COMPUTATION OF CHARACTERISTICS OF C IV TRANSITIONS

In this research, we computed transition probabilities, line strength, and oscillator strengths of more than 5000 transitions in C IV. Very few values of these spectroscopic characteristics were previously known and reported. The calculation method, based on a combination of the weakest bound electron model and numerical approximation, shows reliable values because the correlation between known and calculated values is high. The transition probabilities calculated in this work are compared with known values of the NIST database and those found in literature, and a reasonably good agreement has been observed. The lifetime of Rydberg levels ns, np, nd, nf, ng has been reported up to n = 25. A general sixth-degree polynomial was developed, generating C IV lifetimes with reasonable accuracy. Most of the results presented are new.


INTRODUCTION
The star's atmosphere or other astronomical objects like interstellar nebula have various chemical elements.The chemical composition varies from object to object; astronomers can identify it by recording and measuring the relative amount of electromagnetic radiation emitted by each.Understanding stellar evolution requires precise abundances of various elements, including carbon.The chemical abundance is also vital to understand the complex picture of stars.The study of Excited states of atoms is the foundation of quantum mechanics and has grabbed the focus of scientists for many decades.Many research papers and articles have been published with the application of the transition state of carbon atoms in molecular physics, nanochemistry, medicinal chemistry, environmental chemistry, and material science [1][2][3][4][5][6][7].In 1970, Martinson measured the Mean lives of 16 excited levels in C II -C V with the beam-foil technique and found good agreement for the C III 2s3d 3 D -2s4f 3 F 0 (1923 Å) transition and the C IV 2s 2 S -2p 2 p 0 (1548 Å) transition [8].In 1971, Poulizac also used the beam-foil excitation method to study the carbon spectra between 1100 Å and 7000 Å for C I, C II, C III, C IV, and C V energy ranging from 0.18 to 2.0 MeV [9].In 1979, Ganas used a semiempirical approach with Optical oscillator strengths for excitations from the valence subshell of C (IV) and N (V) and obtained good agreement with experimental data [10].In 1989, Baudinet-Robinet et al. applied the beam-foil-laser method to determine the lifetimes of two levels in multiply ionized carbon atoms and found the results for C Ill 2s3d 1 D, 0.15±0.01ns and C IV 3s 2 S, 0.21±0.02ns.These values are in good agreement with the theoretical predictions.They also determined these lifetimes using the classical (nonselective) beam-foil method and reported ≈ 20% longer than the beam-foil-laser values.These factors limit the accuracy of the lifetime determinations by the beam-foil-laser method [11].In 1996, Gou used the multichannel saddle-point and saddle-point complex-rotation methods for Seven triply excited states of lithium-like beryllium and carbon, using first-order perturbation theory [12].In 1997, Cheng improved the energy levels in neutral carbon using high-resolution infrared solar spectra.The main source is the ATMOS spectrum measured by the Fourier transform spectroscopy technique from 600 to 4800 cm -1 , supplemented by the MARK IV balloon data, covering 4700 to 5700 cm -1 [13].P. Quinet, in 1998, by using the Ritz and the polarization methods, calculated the term energies up to  = 30 and  ≥ 3 in C II, C III, and C IV. His article also reported the predicted wavelengths for these lines of high-nl term energies and the related oscillator strengths [14].Nengwu Zheng et al., in 2001, by employing the WBEPM, computed transition probabilities of C I, C II, C III, and C IV.They calculated the required parameters for the calculation of transition probabilities through a proposed coupled equation which relates the energy and radial expectation value 〈〉 of the Weakest Bound Electron [15].In 2004, Agarwal investigated Energy levels and radiative rates for transitions among the lowest 24 fine structure levels belonging to the 1s 2 nl (n ≤ 5) configurations of C IV using the fully relativistic GRASP code.Additionally, collision strengths for transitions among these levels have been computed over a wide energy range below 28 Ry using the Dirac Atomic R-matrix Code [16].In 2002, Nengwu Zheng and Tao Wang computed the radiative lifetimes, transition probabilities, and oscillator strengths for individual lines of different transitions for atomic carbon and oxygen.In their article, WBEPM theory has been employed for calculations [17].Zheng et al., in 2004, developed a unified WBEPM theory in which they presented the relativistic form of the theory and combined it with the non-relativistic form they proposed earlier.They have employed the newly proposed theory for calculating transition probabilities and F II oscillator strength, carbon atom energy levels, and Ionization potential for oxygen-like ions [18].In 2018, Lischka introduced the progress in time-resolved spectroscopy to explain the characteristic features of excited states accurately.At the same time, the stable molecule's electronic ground state problems can efficiently solve with the implementation of quantum chemical methodology [19].In 2020, Li et al. worked with the multiconfiguration Dirac-Hartree-Fock and Relativistic Configuration Interaction methods for the General-purpose Relativistic Atomic Structure Package GRASP2K to compute the Landé g-factors for states in C I−IV and other atoms.Further, they compared the accuracy of the wave functions for the states and the resulting Landé g-factors' accuracy with the computed excitation energies and energy separations with the National Institute of Standards and Technology (NIST) recommended data [20].In 2022, Whang et al. employed a neural network machine learning method to simulate interatomic potentials for the structural properties of several carbon structures.First-principles Density Functional Theory (DFT) calculations are used to train the potential with a database of crystalline and liquid structures.The excellent accuracy and transferability of the NNP provide a promising tool for accurate atomistic simulations of various carbon materials with faster speed and much lower cost [21].

