration mixing coefficients and contributions for levels in B-like ions. The number in parenthesis refers to the level number.
Table 2. The configuration mixing coefficients and contributions for levels in B-like ions. The number in parenthesis refers to the level number.
Ge
LevelMixContribution
2s22p 2P°1/20.99 (1) + 0.12 (2p3 2P°1/2)0.98 (1) + 0.01 (2p3 2P°1/2)
2s22p 2P°3/20.99 (2) + 0.14 (2p3 2P°3/2)0.98 (2) + 0.02 (2p3 2P°3/2)
2s2p2 4P1/20.94 (3) + 0.30 (9) + 0.13 (7)0.89 (2s2p2 4P1/2) + 0.09 (9) + 0.02 (7)
2s2p2 4P3/2–0.99 (4) + 0.11 (6)+ 0.08 (10)–0.99 (4) + 0.11 (6) + 0.08 (10)
2s2p2 4P5/20.90 (5) – 0.43 (8)0.81 (5) + 0.19 (8)
2s2p2 2D3/20.93 (6) + 0.34 (10) + 0.08 (4)0.87 (6) + 0.12 (10) + 0.01 (4)
2s2p2 2P1/20.84 (7) + 0.47 (9) + 0.27 (3)0.70 (7) + 0.22 (9) + 0.07 (3)
2s2p2 2D5/20.90(8) + 0.43 (5)0.81 (8) + 0.19 (5)
2s2p2 2S1/20.82 (9) – 0.52 (7) – 0.19(3)0.69 (9) + 0.27 (7) + 0.04 (3)
2s2p2 2P3/20.93 (10) – 0.33 (6) – 0.11 (4)0.87 (2s2p2 2P3/2) + 0.11 (6) + 0.01 (4)
Rb
LevelMixContribution
2s22p 2P°1/20.99 (1) + 0.11 (2p3 2P°1/2)0.99 (1) + 0.01(2p3 2P°1/2)
2s22p 2P°3/20.99 (2) + 0.14 (2p3 2P°3/2)0.98 (2) + 0.02 (2p3 2P°3/2)
2s2p2 4P1/20.90 (3) + 0.39 (9) + 0.20 (7)0.80 (3) + 0.15 (9) + 0.04 (7)
2s2p2 4P3/2–0.98 (4) + 0.15 (6) – 0.10 (10)0.97 (4) + 0.02 (6) + 0.01 (10)
2s2p2 4P5/20.82 (5) – 0.57 (8)0.67 (5) + 0.32 (8)
2s2p2 2D3/20.90 (6) + 0.43 (10) +0.09 (4)0.80 (6) + 0.19 (10) +0.01 (4)
2s2p2 2P1/20.85 (7) +0.37 (9) – 0.36 (3)0.73 (7) + 0.14 (9) + 0.13 (3)
2s2p2 2D5/20.82 (8) + 0.57(5)0.67(8) + 0.32 (5)
2s2p2 2S1/20.84 (9) – 0.47 (7) – 0.25 (3)0.77 (9) + 0.22 (7) + 0.07 (3)
2s2p2 2P3/20.89 (10) – 0.41 (6) – 0.16 (4)0.80 (10) + 0.17 (6) + 0.02 (4)
Sr
LevelMixContribution
2s22p 2P°1/20.99 (1) + 0.10 (2p3 2P°1/2)0.99 (1) + 0.01 (2p3 2P°1/2)
2s22p 2P°3/20.99 (2) + 0.13 (2p3 2P°3/2)0.98(2) + 0.02 (2p3 2P°3/2)
2s2p2 4P1/20.89 (3) + 0.41 (9) + 0.22 (7)0.78 (3) + 0.16 (9) + 0.05 (7)
2s2p2 4P3/2–0.98 (4) + 0.16 (6) – 0.11 (10)0.96 (4) + 0.02 (6) – 0.01 (10)
2s2p2 4P5/20.80 (5) – 0.59 (8)0.65(5) + 0.35 (8)
2s2p2 2D3/20.89 (6) – 0.43 (10) – 0.16 (4)0.79 (6) + 0.20 (10) +0.01 (4)
2s2p2 2P1/20.86 (7) – 0.37 (3) + 0.35 (9)0.73 (7) – 0.14 (3) + 0.12 (9)
2s2p2 2D5/20.80 (8) + 0.59 (5)0.65 (8) + 0.35(5)
2s2p2 2S1/20.84 (9) – 0.46 (7) – 0.27 (3)0.71 (9) + 0.21 (7) + 0.07 (3)
