Energy levels, lifetimes, and transition probabilities for Sr XXXII

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Introduction
Nitrogen-like ions have a great interest for the investigation and analysis of high-temperature astrophysical and plasma experiments, so due to its significance, they have attracted a lot of theoretical and experimental attention throughout the years [1][2][3][4][5]. Experimentally, Kaufman et al. [6] measured the wavelengths of the 2s 2 2p 3 − 2s2p 4 and 2s2p 4 − 2p 5 transitions for N-like Cl XI, K XIII, Ca XIV, Sc XV, Ti XVI, and V XVII in laser-produced plasma, and the energies for the 2s 2 2p 3 ( 4 S, 2 D, 2 P), 2s2p 4 ( 4 P, 2 D, 2 S, 2 P), and 2p 5 ( 2 P) have been derived from the measurements. Edlén [7,8] derived the energies of the n = 2 configurations by comparing the observed energy intervals with the theoretical values of Cheng et al. [9] for N-like ions with the atomic number Z ranging from Z = 10 − 36.
A number of approaches have been performed to compute the energies and atomic data for allowed and forbidden transitions in N-like ions. The wavelengths and weighted oscillator strengths for E1 transitions between the levels of the n = 2 configurations [10], as well as the wavelengths and transition probabilities for E2 and M1 transitions within the ground configuration [11], for ions with 10 ≤ Z ≤ 30 , have been estimated using the second-order many-body perturbation theory (MBPT). Chen et al. [12] computed the energy levels, wavelengths, and transition probabilities of E1, M1, E2, and M2 transitions of the n = 2 configurations for Y XXXIII using the MBPT method. Using a combination of configurationinteraction (CI) and the MBPT method, Gu [13] calculated the level energies of the 2s 2 2p 3 , 2s2p 4 , and 2p 5 configurations for ions with Z ≤ 60 . Using the multiconfiguration Dirac-Fock (MCDF) method, Zhang and Sampson [14] Supplementary Information The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s43994-023-00033w.
In the present study, the spectroscopic data of energies, lifetimes, wavelengths, transition probabilities, weighted oscillator strengths, and line strengths for the E1, E2, M1, and M2 transitions between the levels belonging to the n ≤ 3 configurations of Sr XXXII using MCDHF method have been presented.

Method of calculations
The fully relativistic MCDHF approach that implemented in the GRASP2018 atomic structure computer package [25] is used to carry out the present calculations. The MCDHF approach is covered in great detail in Refs. [26,27]. The configuration state functions (CSFs) are expanded in the current computations using the active space approach [28]. These CSFs are formed by single and double (SD) excitations from the multireference (MR) configurations to an active set ( AS ) of orbitals with the principal quantum number 3 ≤ n ≤ 6 and 0 ≤ l ≤ 3 , as well as the configurations generated by one-electron excitation from the 1s subshell are included. The computations take into account the following MR configurations: 2s 2 2p 3 , 2s2p 4 , 2p 5 , 2s 2 2p 2 3l , 2s2p 3 3l , and 2p 4 3l ( l = s, p, d ), and the active set size is defined as follows: The relativistic configuration interaction (RCI) computations incorporating the Breit-interaction (BI) and quantum AS 1 = (3s, 3p, 3d) AS 3 = AS 2 + (5s, 5p, 5d, 5f ) AS 2 = AS 1 + (4s, 4p, 4d, 4f ) AS 4 = AS 3 + (6s, 6p, 6d, 6f ) electrodynamics (QED) effects (vacuum polarization and self-energy) are performed after the self-consistent field (SCF) procedure. In the last stage, the wavefunctions are converted from a jj-coupled CSF basis to an LSJ-coupled CSF basis [29,30]. The total number of CSFs that are included in the calculations in the different layers is presented in Table 1.

Excitation energies and lifetimes
The MCDHF excitation energies ( E MCDHF , in cm −1 ) of the lowest 15 levels belonging to the 2s 2 2p 3 , 2s2p 4 , and 2p 5 configurations of Sr XXXII as a function of the increasing active set ( AS ): MR, AS 1 , AS 2 , AS 3 , and AS 4 are presented in Table 2. When we compared the data from the four active sets AS n ( n = 1 − 4 ), we found that the average values of the absolute difference between the AS n and AS n−1 excitation energies of the 272 levels with the standard deviation are decreased as follows: 5522 ± 3515 cm −1 , 767 ± 688 cm −1 , and 115 ± 110 cm −1 . The largest difference between the AS 3 and AS 4 energies of the lowest 15 levels is 344.8 cm −1 for the 2p 5 2 P o 3/2 level. In Table 3, the present energies for the 2s 2 2p 3 , 2s2p 4 , and 2p 5 configurations have been compared with various theoretical results [13][14][15][16], and the relative energy differences are illustrated graphically in Fig. 1. The comparison demonstrates that the present energies are in full correlation with the CI-MBPT [13] and MCDF [16] values, with the relative difference is about 0.02% from Ref. [16], 0.05% from Ref. [13], and 0.37% from Refs. [14,15]. The comparison demonstrates that the labels of the 2s2p 4 ( 2 S 1/2 and 2 P 1/2 ) levels were switched in Refs. [14][15][16]. Table 4 presents the MCDHF excitation energies ( E MCDHF , in cm −1 ) and lifetimes in the length and velocity forms ( L and V , in s) for the lowest 272 levels belonging to the 2s 2 2p 3 , 2s2p 4 , 2p 5 , 2s 2 2p 2 3l , 2s2p 3 3l , and 2p 4 3l ( l = s, p, d ) configurations from the AS 3 and AS 4 layers, as well as the LS-compositions of the levels from the AS 4 layer in Sr XXXII. All E1, M1, E2, and M2 transition probabilities among the lowest 272 levels are considered in the lifetime calculations. The relative difference between the two sets of