THEORY
Due to its complex nature, the Schrodinger equation for atoms and ions having many electrons is difficult to solve.However, an approximate solution for the hydrogen atom exists, with only one electron in its outermost shell.Like the hydrogen atom, some atoms have only one electron in the outermost shells; thus, the interaction terms are no longer required in the equation for the hydrogen-like atom.Hence Schrodinger equation for hydrogen atoms can be used for such atoms and ions with the approximation that all other electrons in the inner shells together with the nucleus form the core, like the hydrogen atom, e.g., hydrogen-like atoms and ions are Li I, Be II, B III, C IV, N V, and O VI.The theory used in this work is the same as in [22].The Schrodinger equation for hydrogen-like atoms and ions is given by, Here   = (), and () is the radial wavefunction, * ( * ) + () The first term on the right side ( ) is the same as for hydrogen atom, the second term + is the total potential felt by the weakest bound electron.The energy of hydrogen-like atoms and ions is given by, * =  −  and  * =  −  are effective principal and orbital quantum numbers for hydrogen-like atoms and ions. is a quantum defect in principal and orbital quantum numbers (, ).The quantum defect can be expressed as a polynomial in , where  is 1/( −  ), the  is the lowest value of quantum defect.The radial function can be defined as () = ( ) , and can be expressed in terms of associated Laguerre polynomials.
The transition probability  of a transition for spontaneous emission between levels  ,  & ( ,  ) is given as, >  and are energies of upper and lower levels, S is the electric dipole line strength; it is proportional to the dipole matrix element  ( ) which is given as, The lifetime () of Rydberg levels can be found by the following equation; RESULT AND DISCUSSION The Martin formula was used to calculate energies and quantum defects of the Rydberg lithium levels like C IV.These results calculated transition probabilities, oscillator strength, and line strength of five thousand two hundred and fifty transitions.The transition probability mainly depends on the energy difference of the levels involved in the transition and the line strength of the transition.Due to the unavailability of the wavefunction for the atoms and ions, it isn't easy to calculate line strength which depends on the dipole matrix element.However, the Weakest Bound Electron Potential Model (WBEPM) suggests hydrogen-like wavefunction for lithium-like atoms and ions.This wavefunction for C IV was used, and dipole integral was evaluated using the wavefunction of WBEPM; consequently, line strength was evaluated, which was further used in calculating transition probability.The energy levels of ns, np, nd, nf, and ng up to  = 30 have been calculated; using selection rules, more than 5250 transitions in C IV were studied.In Table I, the first column gives the configuration of the upper and lower levels of the transition (nlj).The first letter represents the principal quantum number, the second is the sub-orbital corresponding to the orbital quantum number, and the term in the bracket is the total angular momentum of the level.The second column gives the transition probabilities determined in this work, NIST values, and Zheng's work.The third and fourth columns give oscillator strengths & line strengths determined in this work and NIST values.
Out of these 5250, only 225 transition probabilities are given on the NIST site, the comparison of these transition probabilities with those calculated in this work has a percentage error of less than 1% in most cases, and in a few cases, it is up to 7%.Similar is the case upon comparing the transition probabilities of Zheng's work and this study.However, there is one transition in each comparison with NIST data and Zheng's work, where a large deviation is seen from this work, as mentioned below.