2s2p2 2P3/20.89 (10) – 0.43 (6) – 0.16 (4)0.79 (10) + 0.18 (6) + 0.03 (4)
Ru
LevelMixContribution
2s22p 2P°1/20.99 (1) + 0.08 (2p3 2P°1/2)0.99 (1) + 0.01 (2p3 2P°1/2)
2s22p 2P°3/20.99 (2) + 0.12 (2p3 2P°3/2)0.98(2) + 0.01 (2p3 2P°3/2)
2s2p2 4P1/20.83 (3) + 0.47 (9) + 0.30 (7)0.69 (3) + 0.22 (9) + 0.09 (7)
2s2p2 4P3/2–0.97 (4) + 0.19 (6) – 0.13 (10)0.95 (4) + 0.04 (6) + 0.02 (10)
2s2p2 4P5/20.73(5) – 0.68 (8)0.53 (5) + 0.47 (8)
2s2p2 2D3/20.85 (6) – 0.51 (10) – 0.10 (4)0.73 (6) + 0.26 (10) +0.01 (4)
2s2p2 2P1/20.85 (7) – 0.45 (3) + 0.25 (9)0.73 (7) + 0.20 (3) +0.06 (9)
2s2p2 2D5/20.73(8) + 0.68 (5)0.53 (8) + 0.47 (5)
2s2p2 2S1/20.84 (9) – 0.42 (7) – 0.33 (3)0.71 (9) + 0.18 (7) + 0.11 (3)
2s2p2 2P3/20.85 (10) – 0.48 (6) – 0.21 (4)0.72 (10) + 0.24 (6) + 0.04 (4)
Sn
LevelMixContribution
2s22p 2P°1/20.99 (1) + 0.07 (2p3 2P°1/2)0.99 (1) + 0.00 (2p3 2P°1/2)
2s22p 2P°3/20.99 (2) + 0.11 (2p3 2P°3/2)0.98 (2) + 0.01 (2p3 2P°3/2)
2s2p2 4P1/20.78 (3) + 0.51 (9) + 0.35 (7)0.61 (3) + 0.26 (9) + 0.12 (7)
2s2p2 4P3/2–0.96 (4) + 0.22 (6) – 0.14 (10)0.93 (4) + 0.05 (6) + 0.02 (10)
2s2p2 2D5/2–0.73 (5) + 0.68 (8)0.54 (5) + 0.46 (8)
2s2p2 2D3/20.83 (6) − 0.55 (10) + 0.10 (4)0.68 (6) + 0.31 (10) + 0.01 (4)
2s2p2 2P1/20.85 (7) – 0.50 (3) + 0.18 (9)0.72 (7) + 0.25 (3) + 0.03 (9)
2s2p2 4P5/20.73 (8) + 0.68 (5)0.54 (8) + 0.46 (5)
2s2p2 2S1/20.84 (9) – 0.40 (7) – 0.37 (3)0.70 (9) + 0.16 (7) + 0.14 (3)
2s2p2 2P3/20.82 (10) – 0.52 (6) – 0.24 (4)0.67(10) + 0.27 (6) + 0.06 (4)
Ba
LevelMixContribution
2s22p 2P°1/20.99 (1) + 0.05 (2p3 2P°1/2)0.99 (1) + 0.00(2p3 2P°1/2)
2s22p 2P°3/20.99 (2) + 0.11 (2p3 2P°3/2)0.98 (2) + 0.01 (2p3 2P°3/2)
2s2p2 4P1/20.73 (3) + 0.53 (9) +0.38 (7)0.56 (3) + 0.28 (9) +0.14 (7)
2s2p2 4P3/2–0.96 (4) + 0.27 (6) – 0.16 (10)0.92 (4) +0.05 (6) + 0.02 (10)
2s2p2 2D5/2–0.76(5) + 0.64 (8)0.58 (5) + 0.42 (8)
2s2p2 2D3/20.81 (6) + 0.58 (10) + 0.10 (4)0.65 (6) + 0.33 (10) + 0.01 (4)
2s2p2 2P1/20.84 (7) – 0.52 (3) + 0.13 (9)0.71 (7) + 0.27 (3) + 0.02 (9)
2s2p2 4P5/20.76 (8) + 0.64 (5)0.58 (8) + 0.42 (5)
2s2p2 2S1/20.83 (9) – 0.40 (3) – 0.38 (7)0.70 (9) + 0.16 (3) + 0.14 (7)
2s2p2 2P3/20.80 (10) – 0.54 (6) – 0.26 (4)0.64(10) + 0.29 (6) + 0.07 (4)
Table
Table 3. Convergence of the radiative data and transition energy between the 2s22p 2P°1/2 and 2s2p2 2D3/2 levels of B-like Ge.