Transition data
In Table 5, the current MCDHF wavelengths ( in Å) and transition probabilities ( A in s −1 ) for 2s 2 2p 3 − 2s2p 4 transition array in Sr XXXII are compared with the previous calculations with the MCDF data by Hao et al. [16]. The comparison demonstrates that the current MCDHF wavelengths and transition probabilities match well with the MCDF data [16], where the relative difference is evaluated using the root mean square (RMS%) and is, respectively, 0.14% and 13.9% for wavelengths and transition probabilities. The electric dipole oscillator strengths for transitions among the levels of the 2s 2 2p 3 , 2s2p 4 , and 2p 5 configurations are compared with the MCDF data that calculated by Zhang and Sampson [14]. Figure 3 illustrates the relation between the logarithms of the weighted oscillator strengths of this study, (log 10 (gf TW ) , and the logarithm of the ratio between the weighted oscillator strengths from this study and those from Reference [14], which is log 10 (gf TW ∕gf MCDF ) . Our results and those of MCDF [14] are in good agreement, with a relative difference of up to 26%. Table 6 lists the MCDHF wavelengths ( , in Å), transition probabilities ( A , in s −1 ), weighted oscillator strengths ( gf , dimensionless), and line strengths ( S , in a.u. ) for E1, E2, M1, and M2 transitions between the lowest 272 levels of Sr XXXII. For E1 and E2 transitions, the length and velocity forms of the transition data are also provided. Fig. 1 The relative energy differences between the present MCDHF energies and other theoretical results: CI-MBPT [13] (triangle), MCDF [14,15] (circle), and MCDF [16] (square), for the lowest 15 levels of Sr XXXII Table 4 The present MCDHF energies ( E MCDHF , in cm −1 ) and the length and velocity forms of lifetimes ( , in s) from the AS 3 and AS 4 layers for the lowest 272 levels in Sr XXXII. The LS-compositions from the AS 4 layer is presented in the last column

Accuracy of results
It is feasible to assess the degree of uncertainty of the atomic data for the electric (E1 and E2) and magnetic (M1 and M2) transitions by comparing the results of the two succeeding layers of the current MCDHF calculations ( AS 3 and AS 4 ). In Figs. 4 and 5, the logarithm of the ratio between the length form of S-values from AS 3 and AS 4 layers, log 10 (S AS 3 ∕S AS 4 ) , is displayed against the logarithm of S-values from the AS 3 layer, log 10 (S AS 3 ) , for E1, E2, M1, and M2 transitions. As can be seen, there is a normal distribution of points with scatter that gradually decreases as S AS 3 values increase.
The following steps listed below are taken to estimate the uncertainty for E1, E2, M1 and M2 line strengths: 1. The transitions are divided in groups with different ranges of S AS 3 values. 2. For each group, the roots of the mean squares of the logarithm of the ratio between S AS 3 and S AS 4 values, rms log , and the average value of S AS 3 values ( S mean ) are calculated. 3. The relation rms% = (10 rms log − 1) × 100 is used to determine the corresponding uncertainty percentage. 4. A smooth function is used to display and fit the relationship between the natural logarithms of rms% and S mean . 5. Finally, the absolute value of the actual percentage difference between AS 3 and AS 4 for each transition is compared to the value of the constructed function, and the maximum of the two is selected as an estimate of uncertainty. 6. The National Institute of Standards and Technology (NIST) [31] terminology is used to indicate the uncertainty, with the ranges being AA ≤ 1% , A+ ≤ 2% , A ≤ 3% , B+ ≤ 7% , B ≤ 10% , C+ ≤ 18% , C ≤ 25% , D+ ≤ 40% , D ≤ 50% , and E > 50%.

Conclusion
In the current study, the MCDHF approach was used to compute the energy levels, LS-compositions, and lifetimes for the lowest 272 levels of the 2s 2 2p 3 , 2s2p 4 , 2p 5 , 2s 2 2p 2 3l , 2s2p 3 3l , and 2p 4 3l ( l = s, p, d ) configurations of Sr XXXII. In the computations, the Breit-interaction, vacuum polarization, and self-energy were taken into account. The wavelengths, transition probabilities, weighted oscillator strengths, and line strengths of the E1, E2, M1, and M2 transitions between the levels considered here are also presented.
Comparisons have been made between the current MCDHF results and earlier computations. The current  . 3 The relation between log 10 (gf TW ) and log 10 (gf TW ∕gf MCDF ) for some E1 transitions in Sr XXXII   The uncertainty of E1, E2, M1, and M2 line strengths of Sr XXXII has been evaluated by the comparison between the two layers ( AS 3 and AS 4 ). Our computations produce a large amount of new atomic data. Future fusion plasma studies and other calculations should benefit from the improved data provided in this study.  The log 10 (S AS 3 ∕S AS 4 ) versus log 10 S AS 3 for E1 and E2 transitions Author contributions Conceptualization, software, formal analysis and investigation, data curation, resources, writing-review & editing.
Funding No funding was received to assist with the preparation of this manuscript.
Data availability All data generated and analyzed during this study are included in this article and in its supplementary files.

Conflict of interest
The author has no relevant financial or non-financial interests to disclose.
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Fig. 5
The log 10 (S AS 3 ∕S AS 4 ) versus log 10 S AS 3 for M1 and M2 transitions