The Transition 1s 2 8s -1s 2 2p
The transition probabilities for the transitions 1s 2 8s 3/2 -1s 2 2p 1/2 and 1s 2 8s 1/2 -1s 2 2p 3/2 determined in this work are 5.75×10 7 and 1.15×10 8 , the corresponding values in the NIST data are 1.66×10 9 , and 3.22×10 9 , respectively.A difference of 99% between them can be seen.NIST data classifies these transitions in accuracy code B, which means it has 10% or fewer errors.As mentioned below, Zheng did not measure this transition where a large deviation is seen from this work.

The Transition 1s 2 3d-1s 2 2p
The transition probabilities for the transitions 1s 2 3d 3/2 -1s 2 2p 1/2 determined in this work are 1.52×10 10 , and the corresponding value in Zheng's work is 1.47×10 6 .A big difference is observed between the two, whereas the reported value in the NIST database is close to the value determined in this work (1.46×10 10 ).NIST data classifies it in accuracy code B, which means it has a 10% or less error.The maximum probability is found for the transition 1s 2 3d -1s 2 2p.
Fig. 1 compares transition probabilities calculated in this work and listed in the NIST database.An approximate straight-line graph among transition probabilities between this work and NIST values indicates a good agreement between both.The correlation coefficient between these probabilities is 0.999.
Since many transition probabilities are known, all possible transition probabilities from each level are known; hence equation ( 6) can be used to find the lifetime of the levels.The lifetimes of Rydberg levels 1s 2 ns, 1s 2 np, 1s 2 nd, 1s 2 nf, and 1s 2 ng up to n = 25 have also been determined.Table II gives the values of the lifetime of the corresponding level.A locally developed python program was used to fit a polynomial for each of the known values of lifetimes of Rydberg series.The lifetime for the series ns, np, nd, nf, and ng can be given a function of principal quantum number (n) in the form of a sixth-degree polynomial; the coefficients for the respective series are given in Table III  =  +   +   +   +   +   +     I).That is, most of the values are reported for the first time.The maximum value of transition probabilities does not occur between the two lowest-lying levels, as is the case of the Li atom; instead, it occurs for the transition 1s 2 3d 3/2 -1s 2 2p 1/2 .Most transition probabilities are close to the reported values; a difference up to 7% has been observed in a few cases.A 99.9% correlation is found between calculated and known values of Transition probabilities (see Fig. 1).The comparison of calculated values of oscillator strengths and lines strengths with those listed in NIST shows a good agreement.Only 224 values of transition probabilities, oscillator strengths, and line strengths have been presented in this manuscript; a separate supplementary file contains all the 5250 values.The lifetimes of the first 25 levels of the Rydberg Series ns, np, nd, nf, and ng have also been calculated.A function of principal quantum number can calculate the lifetime; a sixthdegree polynomial gives this function for each Rydberg series for C IV.

Figure 1 .
Figure 1.Plot of transition probabilities listed in the NIST database and corresponding calculated values

CONCLUSION
An extended work has been carried out to determine the transition probabilities, oscillator strengths, and line strength for the transition in Rydberg levels of C IV.Total of 5250 transitions were studied.The calculated values were compared with the reported and NIST database values.The NIST database only contains 224 out of 5250 transitions (see Table

Table I .
List of transition calculated transition probabilities, oscillator strengths, and line strength compared with corresponding values in the NIST database.

Table II .
Lifetimes of Rydberg levels of C IV

Table III .
Coefficients of the sixth-degree polynomial for calculation of lifetimes of C IV series