Table 3. Convergence of the radiative data and transition energy between the 2s22p 2P°1/2 and 2s2p2 2D3/2 levels of B-like Ge.
VV
Active setλ (in Å)AgfSΔE (cm−1)
nBCBCBCSl/Sv
390.352.67E+102.96E+101.31E−011.45E−013.89E−024.31E−020.9021106813
490.652.64E+102.84E+101.30E−011.40E−013.88E−024.18E−020.9281103153
590.682.63E+102.85E+101.30E−011.41E−013.88E−024.20E−020.9241102793
690.682.63E+102.85E+101.30E−011.41E−013.88E−024.20E−020.9241102766
790.682.63E+102.85E+101.30E−011.41E−013.88E−024.20E−020.9241102760
90.73e1102111e
CV
Active setλ (in Å)AgfSΔE (cm−1)
nBCBCBCSl/Sv
390.372.67E+102.83E+101.31E−011.39E−013.89E−024.12E−020.9441106581
490.722.63E+102.68E+101.30E−011.32E−013.87E−023.96E−020.9771102276
590.762.62E+102.69E+101.30E−011.33E−013.87E−023.96E−020.9771101807
690.762.62E+102.69E+101.30E−011.33E−013.87E−023.96E−020.9771101836
790.752.62E+102.69E+101.30E−011.33E−013.87E−023.96E−020.9771101909
90.73e1102111e
e: Ref. [17]
Table
Table 4. Transition data for E1 transitions for B-like ions from 2s22p 2P°1/2, 3/2: lower level i, upper level j, wavelength λ (in Å), transition rate A (s−1), weighted oscillator strength gf, line strength S (in a.u), and accuracy indicator (dT).
Table 4. Transition data for E1 transitions for B-like ions from 2s22p 2P°1/2, 3/2: lower level i, upper level j, wavelength λ (in Å), transition rate A (s−1), weighted oscillator strength gf, line strength S (in a.u), and accuracy indicator (dT).
Ge
Transitionλ (in Å)λc (in Å)
Ref. [17]		AgfSdT
ijBCBCBC
2s2 2p 2P°1/22s2p2 4P1/2174.54174.055.04E+085.43E+084.60E−034.96E−032.64E−032.85E−030.0719
2s2 2p 2P°1/22s2p2 4P3/2135.00134.661.49E+071.57E+071.63E−041.72E−047.24E−057.65E−050.0539
2s2 2p 2P°1/22s2p2 2D3/290.7590.7352.62E+102.69E+101.30E−011.33E−013.87E−023.96E−020.0233
2s2 2p 2P°1/22s2p2 2P1/283.4983.5206.62E+106.75E+101.38E−011.41E−013.80E−023.87E−020.0188
2s2 2p 2P°1/22s2p2 2P3/265.5865.6397.51E+097.58E+091.94E−021.95E−024.18E−034.22E−030.0086
2s2 2p 2P°1/22s2p2 2S1/266.1166.4201.07E+091.09E+091.40E−031.43E−033.06E−043.12E−040.0199
2s2 2p 2P°3/22s2p2 4P1/2359.56357.121.57E+071.73E+076.08E−046.73E−047.20E−047.96E−040.0953
2s2 2p 2P°3/22s2p2 4P3/2224.27223.183.19E+073.62E+079.64E−0410.91E−047.12E−048.06E−040.1170
2s2 2p 2P°3/22s2p2 4P5/2181.58180.884.23E+084.58E+081.25E−021.36E−027.50E−038.12E−030.0763
2s2 2p 2P°3/22s2p2 2D3/2123.90123.831.67E+081.78E+081.54E−031.64E−036.29E−046.70E−040.0612
2s2 2p 2P°3/22s2p2 2P1/2110.75110.771.07E+091.13E+093.95E−034.17E−031.44E−031.52E−030.0543
2s2 2p 2P°3/22s2p2 2D5/2109.30109.049.64E+099.88E+092.39E+002.39E+005.93E−025.93E−020.0008
2s2 2p 2P°3/22s2p2 2S1/282.1182.5745.63E+105.68E+101.14E−011.15E−013.07E−023.10E−020.0090
2s2 2p 2P°3/22s2p2 2P3/281.3081.3707.83E+107.92E+103.10E−013.14E−018.31E−028.40E−020.0111
Rb
Transitionλ(in Å)λc (in Å)
NIST [44]		AgfSdT
ijBCBCBC
2s22p 2P°1/22s2p2 4P1/2136.94137.5711.49E+091.61E+098.40E−039.04E−033.79E−034.07E−030.0702
2s22p 2P°1/22s2p2 4P3/291.2891.7526.38E+076.70E+073.19E−043.35E−049.59E−0510.05E−050.0467
2s22p 2P°1/22s2p2 2D3/265.5165.2365.52E+105.64E+101.42E−011.45E−013.06E−023.13E−020.0216
2s22p 2P°1/22s2p2 2P1/262.6862.3291.09E+111.11E+111.28E−011.30E−012.64E−022.69E−020.0182
2s22p 2P°1/22s2p2 2P3/245.5145.2928.69E+098.74E+091.08E−021.08E−021.62E−031.63E−030.0049
2s22p 2P°1/22s2p2 2S1/245.7345.6377.09E+087.24E+084.44E−044.54E−046.69E−056.84E−050.0215
2s22p 2P°3/22s2p2 4P1/2571.88580.7203.54E+064.19E+063.47E−044.10E−046.53E−047.73E−040.1547
2s22p 2P°3/22s2p2 4P3/2185.14186.8467.22E+078.19E+071.48E−031.68E−039.04E−0410.26E−040.1185
2s22p 2P°3/22s2p2 4P5/2145.13145.4761.07E+091.17E+092.03E−022.21E−029.72E−031.06E−020.0806
2s22p 2P°3/22s2p2 2D3/2102.98102.2298.86E+089.35E+085.63E−035.94E−031.91E−032.01E−030.0522
2s22p 2P°3/22s2p2 2P1/296.1695.2652.38E+092.52E+096.59E−036.98E−032.09E−032.21E−030.0564
2s22p 2P°3/22s2p2 2D5/279.4979.3651.56E+101.60E+108.89E−029.11E−022.32E−022.38E−020.0244
2s22p 2P°3/22s2p2 2S1/261.3061.1069.20E+109.28E+101.04E−011.04E−012.09E−022.11E−020.0092
2s22p 2P°3/22s2p2 2P3/260.9160.4891.35E+111.36E+113.00E−013.03E−016.01E−026.08E−020.0117
Sr
Transitionλ (in Å)λc (in Å)
NIST [44]		AgfSdT
ijBCBCBC
2s22p 2P°1/22s2p2 4P1/2131.09132.01.78E+091.92E+099.18E−039.88E−033.96E−034.26E−030.0704
2s22p 2P°1/22s2p2 4P3/284.4984.98.37E+078.77E+073.58E−043.76E−041.00E−041.04E−040.0460
2s22p 2P°1/22s2p2 2D3/261.4161.26.41E+106.55E+101.45E−011.48E−012.93E−022.99E−020.0214
2s22p 2P°1/22s2p2 2P1/259.1158.81.21E+111.23E+111.26E-011.29E-012.46E-022.51E-020.0180
2s22p 2P°1/22s2p2 2P3/242.2842.18.97E+099.00E+099.62E-039.65E-031.34E-031.34E-030.0037
2s22p 2P°1/22s2p2 2S1/242.4742.46.56E+086.70E+083.54E-043.63E-044.96E-055.07E-050.0222
2s22p 2P°3/22s2p2 4P1/2717.22732.01.72E+062.26E+062.65E-043.48E-046.26E-048.22E-040.2376
2s22p 2P°3/22s2p2 4P3/2178.52180.08.30E+079.51E+071.59E-031.82E-039.32E-041.07E-030.1272
2s22p 2P°3/22s2p2 4P5/2139.42140.01.23E+091.36E+092.16E-022.37E-029.90E-031.09E-020.0913
2s22p 2P°3/22s2p2 2D3/299.5199.01.09E+091.16E+096.48E−036.87E−032.12E−032.25E−030.0563
2s22p 2P°3/22s2p2 2P1/293.6092.82.68E+092.86E+097.04E−037.50E−032.17E−032.31E−030.0621
2s22p 2P°3/22s2p2 2D5/274.3874.31.74E+101.80E+108.69E−028.97E−022.13E−022.19E−020.0312
2s22p 2P°3/22s2p2 2S1/257.7657.61.02E+111.04E+111.02E−011.04E−011.94E−021.98E−020.0156
2s22p 2P°3/22s2p2 2P3/257.4157.11.51E+111.54E+112.99E−013.04E−015.65E−025.75E−020.0172
Ru
Transitionλ(in Å)AgfSdT
ijBCBCBC
2s22p 2P°1/22s2p2 4P1/2103.834.08E+094.40E+091.32E−021.42E−024.50E−034.86E−030.0731
2s22p 2P°1/22s2p2 4P3/253.613.78E+083.93E+086.51E−046.77E−041.15E−041.19E−040.0390
2s22p 2P°1/22s2p2 2D3/241.811.56E+111.60E+111.64E−011.67E−012.26E−022.30E−020.0197
2s22p 2P°1/22s2p2 2P1/241.222.40E+112.44E+111.22E−011.24E−011.66E−021.68E−020.0166
2s22p 2P°1/22s2p2 2P3/227.221.11E+101.11E+104.96E−034.95E−034.44E−044.44E−040.0009
2s22p 2P°1/22s2p2 2S1/227.294.12E+084.25E+089.20E−059.48E−058.26E−068.52E−060.0302
2s22p 2P°3/22s2p2 4P1/2458.212.37E+062.08E+062.99E−042.62E−044.50E−043.95E−040.1240
2s22p 2P°3/22s2p2 4P3/2146.221.66E+081.90E+082.12E−032.43E−031.02E−031.17E−030.1282
2s22p 2P°3/22s2p2 4P5/2113.102.26E+092.50E+092.60E−022.87E−020.97E−021.07E−020.0945
2s22p 2P°3/22s2p2 2D3/282.612.59E+092.75E+091.06E−021.13E−022.89E−033.06E−030.0579
2s22p 2P°3/22s2p2 2P1/280.344.65E+094.98E+099.01E−039.64E−032.38E−032.55E−030.0650
2s22p 2P°3/22s2p2 2D5/249.583.65E+103.73E+108.07E−028.25E−021.32E−021.35E−020.0225
2s22p 2P°3/22s2p2 2S1/240.272.02E+112.04E+119.85E−029.94E−021.30E−021.32E−020.0090
2s22p 2P°3/22s2p2 2P3/240.123.17E+113.21E+113.06E−013.10E−014.04E−024.09E−020.0121
Sn
Transitionλ (in Å)AgfSdT
ijBCBCBC
2s22p 2P°1/22s2p2 4P1/285.337.11E+097.73E+091.55E−021.69E−024.36E−034.74E−030.0801
2s22p 2P°1/22s2p2 4P3/234.671.44E+091.49E+091.04E−031.07E−031.18E−041.22E−040.0335
2s22p 2P°1/22s2p2 2D3/228.643.83E+113.90E+111.88E−011.92E−011.77E−021.81E−020.0184
2s22p 2P°1/22s2p2 2P1/228.545.12E+115.20E+111.25E−011.27E−011.17E−021.19E−020.0154
2s22p 2P°1/22s2p2 2P3/217.701.46E+101.45E+102.75E−032.73E−031.60E−041.59E−040.0049
2s22p 2P°1/22s2p2 2S1/217.732.52E+082.65E+082.37E−052.50E−051.39E−061.46E−060.0492
2s22p 2P°3/22s2p2 4P1/2112.071.11E+081.09E+088.36E−048.18E−043.08E−043.02E−040.0208
2s22p 2P°3/22s2p2 4P3/2121.942.80E+083.27E+082.50E−032.91E−031.00E−031.17E−030.1418
2s22p 2P°3/22s2p2 4P5/232.988.43E+108.61E+108.25E−028.42E−028.96E−039.15E−030.0209
2s22p 2P°3/22s2p2 2D3/270.074.39E+094.71E+091.29E−021.39E−022.98E−033.20E−030.0688
2s22p 2P°3/22s2p2 2P1/269.476.87E+097.44E+099.94E−031.08E−022.27E−032.46E−030.0758
2s22p 2P°3/22s2p2 2D5/294.743.35E+093.76E+092.70E-023.04E-028.43E-039.48E-030.1112
2s22p 2P°3/22s2p2 2S1/227.974.28E+114.32E+111.00E-011.01E-019.24E-039.33E-030.0092
2s22p 2P°3/22s2p2 2P3/227.907.04E+117.12E+113.28E-013.33E-013.02E-023.05E-020.0125
Ba
Transitionλ (in Å)λc (in Å)
NIST [44]		AgfSdT
ijBCBCBC
2s22p 2P°1/22s2p2 4P1/271.7971.6541.06E+101.17E+101.65E−021.80E−023.89E−034.27E−030.0878
2s22p 2P°1/22s2p2 4P3/222.8122.9474.79E+094.93E+091.51E−031.55E−031.14E−041.17E−040.0296
2s22p 2P°1/22s2p2 2D3/219.7919.7699.28E+119.44E+112.18E−012.22E−011.42E−021.44E−020.0175
2s22p 2P°1/22s2p2 2P1/219.8019.7781.14E+121.15E+121.34E−011.36E−018.73E−038.86E−030.0143
2s22p 2P°1/22s2p2 2P3/211.7211.7101.99E+101.97E+101.64E−031.62E−036.31E−056.26E−050.0086
2s22p 2P°1/22s2p2 2S1/211.7311.7301.47E+081.62E+086.08E−066.69E−062.35E−072.58E−070.0909
2s22p 2P°3/22s2p2 4P1/249.9750.1408.67E+088.67E+081.30E−031.30E−032.13E−042.13E−040.0001
2s22p 2P°3/22s2p2 4P3/2103.03103.3164.29E+085.09E+082.73E−033.24E−039.26E−0410.99E−040.1582
2s22p 2P°3/22s2p2 4P5/222.1722.1832.02E+112.07E+118.96E−029.14E−026.54E−036.67E−030.0198
2s22p 2P°3/22s2p2 2D3/260.3159.9346.36E+096.93E+091.39E−021.51E−022.75E−033.00E−030.0822
2s22p 2P°3/22s2p2 2P1/260.4060.0249.31E+0910.19E+091.02E−021.11E−022.02E−032.22E−030.0868
2s22p 2P°3/22s2p2 2D5/280.9280.7364.48E+095.16E+092.64E−023.04E−027.04E−038.09E−030.1307
2s22p 2P°3/22s2p2 2S1/219.4819.4739.37E+119.46E+111.07E−011.08E−016.84E−036.90E−030.0094
2s22p 2P°3/22s2p2 2P3/219.4519.4201.60E+121.63E+123.64E−013.69E−012.33E−022.36E−020.0128

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Khatri, I.; Goyal, A.; Singh, A.K.; Mohan, M. Multi-Configuration Dirac–Hartree–Fock (MCDHF) Calculations for B-Like Ions. Atoms 2016, 4, 13. https://doi.org/10.3390/atoms4020013

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Khatri I, Goyal A, Singh AK, Mohan M. Multi-Configuration Dirac–Hartree–Fock (MCDHF) Calculations for B-Like Ions. Atoms. 2016; 4(2):13. https://doi.org/10.3390/atoms4020013

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Khatri, Indu, Arun Goyal, Avnindra Kumar Singh, and Man Mohan. 2016. "Multi-Configuration Dirac–Hartree–Fock (MCDHF) Calculations for B-Like Ions" Atoms 4, no. 2: 13. https://doi.org/10.3390/atoms4020013

APA Style

Khatri, I., Goyal, A., Singh, A. K., & Mohan, M. (2016). Multi-Configuration Dirac–Hartree–Fock (MCDHF) Calculations for B-Like Ions. Atoms, 4(2), 13. https://doi.org/10.3390/atoms4020013

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